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Developments in the Perturbation Theory of Algebraically Special Spacetimes

Permanent Link: http://ufdc.ufl.edu/UFE0021314/00001

Material Information

Title: Developments in the Perturbation Theory of Algebraically Special Spacetimes
Physical Description: 1 online resource (180 p.)
Language: english
Creator: Price, Lawrence Ray, Jr
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: black, newman, spin
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The detection of gravitational waves is the most exciting prospect for experimental relativity today. With ground based interferometers such as LIGO, VIRGO and GEO online and the space based LISA project in preparation, the experimental apparatus necessary for such work is steadily taking shape. Yet, however capable these experiments are of taking data, the actual detection of gravitational waves relies in a significant way on making sense of the collected signals. Some of the data analysis techniques already in place use knowledge of expected waveforms to aid the search. This is manifested in template based data analysis techniques. For these techniques to be successful, potential sources of gravitational radiation must be identified and the corresponding waveforms for those sources must be computed. It is in this context that black hole perturbation theory has its most immediate consequences. This dissertation presents a new framework for black hole perturbation theory based on the spin coefficient formalism of Geroch, Held and Penrose. The two main components of this framework are a new form for the perturbed Einstein equations and a Maple package, GHPtools, for performing the necessary symbolic computation. This framework provides a powerful tool for performing analyses generally applicable to the entire class of Petrov type D solutions, which include the Kerr and Schwarzschild spacetimes. Several examples of the power and flexibility of the framework are explored. They include a proof of the existence of the radiation gauges of Chrzanowski in Petrov type II spaces as well as a derivation of the Teukolsky-Starobinsky relations that makes no reference to separation of variables. Furthermore, a method of determining the non-radiated multipoles in type D spaces is detailed.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Lawrence Ray Price.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Whiting, Bernard F.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021314:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021314/00001

Material Information

Title: Developments in the Perturbation Theory of Algebraically Special Spacetimes
Physical Description: 1 online resource (180 p.)
Language: english
Creator: Price, Lawrence Ray, Jr
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: black, newman, spin
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The detection of gravitational waves is the most exciting prospect for experimental relativity today. With ground based interferometers such as LIGO, VIRGO and GEO online and the space based LISA project in preparation, the experimental apparatus necessary for such work is steadily taking shape. Yet, however capable these experiments are of taking data, the actual detection of gravitational waves relies in a significant way on making sense of the collected signals. Some of the data analysis techniques already in place use knowledge of expected waveforms to aid the search. This is manifested in template based data analysis techniques. For these techniques to be successful, potential sources of gravitational radiation must be identified and the corresponding waveforms for those sources must be computed. It is in this context that black hole perturbation theory has its most immediate consequences. This dissertation presents a new framework for black hole perturbation theory based on the spin coefficient formalism of Geroch, Held and Penrose. The two main components of this framework are a new form for the perturbed Einstein equations and a Maple package, GHPtools, for performing the necessary symbolic computation. This framework provides a powerful tool for performing analyses generally applicable to the entire class of Petrov type D solutions, which include the Kerr and Schwarzschild spacetimes. Several examples of the power and flexibility of the framework are explored. They include a proof of the existence of the radiation gauges of Chrzanowski in Petrov type II spaces as well as a derivation of the Teukolsky-Starobinsky relations that makes no reference to separation of variables. Furthermore, a method of determining the non-radiated multipoles in type D spaces is detailed.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Lawrence Ray Price.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Whiting, Bernard F.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021314:00001


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a67b06fc412c25d4a26cde40d94631a2
3879fb8d7bb499cf50c8bf61d73c1ccaa372ad42







DEVELOPMENTS IN THE PERTURBATION THEORY OF ALGEBRAICALLY
SPECIAL SPACETIMES


















By
LARRY R. PRICE


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2007



































S2007 Larry R. Price










ACKENOWLED GMENTS

The task of writing acknowledgements necessarily comes the task of forgetting to

acknowledge everyone who deserves it. My apologies to anyone I've forgotten.

First of all, I owe a great deal to my advisor, Bernard Whiting for his patient

guidance and all his support. It has been a pleasure to worth with him for the past five

years.

I would like to thank Steve Detweiler for useful providing useful comments and

perspective throughout the years.

My friends throughout the years deserve a great deal of thanks for making life in

Gainesville bearable: Josh McClellan, Flo Courchay, Wayne Bomstad, Ethan Siegel, Scott

Little, Aaron Manl I1 li li-, lan Vega, K~arthik Shankar and anyone I've forgotten.

I owe a very special thanks to Lisa Danker both for putting up with and making life

easier for me during the creation of this document.

All of my parents-Pam Villa and Larry and Audrey Price-deserve more thanks

than I can give them for their continued support throughout the years.

Finally, thanks go the Alumni fellowship program and Institute for Fundamental

Theory at the University of Florida for financial support over the years.











TABLE OF CONTENTS


page

ACK(NOWLEDGMENTS ......... ... .. :3

ABSTRACT ......... ..... . 6

CHAPTER

1 INTRODUCTION ......... .. .. 8

1.1 Perturbations of Spherically Syninetric Spacetintes .. .. .. 9
1.2 Perturbations of K~err Black Hole Spacetintes ... . .. 1:3
1.3 Metric Perturbations of Black Hole Spacetintes .. .. .. .. 15
1.3.1 Hertz Potentials in Flat space ..... .. 16
1.3.2 The Inversion Problem for Gravity ... .. .. .. 17
1.3.2.1 Ori's construction for K~err ... . .. .. 18
1.3.2.2 Time domain treatment for Schwarzschild .. .. .. .. 19
1.:3.2.3 Working in the R< -~-Wheeler gauge . . 20
1.4 This Work ........... ........... 21

2 NEW TOOLS FOR PERTURBATION THEORY .... .... .. 2:3

2.1 NP .... ........ .......... .. 2:3
2.2 GHP ...... ... .. ....... .... 27
2.3 K~illingf Tensors and Coninuting Operators .... .... :30
2.3.1 Specialization to Petrov Type D .... .... :30
2.3.2 The K~illingf Vectors and Tensor .... ... :31
2.3.3 Coninuting Operators . . .. .. :36
2.4 The Simplified GHP Equations for Type D Backgrounds .. .. .. :38
2.5 Issues of Gauge in Perturbation Theory ... .. .. .. 40
2.6 GHPtools A New Fr-amework for Perturbation Theory .. .. .. .. 42
2.6.1 Einstein's New Clothes . ...... .. 4:3
2.6.2 GHPtools The Details . ..... .. 44

:3 REGGE-WHEELER & TEITKOLSK(Y . ..... 52

:3.1 Parity Decomposition of Spin- and Boost-Weighted Scalars .. .. .. .. 52
:3.2 R< ~- -~-Wheeler ......... . . 56
:3.2.1 The R< -~-Wheeler Gauge . ..... 56
:3.2.2 The R< -~-Wheeler Equation ...... .. 58
:3.3 The Teukolsky Equation ......... .. .. 61
:3.4 Metric Reconstruction from Weyl Scalars .... .... .. 62

4 THE EXISTENCE OF RADIATION GAUGES .... .. .. 66

4.1 The Radiation Gauges ......... .. .. 66
4.2 Imposing the IR G in type II ....... ... .. 69










4.3 Remaining Gauge Freedom ......
4.4 Imposing the IRG in type D ......
4.5 Discussion ......

5 THE TEUK(OLSK(Y-STAROBINSK(Y IDENTITIES . .

6 THE NON-RADIATED MULTIPOLES .....

6.1 Schwarzschild .....
6.1.1 Mass perturbations ......
6.1.2 Angular momentum perturbations .. . .
6.1.2.1 Odd-parity angular momentum perturbations
6.1.2.2 Even-parity dipole perturbations .....
6.2 K~err ...... ...
6.2.1 Mass Perturbations ......
6.2.2 Angular Momentum Perturbations . .
6.2.3 Discussion .....

7 CONCLUSION ......

7.1 Summary .....
7.2 Future Work ........ . .

APPENDIX

A THE GHP RELATIONS .....

B THE PERTURBED EINSTEIN EQUATIONS IN GHP FORM ..

C INTEGRATION A LA HELD ....... .

D SPIN-WEIGHTED SPHERICAL HARMONICS . .

E MAPLE CODE FOR GHPTOOLS .....

REFERENCES ......... . ...

BIOGRAPHICAL SETH .. .. ..........


72
73
76

78

84

91
91
96;
97
99
100
100
104
106

108

108
109


111

113

117

121

123

175

179









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

DEVELOPMENTS IN THE PERTURBATION THEORY OF ALGEBRAICALLY
SPECIAL SPACETIMES

By

Larry R. Price

August 2007

Cl.! I!1-: Bernard Whiting
Major: Physics

The detection of gravitational waves is the most exciting prospect for experimental

relativity tod .v. With ground based interferometers such as LIGO, VIRGO and GEO

online and the space based LISA project in preparation, the experimental apparatus

necessary for such work is steadily taking shape. Yet, however capable these experiments

are of taking data, the actual detection of gravitational waves relies in a significant way

on making sense of the collected signals. Some of the data analysis techniques already

in place use knowledge of expected waveforms to aid the search. This is manifested in

template based data 2.1, ll-k- techniques. For these techniques to be successful, potential

sources of gravitational radiation must be identified and the corresponding waveforms for

those sources must be computed. It is in this context that black hole perturbation theory

has its most immediate consequences.

This dissertation presents a new framework for black hole perturbation theory based

on the spin coefficient formalism of Geroch, Held and Penrose. The two main components

of this framework are a new form for the perturbed Einstein equations and a Maple

package, GHPtools, for performing the necessary symbolic computation. This framework

provides a powerful tool for performing analyses generally applicable to the entire class of

Petrov type D solutions, which include the K~err and Schwarzschild spacetimes.

Several examples of the power and flexibility of the framework are explored. They

include a proof of the existence of the radiation gauges of Cla!. I.1,awski in Petrov type










II spaces as well as a derivation of the Teukolsky-Starobinsky relations that makes

no reference to separation of variables. Furthermore, a method of determining the

non-radiated multipoles in type D spaces is detailed.










CHAPTER 1
INTRODUCTION

Einstein's theory of general relativity, introduced in 1915, to this
one of the final frontiers of fundamental physics. Since its inception progress in the field

has been largely theoretical because of the tremendous difficulty inherent in making

gravitational measurements. In particular, one of the most exciting and fundamental

predictions of general relativity-the existence of gravitational waves-has remained

elusive. Not for long. With ground based interferonieters such as LIGO, VIRGO and GEO

online and the space based LISA mission in preparation, the detection of gravitational

waves is all but inininent. These experiments bring with them the task of analyzing

the data they collect. For some of the promising sources of gravitational waves, the

collision of two black holes, the method of choice for data analysis, known as matched

filtering, requires knowledge of the expected waveforms. In the past two years the field

of numerical relativity has undergone a revolution and promises to provide the most

accurate waveforms for situations involving the collision of two black holes of comparable

nmasses-situations that require the use of full nonlinear general relativity. There is

however, one promising source of gravitational waves that is currently out of reach for

numerical relativity-the situation where the larger black hole is roughly a million times

more massive than the smaller one, known as an extreme mass ratio inspiral, or EAIRI.

This problem lies squarely in the realm of perturbation theory, the subject of the present

work.

In particular, the "solution" of the EAIRI problem requires moving beyond the test

mass approximation of general relativity to describe the motion of the small black hole

(treated as a particle in the spacetinle of the larger black hole because of the huge mass

difference)-one must account for the first order corrections to the motion of the small

black hole, due to self-force. The appropriate equations of motion have been determined in

general by Mino, Sasaki and Tanaka [1] and Quinn and Wald [2] and are referred to as the










MiSaTa~uWa equations. In practice, the more widely used prescription for computing

the self force is due to Detweiler and Whiting [3]. In either case, the fundamental

object of interest is the metric perturbation, hub, introduced by the particle on the

large black hole's spacetime. Therefore the EAIRI problem also requires us to compute

the metric perturbation, before we can compute the self-force on the particle. This is

the piece of the problem to which the present work aims to contribute. Determining

the metric perturbation is a task that depends quite sensitively on the spacetime being

perturbed. For spherically symmetric backgrounds, this problem is well understood and

most of the remaining problems are computational in nature. However, for the more

interesting and .I-r lInIlai--; I I11y relevant situation where the larger black hole is rol Iflrin

our understanding is not quite complete. It is on this more general situation that we

focus. Before we continue, we note that all of the .I-r initsli--;I I11y interesting spacetimes,

including the K~err and Schwarzschild metrics, possess curvature tensors with the same

basic algebraic structure. We will elaborate on this more fully in the next chapter, but for

now we merely point out that these spacetimes belong to the larger class of iy ol l.

special spacetimes.

The remainder of this chapter is devoted to providing a review of the literature [4].

Every attempt has been made to phrase the current discussion in generally accessible

language. Alany of these results will be explored in further detail in later chapters, after

the appropriate formalism has been developed.

1.1 Perturbations of Spherically Symmetric Spacetimes

Historically, the subject of black hole perturbation theory got its start with the

pioneering work of Re -~- and Wheeler [5] (henceforth RW), who provided an analysis

of first order perturbations of the Schwarzschild solution (which was later completed by

Zerilli [6, 7]). The fact that the background is spherically symmetric is crucial to their

analysis. The basics will be presented here. A more complete discussion, in a very different

language, is provided in ('! .pter 3.










Let us begin by considering a small perturbation, hab, of the Schwarzschild geometry.

Thus our spacetime metric is


9ab = gb a Lb,


(11)


where

gfbd ads b I _1)t- 2Ii _'d ~ 28 in2 tii' 2 _2)

is the Schwarzschild metric in Schwarzschild coordinates. Putting Equation 1-1 into the

Einstein equations and keeping only terms linear in hub leads us to the perturbed Einstein

equations :


1 1 1
~ab ~c c b ~a b cc c ~(a b)c + ab ~c c dd ~c d cd) = 0,
2 2 2


(1-3)


where V, is the derivative operator compatible with the background geometry 1-2 and

the indices are raised and lowered with the background metric. Henceforth we will refer to

Sab aS the Einstein tensor, and the expression to the right of it as the Einstein equations

(dropping the qualifier "perturbed" for brevity).

Essentially every perturbative analysis of the Schwarzschild spacetime makes

extensive use of its spherical symmetry. The first step in this direction is to decompose

the components of the metric perturbation into scalar, vector and tensor harmonics.

Heuristically, we write


'U2






respectively and the subscripts


81 S2 'Ul

hab 2 3 'U2
vl v2 t+4

vl v2 t

where s, v and t stand for scalar, vector and tensor,

distinguish between the various scalars and vectors.

Consider the metric of the two-sphere:


_15)


yABd A XB d2 Sin2 d2









Since the usual scalar harmonics, }*1,>, define a complete set of functions on the two-sphere,

we can use them to construct two types of vectors. The first is the so-called even parity

vector defined (up to a constant) byl



where VA is the derivative compatible with Y4s (Equation 1-5). The other vector is the

odd-parity (pseudo-) vector

YBceAB A em, (17)

wher FABis just the standardu Levi-Civita symbol. To define tensor harmonics, we

essentially just take one more derivative of Equations 1-6 and 1-7. The even parity

tensors are given by

VA Blinz, and 34slm,, (1-8)

and the odd-parity (pseudo-) tensor by


714CFCD D B~e (1m


Even parity objects pick up minus signs under a parity transformation (8 i x 8,~

xr + ~) according to (-1) and odd parity objects pick up minus signs according to

(-1) Bl. For this reason the even parity parts are sometimes referred to as "electric" and

the odd parity parts n,! I,e!tic" in the older literature. Because parity is an inherent

syninetry of spherically syninetric backgrounds, it provides a natural way of decoupling

the two degrees of freedom of the gravitational field. Note, however, that parity is not a

good syninetry in even slightly less syninetric spacetintes (e.g. K~err). We will return to



1 The tensor harmonics defined in this chapter are not those generally used, but have
been chosen for their heuristic value. See Thorne's review [8] for the standard tensor
harmonics and their relation to various other representations of the sphere, or Appendix
D for the spin-weighted spherical harmonics which provide another alternative for the
angular decomposition.










this subject in OsI Ilpter 3. Continuing in our cartoon language (Equation 1-4), we now

consider the two sectors of the metric perturbation independently, writing

00v dd yodd


d 0 0 vodd Uodd ( 0

odd ?Ll todd todd

odd ?Ll todd todd

and
81 82 even Deven

even 82 'U3 U]

even even even 84 even

even .. even even S4-

The final step before appealing to the Einstein equations consists of choosing a gauge.

Equation 1-3 is invariant under the transformation


hub i ab ( @)~ab = ab a~b + b a, (1-12)


where (a is an arbitrary vector and 4~ is the Lie derivative. Taking the odd-parity sector

as an example, the R;- ear--Wheeler gauge vector takes the form


(" = (0, 0, AeAB BY~) Im)3


where A is a function chosen so that the odd parity part of the metric perturbation 1-10

takes the form
0 0 0 v dd

0 0 O r

0 00 0

vodd ?L" 0 0.

Similar simplifications arise in the even-parity sector.









Returning to the Einstein equations with this simplified description of the metric

perturbation leads, after some manipulation, to the R;- ea-~~;-Wheeler-Zerilli equations, which

can be compactly described in a single expression, namely

82 o,e d2 o1e
-m -" +-Vo~e(r)qQ = 0, (1-15)


where the letters 'o' and 'e' stand for odd and even, respectively, r* = r + In(-~ 1) just

pushes the horizon out to infinity and ., is the appropriate master variable. Two aspects

of this result are noteworthy: (1) the two degrees of freedom have completely decoupled

and (2) these equations are separable in the Schwarzschild background. These two features

are desirable for any perturbative description of any background spacetime.

1.2 Perturbations of Kerr Black Hole Spacetimes

Unfortunately, the techniques used by RW to obtain a perturbative description of the

Schwarzschild spacetime are of little use when the background geometry possesses only

axial symmetry. Such is the case for the K~err geometry, which describes a rotating black

hole. In Bci-; r-Lindquist coordinates, its metric takes the form


ds2 __ 2Mr,, 4Mar sin2 8P"2 d

P2 22


where p2 __ r2 2 a"COS2 Ha 2 2M~r + a2, M~ is the mass and a = J/M~ is the angular

momentum per mass of the black hole. The spin coefficient formalism of Geroch, Held and

Penrose [9] developed in the next chapter has proved to be fundamental in virtually every

perturbative description of the K~err spacetime.

The first successful perturbation analysis of the K~err geometry was performed

by Teukolsky in a series of papers beginning in 1973 [10-12]. Teukolsky took as his

starting point the perturbed Bianchi identities in a spin coefficient formalism. Each

quantity is perturbed away from its background value and only first order terms are kept.

Equivalently, though with considerably more effort, Teukolsky's result can also be seen as









arising from a wave equation for the perturbed Riemann tensor, using standard methods

[13]. In either case, the result, written here in Bci-;r-Lindquist coordinates, is Teukolsky's

master equation (written here in accord with [14])

d 8 1 8 8 2
d (T2+ 2) +ta_ s(r-M)) -4s(r+iacos0)

8 8 1 8 8
+n" sin2 8 (Sin2 H iS COS 8
8 cos 8 8 cos 8 si2

x As/2 i' __ s~a/2C, s17


where s: = +2 corresponld to the W~eyl scalars ,',, anld I#2-4/3 4, TSpectively. Th'le Weyl

scalars are perturbations of the extremal spin components of the curvature tensor.

The significance of the Weyl scalar ~4 is that far away from the source of gravitational

radiation

~4 N h+ ix, 18

where h+ and hx are the two polarizations of outgoing gravitational radiation in the

transverse traceless gauge. Similar results hold for I',, and incoming radiation. For other

values of s, solutions correspond to fields of other spin: s = 0 is the massless scalar

wave equation, a = +1/2 the Weyl neutrino, a = +1 the Maxwell field, a = +3/2 the

Rarita-Schwinger field, and so on. Note that angular separation necessarily involves time

separation for a / 0.

Separated solutions to Equation 1-17 are of the form ~, = e-iwqeim*,R(r),S(aw,1 8)

(omitting the e, m and w subscripts). The angular functions, sS(aw, 8), are generally

referred to as "spin weighted spheroidal harmonics". In the limit that aw = 0,, s,,em(

reduce to the standard spin weighted spherical harmonics (cf. Appendix D), which are

interrelated by the spin raising and lowering operators, a and 8' [15], developed in the

following chapter. For aw / 0, solutions correspond to functions of different spin weight,

but the ,S(aw, 8) no longer share common eigenvalues. Thus a metric reconstruction

based on spin weight +2 functions would be incompatible with one based on spin weight 0










functions. This incompatibility does not arise for Schwarzschild, where reconstruction from

solutions of the RW equation can translate into comparable metric reconstruction from the

Weyl scalars, since there is a unique way of representing tensors on the sphere.

The spin weighted spherical (and spheroidal) harmonics fail to be defined for e < |8|

and thus the Teukolsky equation can give us no information about the -E = 0, 1 modes.

This is not a surprise since Iel, and #'4 are comporterts of the curvature tensor, which

carries information about the quadrupole (and higher multiple) generated gravitational

waves. In fact, Wald has shown [16] that for vacuum perturbations each of I',, and

('4 is SUffleient to characterize the perturbation of the spacetime, up to shifts in mass

and angular momentum. In Schwarzschild, these lower multiple moments can he

expressed appropriately in terms of spherical harmonics using the RW formalism, but any

comparable expressions for the K~err case would be incompatible with metric coefficients

constructed from spin weight +2 functions (i.e., they would be expressed in different

bases). Yet, these low--A multiple moments are urgently sought, since they convey

information about the energy and both the axial and non-axial components of the angular

momentum of a particle in orbit around the black hole. 1\oreover, in recent calculations

demonstrating the precise relation of the -E = 0, 1 multipoles in Schwarzschild to shifts in

the mass and angular momentum, Detweiler and Poisson [17] emphatically point out that

such shifts are just as important as the radiating multipoles for describing the motion of a

small black hole orbiting a supermassive black hole. The non-radiated multiple moments

are the subject of C'!s Ilter 6.

Solutions of the Teukolsky equation lead quite naturally to metric perturbations

through the use of Hertz potentials which solve Equation 1-17. We now turn our attention

to this subject.

1.3 Metric Perturbations of Black Hole Spacetimes

The first explicit solutions for metric perturbations given in terms of Hertz potentials

were written down by C'!,l~!!. 1,.---1:! [18] and Cohen and K~egeles [19]. This work was










carried out at a time when there was not a strong urge to obtain solutions related to

very specific sources, and so it gave a successful way of creating metric perturbations in

vacuum. Recent interest in EMRIs as a source for gravitational waves has developed a

need for metric perturbations related to known sources, for which curvature perturbations

may be obtained by solving Teukolsky's equation. For this class of problem, the source

is highly localized, and most of the perturbed spacetime can still be treated as vacuum.

We give first a description of solutions to the inversion problem in vacuum, pI lingf special

attention to limitations of each approach. Before we proceed, it will be helpful to give a

brief overview of Hertz potentials in the more immediately familiar context of Maxwell's

equations in flat space. Note that the methods presented here rely crucially on spin

coefficient methods, though we have attempted to keep reference to such methods minimal

for now.

1.3.1 Hertz Potentials in Flat space

To illustrate the essentials of Hertz potential methods we consider the source-free

Maxwell equations in flat spacetime, in essentially the form Cohen and K~egeles attempted

to generalize to curved spacetime [20]:


VaFab = 0 and Eabcd aFed = 0. (1-19)


As usual a vector potential, A,, is introduced and the Lorentz gauge,Vas = 0, is imposed

so that the Maxwell equations lead directly to OA, = 0.

Then a Hertz potential Hab iS introduced via A" = VbHab, Where Hab = -Hba, SO

that the Maxwell field, Fab, iS obtainable by two derivatives of Hab. However, Hab iS Only

defined up to a transformation of the type


Hab Hab + cl~cb + a b ~b a, (1-20)


where Mca"b iS completely antisymmetric and oC" = 0. It is easy to see that in flat

spacetime, where derivatives commute, the transformation Equation 1-20 only changes










40 by the addition of the term V" b b and therefore, in the Lorentz gauge, contributes

nothing to the fields. In practice, Equation 1-20 is used to reduce the Hertz hivector

potential to a single complex (or two real) scalar potential(s). Herein lies the power

of the method. However, moving to curved-space naturally complicates things. While

the wave equations are modified to include curvature pieces, the transformation in

Equation 1-20 is retained (see Cohen and K~egeles [20] and Stewart [21]). As a result,

the field equations are still satisfied and the six components of Hab are still reduced to

two, but the transformation in Equation 1-20 explicitly breaks the Lorentz gauge because

derivatives no longer commute. In this way a new gauge is introduced that brings with it

complications for the inclusion of sources. The necessary and sufficient conditions for the

existence of this gauge are the subject of C'!s Ilter 4.

1.3.2 The Inversion Problem for Gravity

The formulation of the gravitational Hertz potential proceeds analogously to that of

its (flat space) electromagnetic counterpart, with a few differences. For one, the result is a

metric perturbation in one of two complimentary gauges. Additionally, the potential itself

is a solution to the Teukolsky equation for s = +2 (or s = -2; the choice of the sign of

s determines which gauge the metric perturbation is in), though it is not the curvature

perturbation of the metric perturbation it generates. In analogy to the electromagnetic

example above, the components of the metric perturbation are given by two derivatives of

the potential. The natural language in which to express the metric perturbation arising

from the Hertz potential is again the spin coefficient formalism of Newman and Penrose

[22], or its modification due to Geroch, Held and Penrose [9]. Thus we postpone the

formal development of the subject until ('I Ilpter 3, when the necessary formalism is in

place, and instead offer an overview of the general process and documented research on the

topic of reconstructing the metric perturbation from solutions to the Teukolsky equation

(assuming the form of metric perturbation is prescribed), which we will refer to as the

inversion problem.









The problem is that of finding a Hertz potential, given a solution (or both solutions)

to the Teukolsky equation. To make this more precise, we look to the expression of the

curvature perturbations, I',, (s = +2) and #'4 (S = -2), in terms of the Hertz potentials.

If we take the potential to satisfy the s = -2 Teukolsky equation then the perturbation

exists in the in going radiation gauge (IR G) and we have that


2,',, = DDDD[lIRG,] and (1 21)

2p-42;: I[4 IRG 12p- ('281 IRG (1221)

where = [de + s cot 8 i cs e 0,] + is sin 08d and D = A-l[(T2 + 2 t r d + a84] define

derivatives in orthogonall) null directions, p = -(r ia cos 8)-1 and WIRG is the potential.

While for a potential satisfying the s = +2 Teukolsky equation, we have a perturbation in

the outgoing radiation gauge (OR G), where


2p-4 4 = 2 dd 2 ORG] and (1 23)

2,II ,, = ORG 12p- ('2i31 ORG] ) (1-24)

where a = ~pp[(r2 + 2)dt rd + a84] and the complex conjugate of the operator

defined above, are also derivatives in null directions (mutually orthogonal to each other

and those defined by the operators in the IR G). These are the equations we would like

to invert for the potentials WIRG and ~ORG. Once this is done, the potential may then he

used to construct the metric perturbation. We now look at several different approaches to

this problem.

1.3.2.1 Ori's construction for Kerr

In principle, with solutions for the Weyl curvature perturbations on all of spacetime,

one could integrate along null directions to undo the derivatives in Equations 1-21 or

1-22 (or their ORG counterparts). Ori [23] has recently performed this task-integrating

Equation 1-21 in order to find the potential WIRG in terms of I',,










As we noted previously, angular separation is dependent on separation in time

in the K~err spacetime. Ori's analysis therefore takes place in the frequency domain.

Ori's construction is effective in the vacuum situation, for which 9 satisfies Teukolsky's

Equation 1-17 with a = -2, so it does provide a complete solution in the frequency

domain.

For incorporating sources, Ori continues to take Equation 1-21 as correct, where now

I, n is a source-dependent, non-vacuum solution. Equation 1-21 allows the freedom to add

to WIRG any function that is killed by the four derivatives there. Ori utilizes this freedom

to choose functions that reproduce the discontinuity at the source and, by extension,

',, However, Equation 1-17 no longer applies for 9, nor does Equation 1-22 for tb4 in

the form it has here.2 Furthermore, any metric reconstructed in the radiation gauges

is ill-defined at the location of the source, a fact that is proven in ('!s Ilter 4. Moreover,

Ori also finds a problem in the -1, Hol~w regions", which occur wherever null rays (;?i,

incoming from infinity) have been blocked by the source. Apparently, the shadow region

has to be thought of as being identified for each mode independently. For point sources,

discontinuities are thought to develop across the shadow regions, although they have not

been observed in simple flat spacetime model calculations. Nevertheless, no complete

proposal has yet been developed to deal with these expected discontinuities. Earlier work

by Barack and Ori [24] -11--- -; that gauge freedom may phI i- a role in resolving these

1SSUeS.

1.3.2.2 Time domain treatment for Schwarzschild

In a different approach to the inversion problem, Lousto and Whiting [25] have

chosen to work in the time domain. Because of this choice their result is only valid in the




2 The term multiplying W in Equation 5-2 arose by repeated use of the Teukolsky
equation in quite a complicated expression, initially given correctly by Stewart [21], and
also obtainable from the results of C'!s Ilters 2 and 3 here. The full form of the expression
may still apply here.










Schwarzschild background, where angular separation is not dependent on separation in

time. Nevertheless, with the formulation of the Hertz potentials being set in the context

of radiation gauges, Lousto and Whiting were effectively unable to introduce sources into

their treatment. Regardless, several results of their analysis are noteworthy. Similar to

the analysis of R. -~ and Wheeler, Lousto and Whiting made use of angular and parity

decompositions, two features that have eluded application in the K~err background.

One unexpected feature of Lousto and Whiting's work is how algebraically special

frequencies emerge in a fundamental way. Algebraically special solutions arise when one

of I<'n or ~4 is ZeoO While the other is not, and then only for specific (complex) frequencies.

While this is inherently a frequency domain phenomena, it ptil- a crucial role in this time

domain approach. The algebraically special equation here has a source term depending on

the initial data for the Hertz potential-this term effectively corresponds to that which

arises for a Laplace transform. For the Schwarzschild background, all the algebraically

special frequencies are known and the algebraically special solutions have been found

explicitly [26], so the equations for this analysis could be solved by quadrature [25].

Attempts to generalize this technique to the K~err background have to date remained

unsuccessful.

1.3.2.3 Working in the Regge-Wheeler gauge

The RW formalism has been extensively used for Schwarzschild perturbations,

and its implications have been thoroughly investigated. In particular, full sets of gauge

invariant quantities are known, and in chosen cases these have been directly related to the

perturbed Weyl scalars Iel, and #4, Which are naturally gauge invariant [27]. Lousto [28]

has recently chosen to work with such a formulation, rather than with a Hertz potential

formulation. This immediately gives him freedom over gauge choice and it circumvents

the problems previously encountered with the introduction of sources. Having calculated

explicitly the dependence on sources, and knowing also how to represent all relevant










quantities through gauge invariant entities, Lousto thus succeeded in reconstruction

perturbations of Schwarzschild in a way that includes sources.

Lousto actually uses both I',, and #'4 in his construction. For concreteness and for

access to a vast body of prior experience, Lousto also chose to work in a gauge known as

the RW gauge. Note that Equations 1-21 and 1-22 are only valid in the IR G. However,

Ie', and #'4 are eaSily expressible in terms of an arbitrary metric perturbation, which allows

them to be written in terms of the RW variables for any choice of gauge. In the RH

gauge, I',, and #'4 become algebraic in the even parity sector and first order operators in

the odd parity sector. To provide enough conditions to solve for all the components of

the metric perturbation in terms of the Weyl scalars, Lousto must turn to the Einstein

equations (with sources), also in the RW gauge. It is in this way that reconstruction with

sources is accomplished.

The identification of gauge invariant quantities, beyond I',. and #'4, iS Virtually

nonexistent in the K~err spacetime and as pointed out several times before, the angular

decomposition there is not as robust as that available in spherically symmetric backgrounds.

In short, Lousto's work is quite notable for its inclusion of sources, but its reliance on RW

tools and techniques make it difficult to see how to extend the method to the K~err

background.

1.4 This Work

motivated by the success of spin coefficient formalisms in describing perturbations of

type D spacetimes and the incompleteness of current approaches, this dissertation presents

a new framework for perturbation theory that exploits the best features of both standard

treatments of perturbation theory and those based in the methods of a spin coefficient

formalism. As we will see, a natural feature of this formalism is that it applies to general

algebraically special spacetimes with little extra effort. Though our framework is quite

general and provides a new means of understanding perturbations of a wide variety of

spacetimes, we will keep our focus more narrow than that. In particular, the applications










we present here are primarily aimed at providing the missing pieces in the Hertz potential

approach to metric perturbations.

In ('!, Ilter 2, we will develop the formalism necessary for building our framework.

Additionally, the framework will be presented, which includes a new form for the

perturbed Einstein equations as well as a Maple package that aids not only in their

application, but any computation in the formalism of Geroch, Held and Penrose. C'!s Ilter

3 then provides a further discussion of both the RW and Teukolsky formalisms, phrased

in our framework. In ('! .pter 4, the necessary and sufficient conditions for the existence

of the IR G (in a larger class of spacetimes than we consider elsewhere) are determined

with the aid of our form of the Einstein equations. Chapter 5 then uses the IRG metric

perturbation to derive some important relationships between the curvature perturbations

represented by I~,, and ~4, which are of importance for the inversion problem described in

this chapter. Furthermore, this application showcases some of our Maple package's most

useful features. In C'!s Ilter 6 we then present a very different application of our framework

in conjunction with more standard techniques to address the issue of the non-radiated

multipoles .









CHAPTER 2
NEW TOOLS FOR PERTURBATION THEORY

In this chapter we develop the basic formalism we will be working within for the

remainder of this work. We begin with a description of the spin coefficient formalism

of N. i.--us! Ia and Penrose [22] and introduce the modifications of it due to Geroch, Held

and Penrose [9]. Within the latter formalism, we develop the properties of the general

class of spacetimes with which we will be working. Included is a discussion of gauge and

the general framework of relativistic perturbation theory. The chapter ends with the

introduction to the framework we will exploit in subsequent chapters.

2.1 NP

The ?-. i.--us! lIs-Penrose (henceforth NP) formalism has its roots in the spinor

formulation of General Relativity. Despite the great beauty and generality of the spinor

approach, we will approach the subject as a special case of the tetrad formalism. In

this view, the NP formalism is developed by (1) introducing a basis of null vectors for

the spacetime and (2) contracting everything in sight with unique combinations of the

aforementioned basis vectors.

We begin by introducing an orthogonal tetrad of null vectors, 16, n", m" and m", with

la and n" being real and m" and m" being complex conjugates. We will impose a relative

normalization

lnan = -mema = 1, (2-1)

with all other inner products vanishing. As an example to keep in mind, consider an

orthonormal tetrad on Minkowski space, (t", x", y, za), such that t"t, = -x"x, = -y"y,

-zaza = 1. Since the vectors are properly normalized, it is easy to verify that

1 1
la (t" + z), na (t" z")
(2-2)
1 1
m" = (xa iya), m" (Xa iya),









defines a null tetrad. It is important to note that there is some ambiguity implicit in the

above assignment, e.g. we can swap the roles of z' and x" (or y") in the above definitions

without changing the character (real or complex) of the null vectors or modifying their

inner products. We will return to this issue later in this section.

For simplicity, we introduce the following notation for our tetrad (borrowed from

C'I 1.in b I-iekhar [29]):

e*, = (1", na, m", ma),

where the tetrad index (i) = {1, 2, 3, 4} = {1, n, m, m}. In a further attempt to avoid

confusion we'll take spacetime indices from the beginning of the alphabet (a, b, c...) and

tetrad indices from later in the alphabet (i, j, k...). Just as the vector index can be raised

or lowered with the spacetime metric


e" gab = 6i~b and e(ij,gab =6t),

we may introduce a similar object for raising and lowering tetrad indices





For a properly normalized (Equation 2-1) null tetrad

0 1 0 0



0 0 0 -1

0 0 -1 0.

It then follows that we can express our spacetime metric as





where 1(,nb) a ~l~ b + bna).










An important notion in curved space is the connection. In standard tensor language it

makes an appearance through the Clan.!!.!1~ symbols. The equivalents in NP language are

called the spin coefficients and are defined, in general, by


Y(i)(j)(k) = e i) j)v a"(k)b.


(2-4)


It follows from the definition that


(2-5)


There is a total of twelve spin complex coefficients, individually named as follows


is= lamb ab

o- = mamb a 6,

p = mamb a b,

7 = namb a by


S--Ra b anb,

S= --ma b anb,

I- = -ma b anbb,

_T --a b anb,


(2-6)


and
/7 = ~(manb alb me b amb>,

a~ = ~(mamb amb me b anb>,

e = ~(l"Rb alb la b a b>

y = ~(namb amb na b a~b>.

Our e") naturally define four independent, non-commuting directional derivatives


e(i) -- e(i) 8xa


(2-7)


which are also given individual names:


8 m


a = n
8x" '
8


(2-8)










The field equations are obtained from the splitting of the Riemann tensor into a

trace-free part and its traces according to

1 1
Rabcd Cabcd + @ac bd + bd ac gbc ad gad bc) acgbd gbcgad) R. (2-9)
2 2

where Cabcd, abcd, ab and R denote the Weyl tensor, Riemann tensor, Ricci tensor and

Ricci scalar, respectively. Since both the Ricci tensor and the Ricci scalar vanish in the

absence of sources, the Weyl and Riemann tensors are identical in source-free spacetimes.

In that sense the Weyl tensor represents the purely gravitational degrees of freedom.

The Riemann tensor is then expressed purely in terms of the spin coefficients and their

derivatives by contracting all four vector indices with e )'s and making use of the Ricci

identity,

(Ve~b Vb a = Rabcd~d = abcd~d, (2-10)

where vd is an arbitrary vector. In four dimensions the Riemann tensor has twenty

independent components and the Ricci tensor has ten, leaving the Weyl tensor with ten

independent components. In the NP formalism, this translates into five complex scalars:

n = Cabcd a blc d

I = -Cabcd a blc d

',_=-Cabcdla blc d + a b c d), (2-11)

= Cabcdanb cnd

~4 -abcd Ra b c d










The Ricci tensor is represented by the following ten scalars:


1 1
oo0 = R1, 21 -24 ,
2 2

= -(R1 R34), ~02 2 R33,
1 1
oi 2R13, 22 2 -R22, (2-12)
1 1
~12 2 R23, 20 2 -R44 >

to =R14, 1 '
2 24

The field equations then follow from Equations 2-9 and 2-10. A full set of equations for

the NP formalism is composed of the commutators, the equations involving dependence on

matter, and the Bianchi identities. This is given in Appendix A.

2.2 GHP

In 1973 Geroch, Held and Penrose (GHP) [9] introduced some convenient modifications

of the NP formalism. Specifically, they identified the notions of spin and boost weight and

make explicit use of an inherent discrete symmetry of the NP equations.

In the NP formalism, there is an implicit invariance under a certain interchange

of the basis vectors which GHP have built on through the introduction of the prime (')

operation, defined by its action on the tetrad vectors:


(2-13)


A glance at Equations 2-6 and 2-7 -11- -- -; the adoption of a change in notation:





and similarly for the directional derivatives of Equation 2-8


D' = a and 6' = 6. (2-15)










While the metric is invariant under a Lorentz transformation, the tetrad vectors are

not. In the null tetrad formalism, a Lorentz transformation, which in general is described

by six parameters, is broken up into three classes of tetrad rotations. We will consider

only a tetrad rotation of Type III herel In the language of our Minkowski space example,

this amounts to a boost in the z t plane and a rotation in the x y7 plane. Under such a

transformation
za via
(1 v2 1/2 '
to Vza

(1 v2 4

x" = cos Oxa sin Of ,

y"a = sin Oxa + cos Of ~,

which translates to




(2-16)
~ a iB a




where r = J(1 v)/(1 ) The two transformations can be combined into one using

(2 = reie. Then Equation 2-16 may be summarized by



(2-17)



A quantity, X, is then said to be of type {p, q} if, under Equation 2-17, X ("(97.

Alternatively [9], we may ;?i that X possesses spin weight s = (p q)/2 and boost

weight b = (p + q)/2. The p and q values for the tetrad vectors can be read off from

Equation 2-17. They allow one to determine the spin and boost weights of the spin




1 Descriptions of the other types of tetrad rotation can be found in [30] or [29].









coefficients in Equation 2-6, while the spin coefficients in Equation 2-7 have no well

defined spin or boost weight since, under Equation 2-17, they pick up terms involving

derivatives of (. When acting on a quantity of well defined spin and boost weight,

the directional derivatives of Equation 2-8 by themselves also fail to create another

quantity of well defined weight. However, it is possible to combine the spin coefficients

in Equation 2-7 with the action of derivative operators in Equation 2-8 to construct

derivative operators that do produce new quantities with well defined spin and boost

weights. With x taken to be of type {p, q}, we can define these operators as follows:


(2-18)


where P and a are Icelandic characters named "thorn" and "edth''",epciey aho

these derivatives has some well defined type {r, s} in the sense that when they act on a

quantity of type {p, q}, a quantity of type {r + p, a + q} is produced. These new derivative

operators inherit their type from their corresponding tetrad vectors:


(2-19)


It is quite often useful to think of a (P) and 8' (P') as spin (boost) weight raising and

lowering operators, respectively. The derivatives in Equation 2-18 can be combined to

form a covariant derivative operator:



1 1 ao
= V, ( q)nb alb (p q mb amb.
2 2

We note in passing that this definition defines the "GHP connection." Our primary use for

Equation 2-20 will be to express things in GHP language via the replacement V, i 0,

With these definitions, all equations in the NP formalism can be translated into GHP










equations. Note that under prime, {p, q}' { -p, -q}, and under complex conjugation,

{p, q} { q, p}. A basic set of the GHP equations is given in Appendix A.

2.3 Killing Tensors and Commuting Operators

2.3.1 Specialization to Petrov Type D

In this section we provide a brief explanation of why the NP and GHP formalisms

are so specially equipped to handle problems in black hole space-times. For an arbitrary

space-time there are precisely four null vectors, k", that satisfy


kbkekleCabcb~dkyl = 0, (2-21)


where Cabcd is the Weyl tensor introduced in Equation 2-9 and the square brackets []

denote anti-symmetrization. The vectors k" define the so-called principal null directions

of the space-time. For some space-times, one or more of the principal null vectors

coincide. The general classification of space-times based on the number of unique

principal null directions of the Weyl tensor was given in 1954 by Petrov [31] and bears

his name. It turns out that all the black hole solutions of el-r mphli--;cal interest-including

Schwarzschild, K~err and K~err-Newman-are of Petrov type D, meaning they possess

two principal null vectors, each with degeneracy two. According to the Goldberg-Sachs

theorem [32] and its corollaries, for a space-time of type D with 1" and n" aligned along

the principal null directions of the Weyl tensor, the following hold (and reciprocally):


a = s' = a = a' = I<'n = ~1 = i' = ~4 = 0. (2-22)


This is equivalent to the statement that both 1" and n" are both geodesic and shear-free.

Thus, in the NP and GHP formalisms, all black hole space-times are on equal footing. In

the K~err spacetime, the commonly used tetrad (aligned with the principal null directions)









is the so-called K~innersley tetrad [33], which takes the form


l' = ,r 1, 0, (2-23)

Ra 2r 12, -a, 0, a) (2-24)
2 (r2 + 2 COS2 H

meL (ia sin 8, 0, 1, i/ sin 8) (2-25)
Z(r +ia cos 8)

Clearly, Equations 2-22 help simplify the GHP equations tremendously. However,

type D spacetimes are so special that their description in terms of the GHP formalism is

even further simplified. Such simplification is due in large part to the existence of various

objects satisfying suitable generalizations (and specializations) of K~illing's equation.

2.3.2 The Killing Vectors and Tensor

Virtually all of the ... I,!c" that happens when one considers type D spacetimes can

be traced back to the existence of a two-index K~illing spinor. Without delving into the

world of spinors we remark that a two index K~illing spinor [34-36], XAB = X(AB), iS a
solution tO2

VA'(AXBC) = 0, (2-26)

where A and A' are spinor indices and the parentheses denote symmetrization. The first

consequence of the existence of XAB iS that the quantity


(" = VA'Bi~ A __ --/3 qla at /J1 a'I1 am), (2-27)

is a K~illing vector--( satisfies

V(aib) = 0. (2-28)

The proof of this in spinor language can be found in [36], and the GHP expression can

be verified directly by making the replacement V, i 0 and utilizing the expressions in



2 Equation 2-26 is also known as the twister equation, which provides a different means
of understanding its relevance.









Appendix A. Generally speaking, (" is complex, and its real and imaginary parts satisfy

Equation 2-28 independently [36], so all type D spacetimes possess two independent

Killing vectors. These two K~illing vectors each give rise to a constant of motion along

a geodesic. In other words, if u" is tangent to a geodesic (Ub bUa = 0), then (su" is

conserved along u":




=0, (2-29)


where the first term vanishes as a consequence of (K~illing's) Equation 2-28 and the second

because u" is tangent to a geodesic.

In addition to the existence of two K~illingf vectors, the K~illingf spinor also gives rise to

the conformal K~illing tensor [35, 37]:


Pub XABXA'B' -" 2 T -1/3 (lanb) M mbOm)), (2-30)

which also exists in every type D background. The conformal K~illing tensor is alternatively

defined as a solution to

V(cPub)= 09(ab d c)d. (2-31)

Conformal K~illing tensors are useful because they give rise to conserved quantities along

null geodesics. If k" is tangent to a null geodesic (kb bk" = 0 and k'k, = 0) then the

quantity Pubk'kb is COnSerVed along k":


keVe(Pubk'kb) = k'kbkeVePub + 2Pubkek("Vekb)

=k'kbkeV~cPub)

(k, k")kcV bPbc










where we used the fact that k" is tangent to a geodesic in the second line and null in the

fourth line, along with Equation 2-31.

In certain instances we can extend this idea to provide a first integral of the motion

for timelike and spacelike geodesics as well. Such a notion can he realized by defining a

tensor, Kub = K~ab), that satisfies

VK~bc) = 0. (2-32)

A quantity satisfying this relation is called a K~illing-Staeckel tensor. Note that by

definition the metric and symmetric outer products of K~illing vectors both satisfy

Equation 2-32. We reserve the name K~illing-Staeckel tensor for an object that does

not reduce in this way. This is to be distinguished from the antisymmetric K~illing-Yano

tensor satisfying



which can he generally related to the K~illing-Staeckel tensor via Kub ac cYb [38, *

Because we will not make use of K~illing-Yano tensors here, we will follow conventional

language and refer to the K~illing-Staeckel tensor as simply a K~illing tensor. Returning to

the main line of development, given the existence of a K~illing tensor, we can recvele the

argument above (now using Equation 2-32 instead of Equation 2-31) for the conformal

Killing tensor to show that the quantity Kublilb is conSerVed for any it" tangent to a

geodesic, regardless of whether it he timelike, spacelike or null. The question then arises:

When can we find a Kub that satisfies Equation 2-32? To answer this question, we begin

by decomposing the K~illing tensor into its trace-free part and its trace, according to


Kuab = Pub + -Kgab, (2-3:3)


with Pubgub = 0 and K = Kubgub. Using this in (K~illing's) Equation 2-32 and dividing the

resulting expression into trace-free and trace parts gives two equations. The trace-free part

is simply Equation 2-31 and so Pub is the conformal K~illing tensor (as we anticipated with









our notation) which exists in every type D background. The trace part becomes


Va Pub ,bK = 0. (2-34)


The existence of a K satisfying this condition is both necessary and sufficient for the

existence of the K~illing tensor. By making the appropriate substitution (V, 8 ,), using

Equation 2-30 and taking components with respect to the tetrad vectors, we are led to the

followingf:
PK = (I<'_j )-1/3(p +p), S =-<_ -/
(2-35)
P'K = ( 2 2 -1/3(p / pt), S (22-/

By applying all the commutators in Appendix A to K and making use of Equation 2-35,

we arrive at a series of relations which we compactly write (following C'I .!1.4 I-ekhar [29])


p p' 7 '
(2-36)

These integrability conditions are both necessary and sufficient for the existence of a

Kt satisfying Equation 2-34 and thus provide necessary and sufficient conditions for

existence of the K~illing tensor in a type D background. They are satisfied for every

non-accelerating type D spacetime. These relations are the primary result of this section.

It is straightforw~ardl to verify: that K = (e- <;/3 --/3), Whe~re e"ic 1S a phase

factor whose origins will be described below in Equation 2-41. It follows that the K~illing

tensor may be expressed as

Kab~~ ~ 22-/(ab- ic -1/3 F(-ic 13 2ab. (2-37)


Historically, the K~illing tensor was discovered by Carter [40, 41] while considering the

separation of the Hamilton-Jacobi equation in the K~err background. The constant of

motion derived from the K~illing tensor is thus known as the Carter constant.

In a non-acceleratingf spacetime, where the full K~illingf tensor is available, the K~illingf

vector in Equation 2-27 is real up to a complex phase. If we specialize to the K~err










spacetime and the K~innersley tetrad, it takes the value M~-1/3 a, Where t" is the timelike

Killing vector of the K~err spacetime. To see this more generally we need to establish one

more fact. Consider the GHP equation and Bianchi identity:


Pp = p2 (2-38)

P9 2 2p~. (2-39)


We can rewrite Equation 2-39 with the help of Equation 2-38 as


Plnle'_ = 3p

= D(3 In p),


which gives us

2a 3 p, (2-40)

where C is a (possibly complex) function annihilated by P. This is in fact not a proof, but

rather the first step in one. A full proof would consist of showing that this is consistent

with the rest of the GHP equations and Bianchi identities. The coordinate-free integration

technique introduced in OsI Ilpter 5 is ideally suited for this. For now we take it as given

that the Equation 2-40 is true in every type D background, for some compleX3 COnStant,

C. It follows that

-- v- (2-41)
p C'1/3 1~,/3 1~'/3
which defines the phase factor introduced in Equation 2-37. It turns out that in all type

D spacetimes not possessing NUT charge, c = 0. More importantly, we now have the

relations

e~i (2-42)




3 In all type D spacetimes not possessing NUT charge, C is M~, the mass of the
spacetime.









which make it straightforward to see that for spacetimes without acceleration Equation 2-27

is real up to a complex phase (e2ic). NOte als0 that ( ,= -(e. What happened to the other

(linearly independent) K~illing vector? It is given by

Ob a ab 1 / ( 7 -13 ic -1/3 2 b ;*r


[e-'ic -1/3: eCic -:1/32 /'mb- Tmb) (2 -43)

Proving that this expression satisfies K~illingf's equation in general is a bit involved,

and since we'll have no direct use for Equation 2-43 in subsequent chapters, we refer

the interested reader elsewhere [36] for details. Once again, using Equations 2-42, it is

straightforward to see that Equation 2-43 is real up to a phase. Using the K~innersley

tetrad in the K~err spacetime, Equation 2-43 becomes


rib -b ~ b (2-44)


where t" is the timelike K~illing vector and *" is the axial K~illing vector. Because rib is

proportional to a, it clearly vanishes in the Schwarzschild spacetime. This can also been

seen by noting that, in the Schwarzschild spacetime, -r = -r' = 0 and thus comparisons of

Equations 2-27 and 2-43 reveal that the two K~illing vectors are not linearly independent

[42]. In [36] it is shown how one can infer spherical symmetry from this fact.

2.3.3 Commuting Operators

An important property of K~illing vectors is the fact that they commute with all of the

tetrad vectors:

4)gab = 2V(aib) = 0

= 24~(l~anb) m(amb))



where the first line follows from the definition of the K~illing vector and the second and

third from Equation 2-3. By contracting the last line with each of the tetrad vectors and










making further use of Equation 2-32, we establish that


g~l, = 7 ,,~ = 4~m, = 4~m, = 0.


Recall that for any two vectors, A and B, their commutator is given by [A, B] = tAB,

which establishes that the K~illingf vectors of the spacetime commute with all of the tetrad

vectors.

In this light, it is reasonable to expect that we can construct an operator, V,

related to the K~illing vector that commutes with all four of the GHP derivatives.

Because of the fact that spin- and boost-weights enter explicitly into the commutators

(Equations A-1A-3), we would also expect that any such operator would carry spin- and

boost-weight dependence. In fact, such an operator can be constructed. By taking as our

ansatz:

v = ("Be + pA + qB,

and computing all of the commutators, we can find explicit expressions for A and B.

However, this also requires that Equations 2-36 are satisfied, which implies a K~illing

tensor exists. For non-accelerating spacetimes we then have


ii = 2-/(r'd T' 'D pP' +~~1 -2 P +,(-5
2 2p'

where p and q refer to the GHP type of the object being acted on. This result has been

noted by Jeffryes [43], who arrived at it from spinor considerations. If we specialize to

the K~err spacetime and the K~innersley tetrad, it is easy to see that it takes the value

M~-1/3 e + bM~2/3 ~2 + 2 COS2 H-1, Where b is the boost-weight of the quantity being acted

on. Despite this difference between the vector (" and the operator V, we will refer to them

interchangeably as a K~illing vector. Similarly, we can follow the same procedure that led










to Equation 2-45 to obtain a similar operator associated with rl" (Equation 2-43):

P 21/ a -ic -1/3]( l pi -/ 2/


[e-ic -1/3 eic -1/3 2( / /)

+2(pv q)pp' 2-1/3e' 71 --/ -,1/3
(2-46)
2(pT +I q/ '2-136 -13 -/

1pe ,/3i ?-4ic 221/3 1/3 ;2/3 -2/3)

/ 4ic 2 Y/3 1/ -2Y/32/


which also commutes with all four GHP derivations.

On a final note, we remark that in recent work Be o;r [44] obtained an operator

related to K~illing tensor that commutes with the scalar wave equation. The operator

has the feature that it is first order in time. In this context it is tempting to ask if there

exists an operator analogous to those defined for the K~illingf vectors that commutes with

each of the GHP derivatives. The answer is currently unclear and so we leave it for future

investigation.

2.4 The Simplified GHP Equations for Type D Backgrounds

With Equations 2-36 in hand, we are now in a position to completely simplify the

GHP equations for the special case of type D backgrounds. Our starting point is the GHP

equations and Bianchi identities adapted to a Type D background:


Pp = p2 (2-47)

Dr =p~r 7')(2-48)

Sp = -r(p -p) (2-49)

B7 = 72 (2-50)

D'p a'r = pp' -rf 2a (2-51)









P9a 2 3p2 (2-52)




where we have omitted those equations that can he obtained directly by utilizing the

operations of prime and complex conjugation. By applying the commutators to ('2 and

making use of the equations above, we learn that


pp' = P'p (2-54)

8-r' = 8'-r (2-55)

Pr' = B'p. (2-56)


Note that the preceding equations hold for all type D spacetimes. Next we specialize to

non-accelerating spacetimes by making use of Equation 2-36 in the form Tr' = -~ in

Equation 2-56 to obtain

P-r' = 8'p = 2p-r'.(57

Now we compute the commutator [P, D']p and use the GHP equations and the appropriate

version of Equation 2-57 until we arrive at an expression in which the only derivatives are

8'-r and P'p. This expression can then he used with Equations 2-51 and 2-36 to find the

followingf two relations:


P'p =P pp ('+ (T ') -(2(2-58)

a'T =,, T'+p(p' p') + 2 (2-59)

(2-60)


and our task is complete. It is worth pointing out that due to Equations 2-36, these

expressions are not unique. This is a sign that there is some redundancy in the GHP

equations, which is to be expected when we consider such a special class of spacetimes. We

also point out that having expressions for every derivative on every quantity of interest is

sufficient (but not necessary) to completely integrate the background GHP equations. This










is a task that was first performed for the NP equations by K~innersley [33] and later by

Held [45] for the GHP equations. In ChI Ilpter 5, we will discuss the latter of these methods

in more detail.

2.5 Issues of Gauge in Perturbation Theory

One of the most important subtleties associated with perturbation theory in general

relativity is the concept of gauge invariance. The principal of general covariance tells

us that the interesting questions to ask are those that have answers that every observer

agrees upon. In the context of full (non-perturbative, nonlinear) relativity, this is ensured

by focusing on quantities that remain unchanged under coordinate transformations. In

perturbation theory, however, there is a new twist to the problem---there are now two

spacetimes of interest: the unperturbed background spacetime consisting of a manifold,

At with metric gAb (henceforth denoted by (Af, gab)) and the physical spacetime, (Af', 9ub '

that includes both the background and the perturbation. The question of how to relate

perturbations of quantities on (Af', 9u~b) to quantities on (Af, gab) in an unambiguous way

is fundamental for a well defined perturbation theory in general relativity. A complete

analysis, in the context of the GHP formalism, of this question was performed by Stewart

and Walker [46], whose basic results will be developed here. Before we address the

relativistic problem, we very briefly review first-order perturbation theory in a flat

spacetime. In that instance, we think of the quantity of interest, q = q(A), as being

parameterized by some A, so that q(0) corresponds to the unperturbed quantity and q(1)

is the fully perturbed quantity whose first-order perturbations we would like to consider.

It follows from writing q(A) as a Taylor series in A that the first-order perturbation, 6q, is

given by



To adapt this idea to our curved space problem, it helps to think of both the background

and physical spacetimes as members of a one-parameter family of spacetimes, (MAx, 9ub(X)

with A = 1 corresponding to the physical spacetime and A = 0 corresponding to the









background. Now suppose we've identified some geometric quantity of interest (could be

scalar, vector, tensor, etc., for simplicity we write it with no indices), Q = Q(A), and we

are interested in its first order perturbation, 6Q, towards the physical spacetime, evaluated

in the background. Before we can compute anything we must confront the issue of how

to relate quantities on two different curved manifolds. One can imagine introducing a

(suitably well-behaved) vector field, (", that connects points in the physical spacetime to

points in the background. Then, to compute 6Q, we evaluate Q at some point p + 6p in the

physical spacetime, pull the result back along (" to the background spacetime, subtract

from it the value of Q at a point p in the background, divide by 6p and take the limit as

6p 0 The mathematical apparatus for performing this task is the Lie derivative. Thus,

the first order perturbation, 6Q, to a quantity, Q, evaluated in the background spacetime

is given by

6Q = 4Q(A)~= (o (261)

The important point about this prescription is the fact that (" not only fails to be unique,

but there is, in general, no preferred choice for it. A choice of (" is more commonly known

as a choice of gauge. According to Equation 2-61, the difference between 6Q computed

with (" and rl" is given by

sQg sQ, = 4_~,Q,

and so we define 6Q, the gauge transformation of 6Q by


6Q = 6Q1 4Q3. (2-62)


Note that a gauge transformation in this sense represents a change in the way we identify

points in the physical spacetime with points in the background. This is to be distinguished

from a coordinate transformation, which changes the labeling of coordinates in both the

physical and background spacetimes.

The significance of Equation 2-62 is that unless 4~Q = 0 for every (", there is some

ambiguity in identifying the perturbation-we can't differentiate between the contributions










of the perturbation (6Q) and the background (4~Q). Quantities that satisfy 4~Q = 0

for every (0 are therefore called gauge invariant. It is straightforward to see that the

perturbation of Q is gauge invariant if and only if: (1) Q vanishes in the background,

(2) Q is a constant scalar in the background or (3) Q is a constant linear combination of

K~roenecker deltas. This is a result originally due to Sachs [47]. A direct consequence of

this fact is that the metric perturbation, arguably the most fundamental quantity we deal

with, fails to be gauge invariant. Fortunately, type D spacetimes come equipped with two

gauge invariants, I,, and tb4, Which have simple expressions in terms of the components of

the metric perturbation. As we will see, appropriate use of gauge freedom simplifies our

computations tremendously.

2.6 GHPtools A New Framework for Perturbation Theory

With the basic formalism in place, we are ready to present the tools that form the

basis of the subsequent chapters. The motivation for our framework comes from two

places: (1) the desire to take advantage of gauge freedom in standard metric perturbation

theory and (2) the success of the GHP formalism in perturbation theory. As mentioned

in the previous chapter, gauge freedom proved absolutely crucial for the RW analysis

and that of Cohen & K~egeles [20], C!!. I.1, i.---1:! [18], and Stewart [21], and it will

certainly pll li- a central role in any future description of metric perturbations. The

second ingredient, the GHP formalism comes with several advantages. First of all, the

inherent coordinate independence and notational economy makes calculations in general

spacetimes tractable. Furthermore, by virtue of the Goldberg-Sachs theorem, we can deal

with the entire class of type D spacetimes at once. Additionally, spin- and boost- weights

provide useful bookkeeping and, as we'll see, a useful context for understanding the roles

that various quantities pll li-. Last but not least, the use of a spin coefficient formalism has

proved absolutely crucial for studying perturbations of anything other than spherically

symmetric spacetimes. We will put these ideas together to compute the perturbed Einstein

equations in a mixed tetrad-tensor form. This is the heart of our work.









2.6.1 Einstein's New Clothes

The main idea behind our framework is to reorganize the tensors of interest into their

tetrad components. The metric perturbation, for example, has the decomposition


hub = ~'.. "' A b un a~~b + 2hlul(anb) + 2hmmm(,m b)

21hmnja~b) 2htlm76iamb) 2humll~,mb) 2hu-mljamlb) (2-6(3)

+ mmmemb mmmammb,

so that, for example, hit = hub a b. In order for this to be valid within the GHP

formalism, each component of Equation 2-63 must have a well-defined spin- and

boost- weight. Because the background metric (Equation 2-3) is invariant under

a spin-boost (Equation 2-17) it has type {0, 0}, which must also be the type of the

metric perturbation, hab. Therefore the type of the individual components of the metric

perturbation are determined by their tetrad indices:

hit : {2, 2} a {2 2

him : {2, 0} hm : {-2, 0}

him : {0, 2} hm : {0, -2} (2-64)

kmm : {2, -2} kmm : {-2, 2}

hi, : {0,0 kmm,,: {0, 0}.

All of the vectors and tensors we will concern ourselves with can be treated in this way.

It is worthwhile to stop here and take a look at what Equation 2-63 really means.

Comparing with our treatment of Schwarzschild (Equation 1-4), we note that the scalar

parts of the metric are "mixed up" in hu, hin and h,,, all of which have spin weight zero

but differ in boost weight. Similarly, the vector parts are given by him, hm and their

complex conjugates and likewise the tensor pieces are given here by hmm, hmm and hmm.

However, these identifications are completely independent of the background spacetime.

Thus, in a certain sense, Equation 2-63 provides a generalization of the RW mode










decomposition that takes into account both spin- and boost- weight. In the next chapter

we will make some more precise statements in this direction.

Recall our expression for the perturbed Einstein equations:

1 1 1
Sab c ~Cc ab ~a b cc ~c(a b)c gab ~c c dd ~c d cd)
2 2 2

By making the replacement V, i 0 and understanding hab aS referring to the tetrad

components of the metric perturbation given in Equation 2-63, we arrive at the perturbed

Einstein equations in GHP form:

1 1 1
Sab c~c ab -OaOb" cc c8O(a b)c gab c"Oc dd Oc d cd), (2-65)
2 2 2

which (right now, at least) don't look all that different! The tetrad components of

Equation 2-65 for an arbitrary algebraically special background spacetime are given in

Appendix B. Aside from the obvious cosmetic differences, there are several key distinctions

between Equation 2-65 and the standard form of metric perturbation theory worth

pointing out. First of all, our form lacks the background Einstein equations present in the

standard treatment. Taking their place are the background GHP equations and Bianchi

identities. Perhaps more importantly is the inherent coordinate independence. Coupled

with the concepts of spin- and boost-weight, this allows for a certain structural intuition

not present in coordinate based techniques. This point of view will be stressed throughout.

Writing Equation 2-65 is one thing, but actually computing it is another question

entirely, which we now turn our attention to.

2.6.2 GHPtools The Details

To perform such a computation for an arbitrary background spacetime is no small

task, even (or rather especially) in the standard tensor language. For this the aid of Maple

was enlisted. Unfortunately, at the time the computation was performed, there were no

Maple packages available for performing all such computations at the level of generality










required NatUrally, One WaS developed. It has been dubbed GHPtools and the Maple

code for it is the content of Appendix C. The remainder of this chapter is devoted to

explaining its basic use and functionality through a simple Maple worksheet.

Every session begins by invoking GHPtools:

> restart;

> with(GHPtoolsvi);


[BD1, BIlc, BIlp, Bllpc, BIS, BI2c, BI21p, BI-~ ., BIS, BI3c, BI31p, BL,l~ BIg BI~c,
BI~p, BI~pc, COM1l COM~lc, COM~lp, COM~lpc, COME2, COM~2c, COM~2p, COM )..~
COMS3, COM~3c, COM~3p, COM,;).. DGHP, GHP1 GHPlc, GHP1p?, GHPlpc,
GHPB, GHPENP, GHP~c, GHP21p, GHP ).. GHPS, GHP~c, GHP31p, GHP,li..
GHP4 GHP~c, GHP~1p, GHP~pc, GHP5, GHP~c, GHPS1p, GHP~lpc, GHP6,
GHP~c, GHPG1p, GHP~lpc, GHPconj, GHPmult, GHPp~rime, NPconj, NPexp~and,
NPp~rime, comm, ezcomm, fblw.i;:, getpq, schw, idsimp?, idspec, letcon, tetdnK,
tetdnS, tetdnSB, tetupK, tetupS, tetupSB, typed]

To begin with, each variable is directly specified by its usual name. For example

p would be entered in Maple as conjugate(rho). The primed variables have a '1'

appended to the end, so that p' would be entered as conjugate(rhol). The Weyl

scalars are recognized as capital W's with the appropriate number, eg. Psi2. The

derivatives P, 8, P' and 8' are recognized in Maple as th(), eth(), thp() and ethp(),

respectively. GHPtools recognizes the tetrad vectors as labels indicating the position

of the index with the actual index in parentheses. For example la and me would be

input as lup(a) and conjugate(mdn) (c). Finally, GHPtools contains an arbitrary

function, 95 (in Maple: phi), that is quite useful for general calculations. Amongst




4 There is however a series of papers describing rather sophisticated Maple packages
that perform some of the manipulations that we want [48, 49], called GHP and GHPII.
We stress that GHPtools is no way intended to compete with these or any other Maple
packages .










these variables, GHPtools computes the primes and complex conjugates through the

procedures GHPprime () and GHPconj ():

> GHPconj (GHPprime (rho+conjugate (rho)));

pl + pl

> GHPconj (Psi2);



> GHPprime (conjugate(mdn) (a));

mdn(a)

The {p, q} type of any quantity may be obtained by the use of the getpq function,

which returns p and q, in that order:

> getpq(Psi2);

0, O

> getpq(rhol);

-1, -1

> getpq(phi);

1pp, pq

Note that 4 is given the general type pp, qq. Before any computation begins it is often

useful to specify the spacetime in which subsequent computations are to take place by

specifying the value of the global variable spacetime:

> spacetime := typed;


spacetime := {@20 =0, #21 = 0, a = 0, #10 = 0, #22 = 0, II= 0, al = 0, #12 = 0, E = 0,
93 = 0, 94 = 0, 91 = 0, #11 = 0, 90 = 0, N1 = 0, a = 0, 000 = 0, 001 = 0,
002 = 0, O = 0, #12 = 0, #20 = 0, #11 = 0, #21 = 0, #22 = 0, al = 0, a = 0,
a = 0, 002 = 0, #10 = 0, N1 = 0, 90 = 0, 91 = 0, 93 = 0, 94 = 0, 000 = 0,
001 = 0}









Aside from typed, acceptable values for spacetime are flatxyz (11m~l:0,wski space in

Cartesian coordinates, where all the GHP quantities vanish), schw (the Schwarzschild

spacetime, a specialization of type D where -r = -r' = 0 and all other quantities

are real) and none (completely arbitrary spacetime, this is the default if spacetime

is unspecified). The user is free to change the value of spacetime in the middle of

a worksheet and only subsequent evaluations will be affected. For simplicity, we

will henceforth restrict our attention to examples with typed specified. The GHP

equations and Bianchi identities (as well as their primes, complex conjugates and

conjugate primes) are implemented as Maple procedures so that their specification to

the declared spacetime is returned. For example, if typed is specified, then


> GHP1();
> GHPipc () ;

th(p) = p2

thp(pl) = pl2
> BI2() ;
> BI2p();

th(W2) = 3 p 2

thp(W2) = 3 pl 2

The real usefulness of GHPtools comes not from its bookkeeping abilities, but

rather its ability to perform symbolic computations within the GHP formalism.

These abilities begin with the DGHP () procedure, which expands derivatives of objects

occurring in an expression in accordance with the rules of derivations. For example


> expr := rho*rhol + taustaul (conjugate(rho)*rhol)^3 +
> In(conjugate(tau)*conjugate(taul));

exp~r := p pl + -r 71 pl3 p3 n- -rl)

> th(expr);









th(p pl + -71r- pl p~ +In( r71))


> DGHP(%);


th(p) pl + p th(pl) + th(-r) 71l + -r th(-rl) :3 pl2 th(pl) p3 :3 pl3 p2 th(p) +th)

th(-rl)


To date, DGHP () can handle powers and logarithms (the only functions this author

has encountered in the GHP formalism), but the procedure can he easily modified to

accommodate just about any function. Building complicated expressions involving

linear combinations of derivative and multiplicative operators is easily achieved with

the help of the GHPmult () procedure. These expressions can then he expanded with

DGHP(). As an example, consider the expression (P p)4:


> th4phi :=
> GHPmult (th-rho, GHmult th-rho, Homut (h-ro Gult (h-h, h)))

th~phi := th(th(~ p th( ) + p2 _) 2 th( ) p3 ) p th(~ p th( ) + p2 ~
+ p2 ~I-, 3 th( ) + p4
.!:= th(th( ) p ~)
> DGHP(th4phi);

-6 p2 th(p) + 12 p th(p) th( ) + 3 th(p)2 + 4 p th(th(p)) 4 p3 th( ) + p4
th(th(th(p))) 4 th(th(p)) th( ) 6 th(p) th(th( )) + th(th(th(th( ))))
4 p th(th(th( ))) + 6 p2 th(th( ))

Simplifying such expressions is, in the context of type D spacetime without acceleration,

handled by the tdsimp () procedure that substitutes the known values of the

derivatives of the spin coefficients (stored in the globally available list tdspec; such a

procedure can he easily generalized to encompass any spacetime, should the need arise)
into its argument. Thus our previous example simplifies considerably:

> tdsimp (DGHP (th4phi));









th(th(th(th( )))) 4 p th(th(th( )))



Perhaps even more useful is the comm() procedure which commutes derivatives on an

expression. It takes two arguments: the first is the term whose first two derivatives

will be commuted and the second is the expression into which the result will be

substituted. Consider the following examples:


> commute_mel := eth(th(ethp(ethp(phi)))) -
> th(eth(ethp(ethp(phi))));

commute _mel := eth(th(ethp (ethp () )))) th(eth(ethp (ethp () ))))

> DGHP(comm(eth(th(ethp(ethp(phi)))),commuteml)


-p eth(ethp (ethp( ))) + r1 th(ethp(ethp( ))) 2 p -1 ethp(ethp( ))
-p rl1ethp(ethp( )) pq
> commute_me2 := th(th(thp(phi)))-th(thp(th(phi)));

conr;;;n;i. _I,,. :' := th(th(thp( ))) th(thp(th( )))

> tdsimp(comm(thp(th(phi)),commute_me2));


th(eth( )) 7 + eth( ) p- eth( ) p-1 th(eth( )) 71l 2 eth( ) p -1 + th(ethp( )) -r
+ethp() )p r- ethp() p 1-rl-th(ethp( )) 71l- 2ethp( )p -1+ pp th() )7 71
+: pp p r r1- pp p r11-rlpp th() )W2 3pp p W2 +pqth() )771r
+3 pq ~p7 r1 pq p -r171r pq th( ) W2 :3 pq p W2

Computing the perturbed Einstein equations and Weyl scalars necessarily requires the

ability to contract various combinations of the tetrad vectors. This functionality is

provided by the tetcon() procedure, which also takes two arguments. The first is the

expression that contains the uncontracted vectors and the second is a list of the indices

to be contracted over. Take the example of computing the trace of the metric:









> gdn := Idn(a)*ndn(b) + Idn(b)*ndn(a) mdn(a)*conjugate(mdn) (b) -
> mdn(b)*conjugate(mdn) (a);
> gup := subs({1dn=1up, ndn=nup, mdn=mup},gdn);

gdn := Idn(a) ndn(b) + Idn(b) ndn(a) mdn(a) mdn(b) mdn(b) mdn(a)

gup := lup (a) nup (b) + lup (b) nup (a) mup (a) m up (b) mup (b) m up (a)

> tetcon(gdn*gup, {a,b});





Finally, GHPtools provides some functionality for translating expressions into NP

expressions that can subsequently be converted to ordinary coordinate expressions.

This functionality is provide by the aptly named procedure GHP2NP (), which takes as

its input a GHP expression. The functionality provided by the procedure is limited

to expressions involving at most two derivatives. Furthermore, the derivatives must

appear in a specified order according to the following rules: (1) a and 8' must ahr-7i-

appear to the left of P and P', (2) B must appear to the left of a' and (3) P must

appear to the left of P'. Take the following example:



> GHP2NP(th(thp(hln))+eth(ethp(hln)));

DD(a(hln)) + E A(hln) + E A(hln) + 6(6(hln)) + /36(hln) + p316(hln)



In order to aid in the conversion of such quantities into coordinate expressions,

GHPtools contains, as lists of arrays, some commonly used tetrads in the K~err

spacetime. They are: the K~innersley tetrad with indices up tetupK and down tetdnK,

the symmetric tetrad (tetupS, tetdnS) and the symmetric tetrad boosted by a

function B(t, r, 8, 4) and spun by a function S(t, r, 8, 4) (tetupSB, tetdnSB). These

are called simply by invoking their names:









> tetdnK;


mdn = 2 (r a cos(0) I) 2/,lasn) -I(2 2 S 6
r + a cos(0) I 2 r + a cos(0) I

1l r2 2 M\Ir + a2 (2 2 M\Ir + a2) Sin $)2
ndn = 0 -
L2 (r + a cos(0) I) (r a cos(0) I) 2' '2 (r + a cos(0) I) (r a cos(0) I)1 '
Idn ~- (r + a cos(0) I) (r a cos(0) I)r ~


l a sin(0) 1 2-I(2 + 2) Sin 6)
mrdn = 2 0, (r + a cos(0) I) 2/,2
r a cos(0) I 2 r a cos(0) I


Though we may not make explicit reference to it, GHPtools will be (and has been in

earlier parts of this chapter) used extensively both to obtain and verify the result at hand.









CHAPTER 3
REGGE-WHEELER & TEUK(OLSK(Y

As a first application of our framework, we will provide a more detailed discussion of

the R;- -ar--Wheeler and Teukolsky equations. This leads quite naturally to a discussion

of the metric perturbation generated from a Hertz potential, which will phIi-. a ill l.) .r role

in subsequent chapters. Our starting point is a general discussion of parity that does not

assume either spherical symmetry or angular separation from the outset.

3.1 Parity Decomposition of Spin- and Boost-Weighted Scalars

One crucial feature of the R;- ear--Wheeler analysis is the identification of even and

odd-parity modes. In the context of spherically symmetric backgrounds, where angular

dependence can be separated off using spherical harmonics, it is sufficiently simple to

achieve this decomposition by considering the behavior of the spherical harmonics under

a parity transformation directly. For (scalar, vector or tensor) functions defined on more

general 2-surfaces, this task can be cumbersome, if not outright impossible. Furthermore,

narrowing our focus to angular functions obfuscates the fact that there is something more

fundamental happening. It is the goal of this section to provide a more general description

of the parity decomposition, applicable to more general 2-surfaces without appealing to

separation of variables. We will also see that the GHP formalism is uniquely suited to

this description. The decomposition theorems we make use of are proven by Detweiler and

Whiting [50].

Our first assumption is that our spacetime manifold, M~, admits a spacelike, closed

2-surface, S, topologically a 2-sphere, with positive Gaussian curvature and a positive

definite metric given by



where m, and m, are two members of a null tetrad. For a spherically symmetric

background Fab is proportional to the metric of the (round) 2-sphere and m" and m"

can be directly associated with the background metric. More generally, we allow for the










possibility that m" and m" are in fact not the null vectors associated with the background

spacetime, but rather just two (complex) null vectors tangent to S. In that case 1" and n"

will then be identified with the two null directions orthogonal to S. We will use the same

notation regardless of whether the tetrad is globally defined or just in some neighborhood

of S and the application at hand will dictate the appropriate interpretation. The metric's

other role as the projector into S can be realized by simply raising one index


,b --mb ~b (2


For example consider some vector, v", defined in the spacetime:


v" = I 2a" + val"a vmm" v~mm"


The restriction of v" to S is simply given by the projection


b,', vm ~b


which generalizes to n-indexed tensors by projecting each index individually. We can then

carry out covariant differentiation within S by simply taking the full covariant derivative

and projecting back into S with aqb. The final object we introduce is the Levi-Civita

symbol on S, which, in tetrad language takes the following form:


Eab Eluab = i@memb memb). (33)


These are all the tools necessary for what follows.

We begin by considering the projection of vectors defined in the spacetime onto S. To

identify the odd and even parity pieces we start by decomposing a general vector on S


~a = Gab~b even Eab~b odd

-ma(8(eve i'Codd a evn od









where (s,,, and (3,,, are real spin-weight 0 scalars (type {b, b}; b indicating the boost-weight).

Thus, given a quantity with boost-weight b and spin-weight 1, the even parity piece

is simply ifev,, and the odd parity piece is iifor,,/ Similarly the complex conjugate of

such a quantity (same boost weight, but spin-weight -1) has even parity piece a'(ee,,

and odd parity piece -id'(oric. The relative minus sign between an odd-parity object

and its complex conjugate is a possible source of confusion, so we must he careful when

performing parity decompositions.

Symmetric, trace-free two-indexed tensors on S also have a simple parity decomposition.

It is easy to recognize the (two) components of such tensors as spin-weight +2 scalars.

That is, the components are of type {b + 2, b + 2}. We consider the parity decomposition

on S by creating the tensor from a vector on S ta, with boost-weight b and spin-weight 0:


Xab = nc~ckb + bc~cks Jub~cd~ckd, (3-5)


which can in turn he further decomposed into its even and odd parity pieces by applying

Equation 3-4 to yield

Xub = L( (2(cb) 8c~cd Jub~cd~c(~ d)xever, + 2(cib d8c~dhele




which provides us with a means of identifying the even and odd hits of symmetric

trace-free tensors on S. This result generalizes quite easily to n-indexed symmetric

trace-free tensors (with components of spin-weight in and boost-weight b) on S:







1 This agrees with the correspondence between the even and odd parity vector and
tensor spherical harmonics and the spin-weighted spherical harmonics (see Thorne's review
[8] for details) (i.e., the "i" comes along for the ride).










Finally, we remark that scalars naturally arising from contractions of tensors in the

spacetime with various combinations of la and if have no components in S and are thus

all of even-parity. Note that such objects necessarily have zero spin-weight. This provides

enough information to characterize the parity of arbitrary objects.

In practice, we are generally given some spin- and boost-weighted scalar, q' (and/or

its complex conjugate), and we merely want to identify the even- and odd-parity pieces

without explicitly decomposing it according to Equation :37. In this case Equation :37

allows us to do so by simply writing





In the context of a spacetime where 1" and if are fixed by considerations other than being

orthogonal to of and of (e.g. Petrov type D, where we would like them aligned with

the principal null directions), but of and of fail to form a closed 2-surface (the K~err

spacetime provides one such example; this can he seen by noting that B and a' don't

commute), the question arises of whether or not something like Equation :38 is still useful

to consider. It appears so. In such a case the decomposition theorems (the first lines of

Equations :35 and :34) fail to be true, but this isn't a serious issue. Because a~b and Feb

still allow us to decompose tensors into their "proper" and pI-, ud.I" pieces, in place of

Equation :37 we have


70..0, = (-1)nn![n s .. z ,,3 tee, i .,,)+ i ,. .ni,,3 sve + ir ,,) (:39)


where i..~ 1 and "odd" are written in quotes to emphasize the fact that they really

refer to real and imaginary in this context and the bar over tau indicates the proper spin-

and boost-weight. Clearly, Equation :39, lacks the advantage present in Equation :37

of being able to put all of the angular dependence into B and a' and regard the entire

tensor as arising from the two real scalars -rev,, and -r,., Nevertheless it provides a useful

decomposition of spin- and boost-weighted scalars, without separation of variables, that










allows us to use Equation :38 in arbitrary backgrounds. Furthermore in the limit that of

and of become surface-forming (e.g. the a 0 limit of the K~err spacetime), Equation :39

becomes Equation :37. This is one avenue for understanding why parity pt i.is such

an important role in the perturbation theory of spherically symmetric backgrounds.

In the context of null tetrad formalisms we can see the seemingly unmotivated act of

performing parity decomposition, which does not generalize well, as arising from the

quite natural (and perhaps more fundamental) act of separating quantities into their

real and imaginary parts, which is entirely general. In this light, it makes sense that our

attention would be focused on parity because the first perturbative analysis took place

in the spherically symmetric Schwarzschild background in which one cannot differentiate

between the two decompositions but parity has significance there. Regardless, the only use

we make of these results, except for some remarks in ChI Ilpter 5, is below in the case of the

Schwarzschild background where the point is moot.

3.2 Regge-Wheeler

In this section we will provide a perturbative analysis equivalent to that of R< -~-- and

Wheeler for the odd-parity sector of the Schwarzschild spacetime. Though the results are

well known, our methods and language are sufficiently different and original that they shed

some new light on and bring an interesting perspective to the subject. The two keys to

our analysis are essentially the same as those of RW: the parity decomposition and the

RW gauge. Having already discussed the former, we will look now at the latter before

proceeding with the analysis.

3.2.1 The Regge-Wheeler Gauge

R< -~-- and Wheeler describe their gauge choice in terms of the -e-mode decomposition

of a gauge vector. This description is inadequate for our purpose and so our first task is to

translate the RW gauge into mode-independent form. This has been performed by Barack

and Ori [24] who obtained

sin2 Bh a ha~ = 0, (:310)










how = 0, (3-11)

sin 08oe(sin Ohte + 8444)> = 0, (3-12)

sin 08o(sin Ohro + 84&,4) = 0, (3-13)


as the mode-independent expression of the RW gauge. Now we can transform this

description into GHP language. It is a relatively straightforward process now to write the

tetrad components of the metric perturbation (hiz, hi,, etc.) in terms of the coordinate

components of the metric perturbation (htt, h,,, etc.) and invert the relations. With

this knowledge in hand, it becomes evident that Equations 3-10 and 3-11 are simply

combinations of

Amm =0 and hm = 0.

The effect of these conditions is to remove the spin-weight +2 pieces from the metric

perturbation. After a quick look at the coordinate form of the a and 8' operators, we note

that Equations 3-12 and 3-13 are combinations of


B'hlm + c7l;.. = 0 and allt, + B'hm = 0,


which restricts the form of the spin-weight +1 parts of the metric perturbation. Note that

the essence of the RW gauge lies in the fact that all of the information about gravitational

radiation gets pushed into the spin-1 components of the metric perturbation.

In this language, it is natural to generalize these conditions to more general type

D spacetimes on the basis of spin-weight considerations. The spirit of the RW gauge

-II- -_ -r ;that we keep the requirement that no spin 2 components enter the metric

perturbation. The requirement on the spin 1 components is easily generalizable by putting

in pieces proportional to -r and -r' which both vanish in the Schwarzschild background.










The resulting proposal for a generalized RW gaugfe is


hmm = 0,

hmm = 0,
(3-14)
(B + atr + b-r')hlm + (8' + atr + b-r')hlm = 0,

(8' + b-r + a-r')hm + (B + b-r + a-r')hm = 0,

where a and b are (generally complex) constants that must be determined by some

other means. Note that the form of Equations 3-14 is restricted by requiring the gauge

restrictions to be invariant under both prime and complex conjugation. The full utility of

the generalized RW gauge remains to be explored, but it is clear that any simplification it

brings will apply uniformly to all type D spacetimes.

3.2.2 The Regge-Wheeler Equation

With the pieces in place, we turn our attention to the odd-parity perturbations of the

Schwarzschild spacetime. Starting with the description of the background, we have


p = p, p' = p', and '_= ,(3-15)


with all other background quantities vanishing, so the situation is immediately simplified.

Next we proceed with the parity decomposition by writing the components of the metric

perturbation as, for example, him = h +Ib, him h""" ihgg, etc. Note the relative

minus signs between the odd-parity bits and their complex conjugates. From here on

we will specialize to odd-parity and thus drop the "odd" labels and factors of i since no

confusion can arise. With this specialization, our gauge conditions now read:

hmm 0

kmm = 0
(3-16)
a'hlm c7le.. = 0

B'hm cll, = 0.









Putting these simplifications into the (odd-parity) Einstein equations, we see that





is satisfied identically by virtue of the gauge conditions. Furthermore, we have that

SIm = -Stra (and likewise for the nm and uti components), which is no surprise because,

as a tensor, Sub respects the parity decomposition. This leaves us with Sinz, S,m and Enzo.

Starting with the last piece, we can commute the derivatives to write


Enzm = B((p' p')hnz + (P p)hnm>z = 0,


which we can !~Ita.,i Il~e" by peeling off the B to give us


(P' -p')Anz + (P p)h,mz = 0 (:317)

(P' -p')Anz + (P p)h,z, = 0, (:318)


where the second relation follows from complex conjugation of the first (or integration of

Enzo), and we have set the !.Il, I s!i .11.1 <..0!-I .11' to zero for convenience (it would cancel

helow). We now turn our attention to Sim. By successive applications of Equation :317 we

can eliminate all terms involving Ph,tm, arriving at


Sim= (S' 29' +4pP' pp' -4t/'2 Im 2p2 ims

Taking the prime of this (which introduces an overall minus sign because of the parity

decomposition) leads to a similar expression for S,m. Next we take the (sourcefree)

combination

(P' 2p')B'Sim + (P 2p)BS,z~ = 0. (:320)

We can remove from this expression all references to al;,.. and a'h,m: using the gauge

conditions in Equations :316, which, after some serious commuting leads to the quite










simple expression:


{>'9 a'S p'D pb' +t 1 }

This is the R< -~---Wheeler equation. We can clean it up a bit by recognizing the object

being acted on as 2 j"Ll = P17l; p'8'hln,, the odd-parity piece of the perturbation of I<

Furthermore the operator in Equation :321 is the wave operator, 0, in the Schwarzschild

background up to a factor of 1/2. Making these identifications, we now have for the

R< -~---Wheeler equation:

(O + 8t/'2 '--2/3 11L = 0. (:322)

A similar equation for ,: = Im( _') was previously derived by Price [51] (whose

only relation to the present author is Equation :322), who showed that (modulo

angular dependence), Im(t/') is the time derivative of the R< -~---Wheeler variable.

Moreover, without reference to Im _', Jezierski [52] arrived at an equation for odd-parity

perturbations that is essentially identical to Equation :322, though phrased in more

standard language. Additionally, an analysis by Nolan [5:3] who looked at the perturbed

Weyl scalars in terms of gauge invariants of the metric perturbation showed explicitly the

relation between Imt/' and the gauge invariant quantity associated with the RW variable.

Furthermore, Nolan points out that because I', is real in the background, the perturbation

of its imaginary part is, when we restrict our attention to odd-parity, gauge invariant in

the sense discussed in ChI Ilpter 2. Perhaps more surprisingly, Nolan further asserts that

this is true of the perturbations of all the Wevl scalars, which emphasizes the fact that

odd-parity perturbations of spherically symmetric spacetimes are obtainable by virtually

any means.

One thing that sets our treatment of RW apart from others is our sparing use of

spherical symmetry. The only place we make explicit use of it is in Equations :315,

which defines the background GHP quantities. This certainly simplifies the subsequent

calculations considerably, but fails to fully exploit the background symmetry. In










particular, our implementation of the parity decomposition without separation of variables

generalizes quite nicely to spacetimes where parity isn't a good symmetry because we

didn't actually take the step of writing the components of the metric perturbation as

spin-weight 0 scalars with the appropriate number of a's or a''s. The fact that this process

has eluded generalization to the K~err spacetime has more to do with difficulties there than

the particular techniques we utilized, which are fairly general. This stands in contrast to

existing treatments that fully exploit spherical symmetry from the outset and are thus

exclusively applicable in these situations.

The Zerilli equation [7] describing even-parity perturbations of the Schwarzschild

spacetime has so far eluded a direct description in terms of gauge invariant perturbations

of the Weyl scalars. However, the information contained within the Zerilli equation can he

obtained through the metric perturbation that follows from the Teukolsky equation, which

is the focus of the remainder of this chapter.

3.3 The Teukolsky Equation

In contrast to the RW equation, which has it origins in the description of metric

perturbations, the Teukolsky equation [10-12] came directly from considering perturbations

of the Weyl scalars. We, however, are interested in obtaining it directly from the Einstein

equation. Using Teukolsky's expressions for the sources of I',, and #'4, we can obtain this

directly. The sources of the Teukolsky equation are given by


To = (B -r' 4r) [(P 2p)W,> (B ')4]

+(P 4p p)[(B 2r')W,> (P p)luzz], (:32:3)

T4 =('- -47)('-2p), (- 1]

+(p' 4p' ') [(8' 2r)l,, (P' p')luz], (:324)


where To and T4 are the sources for I',, and #'4, Tespectively. 1\aking the replacement

Ib Wab in the equations above leads (after properly rearranging the derivatives with









the help of GHPtools) to the Teukolsky equations. They are


[( -4p )(' ')- 8 4 -f'(' 7) S'_,', = 4xrTo, (3-25)

[(D'~~~~~~~~ p '( ) 8 7 )( )-3']4 T4, (3-26)

where, in terms of the components of the metric perturbation


,,- (8 -T')8 -f')zz (P p)(P p)hmm


[(P P)(B 2r') +(di 7')(> 2p)]him (3-27)




(D' p'( 27)h + (p' 7)(p' 2p')]h~), (-8



and where the parentheses, (), around the tetrad indices denote symmetrization. It

is both interesting and important to note that, in the K~err spacetime, the coordinate

description of Equation 3-26 does not lead to the separable equation discussed in (I Ilpter

1 (Equation 1-17). To obtain a separable equation, an extra factor of --4/3" muSt be

brought in, resulting in the following expression:


[(p' p')(P + 3p) (a' -r)(a + 37r) 3/' _]1' _4/3 4 --4/;:3:34. (3-29)

Below we will see the same expression arising from very different considerations.

3.4 Metric Reconstruction from Weyl Scalars

The solutions of the Teukolsky equation lead quite naturally to a metric perturbation

in several different v- .--s. The original result, due to Cohen and K~egeles [20] used spinor

methods. Shortly after that, C'!,l~!!. i .---1:! [54] obtained essentially the same result

using factorized Green's functions. Some time later, Stewart [21] entered the game and

provided a new derivation rooted in spinor methods. Eventually, Wald [55] introduced a










comparatively simple derivation of the same result. This is the approach we will follow

here.

Wald's method is centered around the notion of adjoints. Consider some linear

differential operator, that takes n-index tensor fields into m-index tensor fields. Its

adjoint, Lt, which takes ni-index tensor fields into n-index tensor fields is defined by


no ...a, (/3)a,...a,, (gtCa)bl...b, /3by...b,, = aa, (3_30)


for some tensor fields c1 "l" and /3bl...b" and some vector field s". If Lt = then L

is self-adjoint. An important property of adjoints is that for two linear operators, L

and M./1 (MZ/)t = M2/t t. Now let 8 = S(hub) denoted the linear Einstein operator,

S the operator that gives either of the Teukolsky equations from 8 (Equation :32:3 or

:324), O = O( ~,, or
and T = T(hub) the operator that acts on the metric perturbation to give I',, or
(Equation :32:3 or :324). Then the Teukolsky equations can he written concisely as


SE = OT. (:3-31)


It follows by taking the adjoint that


'St atS =a ItOt, (:332)


where we have used the fact that the perturbed Einstein equations are self-adjoint. Thus,

if W satisfies OtW = 0, then StW is a solution to the perturbed Einstein equations!

This remarkably simple and elegant result holds for any system having the form of

Equation :3-31, whenever 8 is self-adjoint.

In order to apply this result to the Teukolsky equation we note that scalars are all

self-adjoint and the adjoints of the GHP derivatives are given by









We may express this more concisely by introducing ~D = {D, P', a, a'}, so that





Suppose now that we have a solution to the Teukolsky equation for Ie',, so that 0 is given

by the left side of Equation 3-25 and S is given by the right side of Equation 3-23 (with

Tab replaced with Sab) Wald's method then tells us that if Ot9 = 0, then hub = ISt is

solution to the perturbed Einstein equations. Using Equations 3-33 we can compute StM:




+mamb(P p)(P + 3p)}W + c.c., (3-35)


where we've added the complex conjugate (c.c.) to make the metric perturbation real

and W remains to be specified. Using Equations 3-33, it is clear that the adjoint of

Equation 3-25 is


[(p' p')(P + 3p) (8' r)(B + 37r) 31r'_]W = 0, (3-36)


which is precisely the equation satisfied by ~!4/3 4' (c.f. Equation 3-29), previously

obtained through separability considerations in the K~err spacetime. However, obtaining

Equation 3-36 required no reference to separation of variables in a particular spacetime

and thus applies to all type D spacetimes. It is important to note that although 9 satisfies

the same equation as 4~i/3 4,g 11 iS not1 the perturbation of~ 4 or Ithe metlric~ it generates

(Equation 3-35). In Chapter 5 we will explore W's connection to ~4 more Carefully.

Though the derivation of Equation 3-35 was quite simple, it fails to yield any

information about the gauge in which the metric perturbation exists. In this particular

instance, it is fairly straightforward to verify that the metric perturbation we've been led









to obeys


P~hab = 0, (3-37)

gab ab = 0, (3-38)


which is known in the literature as the ingoing radiation gauge (IRG), an unfortunate

name because ingoing radiation is carried by 1" and Equation 3-37 tells us that the metric

perturbation is completely orthogonal to 16. Thus there is only outgoing radiation in the

ingfoingf radiation gauge! Obtaining the gauge conditions in Equations 3-37 and 3-38 is

more natural in the approaches of Cohen and K~egeles [20] and Stewart [21]. One startling

aspect of the gauge conditions is that there are five of them. This being the case, we must

be concerned about the circumstances under which the metric perturbation in the IRG is

well-defined. This is the subject of the next chapter.

Our derivation began with the Teukolsky equation for Ie',, Had we instead started

with the Teukolsky equation for 4,/3 ~4, We WOuld be led to a metric~ perturbation

in terms of a Hertz potential, 9', that satisfies the Teukolsky equation for Ie',, The

resulting metric perturbation and gauge conditions are then simply the GHP prime of

Equations 3-35, 3-37 and 3-38, respectively. In this case, the metric perturbation exists

in the so-called outgoing radiation gauge (ORG). For the remainder of this work, we will

focus our attention on the IRG metric perturbation, but all the results hold for the ORG

perturbation as well.

On a final note we remark that the Teukolsky equation for Ie',, (Equation 3-25)

actually exists in the more general type II spacetimes, without its companion for ~4- I

this case, Wald's method also leads to metric perturbation (in the IRG, no ORG exists

here), with a potential, 9, satisfying the adjoint of Equation 3-25, which, in this instance,

is not the equation for the perturbation of #4-









CHAPTER 4
THE EXISTENCE OF RADIATION GAUGES

In the previous chapter, it was seen that the perturbations of the Weyl scalars lead

quite naturally to metric perturbations in the radiation gauges, (seemingly over-) specified

by five conditions. In this chapter we will explore the precise circumstances under which

one can impose all five of these conditions. This will require us to examine the perturbed

Einstein tensor, which presents the need to integrate some of the components. For this, we

will appeal to a coordinate-free integration technique based on the GHP formalism, due

to Held [45, 56]. The generality of these methods allow us to prove the result for a much

broader class of spacetimes than we have encountered so far, namely, Petrov type II, which

we will see is the largest class of spacetime for which the radiation gauges are defined. We

begin with a more thorough discussion of the radiation gauges and their origin. 1\ost of

this chapter is taken from published work [57].

4.1 The Radiation Gauges

The in going radiation gauge (IR G) is a crucial ingredient for the reconstruction of

metric perturbations of Petrov type D spacetimes from curvature perturbations. They first

appear, unexplained, in the work of Cohen and K~egeles [58] (for perturbations of Petrov

type II spacetimes) and Cht!. I.!, ~.--- 1:! [54] (who considered perturbations of Petrov type D

spacetimes), but the work that comes closest to our contribution in describing their origin

is that of Stewart [21], again for the more general case of type II spacetimes.

In type II background spacetimes, the IR G is defined by the conditions


P~hub = 0, (4-1)

gab ab = 0, (4-2)









where 1" is aligned with the repeated PND of the background Weyl tensor. If n" rather

than 1" is a repeated PND, we instead define the outgoing radiation gauge (ORG) by


nahab = 0, (4-3)

gab ab = 0. (4-4)


In type II background spacetimes, only one or the other of these options exists (IRG

or ORG), whereas in Petrov type D background spacetimes, there is the possibility of

definingf both gauges. For the remainder of this work we focus on the IRG. Results for the

ORG can be obtained by making the replacement la t qn

Equations 4-1 translate into algebraic conditions on the components of the metric

perturbation. We will refer to the four conditions in (4-1) as the 1- & gauge conditions.l In

terms of the tetrad components of the metric perturbation, these gauge conditions read:


hit = 0, hin = 0, him = 0, him = 0. (4-5)


The condition in Equation 4-2 will be referred to as the trace condition and can be

expressed in terms of the components of the metric perturbation as hi, kmm = 0, which,

when Equation 4-5 is imposed, simply reads


hmm = 0. (4-6)


Because the IRG constitutes a total of five conditions on the metric perturbation, instead

of the four one might expect for a gauge condition, it is necessary to ensure that the

extra condition does not interfere with any physical degree of freedom in the problem,



] Recently, when applied specifically to the Schwarzschild spacetime, these conditions
were given a geometrical interpretation, and referred to as light-cone gauge conditions [59],
though they are not the conditions originally introduced for gravitation with that name
[60]. It may well be that this description is suitable more generally, although presumably
without the specific geometrical interpretation of [59].










such as one coming from a source. The importance of this consideration can be seen

immediately from Equation B-1 of Appendix B, in which every term would be removed

by Equations 4-5 and 4-6, rendering Equation B-1 inoperable whenever it has a non-zero

source. In the next section we will look to the perturbed Einstein equations to determine

the circumstances under which we can safely impose all five of the conditions that

constitute the IRG.

It is useful to note the similarity between the full IRG, (4-1), and the more commonly

known transverse traceless (TT) gauge defined by


V"hab = 0, gab ab = 0, (4-7)


which, at a glance, also appears to be over-specified. In fact, the TT gauge exists for

any vacuum perturbation of an arbitrary, globally hyperbolic, vacuum solution [61],

because imposing the differential part of the gauge does not exhaust all of the available

gauge freedom. Interestingly enough, Stewart's analysis in terms of Hertz potentials

[21] begins by considering a metric perturbation in the TT gauge. However, in order to

construct the curved space analogue of a Hertz potential, he is compelled to perform a

transformation that destroys Equation 4-7 and instead yields a metric perturbation in

the IRG.2 Furthermore it appears that the restriction to type II spacetimes is essential

for Stewart's analysis. From these observations, we expect radiation gauges to exist under

conditions less general than those required for the existence of the TT gauge. At the

same time, we should not be surprised that the IRG inherits the feature of residual gauge

freedom.

Consider a gauge transformation on the metric perturbation generated by a gauge

vector, go. To create a transformed metric in the 1- h gauge, the gauge conditions in




2 See [21] or the electromagnetic example in C'!s Ilter 1 for a more detailed explanation.










Equations 4-5 require

l"(hab ~(a;b)) = 0, (4-8)

where the semicolon denotes the covariant derivative. In terms of components this reads

2P61 = hul,


(4-9)
(P + p)(m + (B + ')ll = him,

(P + P)(m~ + (8' + 7')(: = him.

Similarly, for the trace condition in Equation 4-6 to be satisfied by the gauge transformed

metric, we require

a'(m + am + (P' + pl)(1 + (P + P)(n = hmm. (4-10)

Any extra gauge transformation that satisfies l"~((;b) = 0-solves the homogeneous form

of Equation 4-9preserves the four 1- h gauge conditions in Equations 4-5. This is what

is meant by residual gauge freedom. We will explicitly use this residual gauge freedom to

impose the 1- h and trace conditions simultaneously, thus establishing the IRG. We will

find that some gauge freedom still remains, as explained in Section 4.3.

Now, we turn our attention to the general case of type II background spacetimes.

4.2 Imposing the IRG in type II

In order to show that residual gauge freedom can be used to impose the IRG, we

need to solve for the residual gauge freedom as well as examine any perturbed Einstein

equation that might impede the imposition of the trace condition of the IRG. For this, we

turn to a coordinate-free integration method develop by Held. Rather than give a detailed

explanation, we present the basics and refer the interested reader to the literature for an

in-depth account [45, 46].









The first step is to introduce new derivative operators P/ and 8I = 8 such that they

commute with P when acting on quantities that P annihilates,3


[9, 1 P]xo = [P, 8]xo 0, [P, 8 ]xo 0, (4-11)


where [a, b] denotes the commutator between a and b. The explicit form of the operators

is given in Appendix C. The next step, the heart of Held's method, is to exploit the

GHP equation Pp = p2, and its complex conjugate, Pp = p2, to express everything as

a polynomial in terms of p and p, with coefficients that are annihilated by P. Held's

method is then brought to completion by choosing four independent quantities to

use as coordinates [56, 62]. In this work, we will not take this extra step. For type II

spacetimes (and the accelerating C-metrics), this step has not been carried out, while for

all remaining type D spacetimes, it has been carried through to completion [45, 46].

In a spacetime more general than type II, there is no possibility of having a repeated

PND. When a repeated PND exists, we can appeal to the Goldberg-Sachs theorem [32]

and set is = o- = Wo = ~1 = 0 in Equations B-1-B-7. Following Held's partial integration

of Petrov type II backgrounds [56], we also perform a null rotation (keeping la fixed, but

changing n") to set -r = 0. As a consequence, it follows from the GHP equations that

-r' = 0. Now we are in a position to address the question of when the full IRG can be

imposed. First we apply the 1- & gauge conditions in Equations 4-5 to Equations B-1-B-7.

While most of the perturbed Einstein equations depend on several components of the

metric perturbation, after imposing Equations 4-5, the expression for SiI depends only on

hmm and the ll-component of the perturbed Einstein tensor simply becomes


{(pD p p) + 2pp~hmm {((9 2p)(P + p p)}hmm = 8xri, (4-12)



3 Such quantities are denoted with the degree mark, o, as in Pxo = g









in which the first form indicates that the equation is real, while the second form and its

complex conjugate (which follow from the fact that Pp = p2 and Pp = p2) is the one we

will use to integrate the equation below. If we had not made use of the Goldberg-Sachs

theorem, there would be terms such as o-phmm appearing in Equation 4-12 and our

argument would not hold. We immediately see that 1 = 0 is necessary to satisfy the

trace condition in Equation 4-6. Next we turn our attention to the question of whether

the condition SiI = 0, is sufficient to impose Equation 4-6 using residual gauge freedom.

In order to address this question we will integrate SiI = 0 and the residual gauge

vector, given by the homogeneous form of Equations 4-9. Full integration of the

homogeneous form of Equations 4-9 is carried out in Appendix C, but we will work

through the integration of SiI = 0 here to illustrate the method. We begin by rewriting

Equation 4-12, with the help of Pp = p2 and its complex conjugate, as:


{(9 -,c 2p)( + p p)}h = p'29 kmm ,,)= 0. (4-13)

Integrating once gives



and another integration leads to


hmm= ao lbo(p + p). (4-15)
p 2

However, hmm is, by definition, a real quantity, so we add the complex conjugate and use

be to represent a real quantity in the second term. The final result is that


hmm = aoP + aoP + bo(p + p). (4-16)
p p









Similarly, integration of Equations 4-9, as carried out in Appendix C, leads to the

following solution for the components of the residual gauge vector:

Cl = Clo
11 1\=) 1~ o ~)"
to = 6 O + -+-5(o 40 21
2p p 2
1 (4 17)
(m = _(mo o10




where 920 iS related to the background curvature via 92 20 3~p. In order to use this

residual gauge freedom to impose the full IRG, we return to the gauge transformation for

hmm (Equation 4-10) which becomes, after some manipulation (using Equations C-6C-9

and Equation C-13),

pI ~;6" p ~; 1a, + +(~ )-aa-p' -- '>" n] 4
kmm 8 (moP o+ kopo__g opoa .(-8
p p 2

In this form it is clear that we can impose the trace condition (Equation 4-6) of the full

IRG if we choose our gauge vector so that

-I 1-I --
B (mo + o a ,o( 8- I ploo a on = bo. (4-19)


We have now shown by construction that the condition : = 0 is both necessary and

sufficient for imposing the full IRG in a type II background. We turn next to discussing

the complete extent of the residual gauge freedom in more detail.

4.3 Remaining Gauge Freedom

Although Equations 4-19 involve three real degrees of freedom (ao is complex),

it turns out that only two real degrees of gauge freedom are required to fully remove

any solution of Equation 4-13 for the trace hmm. To see this we introduce the following

identity:

p~~~ p p+p (p + p)Ro, (4-20)
p p p p









which also defines Ro, a quantity annihilated by P. Then we can rewrite Equation 4-16 as


hm =1ao+aoP P a + bo] (p + p). (4-21)
2 pp 2

In a similar fashion, we rewrite Equation 4-18 as



(4-22)
(8~'d + aa' "b" p"o pro a

in which each coefficient in big square brackets is purely real. Now, suppose we have

a particular solution for SiI = 0 (i.e., ao, ao and be are fixed) and our task is to solve

for the components of the gauge vector which removes this solution. By comparing

Equations 4-21 and 4-22 we see that, for any given (mo and (mo, we can fix (to (up to a

solution of D (to = 0) via


p1 0 o (a + ao -I(mo + igo~), (4-23)
2 2

and we can fix (no by setting
1 1, 1~~" -a~0 1
6 o= (o a.)n + be g' g o plol" -t /m m o (4-24)
2 2 2

to completely eliminate the nonzero hmm, thus imposing the full IRG while still leaving

two completely unconstrained degrees of gauge freedom, (mo and (mo. Once in the IRG,

Equations 4-23 and 4-24, with ao, ao and be set to zero and (mo and (mo arbitrary,

give the remaining components of a gauge vector preserving the IRG. It is currently

unclear how to take advantage of this remaining gauge freedom to simplify the analysis of

perturbations in the full IRG.

4.4 Imposing the IRG in type D

Type D background metrics are of considerable theoretical and observational interest

since they include both the Schwarzschild and K~err black hole spacetimes. K~innersley first

obtained all type D metrics by integrating the N. ein-! lIs-Penrose equations [33]. While the









results of the previous section are general enough to encompass the special case of type D

backgrounds, the tetrad choice we made (with -r = 0) is incompatible with the complete

integration of the background field equations which is possible in type D spacetimes [45].

The complete integration requires that each of 1" and n" be aligned with one of the two

PNDs. In that case we can exploit the full power of the Goldbergf-Sachs theorem and its

corollaries to set a = K' = a = a' = ~o = ~1 = ~3 = 4 = 0, while maintaining -r / 0

and -r' / 0. In this section we repeat the previous calculation with this different choice of

tetrad.

The result of integrating SiI = 0 is the same as in the case of a type II background,

given in Equation 4-16. The residual gauge vector, however, now has the following, more

complex, form (details of the integration are given in Appendix C):

Cl = Clo
1 1 1 P1 1\
2 2 2 p2 2
[xo go1 1 1 ~
p p 2 p p

-[7Op(8 UO + so .48' + o (o em p reo (4-25)
o 1 1 o 1 1
p2 P2
1l1
1m=~ ;~ 1 c" o
m mo_ o l o P1
p p n~ i~~ I7:0 il(y&










where the quantities Wo" a o;r and to" determine properties of the background spacetime.4

Now the gauge transformation for hmm becomes

hmm 8 mo aP B + iP m a B
-I1 -I ~
+ p+p ( 8- roa o-A



where we have introduced (note that Bo is purely imaginary)



2 (4-27)
Bo" ={4xo o (~ ) + 5/uo a 2xToin00 oo a .

with c.c. indicating the complex conjugate. Integration of the backgrounds where xo" / 0

and to" / 0 using the Held technique has not made its way into the literature and is

beyond the scope of the present work. As a result, derivatives of xo" and to" appear in

Equations 4-27 but do no harm to our argument. ClIn..-!nig any gauge vector that satisfies


8 (mo p Eo Bo a g@ _.I~+~P C ,o ,)o a nO Ao = bo, (4-28)


will serve to impose the trace condition in the full IRG. Once again we have established

that 1 = 0 is both a necessary and sufficient condition for the existence of the full IRG.

Note that by setting xo" = cto = 0 (i.e., ignoring the C-metrics) in the background,

Ao = Bo = 0, and the result is virtually identical to Equations 4-18 and 4-19. There

is one simplification in that now p'o __ ,o [46]. The full extent of the remaining residual

gauge freedom in Equations 4-28 can be demonstrated along the same lines as used

in Section 4.3. As for the case of a type II background, it resides chiefly in the freely

specifiable (mo and (mo



4 FOT eXample, xo" / 0 leads to the accelerating C-metrics. The condition xo" = 0 implies
to" = 0 and so cto is also related to parameters in the C-metric.









4.5 Discussion

With our new form of the perturbed Einstein equations, use of NP methods has

allowed us to treat the quite general class of type II spacetimes without either choosing

coordinates or finding a metric. In this context, the Held technique has allowed us to

exploit our form of the equations by enabling partial integration in solving SiI = 0 while

investigating the existence of the IRG. Additionally, the Held technique has allowed

us to completely characterize the residual gauge freedom and use it in the radiation

gauge construction. By explicit demonstration, this work establishes our new form of

the perturbed Einstein equations as a powerful tool within perturbation theory, both in

conjunction with the Held technique and otherwise.

For perturbations with 1 = 0, our characterization of the residual gauge freedom

is sufficiently complete that we can explicitly demonstrate the required gauge choice

to remove any non-zero solution for the trace obtained via SiI = 0. Thus, in type II

spacetimes, radiation gauges can be established by a genuine gauge choice, even if only

after a solution of SiI = 0 is chosen.

There are subtle differences between the general type II case and the more restricted

type D case, as there are also in the construction of Hertz potentials for the two cases.

Stewart [21] writes out the type II case rather fully for an IRG. In this case, the

perturbation in 90 is tetrad and gauge invariant, while the potential satisfies the adjoint

(in the sense detailed by Wald [55]) of the s = +2 Teukolsky equation. Remarkably, in the

type D case, this adjoint is actually the s = -2 Teukolsky equation, also satisfied by the

gauge and tetrad invariant perturbation in 94. In the type II case, the adjoint equation

is the same as in type D, but 94 is HO longer tetrad invariant. Compared to the type D

result, the expression for 94 giVen by Stewart has many extra terms depending on W' and

o-', so presumably it does not satisfy the same equation as the potential. As a consequence,

metric reconstruction would be restricted to being built around the perturbation for Wo

(c.f. the comments at the end of OsI Ilpter 3).










In the context of a specific type D background, Wald [63] has argued that mass

and angular montentunt perturbations are not given by any solution to the Teukolsky

equations, and Stewart [21] has shown that these cannot he represented in a radiation

gauge in terms of a potential. What we have done is identify the gauge freedom which

remains in the fully satisfied radiation gauges, neither interfering with the radiation gauge

prescription nor ruling out the possibility of mass and angular montentunt perturbations.

By realizing the explicit construction of the radiation gauges for type II background

spacetintes and by identifying the remaining gauge freedom which they allow, we have, in

a sense, completed a task initially embarked upon by Stewart [21], though in the different

context of Hertz potentials.









CHAPTER 5
THE TEUK(OLSK(Y-STAROBINSK(Y IDENTITIES

Having established the conditions for the existence of the radiation gauges, we will use

the corresponding metric perturbations to establish some useful relationships between the

perturbed Weyl scalars known generally (and quite loosely) as the Teukolsky-St arohinsky

identities. Because Hertz potentials are solutions to the Teukolsky equation, these

identities have immediate relevance for metric reconstruction in the IRG, both in the

time-domain approach of Lousto and Whiting [25] and the frequency domain approach of

Ori [2:3].

The original analysis of Teukolsky [11, 12] was based on the .I- i-i np u'tic form of the

solutions of the separated angular and radial functions in the K~err spacetime as well as a

theorem due to Starohinsky and Churilov [64]. Only later did C'I .!1.4 I-ekhar provide a

full analysis, which is nicely summarized in his book [29]. Our analysis, however, will be

entirely symbolic, involving only GHP quantities. This approach has the advantage not

only of applying to a larger class of spacetimes, but displaying the structure inherent in

the identities in a much more obvious way. A similar analysis of some of the identities we

will discuss was previously undertaken in the NP formalism by Torres del Castillo [65] and

later translated into GHP hy Ortigoza [66]. These prior analyses made use of the most

general type D spacetime and translated back and forth between coordinate-based and

coordinate-free expressions. In contrast, our approach will not make any reference to the

choice of coordinates or a tetrad (other than requiring it to be aligned with the principal

null directions). Because of this, our approach will showcase one of GHPtools' greatest

strengths-the ability to commute several derivatives with relative ease.

Our starting point is the (source-free) IR G metric perturbation given by









As a consequence of Equation :3-35, the actual perturbed Weyl scalars follow directly from

Equations :327 and :328.1 The expressions are at first sight quite complicated, but by

commuting derivatives so that they appear in a standard order and using the fact that the

potential satisfies the Teukolsky equation, they become:


< 94, (5-1)

('414 r_ 4/ /37'a 87a p'P fp 24'2)]W .) (5 2)


The term in square brackets [] in Equation 5-2 is actually just the operator form of the

(generally complex) K~illing vector (acting on 9, which has type {-4,0}) discussed in

C'!s Ilter 2. We can further combine the relations in Equations 5-1 and 5-2 to eliminate

a~ny reference to\ the po~tent~ials. Tphe firt s~tep is to\ act o~n Equaition 5-2 with > #'2-4/3

which gives us

p4 -,-4/:% 4 4 -/ 4 _qpV 53

Commuting the eight derivatives on the first term (using GHPtools, of course) yields the

useful identity

p394 -- 149 __ i14 -- 4!p4W, (5-4)

which we will have occasion to exploit again. Commuting the derivatives in the second

term of Equation 5-3 poses no problem because V commutes with everything. Now it is a

simple matter to identify the resulting expression with the terms in Equations 5-1 and 5-2

to arrive at the following

p4 -4: % 4 1 -4/ 7,, (5-5)


p/4 -,-4/3 __i ~4 --4/:% 4 + V4, (5-6)




i We thanlk Joh~n Friedmanl and Toby Keidl for noting missing factors of in? several
earlier papers. Stewart [21] and C'!,l~!!. 1,.---1:! [18] have these factors correct, the latter
with different sign conventions.









where the second expression follows from taking the prime of the first. We will refer to

these relations as the first form of the Teukolsky-Starobinsky identities. Note that the use

of V as a commuting operator restricts the validity of these relations to non-accelerating

type D metrics. In the an~ ll-k- of Torres del Castillo [65] and Ortigoza [66], where explicit

coordinate expressions were used, Equations 5-5 and 5-6 both appear to be true. This

fact appears to be coincidental since it is unclear how it follows in general from the

fundamental equations of perturbation theory. The remainder of the identities we will

present have not appeared in the literature in this form and we can only claim they hold

for non-accelerating type D spacetimes.

Before we continue, we'll take a look at the content of Equations 5-5 and 5-6 in

the context of th~e Ker~r spacetimre. If wve write i',, ~ R+2(r)S+2(0, Q) and 2 4"/3 4 ,

R-2(r)S-2(0, 4) and understand the time dependence of each to be given by e-ist, then

Equation 5-5 tells us: (1) the result of four radial derivatives on R+2 is proportional to

R-2 and (2) the result of four angular derivatives on S-2 is proportional to S+2. The same

is true of Equation 5-6 with the +'s and -'s swapped. Note that Equations 5-1 and 5-2

(and their primes in the ORG) ;?i essentially the same thing with the subtle difference

that the angular and radial functions are not obviously solutions to the same perturbation.

No such ambiguity arises in Equations 5-5 and 5-6.

Remarkably, we can actually take things a step further and arrive at expressions for
I'n, and ~4 independently. We begin by acting /4 7-4/3 o qain55


p14/4-4/34 -434 pl4 -4/3 14 1,-4/3 i 3Pl4 -4/3 ,,; (5-7)

By recalling that Ir has. th same~,,,. type,. as 2-4/3 i4 '; Car~rieS HO Weight), weit can simply

take the prime and conjugate of Equation 5-4, and use it to commute the derivatives on









the first term on the right as follows


pl4 /-4/3 14 -,-4/3 ; __14 ,-4/p4-4/3pl --/


adL:a124/3:~ 4/ 4/3 -4
= : 314i3" :i 4 2 4 1 2 ( -8


where we made use of Equation 5-6 in the second line and commuted everything through

V in the third line. The second term in Equation 5-7 becomes
4/3','"' 3p474/3
3/4 --' TI l4 II



=3Vij5/4 2 i" 4 +t 9VV 4, (5-9)


where we made use of the complex conjugate of Equation 5-6. Combining these results

gives us

1, 4 p~ 4 / 3 ,4 / 4 4 i 4 3 4 9V V 4 ( 5 -1 0 )


4 --4/3 1 -/ 4 -43 1 -/ <, 9 ,, (5-11)


where we took the prime of the first equation to get the second one "for free." These are

the second form of the Teukolsky-Starobinsky identities. We note in passing that in the

context of the separated solutions of e',, and 2-4/3 4, Ithe relations above allows for the

determination of the magnitude of the proportionality constant relating R+2 and R-2 [29].

Surprisingly, this isn't the end of the story. Recall that in a type D spacetime we also

have at our disposal the outgoing radiation gauge where


21, ,, 84/ 4/399, (5-12)

~4 pl 1 (5-13)









which are easily obtained by taking the prime of Equations 5-1 and 5-2. Note that
whereas 1T satisfc,,~ie the, Teukolskyi. equation; fr (2-,i4/8,:3 4I, SliiSfies the adjoint equation-the

Teukolsky equation for I',, Fr-om the complex conjugate of the preceding equations and

their IR G counterparts, we get the following:


P49 __ 149'31 3 4:399' (514)

P/491 __ 4 2 ~r3'vW (5-15)


the first form of the Teukolsky-Starohinsky relationships for potentials. Note the difference

between the above and Equations 5-5 and 5-6, particularly II theDDII missing facor of 2-/

and the fact that V appears. As with Equations 5-5 and 5-6, we can obtain relations for

each potential individually by acting p/4~~l --4/ onquation 5-14and fur~ther exploitinlg

(the primed conjugate of) Equation 5-4. The result is that

p14 --4/3p49 __14 --4/3 49 4/:39, (5-16)16

p4 --4/3 14' / __4 --4/3 14 / 9V724/39/. (5-17)


We can summarize this last identity by writing


[p/l4 --4/p 4 1,4 --4/ 3 4 4/l~" 31 --4/":3 4, r9} = 01 (5-18)

[4 -4/3p l4 4 --4/3 1, 4 + 9 0 43 -43, .(5-19)


Bardeen has recently pointed out an issue in the standard treatment of the Teukolsky-

Starohinsky identities [67]. In particular, he finds that, in the Schwarzschild

background, there is a hitherto unnoticed relative sign difference between the odd-

and even-parity in the term proportional to 8, (alternatively w when time separation is

performed), which by continuity presumably persists in the K~err background. Bardeen

argues using standard techniques that don't make clear the difference between the ~'s

and their complex conjugates on the right-hand-sides of Equations 5-5 and 5-6. However,

recalling our discussion of parity in Chapter 3, a glance at these equations reveals that










one should in fact expect a relative sign because of the occurrence of 1',, 4 and its complex

conjugate in the same expression. 1\oreover, this must occur even in the K~err spacetinle,

where we have the real-intaginary separation instead of the parity separation. Such a

consideration makes clear the obvious advantage of treating the Teukolsky-Starohinsky

identities in terms of the fundamental GHP quantities. Beginning at this level and then

performing the separation of variables allows for no ambiguity in the resulting expressions.









CHAPTER 6
THE NON-R ADIATED 1\ULTIPOLES

In this chapter we will address the issue of the non-radiated multipoles alluded to in

C'!s Ilter 1. The issue is that the metric constructed from a Hertz potential is incomplete

in the sense that its multiple decomposition necessarily begins at -e = 2 because the

angular dependence of the potential is that of a spin-weight +2 angular function. To see

this explicitly, we focus our attention on the IR G metric perturbation (Equation 3-35) in

the Schwarzschild spacetime, where the potential, 9, can he decomposed into some radial

function, R(r), with exponential time dependence, e-i", and a spin-weight 2 spherical

harmonic, -2 Loz(0, 4) (see Appendix D, for details about the spin-weighted spherical

harmonics). Ignoring the radial and time dependence, we see that the components of the

metric perturbation have angular dependence given by


hit ~ 82-2 at = [(e 1) ( + 1)(e + 2)]1/20Ym ine)

him ~ -2 Bat = [(-e 1)(e + 2)]1/2-1 z,, (6-2)




and similarly for him and hmm. Because the spin-weighted spherical harmonics are

undefined for |8| > -e, the above expressions make it clear that the metric perturbation

in this gauge has no -e = 0, 1 pieces and therefore provides an incomplete description of

the physical spacetime. By continuity, the situation persists in the K~err spacetime. How

incomplete is this description?

For the n, I iR~~ly of this work, we have focused our attention on gravitational

radiation in type D spacetimes. This information is contained in the perturbation of either

I',, or ('4, a Tesult established by Wald [16]. In particular, Wald was able to show that

well-behaved perturbations of I',. and #'4 determine each other and furthermore that either

one characterizes the entire perturbation of the spacetime up to It l i .! perturbations

in mass and angular momentum. With I',, and #'4 determined hv the Hertz potential










(Equations 5-1 and 5-2) this begs the question of why we should concern ourselves with

such trivialities.

The answer is, in part, that these trivial perturbations represent the largest

contribution to the self-force, as shown by Detweiler and Poisson [17]. Although it is

unclear if such contributions persist in all gauge invariant quantities of interest, such as

certain characterizations of the orbital motion of the particle [68], there is in fact a more

compelling reason to be concerned with the non-radiated multipoles. In recent work,

K~eidl, Fr-iedman and Wiseman [69] have looked at the problem of computing the self-force

in a radiation gauge in the context of a static particle in the Schwarzschild spacetime. In

their calculation, they found the perturbations of mass and angular momentum arising

in the construction of a Hertz potential. Thus, although the Hertz potential cannot he

used to determine these perturbations, it must still 1:0.0~.--" about them and they must he

determined by some other means.

In this chapter we will present a general prescription for computing the non-radiated

multipoles. 1\ore specifically, we will consider the problem of computing the shifts in

mass and angular momentum due to a point source in a circular (geodesic), equatorial

orbit around a black hole. Specifically we are after expressions for 61M and 6a, the

shifts in mass and angular momentum, in terms of the orbiting particle's mass, p, and

orbital parameters. The idea is quite simple: match an interior spacetime, (g,, Af-),

to an exterior spacetime, (g91, Af*), differing only in mass and angular momentum,

on a hypersurface (of codimension 1), E,,, containing the perturbation. The basic

conditions for a good matching are (1) that the metric is continuous across 27, and (2)

the first derivatives of the metric are continuous except where the source is infinite. These

conditions are compatible with Israel's quite general junction conditions [70].

Before we can do any ]rce Ib t.11r We must first determine the geometry of 27,. In

spherically symmetric spacetimes, the obvious choice is the simplest-the (round)

2-sphere, as we'll see below in our calculation in Schwarzschild. For the K~err spacetime,










which possesses only axial symmetry, the situation is immensely more complicated. This

issue will be discussed below.

Once we've agreed on a E,, fulfilling our first matching condition requires us to

simply equate the components of the metric (on E,). In other words,


[9a 9 |4 Rb, =0 (6-4)


where |4, indicates the restriction to E,. The only (slight) complication that arises here is

ensuring that there is enough freedom in the metric perturbation to perform the matching.

This will generally require performing a gauge transformation on the interior and exterior

spacetimes. Although this introduces some gauge dependence into the problem, the end

result 6M~ or 6a is in fact gauge invariant, as we will see below.

Imposing the second condition is a bit more involved because of the presence of

the source. By choosing a good matching surface, E,, we can effectively -on! I. out"

the angular dependence of the source. If, for example, E, is a 2-sphere, we can use the

completeness relations to write the angular delta function according to



= 0 m= -

Similar relations hold for complete sets of functions on different closed 2-surfaces. The

source now consists solely of a radial delta function. To handle this, we impose the

perturbed Einstein equations as, for example,


lim Sab dT b dr (6-6)
e-> ro-e o-

where Sab denotes the perturbed Einstein tensor and l~b denotes the stress-energy tensor

of the source and ro is the location of E, as seen from both sides. For a delta function

source due a particle of mass p in a circular equatorial orbit of the K~err spacetime,


lab s 6 rO)b COS 8)6(4 Ot), (6-7)









where u" =. ( 0, 0, J) ;is the four-elocitty of the particle parameterizednr by proper time

(7), ro0 is the radius of the orbit and 02 = ~. For circular equatorial geodesics

r = ro, (6-8)

0 = (6-9)

dt (r, + a2)
r + a(L aE), (6-10)
Odr a
ad4 aT
r = aE + (6-11)
Sd-r a)

with

T = (r, + a2 E aL, (6-12)

where E = E/p and L = L/p are the energy and angular momentum per unit mass,

respectively. We can recover the corresponding result for the Schwarzschild spacetime

by simply taking a 0 Because the integration in Equation 6-6 is purely radial, it

is clear that the only terms that actually participate in the integral on the left side are

those involving two radial derivatives. This is where our form of the perturbed Einstein

equations comes in. While it is generally quite tedious and impractical to compute the

perturbed Einstein tensor for a background more general than Schwarzschild and pick out

the terms involving two derivatives, it is a quite trivial task for the Einstein equations in

GHP form. All we need to do is pick out the pieces involving two of P and P' (a mindless

task with the aid of GHPtools), plug in our favorite tetrad and voila! Note that these

conditions on the second derivatives are generally invariant with respect to choice of

tetrad. Because of this, we will write the jump conditions out in the symmetric tetrad,

which is obtainable from the K~innersley tetrad by a simple spin-hoost (Equation 2-16)










(and thus leaves the PNDs intact). The tetrad is given by:

T2 2


T2 + 2
if=~ \ ?P 0, (6-13)


2/(r + is cos 8) i n 0 sin 8


With this tetrad choice, the radial jump conditions are:


8f2h,war = 1 (6-14)

8f2h,war = 16 ,(6-15)

8?2(hit + h,z~ + 2hl,z 2h,>uz) = 1 n (6-16)

8?2(hira + h,and = 16x 5,,2, (6-17)




where the omitted equations follow by taking the prime and/or complex conjugate of those

listed (the factors of a and P"2 remain unchanged; a feature of the symmetric tetrad),

and it is understood that equality only holds in the sense of Equation 6-6. At a glance

Equations 6-14-618 may appear inconsistent, with the same left-hand-side being equated

to different right-hand-sides. In fact, the circular geodesic nature of u" ensures that this is

not the case.

What we have not yet addressed is the question of what, precisely, we mean by mass

and angular momentum. Suitable definitions arise from the Hamiltonian treatment of

General Relativity initiated by Arnowitt, Deser and Misner [71]. The general idea is

that because Minkowski space provides an unambiguous notion of energy and angular

momentum through time translations and rotations, respectively, we can adapt these

notions to curved spaces if the metric becomes Minkowskian at spacelike infinity. Thus

the ADM definitions require us to restrict our attention to .-i-mptotically flat spacetimes,










spacetimes that become flat near infinity. The most precise definition of .l-i-inia..'l~e

flatness requires a detailed analysis of the conformal structure of spacetime [72], but for

our purposes it will suffice to simply consider the .I- i-n ng .)i oi falloff of the components

of the metric. More precisely, for a set of coordinates (x, y, x) in a metric, gAb, and

r = .1.2 2 2,,~ we required


lim gab 0ab(1
r-oo r(6-19)



These conditions are satisfied hv the Schwarzschild and K~err spacetimes we wish

to consider, but we must he careful to choose an appropriate gauge for the metric

perturbation to ensure that Equations 6-19 are satisfied. Assuming an .I-i-mptotically

flat spacetime, the ADM mass is defined by


Af= lim (bb-Der S (6-20)
16xr stoo

where the symbols need a bit of explanation: we denote the hypersurface of constant t

by E, and its boundary by S. The three-metric on Et is Yub. Then Kub = ub y, b, with

yOb being the metric of flat spacetime (in the same coordinates aS Yub) and a~ = ab ,0 byO"

Additionally, D, is the covariant derivative compatible with ,b, ra is the unit normal

to S, and dS is the surface element on S. For an arbitrary metric perturbation, hub, this

evaluates to

,M = ,itli 2r~rOJ~Z ir sin O, d2d (6-21)




1 In general, having a well-defined angular momentum actually requires a faster falloff
than that given helow. However, because we're restricting our attention to spacetimes with
axial killing vectors, the falloff required for .I-i-inia..'lc flatness is sufficient.









where we've omitted the terms that will vanish in the limit as a result of requiring

.I-noi-nd ic flc atness. Similarly, we define angular momentum by


J = lim (Kb-K b bS,62)


where we have introduced the extrinsic curvature, Kab, of E, and the rotational K~illingf

vector a". For a generic metric perturbation of the K~err spacetime, we have

1 r2xr rx
6J =lim r 7sin 86,4 r2 Sin 8 r 4 ded#. (6-23)


Though these definitions provide the most general prescription for computing the mass

and angular momentum, for stationary and axially symmetric spacetimes (those containing

both timelike and axial K~illing vectors), the K~omar formulae [73] evaluated at infinity

allow us to compute the value of the perturbationS2 of M~ and J. though not the entire

perturbation in the interior and exterior spacetime. The formulae are given by


6M = (Lb- 9a)?a b 3x,~ (6-24)

6J Lb 9b)na b 3x (6-25)


where E is spacelike hypersurface that extends to infinity, n" is the unit normal to it, to

and #" are the timelike and axial K~illing vectors and 2/7d3Z is the volume element on E.

Because our stress-energy tensor is confined to a spacelike hypersurface, Ep, at r = ro, to

compute the ADM mass we must take the limit as To oo. In this limit, with the source

given by Equations 6-7-6-12, the K~omar formulae give (for the K~err spacetime)


sM = pE, (6-26)

6J = p-L. (6-27)



2 We thank John Friedman for -II--- _t h-r;! the use of the K~omar formulae.










These results are to be expected because of the axisymmetric nature of both the

perturbations and the background spacetime. We now turn our attention to the mass

and angular momentum perturbations in the Schwarzschild background.

6.1 Schwarzschild

The Schwarzschild spacetime provides the perfect tested for our technique. 1\oreover,

because of the spherical symmetry of the background, matching the spacetime is quite

straightforward. In this case we can ah-li-s choose the matching hypersurface, Ez>, to be

a (round) 2-sphere and exploit the orthogonality and completeness of the spin-weighted

spherical harmonics to smear out the delta source on Ez,. The only caveat is that we must

choose Ez, outside of the innermost stable circular orbit. If the location of Ez, is ro, then

this amounts to requiring ro > 6Af.

6.1.1 Mass perturbations

Our first task is to construct a suitable description of source-free mass perturbations of

the Schwarzschild spacetime. We will then glue two such spacetimes together, as described

above. We will write the Schwarzschild metric as


d~s2 = d2 f-1 2 ,2 d2 Sin2 8d 2) (628)


where f = 1 2Af/r. According to Birkhoff's theorem, the only static, spherically

symmetric solution to the Einstein equations is the Schwarzschild solution. Thus, we

are assured from the outset that perturbing the mass will simply lead us to another

Schwarzschild spacetime with a mass At + 61f. The nonzero components of the

corresponding metric perturbation are given by

htt -26M
r (6-29y)



which is easily obtained hv linearizing a mass perturbation of Equation 6-28. In order

to characterize mass perturbations more generally, we will introduce more freedom by









performing a gauge transformation. To that end we introduce the gauge vector


I= (PGt, r)e m(8, 4), Q00, r)e m(0, 4), 0, 0), (6-30)


where we've taken a cue from R;--- & Wheeler and decomposed the gauge vector into

spherical harmonics. Note the absence of Co and (4 components in our gauge vector.

We have deliberately omitted these components on the grounds that they interfere

with the form invariance of the metric. In order to determine the functions P(t, r) and

Q(t, r) as well as the appropriate e and m, we'll look at their contribution to the metric

perturbation. Our gauge transformation, (ab = @agb, has the form

-2(fatP +Mlr-2Q _-d, r p d,) -f -1~d

sym 2 f-l(8,Q Mr-2 -1& f-1d f-1 E, 6-1

sym sym 2rQ 0

sym sym sym 2r sin2 8

where "sym" means symmetric and we've dropped the functional dependencies for

simplicity. First, we'll further specialize the gauge transformation by insisting on

preserving the form of Equation 6-28. A consequence of this is that


her = (-fdrP(t, r) + / 8tQ&(t, r)) = 0.


Because Q(t, r) appears in other parts of the metric perturbation, allowing it to carry a

time dependence would destroy the static nature of the perturbation and put us at odds

with Birkhoff's theorem. Therefore we require Q = Q(r), which immediately leads us to

P = P(t). Further consequences of our form invariance requirement are

hts = -f Pdeoh = 0,

hs = -f PdY, = 0,
(6-32)
bro = / 08oh~m = 0,

hts = / 0845me = 0.









Thus we must impose Bo~e~m = ii4Km = 0, which translates into e = m = 0 and

we have established that the angular dependence of the metric perturbation is purely

Yoo(B, 4) = constant. Also, in order to keep the perturbation static, the time dependence

of P(t) must be, at most, linear. Without loss of generality, we set P(t) = a~t. Finally,

our falloff conlditionls inl Equations 6-19 required: Q(r) = 0)) Thus we hlave arrived at
a description of source-free mass perturbations in the Schwarzschild spacetime in a family

of .l-i-!!!!1'1;1 I11y flat gauges that preserve the form of the metric. The physical spacetime

(gab 9 g~bh +ab) has components

2M~Q(r)Yoo 26M~
get = f (1 2a~Yoo) (6-33)

i~( l 2mQYoo + r2 00 nirQ 2rbMlrf) 6


goe= -72 2Q)Yoo (6-35)

g4 = -T2S 2 1- (6-36)

We can give an interpretation to a~ by considering Equation 6-33 with 6M~ = Q = 0, in

which case it is clear that a~ is just a rescaling of the time coordinate.

In order to perform the matching, we need to adapt our generic perturbation to the

interior and exterior spacetimes and choose a particular gauge to perform the matching.

We will begin with the description of the metric on the interior, g~b. Here 6M~ = 0, so

the perturbation is pure gauge. Furthermore, on the interior there is no need to impose

.I-i-inidll'lc flatness. Instead, we will choose Q-(r) so that the interior metric is regular on

the horizon and leave the form of P-(t) untouched. A suitable choice is

P-(t) = a~-t,


&- (> liro 2M~; 67

where r = ro is the location of E, and p is a constant inserted for dimensional reasons and

i > 0. The values of a~- and P will be determined from the jump conditions. Proceeding to









the description of the exterior spacetime, g b, We choose


P+(t) = cft,

&'(>=P ro 2M~(68


where, in anticipation of the nr -, W11111 we've chosen the same dimensional constant, P,

that we used in the description of the interior spacetime and j > 2. With both metrics

specified we now turn our attention to matching the spacetimes.

Because both background metrics are the same, it will suffice to match the perturbations

only. By imposing [hab] = 0, we arrive at three unique conditions:


S+ fo[ca]Yoo + 2 00Yo = 0, (6-39)
ro To

rifo~ Yoo M~[Q]Yoo ro61M = 0, (6-40)
[Q] = 0, (6-41)

where we used fo = f(ro). Our choices for Q+ and Q- (6-38,6-37) ensure that the third

condition is satisfied. We can solve Equations 6-39 and 6-40 to get equations for [c0] and



dQP(i + j)
[0][] (6-42)


6M~ = (ro 2M Yoo = -P(i + j)Yoo, (6-43)


where we've made use of Equations 6-38 and 6-37. Next we will use the jump conditions

to solve for p.

Application of the jump conditions (Equations 6-14-618) is simplified by the

fact that our metric perturbation is pure spin-0. Thus we only need consider the jump

conditions for the spin-0 components of the metric perturbation (hiz, hi,, he, and hmm).

For simplicity we will work with Equation 6-15, though it can be directly verified that the










remaining spin-0 jump conditions all yield the same result. With the source given by the

a 0 limit of Equation 6-7, we have for the tetrad components of the relevant objects:


km ro 2M Yoo (6-44)


& Pr 21Yoo (6-45)
mm 2

fi 8xpl-E
16r 3, s(r ro)6(cos 0)6(4 Ot), (6-46)
a rofo

with all the 6M~ dependence replaced according to Equation 6-43. Imposing Equation 6-6

then leads to

-6_ (cos 8)6(~ Rt),
orJ

2p(i + j) 8xp~E -
ioYo(,)= mx/,O)em0 ) (6-47)
= 0 m= -
where we've decomposed the angular delta functions according to Equation 6-5. We

can eliminate the sum on the right side of Equation 6-48 by multiplying both sides by

Yoo(0, ~), integrating over the sphere and exploiting the orthogonality of the spherical

harmonics. The result is that
(4r) 1/21E
P= -(6-48)
z+j
where we've used Yoo(B, 4) = Yoo(B, 4) = (4xr)-1/2. Finally, We have


[a] =-(6-49)
ro 2M~
sM = pE. (6-50)


These equations complete our construction of the matched spacetime. Note that the

above only restricts the difference between a~ on the interior and exterior. If we recall

Equation 6-41, we see that the same is generally true of Q(r) as well if we drop the

requirements of regularity in the interior and .I--phllcl~ flatness in the exterior.









6.1.2 Angular momentum perturbations

Treating angular momentum perturbations is a bit more involved. One reason for

this is the fact that it inherently changes the form of the metric. From Equation 6-23, it

is clear that our metric perturbation will acquire an At4 component. Realizing this as a

perturbation towards the K~err spacetime, we will write it as

26aM~ sin2 8
has (6-51)

which is just the linearization about a = J/M~ of the corresponding component of

the (background) K~err metric. Because of this, there will be nonzero contributions to

him, hm and their complex conjugates which means that we must now take parity into

consideration. To that end we will introduce a gauge vector with components


it = P(t, 7) m(0, ) (6-52)

(r= Q(t, r) m,(0, ) (6-53)
1 i
le = [R(t, r)--(a +') +S(t, r) (a a')]em(e, ~)
2 2 sm0

=R(t, r) ~t(0, ~) + S(t, r) (6-54)
sin 8
i sin 0
(4= [R(t, r) (a a') S(t, r)> (a + a')]Nem(8, 4)
2 2
=R(t, r)Y m(0 ) Slt, r) sin 0@(0,4) (6-55)

where we've defined Q' =i (8T + T')L = (1m+-1m nd T- = ~(n 8') m =


~(1Ye -1 m,), where 1%m are the spin-weight +1 spherical harmonics discussed in

Appendix D. This form of the gauge vector was obtained by considering (a = (ma + (als

(mh (mm, and making use of the parity decomposition discussed in ChI Ilpter 3. This

makes it easy to see that P, Q and R represent the even-parity degrees of gauge freedom

and S represents the only odd-parity gauge freedom available. A natural question to ask is

what parity the perturbation in Equation 6-51 has. For an answer, we look to the source

terms. A quick computation reveals that m, = Im = m = -Im, from which it follows









that the even-parity (real) parts of the source vanish identically and thus the angular

momentum perturbation is completely odd-parity (imaginary). However, we still have

pieces in the source (such as 1) that contribute to the even parity perturbation. We will

treat each parity individually.

6.1.2.1 Odd-parity angular momentum perturbations

Because our gauge vector only has one nonzero component, our task is greatly

simplified. The contribution of the odd-parity gauge vector to the metric perturbation
takes the form

0 0 -(8tS)(sin 8)-lY- (8tS) sin 0Y+

hO 0 -(8,S -2r-1S)(sin 8)-lY- (8,S -2r-1S) sin 0Y+
sym syvm 0 -S [(sin B)l18Y- + cos 0Yf sinl 08oY ]

sym sym sym 0
(6-56)

where the "-" on hab, referring to the interior spacetime, is to be distinguished from the

-"on Ye which refers to a combination of spin-weight 1 spherical harmonics. In this

situation, we must modify our requirement of form invariance (which is already broken by

the perturbation) to the requirement that only ht- remains nonzero, which preserves the

minimum freedom to match to the exterior. First we set has = 0, which implies me = 0

or 1Yem = -1 m,. This can only hold if m = 0, which means the perturbation is axially

symmetric. Moving on, we turn our attention to eliminating 604. This entails

cos 8Y sin 0804eY =


which has the solution

Ye = 1 og + -1 Yon ~sin 0.

This is just the statement that -e = 1, which is to be expected from the fact that the

perturbation appears in the spin-1 part of the metric. The angular dependence of the









interior perturbation is then characterized by


~1Y10 = r sin 0. (6-57)


Finally, it is easy to set

kg ~8,S 2r-S =0,

by imposing S(t, r) = T2S(t). Note that because of the quadratic dependence on r, we

cannot perform this gauge transformation in the exterior spacetime if we wish to preserve

.I-i-m!hlli' c flatness. This is not a problem because the angular momentum perturbation

provides the necessary freedom for matching. Finally, the piece in h4 is proportional to
the time derivative of S(t, r), which -II__- -is we choose S(t) = yt, to keep the perturbation

static. In summary, we have for the interior and exterior metric perturbations


h.,o =72 Sin 8 11, (6- 58)
26aM~ sin2 8
"t4 (6-59)

with all other components vanishing.

Continuity of the metric perturbation ([hab] = 0) requires

yra =Yl (6-60)
2M~ sin O'

where we've used the equality of azY~o to expand Y,$. As before, the radial jump

conditions will determine y. In this case we'll use the odd-parity (imaginary) part of

Equation 6-17. The relevant tetrad components are given by:

iyr2 1 10
him- him = hm him- 12(-1

&t h = ht h' ir YIO II (6-62)
Im Im am am p~2 1/2 '
752 (Yi"ilG~pL
16xr Tim Ti =r0/ O)s COS 8)6(4 Ot). (6-63)










Imposing the radial jump conditions then gives us

i6y il6xpLZ
f1/2 1'-o 101/2 6CS@ f


OJ l= 0 m= -

After exploiting the orthogonality of the spin-weight 1 spherical harmonics and using

Equation 6-57 we obtain

7 = .(6-64)

It then follows from Equation 6-60 that


6J = 6aM~= pL, (6-65)


which is precisely what is to be expected-all of the angular momentum in the (otherwise

non-rotating) spacetime comes from the angular momentum of the particle. Once again

we can verify directly from Equation 6-23 that we have correctly identified the angular

momentum of the spacetime.

6.1.2.2 Even-parity dipole perturbations

First let's review what we already know: (1) We only need consider the -e = 1

pieces. This was established above when we found the angular momentum to have -e = 1

angular dependence. (2) The even-parity perturbation cannot contribute to the mass or

angular momentum of the spacetime. These perturbations have already been accounted

for. Furthermore, it is clear that in the absence of a source, there would be no e = 1

perturbation in the even parity sector.

In contrast to the situation with mass and angular momentum perturbations, where it

was easy to write down the general form of the perturbations, we have no general form for

the metric perturbation. Without prior knowledge of the perturbation, we must resort to

solving the Einstein equations to determine the perturbation. This has been carried out by

both Zerilli [7]and Detweiler & Poisson [17]. The result is a metric perturbation that can










be written


her pE 2r, sin 8 sin(4 Rt)G(r ro),

her = 2 pE r sin 8 cos(4 Rt)G(r ro),

p-E
bro = ~r, cos v cos(4/ -- Ot6(rV ro0),




Note that the singular nature of this metric perturbation inherently excludes it from

our analysis, as it destroys the continuity of the metric perturbation across E,. It is well

known [7] that the gauge transformation leading to this description can be interpreted as a

transformation from a non-inertial frame tethered to the central black hole to the center of

mass reference frame.

6.2 Kerr

In contrast to the situation in the Schwarzschild background, mass and angular

momentum perturbations in the K~err background are much more complicated. There

is, however, one simplifying feature of the mass and angular momentum perturbations.

Namely, the fact that both perturbations are stationary. Therefore the angular dependence

is not given by the spin-weighted spheroidal harmonics, sS(aw, 8, 4), but rather their

aw = 0 limit-the spin-weighted spherical harmonics.

The primary issue with treating the non-radiated multipoles in the context of

matched spacetimes is the choice of the matching surface, E,. Most of our discussion will

be focused on this issue.

6.2.1 Mass Perturbations

In place of Birkhoff's theorem there is Wald's theorem [16], described earlier, assuring

us that infinitesimal mass perturbations of the K~err solution lead to other K~err solutions

(with infinitesimally different masses, of course) because such perturbations do not

contribute the perturbations of I',, or tb4 (Which we will verify shortly). Thus we have the










nonzero components of the mass perturbation given by


2r6M~
htt= (6-66)
r~2 + 2 cos2 H)
2r(r2 + 2 COS2 )1
her= (6-67)
(r2 2Af~r + a2 2
2ar sin2 81
he (6-68)
r~2 + 2 cos2 H
2a2T Sin4 81
hee = .(6-69)
r2 + 2 COS2 H

Because the calculations in the K~err spacetime are significantly more complicated, we will

take a shortcut to determining the angular dependence of the perturbation by looking at

the tetrad components of the metric perturbation, a result which we will in any case use

shortly. In the symmetric tetrad (Equations 6-13) we have

2r6M~
ha =(6-70)
2r6M~
h,z~ (6-71)


with all other components vanishing. Because both hit and h,z, are spin-weight 0, they

have a natural decomposition into -e = 0, m = 0 scalar (ordinary) spherical harmonics.

Furthermore, utilizing Equation :327 we see that


I~,, = (B r')(B r')hit = 82 11 2r'Bhy, = 0, (6-72)


and similarly for 1/4. Therefore, according to Wald's theorem, we are ensured that

Equations 6-70 and 6-71 are a perturbation towards another K~err solution.

With the angular dependence determined, we are led to consider a gauge vector of the

form

(a = (P(t), Q(r), 0, S(t, 4)), (6-73)









which represents the largest class of gaugfe transformations consistent with form

invariance. This requirement also restricts


S(t, 4) = pt + S(4),


(6-74)


while stationarity again necessitates


P(t) = a~t.


(6-75)


Next we turn our attention to the matching problem.

In order to clarify the issues involved in the matching problem, we'll take a look

at the matching conditions themselves. Suppose we've chosen some E,, but have yet to

specify it explicitly. That is, we have not yet written (or imposed) r = something. The full
set of matching conditions now take the form


(6-76)



(6-77)


(6-78)

(6-79)


htt [Ca](p2 + 2rM~) + 2[Plamr sin2 0 2r61M = 0,

44 : 2 [ca]amr [P] (a2a COS2 H 2 2 + 2) + 2amr)

+2amr dS- 2ar61M = 0,


h,, dQ r6M~
her Q = 0,


& : a2T Sin2 ObM~ (a2 COS2 2 2 2) + 2amr) = 0i, (6-80)

where a = T2 2M~r + a2 and 752 = 2 + 2 COS2 8 aS before and we have imposed the

condition in Equation 6-79 in the others. Note that this reduces to the Schwarzschild

result in Equations 6-39-641 by taking a 0 and setting r = ro. This set of equations









has a solution given by


6M =[] (6-81)


[0]~ =9 (6-82)

[P] [Ca], (6-83)

dS a2a Sin2 8
-[0], (6-84)
d# (T2 82 a2

which is again easily seen to reduce to the Schwarzschild result in the appropriate limit.

From these equations we can see clearly the issues involved in choosing a matching

surface. First, because the left sides of Equations 6-81-684 are all constant, this must be

reflected in the right sides as well, which currently exhibit dependence on both r and 0.

Presumably, some choice of r = r (0) will enforce this, though it is currently unclear what

that choice might be. Note that because of this, r = constant surfaces do not appear to be

good for matching.

What we have encountered appears to be an instance of a longstanding problem

with matching the K~err solution to a source [74, 75]. Namely, there is no known matter

solution that correctly reproduces the multiple structure of the full K~err geometry. In our

problem, we're trying to force the issue by specifying both the metric and the source. On

the other hand, because we're not matching the entire source, which includes quadrupole

and higher moments, but only the non-radiated multipoles that merely take us from one

K~err solution to the next, it is not clear that the matching (in this instance) should fail.

Though we are unable to perform the matching here, we maintain that nothing forbids it.

Most authors faced with this issue turn to the -lei-- rotation" approximation and

keep only terms linear in a. In this approximation the K~err metric can be viewed as the

first order perturbation of the Schwarzschild solution to the K~err solution. That is, the

background is given by Schwarzschild plus a term identical to that in Equation 6-59. It










is no surprise, then, that the resulting background geometry possesses enough spherical

symmetry to allow for a straightforward treatment of the problem. It can he directly

verified that such a procedure would remove the 8 dependence in Equations 6-81-684 and

allow for a matching on r = constant surfaces (which are round 2-spheres in this case).

Because this approach fails to shed new light on the situation in the full K~err spacetime,

we will not follow it here. Instead, we will focus on Equations 6-66-6-69, which we know

to be correct.

Let's review the situation. We have established that the metric perturbation in

Equations 6-66-669 is a perturbation towards another K~err solution with differing

mass. Furthermore, we previously established that 6M~ = pE (Equation 6-26). The

problem is that we are currently unable to perform the matching. In practice, the relevant

portion of the spacetime is the exterior where gravitational radiation and the non-radiated

multipoles are observed far away from the source. Because of this, we contend that

considerations from the K~omar formula and Wald's theorem together provide the correct

perturbation in the exterior spacetime, independently of any matching considerations.

Thus our result is likely useful in the EAIRI problem even though we lack the metric

perturbation everywhere in the spacetime. Moreover, the perturbation is still simple to

interpret and .I-i-.npind'' ;cally flat, so it is amenable to some analysis.

This being the case, we remark that mass perturbations of the K~err background

remain confined to the s = 0 sector of the perturbation. It is likely that this is true in

general (at least in type D), but a general proof of this remains elusive. Furthermore,

contrary to what one might expect in the K~err spacetime, the mass perturbation does

not mix spherical harmonic -modes, but is purely -e = 0. We now turn our attention to

angular momentum perturbations.

6.2.2 Angular Momentum Perturbations

Our lack of success in matching mass perturbations extends to angular momentum

perturbations in precisely the same way, though the expressions involved are more










complicated. This being the case, we will focus our attention on the general features of the

angular momentum perturbation that can he obtained independently of a good matching.

We begin by noting that the nonzero components on the metric perturbation are given by

4M~ar cos2 86a
htt= (6-85)
(r2 2Afr + a2 2'
2a(r2 Sin2 0 + 2rM~ cos2 8)6
her (6-86)
(r2 2Af~r + a2 2
2M~ar sin2 8(2 a2 cos2 8
hte (6-87)
(T2 + 2 COS2 H 2
hoo = -2a Cos2 86a (688)
2a sin2 8 [2 p2 2 a"COS2 :)+3(r. + 2M~ sin2 8 16a
he (6-89)
(r2 + 2 COS2 H 2

The corresponding tetrad components (in the symmetric tetrad) are given by

aba[(r2 a2) Sin2 0 2Afr(cos2 8 p
hit = h,z (6-90)
p"2
aba sin2 8
hi,z (6-91)

aba(cos2 8 p
h,waz= (6-92)

-iba(a2 Iff) Sin2
bi,~ = h,z,>= (6-93)
(r + i cos H) ii
aba sin2 8
h,,n = (6-94)
(r + ia cos 8)2

where we have omitted the complex conjugates. Though it is not immediately obvious,

this perturbation makes no contribution to I~,, or 2/4, enSuring that this is a valid angular

momentum perturbation.

In light of relatively straightforward results for mass perturbations, the nontrivial

form of Equations 6-90-694 comes as a surprise. Unlike mass perturbations, angular

momentum perturbations are not confined to a single s sector, whereas one might expect

them to be exclusively s = 1, as intuition from working in the Schwarzschild background

would lead us to believe. Note that although the perturbation appears in the s = +2

sector of the metric, the vanishing of the s = +2 components of the Weyl curvature keep










the perturbation from contributing to gravitational radiation. More importantly, this is

a sign that our intuition needs adjustment for working in the K~err spacetime. In further

contrast to our prior results, the complicated 8 dependence in the tetrad components

of the metric perturbation leads to mixing of the (spin-weighted) spherical harmonic

e-modes, a complication not previously encountered.

Another surprising feature is the fact that the perturbation is complex and thus

exhibits both types of 1. .. .ty". Although the static nature of the perturbation guarantees

spin-weighted spherical harmonic angular dependence, we must be careful not to speak

of parity in the Schwarzschild sense, but rather the real and imaginary parts of the

perturbation. In any case the implications of this fact are presently unclear and remain to

be determined in future work.

6.2.3 Discussion

In this section we discuss in more detail the possible problems with our matching by

looking more closely at the assumptions that we made. This will lead naturally to ideas

about future work that is beyond our present scope.

First off, one may speculate that our requirement of form invariance is perhaps too

strict to allow for a proper matching. This does not appear to be the case. A result of

Carter [76] implies that, due to stationarity and axial symmetry, the K~err metric (in

Bci-; r-Lindquist coordinates) has precisely the minimum number of nonzero components.

Having established independently that the mass and angular momentum perturbations

preserve theses properties of the background, Carter's result -II_ _- -; that the problem lies

elsewhere .

This leads us to consider whether the introduction of an infinitesimally thin shell of

matter (which is effectively what E, is), necessarily introduces non-K~err perturbations.

A shell (of some currently unspecified shape) would presumably be a differentially

rotating object. It is unclear whether this disrupts the stationarity or axial symmetry

of the exterior spacetime by the introduction of perturbations that we have neglected










to account for. Wald's theorem [16] actually specifies two other types of perturbations

that I',, and d'4 cRallOt ROcount for. perturbations towards the accelerating C-nietrics and

perturbations toward the NITT solution. In the work of K~eidl, et. al. [69], where they

concerned themselves with a static particle in the Schwarzschild geometry, it was found

that the spacetinle on the interior differs front that on the exterior by a perturbation

towards the C-nietrics. This makes physical sense because a static particle is not on a

geodesic of the Schwarzschild spacetime and thus requires acceleration to keep it in place.

Though we have no obvious physical reason to expect these perturbations for circular,

equatorial orbits of the K~err geometry and evidence front the Schwarzschild calculation

-II- -_ -r ;they should not contribute, we have not yet proven a result either way.

Finally, one question that we have overlooked entirely is the question of the stability

of a thin shell. In the Schwarzschild background, this problem has been solved by Brady,

Louko and Poisson [77], who showed that a thin shell is stable and satisfies the dominant

energy condition almost all the way up to the location of the circular photon orbit (located

at r = 3M~). There are no such results to report on for the K~err spacetinte. The closest

thing to a step in this direction is the work of 1\usgrave and Lake [78], who consider the

matching of two K~err spacetintes with different values of mass and angular montentunt.

Unfortunately, these authors were forced to resort to the slow rotation approximation

discussed earlier. Strictly speaking, without knowledge of the existence of a stable shell

of matter sufficiently close to the black hole, we are left to question the validity of our

procedure. This is a problem we leave for future work.










CHAPTER 7
CONCLUSION

7. 1 Summary

Currently, there is much effort being devoted to computing theoretical waveforms for

gravitational wave detection. As a consequence, many researchers are looking for r- .--s

around the longstanding problems inherent in metric reconstruction in the K~err spacetime.

We feel that our new framework for perturbation theory presents a novel and robust tool

for investigation in this area.

First and foremost, by taking advantage of the GHP formalism, our framework

emphasizes and exploits those features common to all black hole spacetimes-their null

structure as manifested in their Petrov type--which, since Teukolsky's derivation of

the equation that bears his name, has been the only proven road to progress in this

difficult subject. Such features have made an appearance through the simplification

in the background GHP equations discussed in (I Ilpter 1. These have lead to useful

simplifications throughout. Besides these features, the built-in concepts of spin- and

boost-weight have allowed us some intuitive insight into the nature of the fundamental

quantities, without resorting to separation of variables.

The creation of GHPtools is the only reason any of this work was feasible in the

first place. Coordinate-independence comes at the price of having to perform many

nontrivial symbolic computations. GHPtools has not only allowed us to perform such

computations, but also to present them in a fully simplified way, bringing some clarity

even to previously known results. This is perhaps most evident in our treatment of the

Teukolsky-Starobinsky identities, where the use of GHPtools masked all of the horrendous

computational complexity involved in their derivation, by providing simple and concise

results in the end.

Furthermore, the coordinate-free nature of our framework has further allowed us to

work in great generality. This was seen in our treatment of the commuting operators of










type D spaces in C'!s Ilter 1. Perhaps the best example of this is our proof of the existence

of radiation gauges in sourcefree regions of spacetime. Our form of the Einstein equations

and Held's integration technique is a powerful combination that allowed us to prove the

result in arbitrary type II backgrounds, where the background integration isn't even

complete.

Finally, our treatment of the non-radiated multipoles demonstrates the power

of our framework when combined with existing techniques. Our results in the K~err

spacetime represent the first attempt at treating this part of the perturbation. Though we

were unable to obtain the description in terms of a matched spacetime, we nevertheless

provided a perturbation suitable for use in metric reconstruction.

7.2 Future Work

For all the generality inherent in the framework we developed, the applications we

presented were narrowly focused around the problem of metric reconstruction in the K~err

spacetime. This leaves many problems to be explored, both within the realm of metric

perturbations of K~err and otherwise. We detail some of these below.

Perhaps most pressing is the generalization of our result for the non-radiated

multipoles in the K~err spacetime to encompass more general orbits. In particular, orbits

not lying in the equatorial plane are of particular interest. Such orbits necessarily contain

off-axis angular momentum, which in turn are widely thought to be related to Carter's

constant (associated with the K~illing tensor). For such orbits the K~omar formulae fail to

completely characterize these off-axis angular momentum components, so it is clear that

we must look elsewhere for a solution. One potential avenue for progress is the Einstein

equations themselves. As we noted in the previous chapter, mass and angular momentum

perturbations are both stationary perturbations with angular dependence characterized

by the spin-weighted spherical harmonics. The simplifications this brings for working with

the Einstein equations is immense and may prove to make the problem tractable, without

recourse to purely numerical methods. In any case, it seems clear that our framework,









either alone or in conjunction with various other techniques, will help to clarify the

problem enormously.

Another avenue worth pursuing is the commuting operator associated with the K~illing

tensor due to Beyer [44] (cf. ('! .pter 1 ). Recall that Beyer's operator commutes with

the scalar wave equation in K~err. It is very tempting to think that such an operator

would exist for the Teukolsky equation as well. The GHP formalism, and GHPtools (of

course), provide the ideal environment in which to study such questions. Furthermore,

in the context of work performed by Jeffryes [79] concerning the implications of the

existence of the K~illing spinor (which includes a discussion of the Teukolsky-Starobinksy

identities), it is natural to think that such an operator may in fact shed some new light

on the Teukolsky-Starobinsky identities in the form presented in ('! .pter 5. Additionally,

the existence of a generalization of Beyer's operator carries with it the possibility of new

decomposition of functions in the K~err spacetime--just as the existence of the K~illing

vectlors and lead to separation in t and cf according to e-ime and e""m* (respectively),l

the eigenfunctions of a generalized Beyer operator may provide a new separation of

variables in the K~err spacetime. This is certainly a possibility worth pursuing.

Finally, both GHPtools and our form of the perturbed Einstein equations are

entirely general and ready for use by researchers interested in more general (or even

more specialized) backgrounds than Petrov type D. In particular, the class of type II

spacetimes seems a likely candidate for further analysis, especially with the aid of the

integration technique of Held. We have only begun to scratch the surface of the wide

v-1I r ii of problems these tools can help solve.




























































(A-10)


APPENDIX A
THE GHP RELATIONS

In this appendix, we give the GHP commutators, field equations and Bianchi

identities, as well as the derivatives of the tetrad vectors. The full set of equations is

obtained by applying to those listed prime, complex conjugation or both. When acting on

a quantity of type {p, q}, the commutators are:


[P, P']


(7 T')8 + (7' f')' p(s'~ Tr'a + 2 11


(A-1)


-q(a'R pfr' + @ol),

(P' p')P + (p p)P' + p(pp'

-q pp'- aa' + ', 1 -I n.







r- 8' = (p p)a ( -r p')

- p's =p2 _/



- P'a -p'a a'p + 'r + k

-- a'-r = pp' + aa' -- -rf -- KK


(A-2)


[a, a']


- ee' + 011 II)


(A-3


The GHP equations are:


Bp

Pp



-r

87

D'p





~ 1 + @ol,

CK 02,

;' I' 2II.


The Bianchi identities are given by:


P 1 a'l',, p@ol + 8 oo


-r'l',, + 4pl 1,' l~_ + -r'@oo 2p~ol

-2a410 + 2n~11 + s@02,










a'l',, 27r' I + 3p 2 2m 3 00p~o

-2-r001 27010o + 2p@l 02~o, (A-11)

2a' I 3-r'a 2 2p 3 ~4 2p'910

+27r'911 + -r'@20 2p@21 22~a, (A-12)

3 'l', 4'' .+ 4 2n' 10 + 2a'@ 1

+p'ao2 2-r021 + 22~. (A-13)


P9 2 8'1 01 00, +p~, 2PII



pi. 2 21~ 20~, a, 28'II



P9 4 .l' 21 +120


Finally, the derivatives of the tetrad vectors are given by


Oalb



BOmb


- a irmb T-mb) na Emb mb)

+ma(@mb ; "IiI.) + Eii, mb)

- a Klb + ~b) na ~b nb)

+m,(p'lb ; "';,.) m('b +Ob)


(A-14)


(A-15)









APPENDIX B
THE PERTURBED EINSTEIN EQUATIONS IN GHP FOR1\

In this appendix we write the components of the perturbed Einstein tensor for an

arbitrary algebraically special (Petrov type II) background. We have assumed the PND is

aligned with 1" and made use of the Goldberg-Sachs theorem. Note that the equations for

Sinz, E~z,> and 8,,2, are complex, so Stra = ir,z and so on:



Sir = {((8' -r')(a -r') + p (P' + p' p') (P p) p' + 92 11

+{-(p + p)(P + p + p) + 4pp~hl,z

+{-(P 3p)(8' --r' + -) + -P- pi'}hi

+{-(P 3p)(B + -r ') + 79- pi)hl,>

+{D(D p p) + 2pp~h,>za, (B-1)





+(8a' -r)(8 r) + p'(P p + p) (P' p')p + 92 + 2pp'}h,z~

+{- (p' + p') (P' + p' + p') + 4 p'p' (8' 2-r) (B 27r)s' }hl,z




+ (D 2p ')s' + '(D' p'r -r'a p ') R'e'hy,>






+(P 2p + p)R' + (8' :3- + -r')a' + B'(a') W:3}hnmz

+{-(a' 2-r)s' a'(P' p' + p')}h,>,>

+{-(a 27)&' a'(P' p' + p')}h,7tr

+{D'(P' p' p') + s'(-r -r') + #'(-r -r') + 2a'a' + 2p'p'}h,>a,, (B-2)
















1.
Ez~~z {p((' pd(l ') + p'(D' p') + (8 2f') +(p' 27)'+2''}z



1.
+ {((D 2p) + p(D -r p)}h r)P+ )- (' r'p- -Pz
1.
+ {-(8' + 2'+)(B 7-7) (8'S + 3rr'P + 377' + 2(8 + -r) '2r)h,




+ (P 2p')p + (9' 2p')p- '( + p)3 p(D' + p')P 92 -2 i


1



+ {(D' 2p)(8' 2') + p(P' + 2p' + 2p') 4p'(D p')


(- (28 2r)' a(P 2p~s + o'( 2r( r)}hir,
1.

(25p 2p)p 2(p 2p)R'+ '7 T)hi,













1.
+-{(( 9 p')(' r'' + F'a) +2pa m
1.
+- { (- + p )8 p) ) (8 1 + 2-' -r) + 2f '(D 2p)h,


- 8 7 7' p 2p'r knr,B4




1.
{ (-a(' p' r) s + a'( + R'e' } ha(r'--r)
1.
+- { (9 p + ) (8 2) p(P' 27' )p + 2'(P p) }h~a- 2z
2('-3rB+-'21-- +4r)--(1-2r),
1.


+-P { (-(' p' p) (a' + ) + (P' r') + (a) p-' + ( 7~ + 2 ')a'



(('- 2p'a' +5 ( 9 2p'f }hy a

+{'(p'~ 2p' -'r') 2-r)r + a'4 Int)hl

+(('- {( ')(> 2p') + -s(B 27 + 27')p + -r('(8 47' + 27) p)~


+-{-r '( 8' 27 + -r ( 2p+ p -2 ('- }hm










+ -((' p') ( pr) + (B -r) r') -r(8' + p) -r( ) + p)2 mm


+{((P-2p)a' +(r + ') + (r -')2 non, (B-6)



1.
{D'(pD p' p') + 2p'p' 'r-7) 'h-7) ,U}i
1.


--{D(B pr + p)r + 2pp')hl,z
1.
- { (9+p' 2p') (8 pr + rp) ( + 2p) + 2p ( + p'8 p')a a'2




t(8'- )(P 2p(a f r') + -r' -(P 2p 2)' f (28 + 4-'p)h

-27-(8 2p)( + 27r'f + pp'}hi 2p-2p 2- +4r'z
1.


1.

- 2f'p' 2r'( p)( r')- 3}him a'h,

-{-(2P 2p(P' 27')p + '( 2p~ 2p)P 2pf + 47' p'(h+ pm

2B7










APPENDIX C:
INTEGRATION A LA HELD

We provide details of the integration that lead to Equation 4-17 and 4-25. As it

turns out, the type II calculation is actually much simpler than the the type D calculation

because it uses a tetrad in which -r = -r' = 0. Therefore we will work out the type D

calculation in detail and the type II result mostly follows by setting certain quantities to

zero, as indicated below.

We will need some results (and their complex conjugates) from the integration of the

type D background:

8 p -o_ o 2 C1
i~p
1 1
o 2 -0 2_
2 2
1 1 1 11
+Tto "~2 2iT 0T~ 0t 2 22p +ToO, (C-2)

-r = -Wo" cto o rpp, (C-3)



92 0 3. (C-5)


As noted in the text, xo" / 0 leads to the accelerating C-metrics, which we include for full

generality. Henceforth the corresponding quantities in type II spacetimes can be obtained

by setting -ro a r = cto 4 0 and Wo 420~1 in the type D result. Thus, in type II




1 This arises from the fact that in type D spacetimes there is only one non-vanishing
Weyl scalar, 92. Ill type II spacetimes, however, both 93 and 94 arT in genOTra alSO
nonzero. Though we do not refer to any of the other Weyl scalars in this work, we would
like maintain agreement with the standard conventions.









spacetimes we have


p' =po 202 6)

-r = 0, (C-7)

7' = 0, (C-8)

92 20 ao3, _9)


the equation for 5 p not following from the limiting process mentioned above. Note

that the quantity B p is never used in any of the integration we perform in the type II

background spacetime. We will also need the definitions of the new operators:

~; 1pW 4 2
P = i-T r( (C-10)
p p 2 p p
+ (C-11)

B (C-12)


where p and q label the GHP type of the quantity being acted on. Additionally, in

Sections 4.2 and 4.4 we make use of the commutator

-; -p' / 1 11 ; p' 1 1 1 -(
[8, 8 ] P' + -i -/ + +~ p +
pp p p 2 p p(C-13)
VCp' 1 1 1/ tr


which is valid in type D and (with -r = 0) type II spacetimes.

We now begin with

DO = 0, (C-14)

which integrates trivially to give

II = Clo. (C-15)

With this information in hand, we can now integrate the equation governing (m:


(P + p)(m + (B + rl)(1 = 0. (C-16)










Rewriting the P piece and using Equation C-11 with p = 1 leads to


19(pim) + -r (1 + poft Cl = 0, (C-17)
p p

which, after substituting Equation C-3, the complex conjugate of Equation C-4 and

Equation C-15 along with some re I1 llpil_ yields





Integration then gives us


m mol~ _I 0 lo a lo _~ ) lo _C19)
p p

and the solution for (m then follows from complex conjugation


m7 = (mo ~o a_ o.(-0
p p

Finally, we are in a position to deal with (n, by writing


P'e + Des + (-r + -r')(m + (-r + -r')(m = 0, (C-21)


in terms of Held's operators (Equations C-1, C-3 and C-4) as

en>(I t~rt-I f- -6
P~+ ~-rllp p1)E
(C-22)
+4P ), ~+ (7 + v'r)(m + (7 + -y')(m = 0.

Substituting Equations C-3, C-4, C-5, C-15, C-19 and C-20, rearranging terms and

letting the dust settle leads to
~;=-p~ 1 1 -a o 1 1\

2 2 pp


,I 1 1, (C- 23)
[xro 8 + cto a or( a +l~ 2xo~mo + 2 o mo
p p
2 2









Integration then results in
1 1= 1 $ P1 1\
2 2 2 pp2/
[xo go ,1 1 1 ~;
-(8+0) -(8 +o) ro+- -+- DIr
p p 2 p p ,
(C 24)

P P
1 1 1l .1


and our task is complete.









APPENDIX D
SPIN-WEIGHTED SPHERICAL HARMONICS

In this appendix, we present the basics of the theory of spin-weighted spherical

harmonics [15, 80]. These functions have a natural place in the GHP formalism and

provide a simple alternative to the more complicated tensor spherical harmonics. The

discussion in this section takes place on the round 2-sphere. In that case, the action of a

on some quantity, X, of spin-weight s is given by


EX = (sin 8)" +i csc 0 (sin 8)-"X, (D-1)


and the action of 8' is


B'X = (sin 8)-" i csc 0 (sin 8)"y. (D-2)


The spin-weighted spherical harmonics, shm(0, ~), are then defined in terms of the

ordinary spherical harmonics by



s em(e, #)= (,)0


but are undefined for |s| > -e. The basic properties of the s m, are easily seen to be


shm, = (-1)m+s-syem, (D-4)

ishmn = (- 8)(c+ s+) s+1Im,, (D-5)

alsT m = (+ sl) (- + 1) s-ibm (D-6)

a'ashem = (e- s)(e+ s + 1) sYem. (D-7)


For each value of s, the spin-weighted spherical harmonics are complete:


s~e(0 4)hm0',#' =6(# #')6(cos 0 cos 0'), (D-8)
= 0 m= -








and orthogfonal:


da sem(, 4)~em(0, ) =barb m.(D-9)










APPENDIX E
MAPLE CODE FOR GHPTOOLS

GHPtoolsv1:= module()

description "GHPtools: Tools for working in the GHP (and NP)

formalism(s).";

export

schw, typed, flatxyz, GHPprime, GHPconj, NPprime, NPconj,

DGHP, NPexpand, ezcomm, comm, getpq, tetcon, GHPmult,

GHP2NP, tetupK, tetdnK, tetupS, tetdnS, tetupSB, tetdnSB,

tdspec, tdsimp, GHP1, GHPip, GHPlc, GHPipc, GHP2, GHP2p,

GHP2c, GHP2pc, GHP3, GHP3p, GHP3c, GHP3pc, GHP4, GHP4p,

GHP4c, GHP4pc, GHP5, GHP5p, GHP5c, GHP5pc, GHP6, GHP~p,

GHP~c, GHP~pc, COM1, COMlp, COMic, COMipc, COM2, COM2p,

COM2c, COM2pc, COM3, COM3p, COM3c, COM3pc, BlI, Blip,

BIlc, BIlpc, BI2, BI2p, BI2c, BI2pc, BI3, BI3p, BI3c,

BI3pc, BI4, BI4p, BI4c, BI4pc;

local

times, ssubs, THORN, THORNP, ETH, ETHP, GHPcomm,

D_delta, D_deltabar, D_DD, D_Delta, GE1, GE2, GE3, GE4,

GE5, GE6;

option

package;



schw:={kappai=0, conjugate(kappal)=0, sigmai=0, conjugate(sigmal)=0,

kappa=0, conjugate(kappa)=0, sigma=0, conjugate(sigma)=0, epsilon=0,

conjugate(epsilon)=0, taul=0, conjugate(taul)=0, tau=0,

conjugate(tau)=0,conjugate(rhol)=rhol, conjugate(rho)=rho, Psil=0,










conjugate(epsiloni)=epsiloni, conjugate(beta)=betal,

conjugate(betal)=betal, Psi0=0, conjugate(Psi0)=0, Psii=0,

conjugate(Psii)=0, conjugate(Psi2)=Psi2, Psi3=0,conjugate(Psi3)=0,

Psi4=0, conjugate(Psi4)=0, Phi00=0, conjugate(Phi00)=0, Phi01=0,

conjugate(Phi01)=0, Phi02=0, conjugate(Phi02)=0, Phil0=0,

conjugate(Phil0)=0, Phill=0, conjugate(Phill)=0, Phil2=0,

conjugate(Phil2)=0, Phi20=0, conjugate(Phi20)=0, Phi21=0,

conjugate(Phi21)=0, Phi22=0, conjugate(Phi22)=0, PI=0,

conjugate(0)=0};



typed:={Psi0=0, conjugate(Psi0)=0, Psi4=0, conjugate(Psi4)=0,

kappai=0, conjugate(kappal)=0, sigmai=0, conjugate(sigmal)=0, kappa=0,

conjugate(kappa)=0, sigma=0, conjugate(sigma)=0, epsilon=0, Psii=0,

conjugate(Psii)=0, Psi3=0,conjugate(Psi3)=0, Phi00=0,

conjugate(Phi00)=0, Phi01=0, conjugate(Phi01)=0, Phi02=0,

conjugate(Phi02)=0, Phil0=0, conjugate(Phil0)=0, Phill=0,

conjugate(Phill)=0, Phil2=0, conjugate(Phil2)=0, Phi20=0,

conjugate(Phi20)=0, Phi21=0, conjugate(Phi21)=0, Phi22=0,

conjugate(Phi22)=0, PI=0, conjugate(0)=0};



flatxyz:={kappai=0, conjugate(kappal)=0, sigmai=0,

conjugate(sigmal)=0, kappa=0, conjugate(kappa)=0, sigma=0,

conjugate(sigma)=0, epsilon=0, conjugate(epsilon)=0, taul=0,

conjugate(taul)=0, tau=0, conjugate(tau)=0,rhol=0, conjugate(rhol)=0,

rho=0, conjugate(rho)=0, Psii=0, epsiloni=0, conjugate(epsiloni)=0,

beta=0, conjugate(beta)=0, betal=0, conjugate(betal)=0, Psi0=0,

conjugate(Psi0)=0, Psii=0, conjugate(Psii)=0, Psi2=0,










conjugate(Psi2)=0, Psi3=0,conjugate(Psi3)=0, Psi4=0,

conjugate(Psi4)=0, Phi00=0, conjugate(Phi00)=0, Phi01=0,

conjugate(Phi01)=0, Phi02=0, conjugate(Phi02)=0, Phil0=0,

conjugate(Phil0)=0, Phill=0, conjugate(Phill)=0, Phil2=0,

conjugate(Phil2)=0, Phi20=0, conjugate(Phi20)=0, Phi21=0,

conjugate(Phi21)=0, Phi22=0, conjugate(Phi22)=0, PI=0,

conjugate(0)=0};



# ssubs performs the simple task of substituting the spacetime list into its

# argument



ssubs := proc(expr)

if evalb(eval(spacetime)= 'spacetime') or

evalb(eval(spacetime)='none') then

return(expr);

else

return(subs(spacetime,expr));

end if;

end proc;



getpq := proc(expr)

local p,q;

if evalb(expr=hnn) then p:=-2; q:=-2

elif evalb(expr=hln) then p:=0; q:=0

elif evalb(expr=hnmb) then p:=-2; q:=0

elif evalb(expr=hnm) then p:=0; q:=-2

elif evalb(expr=hll) then p:=2; q:=2










elif evalb(expr=hlmb) then p:=0; q:=2

elif evalb(expr=hlm) then p:=2; q:=0

elif evalb(expr=hmbmb) then p:=-2; q:=2

elif evalb(expr=hmmb) then p:=0; q:=0

elif evalb(expr=hmm) then p:=2; q:=-2

elif evalb(expr=rho) then p:=1; q:=1

elif evalb(expr=conjugate(rho)) then p:=1; q:=1

elif evalb(expr=rhol) then p:=-1; q:=-1

elif evalb(expr=conjugate(rhol)) then p:=-1; q:=-1

elif evalb(expr=kappa) then p:=3; q:=1

elif evalb(expr=conjugate(kappa)) then p:=1; q:=3

elif evalb(expr=kappal) then p:=-3; q:=-1

elif evalb(expr=conjugate(kappal)) then p:=-1; q:=-3

elif evalb(expr=tau) then p:=1; q:=-1

elif evalb(expr=conjugate(tau)) then p:=-1; q:=1

elif evalb(expr=taul) then p:=-1; q:=1

elif evalb(expr=conjugate(taul)) then p:=1; q:=-1

elif evalb(expr=sigma) then p:=3; q:=-1

elif evalb(expr=conjugate(sigma)) then p:=-1; q:=3

elif evalb(expr=sigmal) then p:=-3; q:=1

elif evalb(expr=conjugate(sigmal)) then p:=1; q:=-3

elif evalb(expr=conjugate(Psi0)) then p:=0; q:=4

elif evalb(expr=conjugate(Psii)) then p:=0; q:=2

elif evalb(expr=conjugate(Psi2)) then p:=0; q:=0

elif evalb(expr=conjugate(Psi3)) then p:=0; q:=-2

elif evalb(expr=conjugate(Psi4)) then p:=0; q:=-4

elif evalb(expr=Psi0) then p:=4; q:=0










elif evalb(expr=Psii) then p:=2; q:=0

elif evalb(expr=Psi2) then p:=0; q:=0

elif evalb(expr=Psi3) then p:=-2; q:=0

elif evalb(expr=Psi4) then p:=-4; q:=0

elif evalb(expr=phi) then p:=pp; q:=pq

elif evalb(expr=conjugate(phi)) then p:=pq; q:=pp

elif evalb(expr=phil) then p:=-pp; q:=-pq

elif evalb(expr=conjugate(phil)) then p:=-pq; q:=-pp

else

p:=UNKNOWN; q:=UNKNOWN

end if;

return(p,q);

end proc;



GHPprime := proc(expr)

return(subs({1dn=ndn, lup=nup, ndn=1dn, nup=1up,

mdn=conjugate(mdn), mup=conjugate(mup), conjugate(mup)=mup,

conjugate(mdn)=mdn, hll=hnn, hnn=hll, hlm=hnmb, hnmb=hlm, hlmb=hnm,

hnm=hlmb, hmm=hmbmb, hmbmb=hmm, th=thp, thp=th, eth=ethp, ethp=eth,

rho=rhol, conjugate(rho)=conjugate(rhol), rhol=rho,

conjugate(rhol)=conjugate(rho), kappa=kappal,

conjugate(kappa)=conjugate(kappal), kappai=kappa,

conjugate(kappal)=conjugate(kappa), tau=taul,

conjugate(tau)=conjugate(taul), taul=tau,

conjugate(taul)=conjugate(tau), sigma=sigmal,

conjugate(sigma)=conjugate(sigmal), sigmai=sigma,

conjugate(sigmal)=conjugate(sigma), epsilon=epsiloni,










conjugate(epsilon)=conjugate(epsiloni), epsiloni=epsilon,

conjugate(epsiloni)=conjugate(epsilon), beta=betal,

conjugate(beta)=conjugate(betal), betal=beta,

conjugate(betal)=conjugate(beta), Psi0=Psi4,

conjugate(Psi0)=conjugate(Psi4), Psi4=Psi0,

conjugate(Psi4)=conjugate(Psi0), Psii=Psi3,

conjugate(Psii)=conjugate(Psi3), Psi3=Psil,

conjugate(Psi3)=conjugate(Psii), Phi00=Phi22, conjugate(Phi00)=Phi22,

Phi01=Phi21, conjugate(Phi01)=Phil2, Phi02=Phi20,

conjugate(Phi02)=Phi02, Phil0=Phil2, conjugate(Phil0)=Phi21,

Phil2=Phil0, conjugate(Phil2)=Phi01, Phi20=Phi02,

conjugate(Phi20)=Phi20, Phi21=Phi01, conjugate(Phi21)=Phi10,

Phi22=Phi00, conjugate(Phi22)=Phi00, phi=phii, phil=phi,

conjugate(phi)=conjugate(phii), conjugate(phii)=conjugate(phi), p=-p,

q=-q, pp=-pp, pq=-pq,hl=hn,hn=hl\},expr));

end proc;



NPprime := proc(expr)

return(GHPprime(expr));

end proc;



GHPconj := proc(expr)

return(subs({mdn=conjugate(mdn), mup=conjugate(mup),

conjugate(mdn)=mdn, conjugate(mup)=mup, hlm=hlmb, hlmb=hlm, hnm=hnmb,

hnmb=hnm, hmm=hmbmb, hmbmb=hmm, ethp=eth,eth=ethp,

beta=conjugate(beta), conjugate(beta)=beta, betai=conjugate(betal),

conjugate(betal)=betal, epsilon=conjugate(epsilon),










conjugate(epsilon)=epsilon, epsiloni=conjugate(epsiloni),

conjugate(epsiloni)=epsiloni, kappa=conjugate(kappa),

conjugate(kappa)=kappa, kappai=conjugate(kappal),

conjugate(kappal)=kappal, sigma=conjugate(sigma),

conjugate(sigma)=sigma, sigmai=conjugate(sigmal),

conjugate(sigmal)=sigmal, rho=conjugate(rho), conjugate(rho)=rho,

rhoi=conjugate(rhol), conjugate(rhol)=rhol, tau=conjugate(tau),

conjugate(tau)=tau, taul=conjugate(taul), conjugate(taul)=taul,

Psi0=conjugate(Psi0), conjugate(Psi0)=Psi0, Psii=conjugate(Psii),

conjugate(Psii)=Psii, Psi2=conjugate(Psi2), conjugate(Psi2)=Psi2,

Psi3=conjugate(Psi3), conjugate(Psi3)=Psi3, Psi4=conjugate(Psi4),

conjugate(Psi4)=Psi4,Phi01=Phil0,Phi10=Ph1Pi2=i0Pi0Pi2P

hil2=Phi21,Phi21=Phil2, phi=conjugate(phi), conjugate(phi)=phi,

phii=conjugate(phii), conjugate(phil)=phii,

p=q,q=p,pp=pq,pq=pp},expr));

end proc;



NPconj := proc(expr)

return(GHPconj(expr))

end proc;



THORN := proc(f)

local i, rest, temp;

if type(f, 'symbol') then return map( th, f)

elif type(f, 'constant') then 0

elif type( f, list ) then map( THORN, f)

elif type( f, set ) then map( THORN, f)










elif type( f, '=' ) then map( THORN, f)

elif type( f, '+' ) then map( THORN, f)

elif type( f, '*' ) then

rest := mul(op(i,f), i=2..nops(f));

THORN(op(1,f))*rest + op(1,f)*THORN(rest);

elif type( f, '^' ) then

op(2,f)*0p(1,f)^(op(2f)-1)*THRNORN(op(,f)

elif type( f, function ) then

if op(0,f) = 'th' then

temp:=THORN(op(f));

return map(THORN, temp);

elif op(0,f) = 'thp' then

return apply(th, f);

elif op(0,f) = 'eth' then

return apply(th, f);

elif op(0,f) = 'ethp' then

return apply(th, f);

elif op(0,f) = 'conjugate' then

return apply(th, f);

elif op(0,f) = 'T' then

return apply(th,f);

elif op(0,f) = 'ln' then

return THORN(op(1,f))/op(1,f);

else

error "routine not built to handle that

function: %1", op(0,f);

end if;










else


error "routine not built to handle that type: \%1",

whattype(f);

end if;

end proc;



THORNP := proc(f)

local i, rest, temp;

if type(f, 'symbol') then return map( thp, f)

elif type(f, 'constant') then 0

elif type( f, list ) then map( THORNP, f)

elif type( f, set ) then map( THORNP, f)

elif type( f, '=' ) then map( THORNP, f)

elif type( f, '+' ) then map( THORNP, f)

elif type( f, '*' ) then

rest := mul(op(i,f), i=2..nops(f));

THORNP(op(1,f))*rest + op(1,f)*THORNP(rest);

elif type( f, '^' ) then

op(2,f)*0p(1,f)^(op(2,f)-1)*THRNPORP(op1,)

elif type( f, function ) then

if op(0,f) = 'th' then

return apply(thp, f);

elif op(0,f) = 'thp' then

temp:=THORNP(op(f));

return map(THORNP, temp);

elif op(0,f) = 'eth' then

return apply(thp, f);










elif op(0,f) = 'ethp' then

return apply(thp, f);

elif op(0,f) = 'conjugate' then

return apply(thp, f);

elif op(0,f) = 'T' then

return apply(thp,f);

elif op(0,f) = 'ln' then

return THORNP(op(1,f))/op(1,f);

else

error "routine not built to handle that

function: %1", op(0,f);

end if;

else

error "routine not built to handle that type: %1",

whattype(f);

end if;

end proc;



ETH := proc(f)

local i, rest, temp;

if type(f, 'symbol') then return map( eth, f)

elif type(f, 'constant') then 0

elif type( f, list ) then map( ETH, f)

elif type( f, set ) then map( ETH, f)

elif type( f, '=' ) then map( ETH, f)

elif type( f, '+' ) then map( ETH, f)

elif type( f, '*' ) then










rest := mul(op(i,f), i=2..nops(f));

ETH(op(1,f))*rest + op(1,f)*ETH(rest);

elif type( f, '^' ) then

op(2,f)*0p(1,f)^(op(2,f)-1)EETHo(op(1,f)

elif type( f, function ) then

if op(0,f) = 'th' then

return apply(eth, f);

elif op(0,f) = 'thp' then

return apply(eth, f);

elif op(0,f) = 'eth' then

temp:=ETH(op(f));

return map(ETH, temp);

elif op(0,f) = 'ethp' then

return apply(eth, f);

elif op(0,f) = 'conjugate' then

return apply(eth, f);

elif op(0,f) = 'T' then

return apply(eth,f);

elif op(0,f) = 'ln' then

return ETH(op(1,f))/op(1,f);

else

error "routine not built to handle that

function: %1", op(0,f);

end if;

else

error "routine not built to handle that type: %1",

whattype(f);










end if;

end proc;



ETHP := proc(f)

local i, rest, temp;

if type(f, 'symbol') then return map( ethp, f)

elif type(f, 'constant') then 0

elif type( f, list ) then map( ETHP, f)

elif type( f, set ) then map( ETHP, f)

elif type( f, '=' ) then map( ETHP, f)

elif type( f, '+' ) then map( ETHP, f)

elif type( f, '*' ) then

rest := mul(op(i,f), i=2..nops(f));

ETHP(op(1,f))*rest + op(1,f)*ETHP(rest);

elif type( f, '^' ) then

op(2,f)*0p(1,f)^(op(2,f)-1)*EETH~P(op(,f)

elif type( f, function ) then

if op(0,f) = 'th' then

return apply(ethp, f);

elif op(0,f) = 'thp' then

return apply(ethp, f);

elif op(0,f) = 'eth' then

return apply(ethp, f);

elif op(0,f) = 'ethp' then

temp:=ETHP(op(f));

return map(ETHP, temp);

elif op(0,f) = 'conjugate' then










return apply(ethp, f);

elif op(0,f) = 'T' then

return apply(ethp,f);

elif op(0,f) = 'ln' then

return ETHP(op(1,f))/op(1,f);

else

error "routine not built to handle that

function: X1", op(0,f);

end if;

else

error "routine not built to handle that type: %1",

whattype(f);

end if;

end proc;



DGHP := proc(expr)

local result;

result:=subs({th=THORN,thp=THORNP,eth=ETHeh=TPsubexr)

return(expand(eval(result)));

end proc;



D_delta := proc(f)

local i, rest, temp;

if type(f, 'symbol') then return map( delta, f)

elif type(f, 'constant') then 0

elif type( f, list ) then map( D_delta, f)

elif type( f, set ) then map( D_delta, f)










elif type( f, '=' ) then map( D_delta, f)

elif type( f, '+' ) then map( D_delta, f)

elif type( f, '*' ) then

rest := mul(op(i,f), i=2..nops(f));

D_delta(op(1,f))*rest + op(1,f)*D_delta(rest);

elif type( f, '^' ) then

op(2,f)*0p(1,f)^(op(2,f)-1)*D_delta(op( 1,)

elif type( f, function ) then

if op(0,f) = 'delta' then

temp:=D_delta(op(f));

return map(D_delta, temp);

elif op(0,f) = 'conjugate(delta)' then

return apply(delta, f);

elif op(0,f) = 'Delta' then

return apply(delta, f);

elif op(0,f) = 'DD' then

return apply(delta, f);

elif op(0,f) = 'conjugate' then

return apply(delta, f);

else

error "routine not built to handle that

function: %1", op(0,f);

end if;

else

error "routine not built to handle that type: %1",

whattype(f);

end if;










end proc;


D_deltabar := proc(f)

local i, rest, temp;

if type(f, 'symbol') then return map( conjugate(delta), f)

elif type(f, 'constant') then 0

elif type( f, list ) then map( D_deltabar, f)

elif type( f, set ) then map( D_deltabar, f)

elif type( f, '=' ) then map( D_deltabar, f)

elif type( f, '+' ) then map( D_deltabar, f)

elif type( f, '*' ) then

rest := mul(op(i,f), i=2..nops(f));

D_deltabar(op(1,f))*rest + op(1,f)*D_deltabar(rest);

elif type( f, '^' ) then

op(2,f)*0p(1,f)^(op(2,f)-1)*D_deltabar(o p(,)

elif type( f, function ) then

if op(0,f) = 'delta' then

return apply(conjugate(delta), f);

elif op(0,f) = 'conjugate(delta)' then

temp:=D_deltabar(op(f));

return map(D_deltabar, temp);

elif op(0,f) = 'Delta' then

return apply(conjugate(delta), f);

elif op(0,f) = 'DD' then

return apply(conjugate(delta), f);

elif op(0,f) = 'conjugate' then

return apply(conjugate(delta), f);










else


error "routine not built to handle that

function: X1", op(0,f);

end if;

else

error "routine not built to handle that type: %1",

whattype(f);

end if;

end proc;



D_DD := proc(f)

local i, rest, temp;

if type(f, 'symbol') then return map( DD, f)

elif type(f, 'constant') then 0

elif type( f, list ) then map( D_DD, f)

elif type( f, set ) then map( D_DD, f)

elif type( f, '=' ) then map( D_DD, f)

elif type( f, '+' ) then map( D_DD, f)

elif type( f, '*' ) then

rest := mul(op(i,f), i=2..nops(f));

D_DD(op(1,f))*rest + op(1,f)*D_DD(rest);

elif type( f, '^' ) then

op(2,f)*0p(1,f)^(op(2,f)-1)*D_DD~o(op(,f)

elif type( f, function ) then

if op(0,f) = 'delta' then

return apply(DD, f);

elif op(0,f) = 'conjugate(delta)' then










return apply(DD, f);

elif op(0,f) = 'Delta' then

return apply(DD, f);

elif op(0,f) = 'DD' then

temp:=D_DD(op(f));

return map(D_DD, temp);

elif op(0,f) = 'conjugate' then

return apply(DD, f);

else

error "routine not built to handle that

function: %1", f;

end if;

else

error "routine not built to handle that type: %1",

whattype(f);

end if;

end proc;



D_Delta := proc(f)

local i, rest, temp;

if type(f, 'symbol') then return map( Delta, f)

elif type(f, 'constant') then 0

elif type( f, list ) then map( D_Delta, f)

elif type( f, set ) then map( D_Delta, f)

elif type( f, '=' ) then map( D_Delta, f)

elif type( f, '+' ) then map( D_Delta, f)

elif type( f, '*' ) then










rest := mul(op(i,f), i=2..nops(f));

D_Delta(op(1,f))*rest + op(1,f)*D_Delta(rest);

elif type( f, '^' ) then

op(2,f)*0p(1,f)^(op(2,f)-1)*D_Delta(op( 1,)

elif type( f, function ) then

if op(0,f) = 'delta' then

return apply(Delta, f);

elif op(0,f) = 'conjugate(delta)' then

return apply(Delta, f);

elif op(0,f) = 'Delta' then

temp:=D_Delta(op(f));

return map(D_Delta, temp);

elif op(0,f) = 'DD' then

return apply(Delta, f);

elif op(0,f) = 'conjugate' then

return apply(Delta, f);

else

error "routine not built to handle that

function: X1", f;

end if;

else

error "routine not built to handle that type: %1",

whattype(f);

end if;

end proc;



NPexpand := proc(expr)










local result;

result:=subs({delta=D_delta,conjugate(dela=detbr

DD=D_DD,Delta=D_Delta},expr);

return(expand(eval(result)));

end proc;



times := proc(x,y)

local i,z,result;

result :=0;

for i from 1 to nops(x) do

if nops(x) <> 1 then z:=op(i,x) else z:=x end if;

if (z='delta' or z='conjugate(delta)' or z='DD' or

z='Delta' or op(0,z)='delta' or

op(0,z)='conjugate(delta)' or op(0,z)='DD' or

op(0,z)='Delta') then result:=result+apply(z,y)

else result:=result+zzy

end if;

end do;

return expand(result);

end proc;



GHP2NP := proc(expr)

local i,result,z,p,q,w;

result :=expr;

for i from 1 to nops(expr) do

w:=0p(i~expr);

if (op(0,w)='*' and op(1,0p(0,0p(nops(w),w))) ='th'










and op(0,0p(1,0p(nops(w),w))) ='th') then

(p,q):=getpq(op(1,0p(1,0p(nops(w),w))) );

result:=result w +

w/op(1,0p(0,0p(nops(w),w)))(op(0,0p(1,0p~np~)w)

(op(1,0p(1,0p(nops(w),w)))))*(times((DD -(p+1)*epsilon

-(q+1)*conjugate(epsilon)),times((DD p*epsilon -

q*conjugate(epsilon)),0p(1,0p(1,0p(nops(w)w)))

elif (op(0,w)='th' and op(0,0p(1,w))='th') then

(p,q):=getpq(op(1,0p(1,w)));

result:=result w + times((DD (p+1)*epsilon

(q+1)*conjugate(epsilon)),times((DD -

p*epsilon q*conjugate(epsilon)),0p(1,0p(1,w))));

elif (op(0,w)='*' and op(1,0p(0,0p(nops(w),w))) ='thp'

and op(0,0p(1,0p(nops(w),w))) ='thp') then

(p,q):=getpq(op(1,0p(1,0p(nops(w),w))) );

result:=result w +

w/op(1,0p(0,0p(nops(w),w)))(op(0,0p(1,0p~np~)w)

(op(1,0p(1,0p(nops(w),w)))))*(times((Delt +

(p-1)*epsiloni+(q-1)*conjugate(epsiloni))imsDet

+ptepsiloni+q*conjugate(epsiloni)),

op(1,0p(1,0p(nops(w),w))))));

elif (op(0,w)='thp' and op(0,0p(1,w))='thp') then

(p,q):=getpq(op(1,0p(1,w)));

result:=result w + times((Delta +

(p-1)*epsiloni+(q-1)*conjugate(epsiloni))imsDet

+ptepsiloni + q*conjugate(epsiloni)),0p(1,0p(1,w))));

elif (op(0,w)='*' and op(1,0p(0,0p(nops(w),w))) ='th'










and op(0,0p(1,0p(nops(w),w))) ='thp') then

(p,q):=getpq(op(1,0p(1,0p(nops(w),w))) );

result:=result w +

w/op(1,0p(0,0p(nops(w),w)))(op(0,0p(1,0p~np~)w)

(op(1,0p(1,0p(nops(w),w)))))*(times((DD -(p-1)*epsilon

-(q-1)*conjugate(epsilon)),times((Delta + p*epsiloni +

q*conjugate(epsiloni)),0p(1,0p(1,0p(nopsw))));

elif (op(0,w)='th' and op(0,0p(1,w))='thp') then

(p,q):=getpq(op(1,0p(1,w)));

result:=result w + times((DD (p-1)*epsilon

(q-1)*conjugate(epsilon)),times((Delta +

p*epsiloni + q*conjugate(epsiloni)),0p(1,0p(1,w))));

elif (op(0,w)='*' and op(1,0p(0,0p(nops(w),w))) ='eth'

and op(0,0p(1,0p(nops(w),w))) ='eth') then

(p,q):=getpq(op(1,0p(1,0p(nops(w),w))) );

result:=result w

+w/op(1,0p(0,0p(nops(w),w)))(op(0,0p(1,0pnpo)w)

(op(1,0p(1,0p(nops(w),w)))))

*(times((delta (p+1)*beta +

(q-1)*conjugate(betal)),times((delta p*beta +

q*conjugate(betal)),op(1,0p(1,0p(nops(w),w)))

elif (op(0,w)='eth' and op(0,0p(1,w))='eth') then

(p,q):=getpq(op(1,0p(1,w)));

result:=result w + times((delta (p+1)*beta

+ (q-1)*conjugate(betal)),times((delta p*beta +

q*conjugate(betal)),op(1,0p(1,w))));

elif (op(0,w)='*' and op(1,0p(0,0p(nops(w),w))) ='eth'










and op(0,0p(1,0p(nops(w),w))) ='th') then

(p,q):=getpq(op(1,0p(1,0p(nops(w),w))) );

result:=result w+

w/op(1,0p(0,0p(nops(w),w)))(op(0,0p(1,0p~np~)w)

(op(1,0p(1,0p(nops(w),w)))))*(times((delt-

(p+1)*beta +(q+1)*conjugate(betal)),times((DD -

p*epsilon-q*conjugate(epsilon)),

op(1,0p(1,0p(nops(w),w))))));

elif (op(0,w)='eth' and op(0,0p(1,w))='th') then

(p,q):=getpq(op(1,0p(1,w)));

result:=result w + times((delta (p+1)*beta

+ (q+1)*conjugate(betal)),times((DD p*epsilon -

q*conjugate(epsilon)),0p(1,0p(1,w))));

elif (op(0,w)='*' and op(1,0p(0,0p(nops(w),w))) ='eth'

and op(0,0p(1,0p(nops(w),w))) ='thp') then

(p,q):=getpq(op(1,0p(1,0p(nops(w),w))) );

result:=result w +

w/op(1,0p(0,0p(nops(w),w)))(op(0,0p(1,0p~np~)w)

(op(1,0p(1,0p(nops(w),w))))) *(times((delta -

(p-1)*beta +(q-1)*conjugate(betal)),times((Delta +

p*epsiloni+q*conjugate(epsiloni)),

op(1,0p(1,0p(nops(w),w))))));

elif (op(0,w)='eth' and op(0,0p(1,w))='thp') then

(p,q):=getpq(op(1,0p(1,w)));

result:=result w + times((delta (p-1)*beta

+ (q-1)*conjugate(betal)),times((Delta + p*epsiloni

+q*conjugate(epsiloni)),0p(1,0p(1,w))));









elif (op(0,w)='*' and op(1,0p(0,0p(nops(w),w))) ='eth'

and op(0,0p(1,0p(nops(w),w))) ='ethp') then\\

(p,q):=getpq(op(1,0p(1,0p(nops(w),w))) );

result :=result-w+w/op(1,0p(0,0p(nops(w),))

(op(0,0p(1,0p(nops(w),w)))(op(1,0p(1,0p(npw))))

*(times((delta (p-1)*beta +(q+1)*conjugate(betal)),

times((conjugate(delta) + p*betal -

q*conjugate(beta)),op(1,0p(1,0p(nops(w),)) )));

elif (op(0,w)='eth' and op(0,0p(1,w))='ethp') then

(p,q):=getpq(op(1,0p(1,w)));

result:=result w + times((delta (p-1)*beta

+ (q+1)*conjugate(betal)),times((conjugate~dla

+ p*betal -q*conjugate(beta)),op(1,0p(1,w))));

elif (op(0,w)='*' and op(1,0p(0,0p(nops(w),w)))

='ethp' and op(0,0p(1,0p(nops(w),w))) ='ethp') then

(p,q):=getpq(op(1,0p(1,0p(nops(w),w))) );

result:=result w +w/op(1,0p(0,0p(nops(w),w)))

(op(0,0p(1,0p(nops(w),w)))(op(1,0p(1,0p(npw))))

*(times((conjugate(delta) + (p-1)*betal

-(q+1)*conjugate(beta)),times((conjugate~dla +

p*betal- q*conjugate(beta)),

op(1,0p(1,0p(nops(w),w))))));

elif (op(0,w)='ethp' and op(0,0p(1,w))='ethp') then

(p,q):=getpq(op(1,0p(1,w)));

result:=result w + times((conjugate(delta) +

(p-1)*betal (q+1)*conjugate(beta)),

times((conjugate(delta) + p*betal-










q*conjugate(beta)),op(1,0p(1,w))));

elif (op(0,w)='*' and op(1,0p(0,0p(nops(w),w)))

='ethp' and op(0,0p(1,0p(nops(w),w))) ='th') then

(p,q):=getpq(op(1,0p(1,0p(nops(w),w))) );
result:=result w +

w/op(1,0p(0,0p(nops(w),w)))(op(0,0p(1,0p~np~)w)

(op(1,0p(1,0p(nops(w),w)))))*(times((conjuaedla

+(p+1)*betal (q+1)*conjugate(beta)),times((DD -

p*epsilon q*conjugate(epsilon)),

op(1,0p(1,0p(nops(w),w))) )));

elif (op(0,w)='ethp' and op(0,0p(1,w))='th') then

(p,q):=getpq(op(1,0p(1,w)));

result:=result w + times((conjugate(delta) +

(p+1)*betal -(q+1)*conjugate(beta)),times((DD -

p*epsilon -q*conjugate(epsilon)),0p(1,0p(1,w))));

elif (op(0,w)='*' and op(1,0p(0,0p(nops(w),w)))

='ethp' and op(0,0p(1,0p(nops(w),w))) ='thp') then

(p,q):=getpq(op(1,0p(1,0p(nops(w),w))) );

result:=result w +

w/op(1,0p(0,0p(nops(w),w)))(op(0,0p(1,0p~np~)w)

(op(1,0p(1,0p(nops(w),w)))))*(times((conjuaedla

+(p-1)*betal (q-1)*conjugate(beta)),times((Delta +

p*epsiloni + q*conjugate(epsiloni)),

op(1,0p(1,0p(nops(w),w))) )));

elif (op(0,w)='ethp' and op(0,0p(1,w))='thp') then

(p,q):=getpq(op(1,0p(1,w)));

result:=result w + times((conjugate(delta) +










(p-1)*betal -(q-1)*conjugate(beta)),times((Delta +

p*epsiloni + q*conjugate(epsiloni)),0p(1,0p(1,w))));

elif(op(0,w)='*' and op(1,0p(0,0p(nops(w),w))) ='th') then

(p,q):=getpq(op(1,0p(nops(w),w)) );

result:=result w +

w/op(0,0p(nops(w),w))(op(1,0p(nops(w),w))

*(times((DD p*epsilon -q*conjugate(epsilon)),

op(1,0p(nops(w),w)) ));

elif(op(0,w)='th') then

(p,q):=getpq(op(1,w));

result:=result w + times((DD p*epsilon -

q*conjugate(epsilon)),0p(1,w));

elif(op(0,w)='*' and op(1,0p(0,0p(nops(w),w))) ='thp') then

(p,q):=getpq(op(1,0p(nops(w),w)) );

result:=result w +

w/op(0,0p(nops(w),w))(op(1,0p(nops(w),w))

*(times((Delta + p*epsiloni+

q*conjugate(epsiloni)),0p(1,0p(nops(w),w)))

elif(op(0,w)='thp') then

(p,q):=getpq(op(1,w));

result:=result w + times((Delta + p*epsiloni

+ q*conjugate(epsiloni)),0p(1,w));

elif(op(0,w)='*' and op(1,0p(0,0p(nops(w),w))) ='eth')

then

(p,q):=getpq(op(1,0p(nops(w),w)) );

result:=result w +

w/op(0,0p(nops(w),w))(op(1,0p(nops(w),w))










*(times((delta p*beta +

q*conjugate(betal)),op(1,0p(nops(w),w))))

elif(op(0,w)='eth') then

(p,q):=getpq(op(1,w));

result:=result w + times((delta p*beta +

q*conjugate(betal)),op(1,w));

elif(op(0,w)='*' and op(1,0p(0,0p(nops(w),w)))

='ethp') then

(p,q):=getpq(op(1,0p(nops(w),w)) );

result:=result w +

w/op(0,0p(nops(w),w))(op(1,0p(nops(w),w))

*(times((conjugate(delta) +p*betal -

q*conjugate(beta)),op(1,0p(nops(w),w))));

elif(op(0,w)='ethp') then

(p,q):=getpq(op(1,w));

result:=result w + times((conjugate(delta) +

p*betal q*conjugate(beta)),op(1,w));

else

result :=result;

end if;

end do;

return(NPexpand(result));

end proc;



GE1:=th(rho)-ethp(kappa)=rho^2+sigmatconugesim)cngaekp)

*tau-taul*kappa+Phi00;

GE2:=th(sigma)-eth(kappa)=sigma*(rho+conugerh)-aptaucng










ate(taul))+Psi0;

GE3:=th(tau)-thp(kappa)=rho*(tau-conjugaetu)+igacougetu

)-taul)+Psil+Phi01;

GE4:=eth(rho)-ethp(sigma)=tau*(rho-conjugerh)+pptcnuaeh

ol)-rhol)-Psil+Phi01;

GE5:=eth(tau)-thp(sigma)=-rhol*sigma-conugesim)rh

+tau^2+kappatconjugate(kappal)+Phi02;

GE6:=thp(rho)-ethp(tau)=rhosconjugate(rho)sgaiml-ucnuge

(tau)-kappa*kappal-Psi2-2*PI;



GHP1 := proc()

return(DGHP(GE1));

end proc;



GHPip := proc()

return(DGHP(GHPprime(GE1)));

end proc;



GHPlc := proc()

return(DGHP(GHPconj(GE1)));

end proc;



GHPlpc := proc()

return(DGHP(GHPconj(GHPprime(GE1))));

end proc;



GHP2 := proc()










return(DGHP(GE2));


end proc;



GHP2p := proc ()

return(DGHP(GHPprime(GE2)));

end proc;



GHP2c := proc ()

return(DGHP (GHPconj (GE2)));

end proc;



GHP2pc:= proc()

return(DGHP (GHPconj (GHPprime (GE2))));

end proc;



GHP3 := proc ()

return(DGHP(GE3));

end proc;



GHP3p := proc ()

return(DGHP(GHPprime(GE3)));

end proc;



GHP3c := proc()

return(DGHP (GHPconj (GE3)));

end proc;










GHP3pc := proc ()

return(DGHP (GHPconj (GHPprime (GE3))));

end proc;



GHP4 := proc ()

return(DGHP(GE4));

end proc;



GHP4p := proc ()

return(DGHP(GHPprime(GE4)));

end proc;



GHP4c := proc ()

return(DGHP (GHPconj (GE4)));

end proc;



GHP4pc := proc ()

return(DGHP (GHPconj (GHPprime (GE4))));

end proc;



GHP5 := proc ()

return(DGHP(GE5));

end proc;



GHP5p := proc ()

return(DGHP(GHPprime(GE5)));

end proc;












GHP5c := proc ()

return(DGHP (GHPconj (GE5)));

end proc;



GHP5pc := proc ()

return(DGHP (GHPconj (GHPprime (GE5))));

end proc;



GHP6 := proc ()

return(DGHP(GE6));

end proc;



GHP~p := proc()

return(DGHP(GHPprime(GE6)));

end proc;



GHP~c := proc()

return(DGHP (GHPconj (GE6)));

end proc;



GHP~pc := proc ()

return(DGHP (GHPconj (GHPprime (GE6))));

end proc;



COM1 := proc ()










return(DGHP(th(thp(z))-thp(th(z))=(conjugetu)au)thz

+(tau-conjugate(taul))*ethp(z)-p*(kappalkpp-utu+PiPhlPIz

-q*(conjugate(kappal)*conjugate(kappa)-cougetu)cngaeau+

conjugate(Psi2)+Phill-PI)*z));

end proc;



COMlp := proc()

return(GHPprime(COM1()));

end proc;



COMic := proc()

return(GHPconj(COMi()));

end proc;



COMlpc := proc()

return(GHPconj(GHPprime(COM1())));

end proc;



COM2 := proc()



return(DGHP(th(eth(z))-eth(th(z))=conjugaeroetz)sgathz-

conjugate(taul)*th(z)-kappatthp(z)-p*(rhokpp-ulig+Pi)z

q*(conjugate(sigmal)*conjugate(kappa)-conuaerocnugetu)

+Phi01)*z));

end proc;



COM2p := proc()










return(GHPprime(COM2()));


end proc;



COM2c := proc()

return(GHPconj(COM2()));

end proc;



COM2pc := proc()

return(GHPconj(GHPprime(COM2())));

end proc;



COM3 := proc()



return(DGHP(eth(ethp(z))-ethp(eth(z))=(-rhicnuaero)*hz

+(rho-conjugate(rho))*thp(z)+p*(rho*rhol-imsga+P2PhlPIz

-q*(conjugate(rho)*conjugate(rhol)-conjugesim)cnuaeima+

conjugate(Psi2)-Phill-PI)*z));

end proc;



COM3p := proc()

return(GHPprime(COM3()));

end proc;



COM3c := proc()

return(GHPconj(COM3()));

end proc;










COM3pc := proc()

return(GHPconj(GHPprime(COM3())));

end proc;



BlI := proc()



return(DGHP(th(Psil)-ethp(Psi0)-th(Phi01)ehPi0=tuti+4ro

Psil-3*kappa*Psi2+conjugate(taul)*Phi00-2onuaer)Ph1

-2*sigma*Phil0+2*kappa*Phil1+conjugate(kap)Pi2;

end proc;



BIlp := proc()

return(GHPprime(BI1()));

end proc;



Blic := proc()

return(GHPconj(BIl()));

end proc;



BIlpc:= proc()

return(GHPconj(GHPprime(BI1())));

end proc;



BI2 := proc()



return(DGHP(th(Psi2)-ethp(Psii)-ethp(Phi1)hpPi0+thI=sgl

*Psi0-2*taul*Psil+3*rho*Psi2-2*kappa*Psi3+ojgt~hl*h0










-2*conjugate(tau)*Phi01-2*tau*Phi10+2*rho*h1+ojgt~im)Pi2)

end proc;



BI2p := proc()

return(GHPprime(BI2()));

end proc;



BI2c := proc()

return(GHPconj(BI2()));

end proc;



BI2pc:= proc()

return(GHPconj(GHPprime(BI2())));

end proc;



BI3 := proc()



return(DGHP(th(Psi3)-ethp(Psi2)-th(Phi21)ehPi0-etpI=2igl

*Psil-3*taul*Psi2+2*rho*Psi3-kappa*Psi4-2roPh0

+2*taul*Phil1+conjugate(taul)*Phi20-2*conuaeroPh2

+conjugate(kappa)*Phi22));

end proc;



BI3p := proc()

return(GHPprime(BI3()));

end proc;










BI3c := proc()

return(GHPconj(BI3()));

end proc;



BI3pc:= proc()

return(GHPconj (GHPprime (BI3())));

end proc;



BI4 := proc()



return(DGHP(th(Psi4)-ethp(Psi3)-ethp(Phi2)tpPi0=sgmls2-

*taul*Psi3+rho*Psi4-2*kappal*Phil0+2*sigma*hl

+conjugate(rhol)*Phi20-2*conjugate(tau)*Ph2+ojgt~im)Pi2)

end proc;



BI4p := proc ()

return (GHPprime (BI4 ())) ;

end proc;



BI4c := proc ()

return(GHPconj (BI4()));

end proc;



BI4pc:= proc()

return (GHPconj (GHPprime (BI4 ())));

end proc;










#ezcomm forms the basis of comm

ezcomm := proc(expr,sexpr)

local di, d2, lo, comm, P, Q;

# first do some parsing of the expression get the derivatives to

commute (di,d2) and the leftover (lo)

dl:=op(0,expr);

d2:=0p(0,0p(1,expr));

lo:=0p(1,0p(1,expr));

# return an error if we don't get the expected input

if ((dl='symbol') or (d2='symbol')) then

error "Not enough derivatives in X1 to commute",

expr;

elif (dl=d2) then

error "No commutin' to be done here! X1", expr;

end if;

# now figure out which commutator we need

if (dl='th') and (d2='thp') then

comm:=th(thp(z))=solve(COMi(),th(thp(z)))

elif (dl='thp') and (d2='th') then

comm:=thp(th(z))=solve(COMi(),thp(th(z)))

elif (dl='th') and (d2='eth') then

comm:=th(eth(z))=solve(COM2(),th(eth(z)))

elif (dl='eth') and (d2='th') then

comm:=eth(th(z))=solve(COM2(),eth(th(z)))

elif (dl='th') and (d2='ethp') then

comm:=th(ethp(z))=solve(COM2c(),th(ethp(z))

elif (dl='ethp') and (d2='th') then










comm:=ethp(th(z))=solve(COM2c(),ethp(th(z))

elif (dl='thp') and (d2='ethp') then

comm:=thp(ethp(z))=solve(COM2p(),thp(ethp~))

elif (dl='ethp') and (d2='thp') then

comm:=ethp(thp(z))=solve(COM2p(),ethp(thp~))

elif (dl='thp') and (d2='eth') then

comm:=thp(eth(z))=solve(COM2pc(),thp(ethz);

elif (dl='eth') and (d2='thp') then

comm:=eth(thp(z))=solve(COM2pc(),eth(thpz);

elif (dl='eth') and (d2='ethp') then

comm:=eth(ethp(z))=solve(COM3(),eth(ethpz);

elif (dl='ethp') and (d2='eth') then

comm:=ethp(eth(z))=solve(COM3(),ethp(ethz);

else error "Can't commute X1 and X2", di, d2;

end if;

# add up p and q values from the components of the metric perturbation

P:=0 + 2*(numboccur(10,hll) + numboccur(lo,hlm) +

numboccur(lo,hmm)) 2*(numboccur(10,hnn) + numboccur(lo,hnmb) +

numboccur(lo,hmbmb)) + numboccur(lo,th) + numboccur(lo,eth) -

numboccur(lo,thp) numboccur(lo,ethp);

Q:=0 + 2*(numboccur(10,hll) + numboccur(lo,hlmb) +

numboccur(lo,hmbmb)) 2*(numboccur(10,hnn) + numboccur(lo,hnm) +

numboccur(lo,hmm)) + numboccur(lo,th) + numboccur(lo,ethp) -

numboccur(lo,thp) numboccur(lo,eth);

# now add up p and q values from all other objects

# this is where we can modify the procedure to recognize new things

if (has(lo,rho) and not(has(lo,conjugate(rho)))) then










P:=P+1;

Q:=Q+1;

elif has(10,conjugate(rho)) then

P:=P+1;

Q:=Q+1;

elif (has(lo,rhol) and not(has(10,conjugate(rhol)))) then

P:=P-1;

Q:=Q-1;

elif has(10,conjugate(rhol)) then

P:=P-1;

Q:=Q-1;

elif (has(lo,kappa) and not(has(10,conjugate(kappa)))) then

P:=P+3;

Q:=Q+1;

elif has(10,conjugate(kappa)) then

P:=P+1;

Q:=Q+3;

elif (has(lo,kappal) and not(has(10,conjugate(kappal)))) then

P:=P-3;

Q:=Q-1;

elif has(10,conjugate(kappal)) then

P:=P-1;

Q:=Q-3;

elif (has(lo,tau) and not(has(10,conjugate(tau)))) then

P:=P+1;

Q:=Q-1;

elif has(lo,conjugate(tau)) then










P:=P-1;

Q:=Q+1;

elif (has(lo,taul) and not(has(10,conjugate(taul)))) then

P:=P-1;

Q:=Q+1;

elif has(10,conjugate(taul)) then

P:=P+1;

Q:=Q-1;

elif (has(lo,sigma) and not(has(10,conjugate(sigma)))) then

P:=P+3;

Q:=Q-1;

elif has(10,conjugate(sigma)) then

P:=P-1;

Q:=Q+3;

elif (has(lo,sigmal) and not(has(10,conjugate(sigmal)))) then

P:=P-3;

Q:=Q+1;

elif has(10,conjugate(sigmal)) then

P:=P+1;

Q:=Q-3;

elif (has(10,Psi0) and not(has(10,conjugate(Psi0)))) then

P:=P+4;

elif has(10,conjugate(Psi0)) then

Q:=Q+4;

elif (has(lo,Psil) and not(has(10,conjugate(Psii)))) then

P:=P+2;

elif has(lo,conjugate(Psii)) then










Q:=Q+2;

elif (has (lo, Psi3) and not (has (lo, conjugate (Psi3)))) then

P:=P-2;

elif has (lo, conjugate (Psi3)) then

Q:=Q-2;

elif (has (lo, Psi4) and not (has (lo, conjugate (Psi4)))) then

P:=P-4;

elif has (lo, conjugate (Psi4)) then

Q:=Q-4;

elif has (lo, xi_1) then

P:=P+1;

Q:=Q+1;

elif has (lo, xi_n) then

P:=P-1;

Q:=Q-1;

elif has (lo, xi_m) then

P:=P+1;

Q:=Q-1;

elif has (lo, xi_mb) then

P:=P-1;

Q:=Q+1;

elif (has (lo,phi) and not (has (lo, conjugate (phi)))) then

P:=P+pp;

Q:=Q+pq;

elif has (lo, conjugate (phi)) then

P:=P+pq;

Q:=Q+pp;










elif (has(lo,phil) and not(has(10,conjugate(phil)))) then

P:=P-pp;

Q:=Q-pq;

elif has(10,conjugate(phil)) then

P:=P-pq;

Q:=Q-pp;

elif (has(lo,chil) and not(has(10,conjugate(chil)))) then

P:=P+4;

elif has(10,conjugate(chil)) then

Q:=Q+4;

elif (has(10,chi2) and not(has(10,conjugate(chi2)))) then

P:=P-4;

elif has(10,conjugate(chi2)) then

Q:=Q-4;

elif (has(lo,omegal) and not(has(10,conjugate(omegal)))) then

P:=P+4;

elif has(10,conjugate(omegal)) then

Q:=Q+4;

elif (has(10,omega2) and not(has(10,conjugate(omega2)))) then

P:=P-4;

elif has(10,conjugate(omega2)) then

Q:=Q-4;

elif (has(lo,etal) and not(has(10,conjugate(etal)))) then

P:=P+4;

elif has(10,conjugate(etal)) then

Q:=Q+4;

elif (has(10,eta2) and not(has(10,conjugate(eta2)))) then










P:=P-4;

elif has (lo, conjugate (eta2)) then

Q:=Q-4;

elif (has (lo, xil) and not (has (lo, conjugate (xii)))) then

P:=P+4;

elif has (lo, conjugate (xii)) then

Q:=Q+4;

elif (has (lo, xi2) and not (has (lo, conjugate (xi2)))) then

P:=P-4;

elif has (lo, conjugate (xi2)) then

Q:=Q-4;

elif has(lo,h) then

P:=P+0;

Q:=Q+0;

elif has(lo,hl) then

P:=P+1;

Q:=Q+1;

elif has(lo,hn) then

P:=P-1;

Q:=Q-1;

end if ;

return(DGHP(subs(subs({p=P,q=Q,z=10},commsep);

end proc;



GHPcomm := proc (whichcom, solvef or, whichvar)

local a,b;

(a,b) :=getpq(whichvar);










return(solvefor=solve(subs({z=whichvar,paqb}

whichcom),solvefor));

end proc;



# comm applies ezcomm until a given expression is completely commuted



comm:=proc(exprl,expr2)

local ans;

ans:=expr2;

while has(ans,expri) do

ans:=DGHP(ezcomm(expri,ans));

end do;

return(ans)

end proc;



# tetcon is an exercise in working around maple; it essentially wor



tetcon := proc(expr,indes)

local i, lexpr, lup_pieces, nup_pieces, mup_pieces,

abup_pieces, zi, z2, z3, z4, z5, z6, z7, z8, z9, z10, zil, zi2,

01, 01a, 02, 02a, 03, 03a, 04, 04a;

lexpr:=expand(expr);

for i in indcs do

lup_pieces := select(has,1expr,1up(i))+xxxyyyzzz;

if expand(1up_pieces-xxxyyyzzz) <> 0 then

zl:=select(has,1up_pieces,1dn(i));

z2:=select(has,1up_pieces,mdn(i));










z3:=select(has,1up_pieces,conjugate(mdn)(i)

01:=select(has,1up_pieces,ndn(i));

01a:=expand(ol/(1up(i)*ndn(i)));

else

zl:=0; z2:=0; z3:=0; 01:=0; 01a:=0;

end if;

nup_pieces := select(has,1expr,nup(i))+xxxyyyzzz;

if expand(nup_pieces-xxxyyyzzz) <> 0 then

z4:=select(has,nup_pieces,ndn(i));

z5:=select(has,nup_pieces,mdn(i));

z6:=select(has,nup_pieces,conjugate(mdn)(i)

02:=select(has,nup_pieces,1dn(i));

02a:=expand(o2/(nup(i)*1dn(i)));

else

z4:=0; z5:=0; z6:=0; 02:=0; 02a:=0;

end if;

mup_pieces := select(has,1expr, mup(i))+xxxyyyzzz;

if expand(mup_pieces-xxxyyyzzz) <> 0 then

z7:=select(has,mup_pieces,1dn(i));

z8:=select(has,mup_pieces,ndn(i));

z9:=select(has,mup_pieces,mdn(i));

03:=select(has,mup_pieces,conjugate(mdn)(i)

03a:=expand(o3/(mup(i)*conjugate(mdn)(i)))

else

z7:=0; z8:=0; z9:=0; 03:=0; 03a:=0;

end if;

mbup_pieces := select(has,1expr,conjugate(mup)(i))+xxxyyz;










if expand(mbup_pieces-xxxyyyzzz) <> 0 then

zl0:=select(has,mbup_pieces,1dn(i));

zil:=select(has,mbup_pieces,ndn(i));

zl2:=select(has,mbup_pieces,conjugate(mdn)i)

04:=select(has,mbup_pieces,mdn(i));

04a:=expand(o4/(conjugate(mup)(i)*mdn(i)))

else


z10:=0; zil:=0; zi2:=0; 04:=0; 04a:=0;


end if;

lexpr:=expand(lexpr (zi+z2+z3+z4+z5+z6+z7+z8+z9

+z10+z11+zi2 + 01+02+03+04)+ 01a+02a-03a-04a)

end do;

return(expand(lexpr));

end proc;



GHPmult := proc(x,y)

local i,z,result;

result :=0;

for i from 1 to nops(x) do

if nops(x) <> 1 then z:=op(i,x) else z:=x end if;

if (z='th' or z='thp' or z='eth' or z='ethp' or

op(0,z)='th' or op(0,z)='thp' or op(0,z)='eth' or

op(0,z)='ethp') then

result :=result+apply(z,y)

else result:=result+zzy

end if;


end do;










return expand(result);


end proc;



tdspec :=

DGHP({th(pp)=0,eth(pp)=0,thp(pp)=0,ethp( p)0hp)=ehp)0,pp)0

,ethp(pq)=0,eth(rho) = rhostau-tausconjugate(rho), th(rho) = rho^2,

thp(rhol) = rhol^2, th(conjugate(rho)) = conjugate(rho)^2,

thp(conjugate(rhol)) = conjugate(rhol)^2, th(tau)=

rho*tau-rhosconjugate(taul), thp(taul)=

rhol*taul-rhoisconjugate(tau), th(conjugate(tau)) =

conjugate(rho)*c onjugate(tau)-conjugate( rotal

thp(conjugate(taul)) =

conjugate(rhol)*conjugate(taul)-conjugatero)tu ethp(rhol)=

rhoistaul-taulaconjugate(rhol), ethp(conjugate(rho))=

conjugate(rho)*c onjugate(tau)-conjugate( ta)ro eth(conjugate(rhol))

= conjugate(rhol)*conjugate(taul)-conjugate~al*hl eth(tau) =

tau^2, ethp(taul) = taul^2, ethp(conjugate(tau)) = conjugate(tau)^2,

eth(conjugate(taul)) = conjugate(taul)^2, th(Psi2) = 3*rho*Psi2,

thp(Psi2) = 3*rhol*Psi2, th(conjugate(Psi2))=

3*conjugate(rho)*conjugate(Psi2), thp(conjugate(Psi2))=

3*conjugate(rhol)*conjugate(Psi2), ethp(Psi2) = 3*taul*Psi2, eth(Psi2)

= 3*tau*Psi2, eth(conjugate(Psi2)) =

3*conjugate(taul)*conjugate(Psi2), ethp(conjugate(Psi2))=

3*conjugate(tau)*conjugate(Psi2), ethp(rho) = 2*rho*taul,

eth(conjugate(rho)) = 2*conjugate(rho)*conjugate(taul), eth(rhol)=

2*rhol*tau, ethp(conjugate(rhol)) = 2*conjugate(rhol)*conjugate(tau),

th(taul) = 2*rho*taul, thp(rho) =










rho*rhol-taulaconjugate(taul)+tau*taul-1/l hsojgt(s2/ojg

te(rho)-1/2*Psi2, th(conjugate(taul))=

2*conjugate(rho)*conjugate(taul), thp(tau) = 2*rhol*tau,

thp(conjugate(tau)) = 2*conjugate(rhol)*conjugate(tau),

thp(conjugate(rho)) =

conjugate(rho)*conjugate(rhol)-taulacnuatcal)cnugtnau*o

jugate(taul)-1/2*conjugate(rho)*Psi2/rho-12cnuaePi)

ethp(tau)=

rho*rhol+tau*taul-1/2*rhosconjugate(Psi2)cnuaero+/Ps2ho

conjugate(rhol), eth(conjugate(tau)) =

conjugate(rho)*conjugate(rhol)+conjugatetu)cngaeau-12on

ugate(rho)*Psi2/rho+1/2*conjugate(Psi2)-rocnugerh) th(rhol)



rho*rhol-taulaconjugate(taul)+tau*taul-1/l hsojgt(s2/ojg

te(rho)-1/2*Psi2, th(conjugate(rhol)) =

conjugate(rho)*conjugate(rhol)-taulacnuatcal)cnugtnau*o

jugate(taul)-1/2*conjugate(rho)*Psi2/rho-12cnuaePi)

eth(taul)=

rho*rhol+tau*taul-1/2*rhosconjugate(Psi2)cnuaero+/Ps2ho

conjugate(rhol), ethp(conjugate(taul))=

conjugate(rho)*conjugate(rhol)+conjugatetu)cngaeau-12on

ugate(rho)*Psi2/rho+1/2*conjugate(Psi2)-rhicnuaero\)



tdsimp := proc(expr)



return(DGHP(subs(tdspec,DGHP(subs(tdspecDGPsbtdpcGHsusd

spec,DGHP(subs(tdspec,DGHP(subs(tdspec,DGPsbtdpcer)))))














end proc;


tetupK := {1up = vector([(r^2+a^2)/(r^2-2*M*r+a^2), 1, 0,

a/(r^2-2*M*r+a^2)]), mup =

vector([1/2*Itatsin(theta)*2^(1/2)/(r+I*acstea) 0,

1/2*2^ (1/2)/(r+I*atcos(theta)),

1/2*I*2^(1/2)/(sin(theta)*(r+Ita*cos(thet)]) nup=

vector([1/2*(r^2+a^2)/((r+I*atcstea) rIacos(theta)))

-1/2*(r^2-2*M*r+a^2)/((r+I*atcstea)*rIacos(theta))) 0,

1/2*a/((r+I*atcos(theta))*(r-Ita*cos(thet)]) conjugate(mup) =

vector([-1/2*Itatsin(theta)*2^(1/2)/(r-Itacstea) 0,

1/2*2^ (1/2)/(r-Itatcos(theta)),

-1/2*I*2^(1/2)/(sin(theta)*(r-Ita*cos(thet)))}



tetdnK := \{mdn =

vector([1/2*Itatsin(theta)*2^(1/2)/(r+I*acstea) 0,

-1/2*(r-Itatcos(theta))*2^(1/2),

-1/2*I*(r^2+a^2)*sin(theta)*2^(1/2)/(r+I~acstea)) ndn=

vector([1/2*(r^2-2*M*r+a^2)/((r+I*atcos(tea)(-acotha))

1/2, 0,

-1/2*a*(r^2-2*M*r+a^2)*sin(theta)^2/((r+Iacste))(-acoth

ta)))]), Idn = vector([1,

-(r+I*atcos(theta))*(r-Itatcos(theta))/(r^-*~~^) 0,

-atsin(theta)^2]), conjugate(mdn) =

vector([-1/2*Itatsin(theta)*2^(1/2)/(r-Itacstea) 0,










-1/2*(r+I*atcos(theta))*2^(1/2),

1/2*I*(r^2+a^2)*sin(theta)*2^(1/2)/(r-I* acstea))}



tetupS := \{1up=

vector([1/2*(r^2+a^2)*2^(1/2)/((r^2-2*M* ra2(+Iacste))r-

*ascos(theta)))^(1/2),

1/2*2^ (1/2)* ((r^2-2*M*r+a^2)/((r+I*a tcstet)*rItooshea))



1/2*a*2^(1/2)/((r^2-2*M*r+a^2)*(r+I*atcostea)(-acotha)^

(1/2)]), nup =

vector([1/2*(r^2+a^2)*2^(1/2)/((r^2-2*M* ra2(+Iacste))r-

*ascos(theta)))^(1/2),

-1/2*2^(1/2)*((r ^2-2*M*r+a^2)/((r+I*a tcstea)*rIacsoht))



1/2*a*2^(1/2)/((r^2-2*M*r+a^2)*(r+I*atcostea)(-acotha)^

(1/2)]), mup = vector([1/2*Itatsin(theta)*2^(1/2)/(r+I*acste))

0, 1/2*2^(1/2)/(r+I*atcos(theta)),

1/2*I*2^(1/2)/(sin(theta)*(r+Ita*cos(thet)]) conjugate(mup)=

vector([-1/2*Itatsin(theta)*2^(1/2)/(r-Itacstea) 0,

1/2*2^ (1/2)/(r-Itatcos(theta)),

-1/2*I*2^(1/2)/(sin(theta)*(r-Ita*cos(thet)))}



tetdnS := \{mdn=

vector([1/2*Itatsin(theta)*2^(1/2)/(r+I*acstea) 0,

-1/2*(r-Itatcos(theta))*2^(1/2),

-1/2*I*(r^2+a^2)*sin(theta)*2^(1/2)/(r+I~acstea))

conjugate(mdn) =










vector([-1/2*Itatsin(theta)*2^(1/2)/(r-Itacstea) 0,

-1/2*(r+I*atcos(theta))*2^(1/2),

1/2*I*(r^2+a^2)*sin(theta)*2^(1/2)/(r-I* acstea)) ndn=

vector([1/2*2^(1/2)*((r^2-2*M*r+a^2)/((r+acotha)(rIaosh



1/2*2^ (1/2)* ((r+I*atcos(theta))*(r-Itatcotha)/r22Mr^)^(

/2), 0,

-1/2*atsin(theta)^2*2^(1/2)*((r^2-2*M*r+a2/(+acotha)(rI

a*cos(theta))))^(1/2)]), Idn=

vector([1/2*2^(1/2)*((r^2-2*M*r+a^2)/((r+acotha)(rIaosh



-1/2*2^(1/2)*((r+I*atcos(theta))*(r-Ita*cstea)(^-*~~^)^

1/2), 0,

-1/2*atsin(theta)^2*2^(1/2)*((r^2-2*M*r+a2/(+acotha)(rI

a*cos(theta))))^(1/2>)])\;



tetupSB := \{1up = vector([1/2*B(t, r, theta,

phi)*(r^2+a^2)*2^(1/2)/((r^2-2*M*r+a^2)* (+acotha)(rIaos

theta)))^(1/2), 1/2*B(t, r, theta,

phi)*2^(1/2)*((r ^2-2*M*r+a^2)/((r+I*a tcstea)*rIacsoht))

^(1/2), 0, 1/2*B(t, r, theta,

phi)*a*2^(1/2)/((r^2-2*M*r+a^2)*(r+I*atcotha)(rIacshe))

^(1/2)]), nup = vector([1/2*(r^2+a^2)*2^(1/2)/(B(t, r, theta,

phi)*((r^2-2*M*r+a^2)*(r+I*atcstea)*rIacos(theta)))^12)

-1/2*2^(1/2)*((r ^2-2*M*r+a^2)/((r+I*a tcstea)*rIacsoht))

^(1/2)/B(t, r, theta, phi), 0, 1/2*a*2^(1/2)/(B(t, r, theta,

phi)*((r^2-2*M*r+a^2)*(r+I*atcstea)*rIacos(theta)))^12))










conjugate(mup) = vector([-1/2*Itexp(-I*S(t, r, theta,

phi))*atsin(theta)*2^(1/2)/(r-Ita*cos(thea) 0, 1/2*exp(-I*S(t, r,

theta, phi))*2^(1/2)/(r-Itatcos(theta)), -1/2*Itexp(-I*S(t, r, theta,

phi))*2^(1/2)/(sin(theta)*(r-Ita*cos(thet)]) mup=

vector([1/2*Itexp(I*S(t, r, theta,

phi))*atsin(theta)*2^(1/2)/(r+Ita*cos(thea) 0, 1/2*exp(I*S(t, r,

theta, phi))*2^(1/2)/(r+I*atcos(theta)), 1/2*Itexp(I*S(t, r, theta,

phi))*2^(1/2)/(sin(theta)*(r+Ita*cos(thet)]\}



tetdnSB := \{conjugate(mdn) = vector([-1/2*Itexp(-I*S(t, r, theta,

phi))*atsin(theta)*2^(1/2)/(r-Ita*cos(thea) 0, -1/2*exp(-I*S(t, r,

theta, phi))*(r+I*atcos(theta))*2^(1/2), 1/2*Itexp(-I*S(t, r, theta,

phi))*(r^2+a^2)*sin(theta)*2^(1/2)/(r-I* acstea)) mdn=

vector([1/2*Itexp(I*S(t, r, theta,

phi))*atsin(theta)*2^(1/2)/(r+Ita*cos(thea) 0, -1/2*exp(I*S(t, r,

theta, phi))*(r-Itatcos(theta))*2^(1/2), -1/2*Itexp(I*S(t, r, theta,

phi))*(r^2+a^2)*sin(theta)*2^(1/2)/(r+I*acste)]) ndn =

vector([1/2*2^(1/2)*((r^2-2*M*r+a^2)/((r+acotha)(rIaosh

eta))))^(1/2)/B(t, r, theta, phi),

1/2*2^ (1/2)* ((r+I*atcos(theta))*(r-Itatcotha)/r22Mr^)^(

/2)/B(t, r, theta, phi), 0,

-1/2*atsin(theta)^2*2^(1/2)*((r^2-2*M*r+a2/(+acotha)(rI

a*cos(theta))))^(1/2)/B(t, r, theta, phi)]), Idn = vector([1/2*B(t, r,

theta,

phi)*2^(1/2)*((r ^2-2*M*r+a^2)/((r+I*a tcstea)*rIacsoht))

^(1/2), -1/2*B(t, r, theta,

phi)*2^(1/2)*((r+I*atcos(theta))*(r-Ita*cstea)(^-*~~^)^










1/2), 0, -1/2*B(t, r, theta,

phi)*atsin(theta) ^2*2^ (1/2)*((r^2-2*M*r+a2/(rIacstha) r-

a*cos (theta))))^ (1/2>)])\;



end module:









REFERENCES

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Press, Ney York, 1984.









BIOGRAPHICAL SKETCH

Larry was born in 1978, in El Paso, Texas. He is the eldest child of (the elder) Larry

Price and Pamela Villa. At last count, he has approximately 6 siblings.

Fr-om the ages of about five to twelve, he attended a funny sort of school where the

students were all forced to dress the same and gather on Fil 1 .- to listen to a man in

a dress read from a big book. He was treated well there, but his entry into the Texas

public school system in the fifth grade proved to be a good move. In middle school, Larry

realized he understood algebra much better than his teacher (who happened to also be

the school's basketball coach), a point that he made clear in class at every opportunity. It

goes without ?iing that his initial desire to publicly humiliate jocks subsequently grew

into a much deeper interest in mathematics and physics. These interests were furthered

in high school, where Larry explored other areas as well. Among these is the theater. Few

people are aware that Larry has performed in leading roles in several musicals, as well as

an operetta.

Upon graduating high school in 1997, Larry decided that it would be best to get as

far .li.-- ., from El Paso as he could. To this end, he attended a small liberal arts school

named Reed College in Portland, Oregon, where he spent some of the best years of his life.

Reed provided a valuable opportunity for Larry to further pursue the sciences and read

some really great books at the same time. It also gave him the opportunity to interact

with many interesting people from widely different backgrounds. It was there that Larry

came in contact with Nick Wheeler, a truly unique individual who remains a trusted

mentor. Alas, all good things must come to an end, and so Larry graduated from Reed

with a B.A. in physics in 2001.

With his path uncertain at the time, Larry decided to stay in Portland for the

following year. There Larry tried his hand as a computational chemist for Schroidinger,

Inc. The people there were fantastic and the p l.lllhacks weren't bad, but he need more










from his work. Graduate school seemed like a good remedy. This is how Larry came to

Florida.

Larry entered the graduate program at the University of Florida in 2002. There he

had the good fortune to work with Bernard Whiting, who introduced Larry to the subject

of general relativity. To Larry's surprise, he finished the doctoral program in five years,

graduating in the summer of 2007. What the future holds for Larry is uncertain. What

is certain is that in the fall Larry will continue his tour of the country in Milwaukee,

Wisconsin, where he has accepted a postdoctoral position at the University of Wisconsin.





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Thetaskofwritingacknowledgementsnecessarilycomesthetaskofforgettingtoacknowledgeeveryonewhodeservesit.MyapologiestoanyoneI'veforgotten.Firstofall,Ioweagreatdealtomyadvisor,BernardWhitingforhispatientguidanceandallhissupport.Ithasbeenapleasuretoworthwithhimforthepastveyears.IwouldliketothankSteveDetweilerforusefulprovidingusefulcommentsandperspectivethroughouttheyears.MyfriendsthroughouttheyearsdeserveagreatdealofthanksformakinglifeinGainesvillebearable:JoshMcClellan,FloCourchay,WayneBomstad,EthanSiegel,ScottLittle,AaronManalaysay,IanVega,KarthikShankarandanyoneI'veforgotten.IoweaveryspecialthankstoLisaDankerbothforputtingupwithandmakinglifeeasierformeduringthecreationofthisdocument.Allofmyparents|PamVillaandLarryandAudreyPrice|deservemorethanksthanIcangivethemfortheircontinuedsupportthroughouttheyears.Finally,thanksgotheAlumnifellowshipprogramandInstituteforFundamentalTheoryattheUniversityofFloridafornancialsupportovertheyears. 3

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page ACKNOWLEDGMENTS ................................. 3 ABSTRACT ........................................ 6 CHAPTER 1INTRODUCTION .................................. 8 1.1PerturbationsofSphericallySymmetricSpacetimes ............. 9 1.2PerturbationsofKerrBlackHoleSpacetimes ................. 13 1.3MetricPerturbationsofBlackHoleSpacetimes ................ 15 1.3.1HertzPotentialsinFlatspace ..................... 16 1.3.2TheInversionProblemforGravity ................... 17 1.3.2.1Ori'sconstructionforKerr .................. 18 1.3.2.2TimedomaintreatmentforSchwarzschild ......... 19 1.3.2.3WorkingintheRegge-Wheelergauge ............ 20 1.4ThisWork .................................... 21 2NEWTOOLSFORPERTURBATIONTHEORY ................. 23 2.1NP ........................................ 23 2.2GHP ....................................... 27 2.3KillingTensorsandCommutingOperators .................. 30 2.3.1SpecializationtoPetrovTypeD .................... 30 2.3.2TheKillingVectorsandTensor .................... 31 2.3.3CommutingOperators ......................... 36 2.4TheSimpliedGHPEquationsforTypeDBackgrounds .......... 38 2.5IssuesofGaugeinPerturbationTheory .................... 40 2.6GHPtools-ANewFrameworkforPerturbationTheory ........... 42 2.6.1Einstein'sNewClothes ......................... 43 2.6.2GHPtools-TheDetails ......................... 44 3REGGE-WHEELER&TEUKOLSKY ....................... 52 3.1ParityDecompositionofSpin-andBoost-WeightedScalars ......... 52 3.2Regge-Wheeler ................................. 56 3.2.1TheRegge-WheelerGauge ....................... 56 3.2.2TheRegge-WheelerEquation ..................... 58 3.3TheTeukolskyEquation ............................ 61 3.4MetricReconstructionfromWeylScalars ................... 62 4THEEXISTENCEOFRADIATIONGAUGES .................. 66 4.1TheRadiationGauges ............................. 66 4.2ImposingtheIRGintypeII .......................... 69 4

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........................... 72 4.4ImposingtheIRGintypeD .......................... 73 4.5Discussion .................................... 76 5THETEUKOLSKY-STAROBINSKYIDENTITIES ................ 78 6THENON-RADIATEDMULTIPOLES ...................... 84 6.1Schwarzschild .................................. 91 6.1.1Massperturbations ........................... 91 6.1.2Angularmomentumperturbations ................... 96 6.1.2.1Odd-parityangularmomentumperturbations ....... 97 6.1.2.2Even-paritydipoleperturbations .............. 99 6.2Kerr ....................................... 100 6.2.1MassPerturbations ........................... 100 6.2.2AngularMomentumPerturbations ................... 104 6.2.3Discussion ................................ 106 7CONCLUSION .................................... 108 7.1Summary .................................... 108 7.2FutureWork ................................... 109 APPENDIX ATHEGHPRELATIONS .............................. 111 BTHEPERTURBEDEINSTEINEQUATIONSINGHPFORM ......... 113 CINTEGRATIONALAHELD ............................ 117 DSPIN-WEIGHTEDSPHERICALHARMONICS .................. 121 EMAPLECODEFORGHPTOOLS ......................... 123 REFERENCES ....................................... 175 BIOGRAPHICALSKETCH ................................ 179 5

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Thedetectionofgravitationalwavesisthemostexcitingprospectforexperimentalrelativitytoday.WithgroundbasedinterferometerssuchasLIGO,VIRGOandGEOonlineandthespacebasedLISAprojectinpreparation,theexperimentalapparatusnecessaryforsuchworkissteadilytakingshape.Yet,howevercapabletheseexperimentsareoftakingdata,theactualdetectionofgravitationalwavesreliesinasignicantwayonmakingsenseofthecollectedsignals.Someofthedataanalysistechniquesalreadyinplaceuseknowledgeofexpectedwaveformstoaidthesearch.Thisismanifestedintemplatebaseddataanalysistechniques.Forthesetechniquestobesuccessful,potentialsourcesofgravitationalradiationmustbeidentiedandthecorrespondingwaveformsforthosesourcesmustbecomputed.Itisinthiscontextthatblackholeperturbationtheoryhasitsmostimmediateconsequences. ThisdissertationpresentsanewframeworkforblackholeperturbationtheorybasedonthespincoecientformalismofGeroch,HeldandPenrose.ThetwomaincomponentsofthisframeworkareanewformfortheperturbedEinsteinequationsandaMaplepackage,GHPtools,forperformingthenecessarysymboliccomputation.ThisframeworkprovidesapowerfultoolforperforminganalysesgenerallyapplicabletotheentireclassofPetrovtypeDsolutions,whichincludetheKerrandSchwarzschildspacetimes. Severalexamplesofthepowerandexibilityoftheframeworkareexplored.TheyincludeaproofoftheexistenceoftheradiationgaugesofChrzanowskiinPetrovtype 6

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7

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Einstein'stheoryofgeneralrelativity,introducedin1915,tothisdayremainsasoneofthenalfrontiersoffundamentalphysics.Sinceitsinceptionprogressintheeldhasbeenlargelytheoreticalbecauseofthetremendousdicultyinherentinmakinggravitationalmeasurements.Inparticular,oneofthemostexcitingandfundamentalpredictionsofgeneralrelativity|theexistenceofgravitationalwaves|hasremainedelusive.Notforlong.WithgroundbasedinterferometerssuchasLIGO,VIRGOandGEOonlineandthespacebasedLISAmissioninpreparation,thedetectionofgravitationalwavesisallbutimminent.Theseexperimentsbringwiththemthetaskofanalyzingthedatatheycollect.Forsomeofthepromisingsourcesofgravitationalwaves,thecollisionoftwoblackholes,themethodofchoicefordataanalysis,knownasmatchedltering,requiresknowledgeoftheexpectedwaveforms.Inthepasttwoyearstheeldofnumericalrelativityhasundergonearevolutionandpromisestoprovidethemostaccuratewaveformsforsituationsinvolvingthecollisionoftwoblackholesofcomparablemasses|situationsthatrequiretheuseoffullnonlineargeneralrelativity.Thereishowever,onepromisingsourceofgravitationalwavesthatiscurrentlyoutofreachfornumericalrelativity|thesituationwherethelargerblackholeisroughlyamilliontimesmoremassivethanthesmallerone,knownasanextrememassratioinspiral,orEMRI.Thisproblemliessquarelyintherealmofperturbationtheory,thesubjectofthepresentwork. Inparticular,the\solution"oftheEMRIproblemrequiresmovingbeyondthetestmassapproximationofgeneralrelativitytodescribethemotionofthesmallblackhole(treatedasaparticleinthespacetimeofthelargerblackholebecauseofthehugemassdierence)|onemustaccountfortherstordercorrectionstothemotionofthesmallblackhole,duetoself-force.TheappropriateequationsofmotionhavebeendeterminedingeneralbyMino,SasakiandTanaka[ 1 ]andQuinnandWald[ 2 ]andarereferredtoasthe 8

PAGE 9

3 ].Ineithercase,thefundamentalobjectofinterestisthemetricperturbation,hab,introducedbytheparticleonthelargeblackhole'sspacetime.ThereforetheEMRIproblemalsorequiresustocomputethemetricperturbation,beforewecancomputetheself-forceontheparticle.Thisisthepieceoftheproblemtowhichthepresentworkaimstocontribute.Determiningthemetricperturbationisataskthatdependsquitesensitivelyonthespacetimebeingperturbed.Forsphericallysymmetricbackgrounds,thisproblemiswellunderstoodandmostoftheremainingproblemsarecomputationalinnature.However,forthemoreinterestingandastrophysicallyrelevantsituationwherethelargerblackholeisrotating,ourunderstandingisnotquitecomplete.Itisonthismoregeneralsituationthatwefocus.Beforewecontinue,wenotethatalloftheastrophysicallyinterestingspacetimes,includingtheKerrandSchwarzschildmetrics,possesscurvaturetensorswiththesamebasicalgebraicstructure.Wewillelaborateonthismorefullyinthenextchapter,butfornowwemerelypointoutthatthesespacetimesbelongtothelargerclassofalgebraicallyspecialspacetimes. Theremainderofthischapterisdevotedtoprovidingareviewoftheliterature[ 4 ].Everyattempthasbeenmadetophrasethecurrentdiscussioningenerallyaccessiblelanguage.Manyoftheseresultswillbeexploredinfurtherdetailinlaterchapters,aftertheappropriateformalismhasbeendeveloped. 5 ](henceforthRW),whoprovidedananalysisofrstorderperturbationsoftheSchwarzschildsolution(whichwaslatercompletedbyZerilli[ 6 7 ]).Thefactthatthebackgroundissphericallysymmetriciscrucialtotheiranalysis.Thebasicswillbepresentedhere.Amorecompletediscussion,inaverydierentlanguage,isprovidedinChapter3. 9

PAGE 10

where rdt212M r1dr2r2(d2+sin2d2)(1{2) istheSchwarzschildmetricinSchwarzschildcoordinates.PuttingEquation 1{1 intotheEinsteinequationsandkeepingonlytermslinearinhableadsustotheperturbedEinsteinequations: 2rcrchab1 2rarbhcc+rcr(ahb)c+1 2gab(rcrchddrcrdhcd)=0;(1{3) whereraisthederivativeoperatorcompatiblewiththebackgroundgeometry 1{2 andtheindicesareraisedandloweredwiththebackgroundmetric.HenceforthwewillrefertoEabastheEinsteintensor,andtheexpressiontotherightofitastheEinsteinequations(droppingthequalier\perturbed"forbrevity). EssentiallyeveryperturbativeanalysisoftheSchwarzschildspacetimemakesextensiveuseofitssphericalsymmetry.Therststepinthisdirectionistodecomposethecomponentsofthemetricperturbationintoscalar,vectorandtensorharmonics.Heuristically,wewrite wheres;vandtstandforscalar,vectorandtensor,respectivelyandthesubscriptsdistinguishbetweenthevariousscalarsandvectors. Considerthemetricofthetwo-sphere: 10

PAGE 11

whererAisthederivativecompatiblewithAB(Equation 1{5 ).Theothervectoristheodd-parity(pseudo-)vector whereABisjustthestandardLevi-Civitasymbol.Todenetensorharmonics,weessentiallyjusttakeonemorederivativeofEquations 1{6 and 1{7 .Theevenparitytensorsaregivenby andtheodd-parity(pseudo-)tensorby Evenparityobjectspickupminussignsunderaparitytransformation(!;!+)accordingto(1)`,andoddparityobjectspickupminussignsaccordingto(1)`+1.Forthisreasontheevenparitypartsaresometimesreferredtoas\electric"andtheoddparityparts\magnetic"intheolderliterature.Becauseparityisaninherentsymmetryofsphericallysymmetricbackgrounds,itprovidesanaturalwayofdecouplingthetwodegreesoffreedomofthegravitationaleld.Note,however,thatparityisnotagoodsymmetryinevenslightlylesssymmetricspacetimes(e.g.Kerr).Wewillreturnto 8 ]forthestandardtensorharmonicsandtheirrelationtovariousotherrepresentationsofthesphere,orAppendixDforthespin-weightedsphericalharmonicswhichprovideanotheralternativefortheangulardecomposition. 11

PAGE 12

1{4 ),wenowconsiderthetwosectorsofthemetricperturbationindependently,writing and ThenalstepbeforeappealingtotheEinsteinequationsconsistsofchoosingagauge.Equation 1{3 isinvariantunderthetransformation whereaisanarbitraryvectorand$istheLiederivative.Takingtheodd-paritysectorasanexample,theRegge-Wheelergaugevectortakestheform whereisafunctionchosensothattheoddparitypartofthemetricperturbation 1{10 takestheform Similarsimplicationsariseintheeven-paritysector. 12

PAGE 13

wheretheletters'o'and'e'standforoddandeven,respectively,r=r+ln(r where~2=r2+a2cos2,=r22Mr+a2,Misthemassanda=J=Mistheangularmomentumpermassoftheblackhole.ThespincoecientformalismofGeroch,HeldandPenrose[ 9 ]developedinthenextchapterhasprovedtobefundamentalinvirtuallyeveryperturbativedescriptionoftheKerrspacetime. TherstsuccessfulperturbationanalysisoftheKerrgeometrywasperformedbyTeukolskyinaseriesofpapersbeginningin1973[ 10 { 12 ].TeukolskytookashisstartingpointtheperturbedBianchiidentitiesinaspincoecientformalism.Eachquantityisperturbedawayfromitsbackgroundvalueandonlyrstordertermsarekept.Equivalently,thoughwithconsiderablymoreeort,Teukolsky'sresultcanalsobeseenas 13

PAGE 14

13 ].Ineithercase,theresult,writtenhereinBoyer-Lindquistcoordinates,isTeukolsky'smasterequation(writtenhereinaccordwith[ 14 ]) @r@ @r1 n(r2+a2)@ @t+a@ @s(rM)o24s(r+iacos)@ @t+@ @cossin2@ @cos+1 sin2nasin2@ @t+@ @+iscoso2)s=2s=4s=2Ts; wheres=2correspondtotheWeylscalars0and4=324,respectively.TheWeylscalarsareperturbationsoftheextremalspincomponentsofthecurvaturetensor.ThesignicanceoftheWeylscalar4isthatfarawayfromthesourceofgravitationalradiation whereh+andharethetwopolarizationsofoutgoinggravitationalradiationinthetransversetracelessgauge.Similarresultsholdfor0andincomingradiation.Forothervaluesofs,solutionscorrespondtoeldsofotherspin:s=0isthemasslessscalarwaveequation,s=1=2theWeylneutrino,s=1theMaxwelleld,s=3=2theRarita-Schwingereld,andsoon.Notethatangularseparationnecessarilyinvolvestimeseparationfora6=0. SeparatedsolutionstoEquation 1{17 areoftheforms=ei!teimsR(r)sS(a!;)(omittingthe`,mand!subscripts).Theangularfunctions,sS(a!;),aregenerallyreferredtoas\spinweightedspheroidalharmonics".Inthelimitthata!=0,sS`m()reducetothestandardspinweightedsphericalharmonics(cf.AppendixD),whichareinterrelatedbythespinraisingandloweringoperators,and0[ 15 ],developedinthefollowingchapter.Fora!6=0,solutionscorrespondtofunctionsofdierentspinweight,butthesS(a!;)nolongersharecommoneigenvalues.Thusametricreconstructionbasedonspinweight2functionswouldbeincompatiblewithonebasedonspinweight0 14

PAGE 15

Thespinweightedspherical(andspheroidal)harmonicsfailtobedenedfor`
PAGE 16

20 ]: Asusualavectorpotential,Aa,isintroducedandtheLorentzgauge,raAa=0,isimposedsothattheMaxwellequationsleaddirectlyto2Aa=0. ThenaHertzpotentialHabisintroducedviaAa=rbHab,whereHab=Hba,sothattheMaxwelleld,Fab,isobtainablebytwoderivativesofHab.However,Habisonlydeneduptoatransformationofthetype whereMcabiscompletelyantisymmetricand2Ca=0.Itiseasytoseethatinatspacetime,wherederivativescommute,thetransformationEquation 1{20 onlychanges 16

PAGE 17

1{20 isusedtoreducetheHertzbivectorpotentialtoasinglecomplex(ortworeal)scalarpotential(s).Hereinliesthepowerofthemethod.However,movingtocurved-spacenaturallycomplicatesthings.Whilethewaveequationsaremodiedtoincludecurvaturepieces,thetransformationinEquation 1{20 isretained(seeCohenandKegeles[ 20 ]andStewart[ 21 ]).Asaresult,theeldequationsarestillsatisedandthesixcomponentsofHabarestillreducedtotwo,butthetransformationinEquation 1{20 explicitlybreakstheLorentzgaugebecausederivativesnolongercommute.Inthiswayanewgaugeisintroducedthatbringswithitcomplicationsfortheinclusionofsources.ThenecessaryandsucientconditionsfortheexistenceofthisgaugearethesubjectofChapter4. 22 ],oritsmodicationduetoGeroch,HeldandPenrose[ 9 ].ThuswepostponetheformaldevelopmentofthesubjectuntilChapter3,whenthenecessaryformalismisinplace,andinsteadoeranoverviewofthegeneralprocessanddocumentedresearchonthetopicofreconstructingthemetricperturbationfromsolutionstotheTeukolskyequation(assumingtheformofmetricperturbationisprescribed),whichwewillrefertoastheinversionproblem. 17

PAGE 18

20=DDDDIRG;and (1{21) 244=1 4LLLLIRG1232@tIRG; whereL=[@+scoticsc@']+iasin@tandD=1[(r2+a2)@t+@r+a@]denederivativesin(orthogonal)nulldirections,=(riacos)1andIRGisthepotential.Whileforapotentialsatisfyingthes=+2Teukolskyequation,wehaveaperturbationintheoutgoingradiationgauge(ORG),where 244=2bbbb2ORG;and (1{23) 20=1 4LLLLORG+1232@tORG; whereb=1 2[(r2+a2)@t@r+a@]andL,thecomplexconjugateoftheoperatordenedabove,arealsoderivativesinnulldirections(mutuallyorthogonaltoeachotherandthosedenedbytheoperatorsintheIRG).ThesearetheequationswewouldliketoinvertforthepotentialsIRGandORG.Oncethisisdone,thepotentialmaythenbeusedtoconstructthemetricperturbation.Wenowlookatseveraldierentapproachestothisproblem. 1{21 or 1{22 (ortheirORGcounterparts).Ori[ 23 ]hasrecentlyperformedthistask|integratingEquation 1{21 inordertondthepotentialIRGintermsof0. 18

PAGE 19

1{17 withs=2,soitdoesprovideacompletesolutioninthefrequencydomain. Forincorporatingsources,OricontinuestotakeEquation 1{21 ascorrect,wherenow0isasource-dependent,non-vacuumsolution.Equation 1{21 allowsthefreedomtoaddtoIRGanyfunctionthatiskilledbythefourderivativesthere.Oriutilizesthisfreedomtochoosefunctionsthatreproducethediscontinuityatthesourceand,byextension,0.However,Equation 1{17 nolongerappliesfor,nordoesEquation 1{22 for4intheformithashere. 24 ]suggeststhatgaugefreedommayplayaroleinresolvingtheseissues. 25 ]havechosentoworkinthetimedomain.Becauseofthischoicetheirresultisonlyvalidinthe 5{2 arosebyrepeateduseoftheTeukolskyequationinquiteacomplicatedexpression,initiallygivencorrectlybyStewart[ 21 ],andalsoobtainablefromtheresultsofChapters2and3here.Thefullformoftheexpressionmaystillapplyhere. 19

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OneunexpectedfeatureofLoustoandWhiting'sworkishowalgebraicallyspecialfrequenciesemergeinafundamentalway.Algebraicallyspecialsolutionsarisewhenoneof0or4iszerowhiletheotherisnot,andthenonlyforspecic(complex)frequencies.Whilethisisinherentlyafrequencydomainphenomena,itplaysacrucialroleinthistimedomainapproach.ThealgebraicallyspecialequationherehasasourcetermdependingontheinitialdatafortheHertzpotential|thistermeectivelycorrespondstothatwhicharisesforaLaplacetransform.FortheSchwarzschildbackground,allthealgebraicallyspecialfrequenciesareknownandthealgebraicallyspecialsolutionshavebeenfoundexplicitly[ 26 ],sotheequationsforthisanalysiscouldbesolvedbyquadrature[ 25 ].AttemptstogeneralizethistechniquetotheKerrbackgroundhavetodateremainedunsuccessful. 27 ].Lousto[ 28 ]hasrecentlychosentoworkwithsuchaformulation,ratherthanwithaHertzpotentialformulation.Thisimmediatelygiveshimfreedomovergaugechoiceanditcircumventstheproblemspreviouslyencounteredwiththeintroductionofsources.Havingcalculatedexplicitlythedependenceonsources,andknowingalsohowtorepresentallrelevant 20

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Loustoactuallyusesboth0and4inhisconstruction.Forconcretenessandforaccesstoavastbodyofpriorexperience,LoustoalsochosetoworkinagaugeknownastheRWgauge.NotethatEquations 1{21 and 1{22 areonlyvalidintheIRG.However,0and4areeasilyexpressibleintermsofanarbitrarymetricperturbation,whichallowsthemtobewrittenintermsoftheRWvariablesforanychoiceofgauge.IntheRWgauge,0and4becomealgebraicintheevenparitysectorandrstorderoperatorsintheoddparitysector.ToprovideenoughconditionstosolveforallthecomponentsofthemetricperturbationintermsoftheWeylscalars,LoustomustturntotheEinsteinequations(withsources),alsointheRWgauge.Itisinthiswaythatreconstructionwithsourcesisaccomplished. Theidenticationofgaugeinvariantquantities,beyond0and4,isvirtuallynonexistentintheKerrspacetimeandaspointedoutseveraltimesbefore,theangulardecompositionthereisnotasrobustasthatavailableinsphericallysymmetricbackgrounds.Inshort,Lousto'sworkisquitenotableforitsinclusionofsources,butitsrelianceonRWtoolsandtechniquesmakeitdiculttoseehowtoextendthemethodtotheKerrbackground. 21

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InChapter2,wewilldeveloptheformalismnecessaryforbuildingourframework.Additionally,theframeworkwillbepresented,whichincludesanewformfortheperturbedEinsteinequationsaswellasaMaplepackagethataidsnotonlyintheirapplication,butanycomputationintheformalismofGeroch,HeldandPenrose.Chapter3thenprovidesafurtherdiscussionofboththeRWandTeukolskyformalisms,phrasedinourframework.InChapter4,thenecessaryandsucientconditionsfortheexistenceoftheIRG(inalargerclassofspacetimesthanweconsiderelsewhere)aredeterminedwiththeaidofourformoftheEinsteinequations.Chapter5thenusestheIRGmetricperturbationtoderivesomeimportantrelationshipsbetweenthecurvatureperturbationsrepresentedby0and4,whichareofimportancefortheinversionproblemdescribedinthischapter.Furthermore,thisapplicationshowcasessomeofourMaplepackage'smostusefulfeatures.InChapter6wethenpresentaverydierentapplicationofourframeworkinconjunctionwithmorestandardtechniquestoaddresstheissueofthenon-radiatedmultipoles. 22

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Inthischapterwedevelopthebasicformalismwewillbeworkingwithinfortheremainderofthiswork.WebeginwithadescriptionofthespincoecientformalismofNewmanandPenrose[ 22 ]andintroducethemodicationsofitduetoGeroch,HeldandPenrose[ 9 ].Withinthelatterformalism,wedevelopthepropertiesofthegeneralclassofspacetimeswithwhichwewillbeworking.Includedisadiscussionofgaugeandthegeneralframeworkofrelativisticperturbationtheory.Thechapterendswiththeintroductiontotheframeworkwewillexploitinsubsequentchapters. Webeginbyintroducinganorthogonaltetradofnullvectors,la;na;maandma,withlaandnabeingrealandmaandmabeingcomplexconjugates.Wewillimposearelativenormalization withallotherinnerproductsvanishing.Asanexampletokeepinmind,consideranorthonormaltetradonMinkowskispace,(ta;xa;ya;za),suchthattata=xaxa=yaya=zaza=1.Sincethevectorsareproperlynormalized,itiseasytoverifythat 23

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Forsimplicity,weintroducethefollowingnotationforourtetrad(borrowedfromChandrasekhar[ 29 ]):ea(i)=(la;na;ma;ma); 2{1 )nulltetrad(i)(j)=(i)(j)=0BBBBBBB@0100100000010010:1CCCCCCCA wherel(anb)=1 2(lanb+lbna). 24

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Itfollowsfromthedenitionthat Thereisatotaloftwelvespincomplexcoecients,individuallynamedasfollows and 2(manbralbmambramb);=1 2(mambrambmalbranb);=1 2(lanbralblambramb);=1 2(nambrambnalbranb):(2{7) Ourea(i)naturallydenefourindependent,non-commutingdirectionalderivativese(i)ea(i)@ @xa; @xa;=ma@ @xa;=na@ @xa;=ma@ @xa:(2{8) 25

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2(gacRbd+gbdRacgbcRadgadRbc)1 2(gacgbdgbcgad)R:(2{9) whereCabcd;Rabcd;RabandRdenotetheWeyltensor,Riemanntensor,RiccitensorandRicciscalar,respectively.SinceboththeRiccitensorandtheRicciscalarvanishintheabsenceofsources,theWeylandRiemanntensorsareidenticalinsource-freespacetimes.InthatsensetheWeyltensorrepresentsthepurelygravitationaldegreesoffreedom.TheRiemanntensoristhenexpressedpurelyintermsofthespincoecientsandtheirderivativesbycontractingallfourvectorindiceswithea(i)'sandmakinguseoftheRicciidentity, (rarbrbra)vk=Rabcdvd=Rabcdvd;(2{10) wherevdisanarbitraryvector.InfourdimensionstheRiemanntensorhastwentyindependentcomponentsandtheRiccitensorhasten,leavingtheWeyltensorwithtenindependentcomponents.IntheNPformalism,thistranslatesintovecomplexscalars: 2Cabcd(lanblcnd+lanbmcmd);3=Cabcdlanbmcnd;4=Cabcdnambncmd:(2{11) 26

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00=1 2R11;11=1 4(R12+R34);01=1 2R13;12=1 2R23;10=1 2R14;21=1 2R24;02=1 2R33;22=1 2R22;20=1 2R44;=1 24R:(2{12) TheeldequationsthenfollowfromEquations 2{9 and 2{10 .AfullsetofequationsfortheNPformalismiscomposedofthecommutators,theequationsinvolvingdependenceonmatter,andtheBianchiidentities.ThisisgiveninAppendixA. 9 ]introducedsomeconvenientmodicationsoftheNPformalism.Specically,theyidentiedthenotionsofspinandboostweightandmakeexplicituseofaninherentdiscretesymmetryoftheNPequations. IntheNPformalism,thereisanimplicitinvarianceunderacertaininterchangeofthebasisvectorswhichGHPhavebuiltonthroughtheintroductionoftheprime(0)operation,denedbyitsactiononthetetradvectors: (la)0=na;(ma)0=ma;(na)0=la;(ma)0=ma:(2{13) AglanceatEquations 2{6 and 2{7 suggeststheadoptionofachangeinnotation: andsimilarlyforthedirectionalderivativesofEquation 2{8 27

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2;~xa=cosxasinya;~ya=sinxa+cosya; ~la=rla;~na=r1na;~ma=eima;~ma=eima;(2{16) wherer=p 2{16 maybesummarizedby Aquantity,,isthensaidtobeoftypefp;qgif,underEquation 2{17 ,!pq.Alternatively[ 9 ],wemaysaythatpossessesspinweights=(pq)=2andboostweightb=(p+q)=2.ThepandqvaluesforthetetradvectorscanbereadofromEquation 2{17 .Theyallowonetodeterminethespinandboostweightsofthespin 30 ]or[ 29 ]. 28

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2{6 ,whilethespincoecientsinEquation 2{7 havenowelldenedspinorboostweightsince,underEquation 2{17 ,theypickuptermsinvolvingderivativesof.Whenactingonaquantityofwelldenedspinandboostweight,thedirectionalderivativesofEquation 2{8 bythemselvesalsofailtocreateanotherquantityofwelldenedweight.However,itispossibletocombinethespincoecientsinEquation 2{7 withtheactionofderivativeoperatorsinEquation 2{8 toconstructderivativeoperatorsthatdoproducenewquantitieswithwelldenedspinandboostweights.Withtakentobeoftypefp;qg,wecandenetheseoperatorsasfollows: whereandareIcelandiccharactersnamed\thorn"and\edth",respectively.Eachofthesederivativeshassomewelldenedtypefr;sginthesensethatwhentheyactonaquantityoftypefp;qg,aquantityoftypefr+p;s+qgisproduced.Thesenewderivativeoperatorsinherittheirtypefromtheircorrespondingtetradvectors: Itisquiteoftenusefultothinkof()and0(0)asspin(boost)weightraisingandloweringoperators,respectively.ThederivativesinEquation 2{18 canbecombinedtoformacovariantderivativeoperator: a=la0+nama0ma=ra1 2(p+q)nbralb+1 2(pq)mbramb:(2{20) Wenoteinpassingthatthisdenitiondenesthe\GHPconnection."OurprimaryuseforEquation 2{20 willbetoexpressthingsinGHPlanguageviathereplacementra!a.Withthesedenitions,allequationsintheNPformalismcanbetranslatedintoGHP 29

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2.3.1SpecializationtoPetrovTypeD whereCabcdistheWeyltensorintroducedinEquation 2{9 andthesquarebrackets[]denoteanti-symmetrization.Thevectorskadenetheso-calledprincipalnulldirectionsofthespace-time.Forsomespace-times,oneormoreoftheprincipalnullvectorscoincide.Thegeneralclassicationofspace-timesbasedonthenumberofuniqueprincipalnulldirectionsoftheWeyltensorwasgivenin1954byPetrov[ 31 ]andbearshisname.Itturnsoutthatalltheblackholesolutionsofastrophysicalinterest|includingSchwarzschild,KerrandKerr-Newman|areofPetrovtypeD,meaningtheypossesstwoprincipalnullvectors,eachwithdegeneracytwo.AccordingtotheGoldberg-Sachstheorem[ 32 ]anditscorollaries,foraspace-timeoftypeDwithlaandnaalignedalongtheprincipalnulldirectionsoftheWeyltensor,thefollowinghold(andreciprocally): Thisisequivalenttothestatementthatbothlaandnaarebothgeodesicandshear-free.Thus,intheNPandGHPformalisms,allblackholespace-timesareonequalfooting.IntheKerrspacetime,thecommonlyusedtetrad(alignedwiththeprincipalnulldirections) 30

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33 ],whichtakestheform 2(r2+a2cos2)r2+a2;;0;a; Clearly,Equations 2{22 helpsimplifytheGHPequationstremendously.However,typeDspacetimesaresospecialthattheirdescriptionintermsoftheGHPformalismisevenfurthersimplied.Suchsimplicationisdueinlargeparttotheexistenceofvariousobjectssatisfyingsuitablegeneralizations(andspecializations)ofKilling'sequation. 34 { 36 ],AB=(AB),isasolutionto whereAandA0arespinorindicesandtheparenthesesdenotesymmetrization.TherstconsequenceoftheexistenceofABisthatthequantity isaKillingvector|asatises Theproofofthisinspinorlanguagecanbefoundin[ 36 ],andtheGHPexpressioncanbeverieddirectlybymakingthereplacementra!aandutilizingtheexpressionsin 2{26 isalsoknownasthetwistorequation,whichprovidesadierentmeansofunderstandingitsrelevance. 31

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2{28 independently[ 36 ],soalltypeDspacetimespossesstwoindependentKillingvectors.ThesetwoKillingvectorseachgiverisetoaconstantofmotionalongageodesic.Inotherwords,ifuaistangenttoageodesic(ubrbua=0),thenauaisconservedalongua: wherethersttermvanishesasaconsequenceof(Killing's)Equation 2{28 andthesecondbecauseuaistangenttoageodesic. InadditiontotheexistenceoftwoKillingvectors,theKillingspinoralsogivesrisetotheconformalKillingtensor[ 35 37 ]: 2(22)1=3(l(anb)+m(amb));(2{30) whichalsoexistsineverytypeDbackground.TheconformalKillingtensorisalternativelydenedasasolutionto 3g(abrdPc)d:(2{31) ConformalKillingtensorsareusefulbecausetheygiverisetoconservedquantitiesalongnullgeodesics.Ifkaistangenttoanullgeodesic(kbrbka=0andkaka=0)thenthequantityPabkakbisconservedalongka: 3(kaka)kcrbPbc=0;

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2{31 Incertaininstanceswecanextendthisideatoprovidearstintegralofthemotionfortimelikeandspacelikegeodesicsaswell.Suchanotioncanberealizedbydeningatensor,Kab=K(ab),thatsatises AquantitysatisfyingthisrelationiscalledaKilling-Staeckeltensor.NotethatbydenitionthemetricandsymmetricouterproductsofKillingvectorsbothsatisfyEquation 2{32 .WereservethenameKilling-Staeckeltensorforanobjectthatdoesnotreduceinthisway.ThisistobedistinguishedfromtheantisymmetricKilling-Yanotensorsatisfyingr(aYb)c=0; 38 39 ].BecausewewillnotmakeuseofKilling-Yanotensorshere,wewillfollowconventionallanguageandrefertotheKilling-StaeckeltensorassimplyaKillingtensor.Returningtothemainlineofdevelopment,giventheexistenceofaKillingtensor,wecanrecycletheargumentabove(nowusingEquation 2{32 insteadofEquation 2{31 )fortheconformalKillingtensortoshowthatthequantityKabuaubisconservedforanyuatangenttoageodesic,regardlessofwhetheritbetimelike,spacelikeornull.Thequestionthenarises:WhencanwendaKabthatsatisesEquation 2{32 ?Toanswerthisquestion,webeginbydecomposingtheKillingtensorintoitstrace-freepartanditstrace,accordingto 4Kgab;(2{33) withPabgab=0andK=Kabgab.Usingthisin(Killing's)Equation 2{32 anddividingtheresultingexpressionintotrace-freeandtracepartsgivestwoequations.Thetrace-freepartissimplyEquation 2{31 andsoPabistheconformalKillingtensor(asweanticipatedwith 33

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4rbK=0:(2{34) TheexistenceofaKsatisfyingthisconditionisbothnecessaryandsucientfortheexistenceoftheKillingtensor.Bymakingtheappropriatesubstitution(ra!a),usingEquation 2{30 andtakingcomponentswithrespecttothetetradvectors,weareledtothefollowing: ByapplyingallthecommutatorsinAppendixAtoKandmakinguseofEquation 2{35 ,wearriveataseriesofrelationswhichwecompactlywrite(followingChandrasekhar[ 29 ])as TheseintegrabilityconditionsarebothnecessaryandsucientfortheexistenceofaKsatisfyingEquation 2{34 andthusprovidenecessaryandsucientconditionsforexistenceoftheKillingtensorinatypeDbackground.Theyaresatisedforeverynon-acceleratingtypeDspacetime.Theserelationsaretheprimaryresultofthissection.ItisstraightforwardtoverifythatK=1 2(e2ic2=32+e2ic2=32),wheree2icisaphasefactorwhoseoriginswillbedescribedbelowinEquation 2{41 .ItfollowsthattheKillingtensormaybeexpressedas 8(eic1=32+eic1=32)2gab:(2{37) Historically,theKillingtensorwasdiscoveredbyCarter[ 40 41 ]whileconsideringtheseparationoftheHamilton-JacobiequationintheKerrbackground.TheconstantofmotionderivedfromtheKillingtensoristhusknownastheCarterconstant. Inanon-acceleratingspacetime,wherethefullKillingtensorisavailable,theKillingvectorinEquation 2{27 isrealuptoacomplexphase.IfwespecializetotheKerr 34

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WecanrewriteEquation 2{39 withthehelpofEquation 2{38 as whereCisa(possiblycomplex)functionannihilatedby.Thisisinfactnotaproof,butrathertherststepinone.AfullproofwouldconsistofshowingthatthisisconsistentwiththerestoftheGHPequationsandBianchiidentities.Thecoordinate-freeintegrationtechniqueintroducedinChapter5isideallysuitedforthis.FornowwetakeitasgiventhattheEquation 2{40 istrueineverytypeDbackground,forsomecomplex whichdenesthephasefactorintroducedinEquation 2{37 .ItturnsoutthatinalltypeDspacetimesnotpossessingNUTcharge,c=0.Moreimportantly,wenowhavetherelations 35

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2{27 isrealuptoacomplexphase(e2ic).Notealsothat0a=a.Whathappenedtotheother(linearlyindependent)Killingvector?Itisgivenby 81=32n[eic1=32eic1=32]2(0lbnb)[eic1=32+eic1=32]2(0mbmb)o: ProvingthatthisexpressionsatisesKilling'sequationingeneralisabitinvolved,andsincewe'llhavenodirectuseforEquation 2{43 insubsequentchapters,werefertheinterestedreaderelsewhere[ 36 ]fordetails.Onceagain,usingEquations 2{42 ,itisstraightforwardtoseethatEquation 2{43 isrealuptoaphase.UsingtheKinnersleytetradintheKerrspacetime,Equation 2{43 becomes Mb;(2{44) wheretaisthetimelikeKillingvectorandaistheaxialKillingvector.Becausebisproportionaltoa,itclearlyvanishesintheSchwarzschildspacetime.Thiscanalsobeenseenbynotingthat,intheSchwarzschildspacetime,=0=0andthuscomparisonsofEquations 2{27 and 2{43 revealthatthetwoKillingvectorsarenotlinearlyindependent[ 42 ].In[ 36 ]itisshownhowonecaninfersphericalsymmetryfromthisfact. 2{3 .Bycontractingthelastlinewitheachofthetetradvectorsand 36

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2{32 ,weestablishthat$la=$na=$ma=$ma=0: Inthislight,itisreasonabletoexpectthatwecanconstructanoperator,V,relatedtotheKillingvectorthatcommuteswithallfouroftheGHPderivatives.Becauseofthefactthatspin-andboost-weightsenterexplicitlyintothecommutators(Equations A{1 { A{3 ),wewouldalsoexpectthatanysuchoperatorwouldcarryspin-andboost-weightdependence.Infact,suchanoperatorcanbeconstructed.Bytakingasouransatz:V=aa+pA+qB; 2{36 aresatised,whichimpliesaKillingtensorexists.Fornon-acceleratingspacetimeswethenhave 22p+ wherepandqrefertotheGHPtypeoftheobjectbeingactedon.ThisresulthasbeennotedbyJeryes[ 43 ],whoarrivedatitfromspinorconsiderations.IfwespecializetotheKerrspacetimeandtheKinnersleytetrad,itiseasytoseethatittakesthevalueM1=3@t+bM2=3(r2+a2cos2)1,wherebistheboost-weightofthequantitybeingactedon.DespitethisdierencebetweenthevectoraandtheoperatorV,wewillrefertotheminterchangeablyasaKillingvector.Similarly,wecanfollowthesameprocedurethatled 37

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2{45 toobtainasimilaroperatorassociatedwitha(Equation 2{43 ): 81=32n[eic1=32eic1=32]2(00)[eic1=32+eic1=32]2(00)+2(pq)01=32(e2ic1=321=32)2(p+q)01=32(e2ic1=32+1=32)1 2pe2ic1=32(e4ic21=321=322=322=32)1 2q1=32(e4ic21=321=322=322=32)o;(2{46) whichalsocommuteswithallfourGHPderivations. Onanalnote,weremarkthatinrecentworkBeyer[ 44 ]obtainedanoperatorrelatedtoKillingtensorthatcommuteswiththescalarwaveequation.Theoperatorhasthefeaturethatitisrstorderintime.InthiscontextitistemptingtoaskifthereexistsanoperatoranalogoustothosedenedfortheKillingvectorsthatcommuteswitheachoftheGHPderivatives.Theansweriscurrentlyunclearandsoweleaveitforfutureinvestigation. 2{36 inhand,wearenowinapositiontocompletelysimplifytheGHPequationsforthespecialcaseoftypeDbackgrounds.OurstartingpointistheGHPequationsandBianchiidentitiesadaptedtoaTypeDbackground: (2{48) (2{49) 38

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wherewehaveomittedthoseequationsthatcanbeobtaineddirectlybyutilizingtheoperationsofprimeandcomplexconjugation.Byapplyingthecommutatorsto2andmakinguseoftheequationsabove,welearnthat NotethattheprecedingequationsholdforalltypeDspacetimes.Nextwespecializetonon-acceleratingspacetimesbymakinguseofEquation 2{36 intheform0= 2{56 toobtain Nowwecomputethecommutator[;0]andusetheGHPequationsandtheappropriateversionofEquation 2{57 untilwearriveatanexpressioninwhichtheonlyderivativesare0and0.ThisexpressioncanthenbeusedwithEquations 2{51 and 2{36 tondthefollowingtworelations: 22 22 (2{60) andourtaskiscomplete.ItisworthpointingoutthatduetoEquations 2{36 ,theseexpressionsarenotunique.ThisisasignthatthereissomeredundancyintheGHPequations,whichistobeexpectedwhenweconsidersuchaspecialclassofspacetimes.Wealsopointoutthathavingexpressionsforeveryderivativeoneveryquantityofinterestissucient(butnotnecessary)tocompletelyintegratethebackgroundGHPequations.This 39

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33 ]andlaterbyHeld[ 45 ]fortheGHPequations.InChapter5,wewilldiscussthelatterofthesemethodsinmoredetail. 46 ],whosebasicresultswillbedevelopedhere.Beforeweaddresstherelativisticproblem,weverybrieyreviewrst-orderperturbationtheoryinaatspacetime.Inthatinstance,wethinkofthequantityofinterest,q=q(),asbeingparameterizedbysome,sothatq(0)correspondstotheunperturbedquantityandq(1)isthefullyperturbedquantitywhoserst-orderperturbationswewouldliketoconsider.Itfollowsfromwritingq()asaTaylorseriesinthattherst-orderperturbation,q,isgivenbyq=dq() 40

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Theimportantpointaboutthisprescriptionisthefactthatanotonlyfailstobeunique,butthereis,ingeneral,nopreferredchoiceforit.Achoiceofaismorecommonlyknownasachoiceofgauge.AccordingtoEquation 2{61 ,thedierencebetweenQcomputedwithaandaisgivenbyQQ=$Q; ~Q=Q$Q:(2{62) Notethatagaugetransformationinthissenserepresentsachangeinthewayweidentifypointsinthephysicalspacetimewithpointsinthebackground.Thisistobedistinguishedfromacoordinatetransformation,whichchangesthelabelingofcoordinatesinboththephysicalandbackgroundspacetimes. ThesignicanceofEquation 2{62 isthatunless$Q=0foreverya,thereissomeambiguityinidentifyingtheperturbation|wecan'tdierentiatebetweenthecontributions 41

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47 ].Adirectconsequenceofthisfactisthatthemetricperturbation,arguablythemostfundamentalquantitywedealwith,failstobegaugeinvariant.Fortunately,typeDspacetimescomeequippedwithtwogaugeinvariants,0and4,whichhavesimpleexpressionsintermsofthecomponentsofthemetricperturbation.Aswewillsee,appropriateuseofgaugefreedomsimpliesourcomputationstremendously. 20 ],Chrzanowski[ 18 ],andStewart[ 21 ],anditwillcertainlyplayacentralroleinanyfuturedescriptionofmetricperturbations.Thesecondingredient,theGHPformalismcomeswithseveraladvantages.Firstofall,theinherentcoordinateindependenceandnotationaleconomymakescalculationsingeneralspacetimestractable.Furthermore,byvirtueoftheGoldberg-Sachstheorem,wecandealwiththeentireclassoftypeDspacetimesatonce.Additionally,spin-andboost-weightsprovideusefulbookkeepingand,aswe'llsee,ausefulcontextforunderstandingtherolesthatvariousquantitiesplay.Lastbutnotleast,theuseofaspincoecientformalismhasprovedabsolutelycrucialforstudyingperturbationsofanythingotherthansphericallysymmetricspacetimes.WewillputtheseideastogethertocomputetheperturbedEinsteinequationsinamixedtetrad-tensorform.Thisistheheartofourwork. 42

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sothat,forexample,hll=hablalb.InorderforthistobevalidwithintheGHPformalism,eachcomponentofEquation 2{63 musthaveawell-denedspin-andboost-weight.Becausethebackgroundmetric(Equation 2{3 )isinvariantunderaspin-boost(Equation 2{17 )ithastypef0;0g,whichmustalsobethetypeofthemetricperturbation,hab.Thereforethetypeoftheindividualcomponentsofthemetricperturbationaredeterminedbytheirtetradindices: Allofthevectorsandtensorswewillconcernourselveswithcanbetreatedinthisway. ItisworthwhiletostophereandtakealookatwhatEquation 2{63 reallymeans.ComparingwithourtreatmentofSchwarzschild(Equation 1{4 ),wenotethatthescalarpartsofthemetricare\mixedup"inhll;hlnandhnn,allofwhichhavespinweightzerobutdierinboostweight.Similarly,thevectorpartsaregivenbyhlm;hnmandtheircomplexconjugatesandlikewisethetensorpiecesaregivenherebyhmm;hmmandhmm.However,theseidenticationsarecompletelyindependentofthebackgroundspacetime.Thus,inacertainsense,Equation 2{63 providesageneralizationoftheRWmode 43

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RecallourexpressionfortheperturbedEinsteinequations:Eab=1 2rcrchab1 2rarbhcc+rcr(ahb)c+1 2gab(rcrchddrcrdhcd): 2{63 ,wearriveattheperturbedEinsteinequationsinGHPform: 2cchab1 2abhcc+c(ahb)c+1 2gab(cchddcdhcd);(2{65) which(rightnow,atleast)don'tlookallthatdierent!ThetetradcomponentsofEquation 2{65 foranarbitraryalgebraicallyspecialbackgroundspacetimearegiveninAppendixB.Asidefromtheobviouscosmeticdierences,thereareseveralkeydistinctionsbetweenEquation 2{65 andthestandardformofmetricperturbationtheoryworthpointingout.Firstofall,ourformlacksthebackgroundEinsteinequationspresentinthestandardtreatment.TakingtheirplacearethebackgroundGHPequationsandBianchiidentities.Perhapsmoreimportantlyistheinherentcoordinateindependence.Coupledwiththeconceptsofspin-andboost-weight,thisallowsforacertainstructuralintuitionnotpresentincoordinatebasedtechniques.Thispointofviewwillbestressedthroughout. WritingEquation 2{65 isonething,butactuallycomputingitisanotherquestionentirely,whichwenowturnourattentionto. 44

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EverysessionbeginsbyinvokingGHPtools:> Tobeginwith,eachvariableisdirectlyspeciedbyitsusualname.ForexamplewouldbeenteredinMapleasconjugate(rho).Theprimedvariableshavea`1'appendedtotheend,sothat0wouldbeenteredasconjugate(rho1).TheWeylscalarsarerecognizedascapital'swiththeappropriatenumber,e.g.Psi2.Thederivatives,,0and0arerecognizedinMapleasth(),eth(),thp()andethp(),respectively.GHPtoolsrecognizesthetetradvectorsaslabelsindicatingthepositionoftheindexwiththeactualindexinparentheses.Forexamplelaandmcwouldbeinputaslup(a)andconjugate(mdn)(c).Finally,GHPtoolscontainsanarbitraryfunction,(inMaple:phi),thatisquiteusefulforgeneralcalculations.Amongst 48 49 ],calledGHPandGHPII.WestressthatGHPtoolsisnowayintendedtocompetewiththeseoranyotherMaplepackages. 45

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Thefp;qgtypeofanyquantitymaybeobtainedbytheuseofthegetpqfunction,whichreturnspandq,inthatorder:> 1=0; =0; =0; 1=0;

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TherealusefulnessofGHPtoolscomesnotfromitsbookkeepingabilities,butratheritsabilitytoperformsymboliccomputationswithintheGHPformalism.TheseabilitiesbeginwiththeDGHP()procedure,whichexpandsderivativesofobjectsoccurringinanexpressioninaccordancewiththerulesofderivations.Forexample> 1)>

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1))> Todate,DGHP()canhandlepowersandlogarithms(theonlyfunctionsthisauthorhasencounteredintheGHPformalism),buttheprocedurecanbeeasilymodiedtoaccommodatejustaboutanyfunction.BuildingcomplicatedexpressionsinvolvinglinearcombinationsofderivativeandmultiplicativeoperatorsiseasilyachievedwiththehelpoftheGHPmult()procedure.TheseexpressionscanthenbeexpandedwithDGHP().Asanexample,considertheexpression()4:> Simplifyingsuchexpressionsis,inthecontextoftypeDspacetimewithoutacceleration,handledbythetdsimp()procedurethatsubstitutestheknownvaluesofthederivativesofthespincoecients(storedinthegloballyavailablelisttdspec;suchaprocedurecanbeeasilygeneralizedtoencompassanyspacetime,shouldtheneedarise)intoitsargument.Thusourpreviousexamplesimpliesconsiderably:>

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Perhapsevenmoreusefulisthecomm()procedurewhichcommutesderivativesonanexpression.Ittakestwoarguments:therstisthetermwhosersttwoderivativeswillbecommutedandthesecondistheexpressionintowhichtheresultwillbesubstituted.Considerthefollowingexamples:> me1:=eth(th(ethp(ethp())))th(eth(ethp(ethp())))> 1ethp(ethp()) 1ethp(ethp())pq> me2:=th(th(thp()))th(thp(th()))> eth() 1th(ethp()) 1+ppth()1+3pp1pp1 1+3pq 1pq 1 3pq ComputingtheperturbedEinsteinequationsandWeylscalarsnecessarilyrequirestheabilitytocontractvariouscombinationsofthetetradvectors.Thisfunctionalityisprovidedbythetetcon()procedure,whichalsotakestwoarguments.Therstistheexpressionthatcontainstheuncontractedvectorsandthesecondisalistoftheindicestobecontractedover.Taketheexampleofcomputingthetraceofthemetric: 49

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Finally,GHPtoolsprovidessomefunctionalityfortranslatingexpressionsintoNPexpressionsthatcansubsequentlybeconvertedtoordinarycoordinateexpressions.ThisfunctionalityisprovidebytheaptlynamedprocedureGHP2NP(),whichtakesasitsinputaGHPexpression.Thefunctionalityprovidedbytheprocedureislimitedtoexpressionsinvolvingatmosttwoderivatives.Furthermore,thederivativesmustappearinaspeciedorderaccordingtothefollowingrules:(1)and0mustalwaysappeartotheleftofand0,(2)mustappeartotheleftof0and(3)mustappeartotheleftof0.Takethefollowingexample:> (hln)+ Inordertoaidintheconversionofsuchquantitiesintocoordinateexpressions,GHPtoolscontains,aslistsofarrays,somecommonlyusedtetradsintheKerrspacetime.Theyare:theKinnersleytetradwithindicesuptetupKanddowntetdnK,thesymmetrictetrad(tetupS,tetdnS)andthesymmetrictetradboostedbyafunctionB(t;r;;)andspunbyafunctionS(t;r;;)(tetupSB,tetdnSB).Thesearecalledsimplybyinvokingtheirnames: 50

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2Iasin()p 2(racos()I)p 2I(r2+a2)sin()p 2r22Mr+a2 2;0;1 2a(r22Mr+a2)sin()2 2Iasin()p 2(r+acos()I)p 2I(r2+a2)sin()p 51

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Asarstapplicationofourframework,wewillprovideamoredetaileddiscussionoftheRegge-WheelerandTeukolskyequations.ThisleadsquitenaturallytoadiscussionofthemetricperturbationgeneratedfromaHertzpotential,whichwillplayamajorroleinsubsequentchapters.Ourstartingpointisageneraldiscussionofparitythatdoesnotassumeeithersphericalsymmetryorangularseparationfromtheoutset. 50 ]. Ourrstassumptionisthatourspacetimemanifold,M,admitsaspacelike,closed2-surface,S,topologicallya2-sphere,withpositiveGaussiancurvatureandapositivedenitemetricgivenby wheremaandmaaretwomembersofanulltetrad.Forasphericallysymmetricbackgroundabisproportionaltothemetricofthe(round)2-sphereandmaandmacanbedirectlyassociatedwiththebackgroundmetric.Moregenerally,weallowforthe 52

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Forexampleconsidersomevector,va,denedinthespacetime:va=vlna+vnlavmmavmma: Theseareallthetoolsnecessaryforwhatfollows. WebeginbyconsideringtheprojectionofvectorsdenedinthespacetimeontoS.ToidentifytheoddandevenparitypieceswestartbydecomposingageneralvectoronS (3{4) 53

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Symmetric,trace-freetwo-indexedtensorsonSalsohaveasimpleparitydecomposition.Itiseasytorecognizethe(two)componentsofsuchtensorsasspin-weight2scalars.Thatis,thecomponentsareoftypefb2;b2g.WeconsidertheparitydecompositiononSbycreatingthetensorfromavectoronS,a,withboost-weightbandspin-weight0: whichcaninturnbefurtherdecomposedintoitsevenandoddparitypiecesbyapplyingEquation 3{4 toyield whichprovidesuswithameansofidentifyingtheevenandoddbitsofsymmetrictrace-freetensorsonS.Thisresultgeneralizesquiteeasilyton-indexedsymmetrictrace-freetensors(withcomponentsofspin-weightnandboost-weightb)onS: 8 ]fordetails)(i.e.,the\i"comesalongfortheride). 54

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Inpractice,wearegenerallygivensomespin-andboost-weightedscalar,(and/oritscomplexconjugate),andwemerelywanttoidentifytheeven-andodd-paritypieceswithoutexplicitlydecomposingitaccordingtoEquation 3{7 .InthiscaseEquation 3{7 allowsustodosobysimplywriting Inthecontextofaspacetimewherelaandnaarexedbyconsiderationsotherthanbeingorthogonaltomaandma(e.g.PetrovtypeD,wherewewouldlikethemalignedwiththeprincipalnulldirections),butmaandmafailtoformaclosed2-surface(theKerrspacetimeprovidesonesuchexample;thiscanbeseenbynotingthatand0don'tcommute),thequestionarisesofwhetherornotsomethinglikeEquation 3{8 isstillusefultoconsider.Itappearsso.Insuchacasethedecompositiontheorems(therstlinesofEquations 3{5 and 3{4 )failtobetrue,butthisisn'taseriousissue.Becauseabandabstillallowustodecomposetensorsintotheir\proper"and\pseudo"pieces,inplaceofEquation 3{7 wehave where\even"and\odd"arewritteninquotestoemphasizethefactthattheyreallyrefertorealandimaginaryinthiscontextandthebarovertauindicatestheproperspin-andboost-weight.Clearly,Equation 3{9 ,lackstheadvantagepresentinEquation 3{7 ofbeingabletoputalloftheangulardependenceintoand0andregardtheentiretensorasarisingfromthetworealscalarsevenandodd.Neverthelessitprovidesausefuldecompositionofspin-andboost-weightedscalars,withoutseparationofvariables,that 55

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3{8 inarbitrarybackgrounds.Furthermoreinthelimitthatmaandmabecomesurface-forming(e.g.thea!0limitoftheKerrspacetime),Equation 3{9 becomesEquation 3{7 .Thisisoneavenueforunderstandingwhyparityplayssuchanimportantroleintheperturbationtheoryofsphericallysymmetricbackgrounds.Inthecontextofnulltetradformalismswecanseetheseeminglyunmotivatedactofperformingparitydecomposition,whichdoesnotgeneralizewell,asarisingfromthequitenatural(andperhapsmorefundamental)actofseparatingquantitiesintotheirrealandimaginaryparts,whichisentirelygeneral.Inthislight,itmakessensethatourattentionwouldbefocusedonparitybecausetherstperturbativeanalysistookplaceinthesphericallysymmetricSchwarzschildbackgroundinwhichonecannotdierentiatebetweenthetwodecompositionsbutparityhassignicancethere.Regardless,theonlyusewemakeoftheseresults,exceptforsomeremarksinChapter5,isbelowinthecaseoftheSchwarzschildbackgroundwherethepointismoot. 24 ]whoobtained sin2hh=0;(3{10) 56

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sin@(sinht+@ht)=0; sin@(sinhr+@hr)=0; asthemode-independentexpressionoftheRWgauge.NowwecantransformthisdescriptionintoGHPlanguage.Itisarelativelystraightforwardprocessnowtowritethetetradcomponentsofthemetricperturbation(hll,hln,etc.)intermsofthecoordinatecomponentsofthemetricperturbation(htt,hrr,etc.)andinverttherelations.Withthisknowledgeinhand,itbecomesevidentthatEquations 3{10 and 3{11 aresimplycombinationsofhmm=0andhmm=0: 3{12 and 3{13 arecombinationsof0hlm+hlm=0andhnm+0hnm=0; Inthislanguage,itisnaturaltogeneralizetheseconditionstomoregeneraltypeDspacetimesonthebasisofspin-weightconsiderations.ThespiritoftheRWgaugesuggeststhatwekeeptherequirementthatnospin2componentsenterthemetricperturbation.Therequirementonthespin1componentsiseasilygeneralizablebyputtinginpiecesproportionaltoand0whichbothvanishintheSchwarzschildbackground. 57

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whereaandbare(generallycomplex)constantsthatmustbedeterminedbysomeothermeans.NotethattheformofEquations 3{14 isrestrictedbyrequiringthegaugerestrictionstobeinvariantunderbothprimeandcomplexconjugation.ThefullutilityofthegeneralizedRWgaugeremainstobeexplored,butitisclearthatanysimplicationitbringswillapplyuniformlytoalltypeDspacetimes. withallotherbackgroundquantitiesvanishing,sothesituationisimmediatelysimplied.Nextweproceedwiththeparitydecompositionbywritingthecomponentsofthemetricperturbationas,forexample,hlm=hevenlm+ihoddlm,hlm=hevenlmihoddlm,etc.Notetherelativeminussignsbetweentheodd-paritybitsandtheircomplexconjugates.Fromhereonwewillspecializetoodd-parityandthusdropthe\odd"labelsandfactorsofisincenoconfusioncanarise.Withthisspecialization,ourgaugeconditionsnowread: 58

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(00)hlm+()hnm=0 (3{17) (00)hlm+()hnm=0; wherethesecondrelationfollowsfromcomplexconjugationoftherst(orintegrationofEmm),andwehavesetthe\integrationconstant"tozeroforconvenience(itwouldcancelbelow).WenowturnourattentiontoElm.BysuccessiveapplicationsofEquation 3{17 wecaneliminatealltermsinvolvinghnm,arrivingat 2n(020+402042)hlm22hnmo(3{19) Takingtheprimeofthis(whichintroducesanoverallminussignbecauseoftheparitydecomposition)leadstoasimilarexpressionforEnm.Nextwetakethe(sourcefree)combination (020)0Elm+(2)Enm=0:(3{20) Wecanremovefromthisexpressionallreferencestohlmand0hnmusingthegaugeconditionsinEquations 3{16 ,which,aftersomeseriouscommutingleadstothequite 59

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ThisistheRegge-Wheelerequation.Wecancleanitupabitbyrecognizingtheobjectbeingactedonas2_odd2=hnm00hlm,theodd-paritypieceoftheperturbationof2.FurthermoretheoperatorinEquation 3{21 isthewaveoperator,2,intheSchwarzschildbackgrounduptoafactorof1=2.Makingtheseidentications,wenowhavefortheRegge-Wheelerequation: (2+82)2=32_odd2=0:(3{22) Asimilarequationfor_odd2=Im(_2)waspreviouslyderivedbyPrice[ 51 ](whoseonlyrelationtothepresentauthorisEquation 3{22 ),whoshowedthat(moduloangulardependence),Im(_2)isthetimederivativeoftheRegge-Wheelervariable.Moreover,withoutreferencetoIm(_2)Jezierski[ 52 ]arrivedatanequationforodd-parityperturbationsthatisessentiallyidenticaltoEquation 3{22 ,thoughphrasedinmorestandardlanguage.Additionally,ananalysisbyNolan[ 53 ]wholookedattheperturbedWeylscalarsintermsofgaugeinvariantsofthemetricperturbationshowedexplicitlytherelationbetweenIm_2andthegaugeinvariantquantityassociatedwiththeRWvariable.Furthermore,Nolanpointsoutthatbecause2isrealinthebackground,theperturbationofitsimaginarypartis,whenwerestrictourattentiontoodd-parity,gaugeinvariantinthesensediscussedinChapter2.Perhapsmoresurprisingly,NolanfurtherassertsthatthisistrueoftheperturbationsofalltheWeylscalars,whichemphasizesthefactthatodd-parityperturbationsofsphericallysymmetricspacetimesareobtainablebyvirtuallyanymeans. OnethingthatsetsourtreatmentofRWapartfromothersisoursparinguseofsphericalsymmetry.TheonlyplacewemakeexplicituseofitisinEquations 3{15 ,whichdenesthebackgroundGHPquantities.Thiscertainlysimpliesthesubsequentcalculationsconsiderably,butfailstofullyexploitthebackgroundsymmetry.In 60

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TheZerilliequation[ 7 ]describingeven-parityperturbationsoftheSchwarzschildspacetimehassofareludedadirectdescriptionintermsofgaugeinvariantperturbationsoftheWeylscalars.However,theinformationcontainedwithintheZerilliequationcanbeobtainedthroughthemetricperturbationthatfollowsfromtheTeukolskyequation,whichisthefocusoftheremainderofthischapter. 10 { 12 ]camedirectlyfromconsideringperturbationsoftheWeylscalars.We,however,areinterestedinobtainingitdirectlyfromtheEinsteinequation.UsingTeukolsky'sexpressionsforthesourcesof0and4,wecanobtainthisdirectly.ThesourcesoftheTeukolskyequationaregivenby whereT0andT4arethesourcesfor0and4,respectively.MakingthereplacementTab=1 8Eabintheequationsaboveleads(afterproperlyrearrangingthederivativeswith 61

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[(4)(00)(40)(00)32]0=4T0; [(0400)()(040)()32]4=4T4; where,intermsofthecomponentsofthemetricperturbation 2((0)(0)hll+()()hmm[()(20)+(0)(2)]h(lm)); 2((0)(0)hnn+(00)(00)hmm[(00)(02)+(0)(020)]h(nm)); andwheretheparentheses,(),aroundthetetradindicesdenotesymmetrization.Itisbothinterestingandimportanttonotethat,intheKerrspacetime,thecoordinatedescriptionofEquation 3{26 doesnotleadtotheseparableequationdiscussedinChapter1(Equation 1{17 ).Toobtainaseparableequation,anextrafactorof4=32mustbebroughtin,resultinginthefollowingexpression: [(00)(+3)(0)(+3)32]4=324=44=32T4:(3{29) Belowwewillseethesameexpressionarisingfromverydierentconsiderations. 20 ]usedspinormethods.Shortlyafterthat,Chrzanowski[ 54 ]obtainedessentiallythesameresultusingfactorizedGreen'sfunctions.Sometimelater,Stewart[ 21 ]enteredthegameandprovidedanewderivationrootedinspinormethods.Eventually,Wald[ 55 ]introduceda 62

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Wald'smethodiscenteredaroundthenotionofadjoints.Considersomelineardierentialoperator,L,thattakesn-indextensoreldsintom-indextensorelds.Itsadjoint,Ly,whichtakesm-indextensoreldsinton-indextensoreldsisdenedby forsometensoreldsa1:::amandb1:::bnandsomevectoreldsa.IfLy=L,thenLisself-adjoint.Animportantpropertyofadjointsisthatfortwolinearoperators,LandM,(LM)y=MyLy.NowletE=E(hab)denotedthelinearEinsteinoperator,StheoperatorthatgiveseitheroftheTeukolskyequationsfromE(Equation 3{23 or 3{24 ),O=O(0or4)thesource-freeTeukolskyoperator(Equation 3{25 or 3{29 )andT=T(hab)theoperatorthatactsonthemetricperturbationtogive0or4(Equation 3{23 or 3{24 ).ThentheTeukolskyequationscanbewrittenconciselyas Itfollowsbytakingtheadjointthat wherewehaveusedthefactthattheperturbedEinsteinequationsareself-adjoint.Thus,ifsatisesOy=0,thenSyisasolutiontotheperturbedEinsteinequations!ThisremarkablysimpleandelegantresultholdsforanysystemhavingtheformofEquation 3{31 ,wheneverEisself-adjoint. InordertoapplythisresulttotheTeukolskyequationwenotethatscalarsareallself-adjointandtheadjointsoftheGHPderivativesaregivenby 63

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SupposenowthatwehaveasolutiontotheTeukolskyequationfor0,sothatOisgivenbytheleftsideofEquation 3{25 andSisgivenbytherightsideofEquation 3{23 (withTabreplacedwithEab).Wald'smethodthentellsusthatifOy=0,thenhab=SyisasolutiontotheperturbedEinsteinequations.UsingEquations 3{33 wecancomputeSy: wherewe'veaddedthecomplexconjugate(c.c.)tomakethemetricperturbationrealandremainstobespecied.UsingEquations 3{33 ,itisclearthattheadjointofEquation 3{25 is [(00)(+3)(0)(+3)32]=0;(3{36) whichispreciselytheequationsatisedby4=324(c.f.Equation 3{29 ),previouslyobtainedthroughseparabilityconsiderationsintheKerrspacetime.However,obtainingEquation 3{36 requirednoreferencetoseparationofvariablesinaparticularspacetimeandthusappliestoalltypeDspacetimes.Itisimportanttonotethatalthoughsatisesthesameequationas4=324,itisnottheperturbationof4forthemetricitgenerates(Equation 3{35 ).InChapter5wewillexplore'sconnectionto4morecarefully. ThoughthederivationofEquation 3{35 wasquitesimple,itfailstoyieldanyinformationaboutthegaugeinwhichthemetricperturbationexists.Inthisparticularinstance,itisfairlystraightforwardtoverifythatthemetricperturbationwe'vebeenled 64

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whichisknownintheliteratureastheingoingradiationgauge(IRG),anunfortunatenamebecauseingoingradiationiscarriedbylaandEquation 3{37 tellsusthatthemetricperturbationiscompletelyorthogonaltola.Thusthereisonlyoutgoingradiationintheingoingradiationgauge!ObtainingthegaugeconditionsinEquations 3{37 and 3{38 ismorenaturalintheapproachesofCohenandKegeles[ 20 ]andStewart[ 21 ].Onestartlingaspectofthegaugeconditionsisthatthereareveofthem.Thisbeingthecase,wemustbeconcernedaboutthecircumstancesunderwhichthemetricperturbationintheIRGiswell-dened.Thisisthesubjectofthenextchapter. OurderivationbeganwiththeTeukolskyequationfor0.HadweinsteadstartedwiththeTeukolskyequationfor4=324,wewouldbeledtoametricperturbationintermsofaHertzpotential,0,thatsatisestheTeukolskyequationfor0.TheresultingmetricperturbationandgaugeconditionsarethensimplytheGHPprimeofEquations 3{35 3{37 and 3{38 ,respectively.Inthiscase,themetricperturbationexistsintheso-calledoutgoingradiationgauge(ORG).Fortheremainderofthiswork,wewillfocusourattentionontheIRGmetricperturbation,butalltheresultsholdfortheORGperturbationaswell. OnanalnoteweremarkthattheTeukolskyequationfor0(Equation 3{25 )actuallyexistsinthemoregeneraltypeIIspacetimes,withoutitscompanionfor4.Inthiscase,Wald'smethodalsoleadstometricperturbation(intheIRG;noORGexistshere),withapotential,,satisfyingtheadjointofEquation 3{25 ,which,inthisinstance,isnottheequationfortheperturbationof4. 65

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Inthepreviouschapter,itwasseenthattheperturbationsoftheWeylscalarsleadquitenaturallytometricperturbationsintheradiationgauges,(seeminglyover-)speciedbyveconditions.Inthischapterwewillexploretheprecisecircumstancesunderwhichonecanimposeallveoftheseconditions.ThiswillrequireustoexaminetheperturbedEinsteintensor,whichpresentstheneedtointegratesomeofthecomponents.Forthis,wewillappealtoacoordinate-freeintegrationtechniquebasedontheGHPformalism,duetoHeld[ 45 56 ].Thegeneralityofthesemethodsallowustoprovetheresultforamuchbroaderclassofspacetimesthanwehaveencounteredsofar,namely,PetrovtypeII,whichwewillseeisthelargestclassofspacetimeforwhichtheradiationgaugesaredened.Webeginwithamorethoroughdiscussionoftheradiationgaugesandtheirorigin.Mostofthischapteristakenfrompublishedwork[ 57 ]. 58 ](forperturbationsofPetrovtypeIIspacetimes)andChrzanowski[ 54 ](whoconsideredperturbationsofPetrovtypeDspacetimes),buttheworkthatcomesclosesttoourcontributionindescribingtheiroriginisthatofStewart[ 21 ],againforthemoregeneralcaseoftypeIIspacetimes. IntypeIIbackgroundspacetimes,theIRGisdenedbytheconditions 66

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IntypeIIbackgroundspacetimes,onlyoneortheotheroftheseoptionsexists(IRGorORG),whereasinPetrovtypeDbackgroundspacetimes,thereisthepossibilityofdeningbothgauges.FortheremainderofthisworkwefocusontheIRG.ResultsfortheORGcanbeobtainedbymakingthereplacementla$na. Equations 4{1 translateintoalgebraicconditionsonthecomponentsofthemetricperturbation.Wewillrefertothefourconditionsin( 4{1 )asthelhgaugeconditions. TheconditioninEquation 4{2 willbereferredtoasthetraceconditionandcanbeexpressedintermsofthecomponentsofthemetricperturbationashlnhmm=0;which,whenEquation 4{5 isimposed,simplyreads BecausetheIRGconstitutesatotalofveconditionsonthemetricperturbation,insteadofthefouronemightexpectforagaugecondition,itisnecessarytoensurethattheextraconditiondoesnotinterferewithanyphysicaldegreeoffreedomintheproblem, 59 ],thoughtheyarenottheconditionsoriginallyintroducedforgravitationwiththatname[ 60 ].Itmaywellbethatthisdescriptionissuitablemoregenerally,althoughpresumablywithoutthespecicgeometricalinterpretationof[ 59 ]. 67

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B{1 ofAppendixB,inwhicheverytermwouldberemovedbyEquations 4{5 and 4{6 ,renderingEquation B{1 inoperablewheneverithasanon-zerosource.InthenextsectionwewilllooktotheperturbedEinsteinequationstodeterminethecircumstancesunderwhichwecansafelyimposeallveoftheconditionsthatconstitutetheIRG. ItisusefultonotethesimilaritybetweenthefullIRG,( 4{1 ),andthemorecommonlyknowntransversetraceless(TT)gaugedenedby which,ataglance,alsoappearstobeover-specied.Infact,theTTgaugeexistsforanyvacuumperturbationofanarbitrary,globallyhyperbolic,vacuumsolution[ 61 ],becauseimposingthedierentialpartofthegaugedoesnotexhaustalloftheavailablegaugefreedom.Interestinglyenough,Stewart'sanalysisintermsofHertzpotentials[ 21 ]beginsbyconsideringametricperturbationintheTTgauge.However,inordertoconstructthecurvedspaceanalogueofaHertzpotential,heiscompelledtoperformatransformationthatdestroysEquation 4{7 andinsteadyieldsametricperturbationintheIRG. Consideragaugetransformationonthemetricperturbationgeneratedbyagaugevector,a.Tocreateatransformedmetricinthelhgauge,thegaugeconditionsin 21 ]ortheelectromagneticexampleinChapter1foramoredetailedexplanation. 68

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4{5 require wherethesemicolondenotesthecovariantderivative.Intermsofcomponentsthisreads 2l=hll;0l+n+(+0)m+(+0)m=hln;(+)m+(+0)l=hlm;(+)m+(0+0)l=hlm:(4{9) Similarly,forthetraceconditioninEquation 4{6 tobesatisedbythegaugetransformedmetric,werequire Anyextragaugetransformationthatsatisesla(a;b)=0|solvesthehomogeneousformofEquation 4{9 |preservesthefourlhgaugeconditionsinEquations 4{5 .Thisiswhatismeantbyresidualgaugefreedom.Wewillexplicitlyusethisresidualgaugefreedomtoimposethelhandtraceconditionssimultaneously,thusestablishingtheIRG.Wewillndthatsomegaugefreedomstillremains,asexplainedinSection 4.3 Now,weturnourattentiontothegeneralcaseoftypeIIbackgroundspacetimes. 45 46 ]. 69

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[;~0]x=0;[;~]x=0;[;~0]x=0;(4{11) where[a;b]denotesthecommutatorbetweenaandb.TheexplicitformoftheoperatorsisgiveninAppendixC.Thenextstep,theheartofHeld'smethod,istoexploittheGHPequation=2,anditscomplexconjugate,=2,toexpresseverythingasapolynomialintermsofand,withcoecientsthatareannihilatedby.Held'smethodisthenbroughttocompletionbychoosingfourindependentquantitiestouseascoordinates[ 56 62 ].Inthiswork,wewillnottakethisextrastep.FortypeIIspacetimes(andtheacceleratingC-metrics),thisstephasnotbeencarriedout,whileforallremainingtypeDspacetimes,ithasbeencarriedthroughtocompletion[ 45 46 ]. InaspacetimemoregeneralthantypeII,thereisnopossibilityofhavingarepeatedPND.WhenarepeatedPNDexists,wecanappealtotheGoldberg-Sachstheorem[ 32 ]andset==0=1=0inEquationsB-1{B-7.FollowingHeld'spartialintegrationofPetrovtypeIIbackgrounds[ 56 ],wealsoperformanullrotation(keepinglaxed,butchangingna)toset=0.Asaconsequence,itfollowsfromtheGHPequationsthat0=0.NowweareinapositiontoaddressthequestionofwhenthefullIRGcanbeimposed.FirstweapplythelhgaugeconditionsinEquations 4{5 toEquationsB-1{B-7.WhilemostoftheperturbedEinsteinequationsdependonseveralcomponentsofthemetricperturbation,afterimposingEquations 4{5 ,theexpressionforElldependsonlyonhmmandthell-componentoftheperturbedEinsteintensorsimplybecomes 70

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4{12 andourargumentwouldnothold.WeimmediatelyseethatTll=0isnecessarytosatisfythetraceconditioninEquation 4{6 .NextweturnourattentiontothequestionofwhethertheconditionEll=0,issucienttoimposeEquation 4{6 usingresidualgaugefreedom. InordertoaddressthisquestionwewillintegrateEll=0andtheresidualgaugevector,givenbythehomogeneousformofEquations 4{9 .FullintegrationofthehomogeneousformofEquations 4{9 iscarriedoutinAppendixC,butwewillworkthroughtheintegrationofEll=0heretoillustratethemethod.WebeginbyrewritingEquation 4{12 ,withthehelpof=2anditscomplexconjugate,as: 3 Integratingoncegives andanotherintegrationleadsto +1 2b(+):(4{15) However,hmmis,bydenition,arealquantity,soweaddthecomplexconjugateandusebtorepresentarealquantityinthesecondterm.Thenalresultisthat +b(+):(4{16) 71

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4{9 ,ascarriedoutinAppendixC,leadstothefollowingsolutionforthecomponentsoftheresidualgaugevector: 21 ~0l+1 2(2+2)l;m=1 m~l;m=1 where2isrelatedtothebackgroundcurvaturevia2=23.InordertousethisresidualgaugefreedomtoimposethefullIRG,wereturntothegaugetransformationforhmm(Equation 4{10 )whichbecomes,aftersomemanipulation(usingEquations C{6 { C{9 andEquation C{13 ), h~m+~0li+(+)[1 2(~0~+~~000)l+n]:(4{18) Inthisformitisclearthatwecanimposethetracecondition(Equation 4{6 )ofthefullIRGifwechooseourgaugevectorsothat ~0m+~0l=a;1 2(~0~+~~000)l+n=b:(4{19) WehavenowshownbyconstructionthattheconditionTll=0isbothnecessaryandsucientforimposingthefullIRGinatypeIIbackground.Weturnnexttodiscussingthecompleteextentoftheresidualgaugefreedominmoredetail. 4{19 involvethreerealdegreesoffreedom(aiscomplex),itturnsoutthatonlytworealdegreesofgaugefreedomarerequiredtofullyremoveanysolutionofEquation 4{13 forthetracehmm.Toseethisweintroducethefollowingidentity: =(+)1 1 72

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4{16 as 2(a+a) +[1 2(aa)+b](+):(4{21) Inasimilarfashion,werewriteEquation 4{18 as 2(~0m+~m)+~0li +h1 2(~0m~m)1 2(~0~+~~000)l+ni(+);(4{22) inwhicheachcoecientinbigsquarebracketsispurelyreal.Now,supposewehaveaparticularsolutionforEll=0(i.e.,a,aandbarexed)andourtaskistosolveforthecomponentsofthegaugevectorwhichremovesthissolution.BycomparingEquations 4{21 and 4{22 weseethat,foranygivenmandm,wecanxl(uptoasolutionof~0l=0)via ~0l=1 2(a+a)1 2(~0m+~m);(4{23) andwecanxnbysetting 2(aa)+b+1 2(~0~+~~000)l1 2(~0m~m);(4{24) tocompletelyeliminatethenonzerohmm,thusimposingthefullIRGwhilestillleavingtwocompletelyunconstraineddegreesofgaugefreedom,mandm.OnceintheIRG,Equations 4{23 and 4{24 ,witha,aandbsettozeroandmandmarbitrary,givetheremainingcomponentsofagaugevectorpreservingtheIRG.ItiscurrentlyunclearhowtotakeadvantageofthisremaininggaugefreedomtosimplifytheanalysisofperturbationsinthefullIRG. 33 ].Whilethe 73

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45 ].ThecompleteintegrationrequiresthateachoflaandnabealignedwithoneofthetwoPNDs.InthatcasewecanexploitthefullpoweroftheGoldberg-Sachstheoremanditscorollariestoset=0==0=0=1=3=4=0,whilemaintaining6=0and06=0.Inthissectionwerepeatthepreviouscalculationwiththisdierentchoiceoftetrad. TheresultofintegratingEll=0isthesameasinthecaseofatypeIIbackground,giveninEquation 4{16 .Theresidualgaugevector,however,nowhasthefollowing,morecomplex,form(detailsoftheintegrationaregiveninAppendixC): 2l+1 2l+l+1 2l1 2+h 21 ~0l[(~+)+(~0+)]l+m m1 2m1 m1 l1 +l(~0+)l;(4{25) 74

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h~m+~0l+Bi+(+)h1 2(~0~+~~000)l+nAi;(4{26) wherewehaveintroduced(notethatBispurelyimaginary) 2f2~+~()+gl+c.c.;B=1 4f4~+~()+52glc.c.;(4{27) withc.c.indicatingthecomplexconjugate.Integrationofthebackgroundswhere6=0and6=0usingtheHeldtechniquehasnotmadeitswayintotheliteratureandisbeyondthescopeofthepresentwork.Asaresult,derivativesofandappearinEquations 4{27 butdonoharmtoourargument.Choosinganygaugevectorthatsatises ~0m+~0lB=a;1 2(~0~+~~000)l+nA=b;(4{28) willservetoimposethetraceconditioninthefullIRG.OnceagainwehaveestablishedthatTll=0isbothanecessaryandsucientconditionfortheexistenceofthefullIRG.Notethatbysetting==0(i.e.,ignoringtheC-metrics)inthebackground,A=B=0,andtheresultisvirtuallyidenticaltoEquations 4{18 and 4{19 .Thereisonesimplicationinthatnow0=0[ 46 ].ThefullextentoftheremainingresidualgaugefreedominEquations 4{28 canbedemonstratedalongthesamelinesasusedinSection 4.3 .AsforthecaseofatypeIIbackground,itresideschieyinthefreelyspeciablemandm. 75

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ForperturbationswithTll=0,ourcharacterizationoftheresidualgaugefreedomissucientlycompletethatwecanexplicitlydemonstratetherequiredgaugechoicetoremoveanynon-zerosolutionforthetraceobtainedviaEll=0.Thus,intypeIIspacetimes,radiationgaugescanbeestablishedbyagenuinegaugechoice,evenifonlyafterasolutionofEll=0ischosen. TherearesubtledierencesbetweenthegeneraltypeIIcaseandthemorerestrictedtypeDcase,astherearealsointheconstructionofHertzpotentialsforthetwocases.Stewart[ 21 ]writesoutthetypeIIcaseratherfullyforanIRG.Inthiscase,theperturbationin0istetradandgaugeinvariant,whilethepotentialsatisestheadjoint(inthesensedetailedbyWald[ 55 ])ofthes=+2Teukolskyequation.Remarkably,inthetypeDcase,thisadjointisactuallythes=2Teukolskyequation,alsosatisedbythegaugeandtetradinvariantperturbationin4.InthetypeIIcase,theadjointequationisthesameasintypeD,but4isnolongertetradinvariant.ComparedtothetypeDresult,theexpressionfor4givenbyStewarthasmanyextratermsdependingon0and0,sopresumablyitdoesnotsatisfythesameequationasthepotential.Asaconsequence,metricreconstructionwouldberestrictedtobeingbuiltaroundtheperturbationfor0(c.f.thecommentsattheendofChapter3). 76

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63 ]hasarguedthatmassandangularmomentumperturbationsarenotgivenbyanysolutiontotheTeukolskyequations,andStewart[ 21 ]hasshownthatthesecannotberepresentedinaradiationgaugeintermsofapotential.Whatwehavedoneisidentifythegaugefreedomwhichremainsinthefullysatisedradiationgauges,neitherinterferingwiththeradiationgaugeprescriptionnorrulingoutthepossibilityofmassandangularmomentumperturbations.ByrealizingtheexplicitconstructionoftheradiationgaugesfortypeIIbackgroundspacetimesandbyidentifyingtheremaininggaugefreedomwhichtheyallow,wehave,inasense,completedataskinitiallyembarkeduponbyStewart[ 21 ],thoughinthedierentcontextofHertzpotentials. 77

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Havingestablishedtheconditionsfortheexistenceoftheradiationgauges,wewillusethecorrespondingmetricperturbationstoestablishsomeusefulrelationshipsbetweentheperturbedWeylscalarsknowngenerally(andquiteloosely)astheTeukolsky-Starobinskyidentities.BecauseHertzpotentialsaresolutionstotheTeukolskyequation,theseidentitieshaveimmediaterelevanceformetricreconstructionintheIRG,bothinthetime-domainapproachofLoustoandWhiting[ 25 ]andthefrequencydomainapproachofOri[ 23 ]. TheoriginalanalysisofTeukolsky[ 11 12 ]wasbasedontheasymptoticformofthesolutionsoftheseparatedangularandradialfunctionsintheKerrspacetimeaswellasatheoremduetoStarobinskyandChurilov[ 64 ].OnlylaterdidChandrasekharprovideafullanalysis,whichisnicelysummarizedinhisbook[ 29 ].Ouranalysis,however,willbeentirelysymbolic,involvingonlyGHPquantities.Thisapproachhastheadvantagenotonlyofapplyingtoalargerclassofspacetimes,butdisplayingthestructureinherentintheidentitiesinamuchmoreobviousway.AsimilaranalysisofsomeoftheidentitieswewilldiscusswaspreviouslyundertakenintheNPformalismbyTorresdelCastillo[ 65 ]andlatertranslatedintoGHPbyOrtigoza[ 66 ].TheseprioranalysesmadeuseofthemostgeneraltypeDspacetimeandtranslatedbackandforthbetweencoordinate-basedandcoordinate-freeexpressions.Incontrast,ourapproachwillnotmakeanyreferencetothechoiceofcoordinatesoratetrad(otherthanrequiringittobealignedwiththeprincipalnulldirections).Becauseofthis,ourapproachwillshowcaseoneofGHPtools'greateststrengths{theabilitytocommuteseveralderivativeswithrelativeease. Ourstartingpointisthe(source-free)IRGmetricperturbationgivenbyhab=flalb()(+3)l(amb)[(+)(+3)+(+0)(+3)]+mamb()(+3)g+c.c.: 3{35 ) 78

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3{35 ,theactualperturbedWeylscalarsfollowdirectlyfromEquations 3{27 and 3{28 24; 2n0434=32[1=32(000+022)]o: Theterminsquarebrackets[]inEquation 5{2 isactuallyjusttheoperatorformofthe(generallycomplex)Killingvector(actingon,whichhastypef4;0g)discussedinChapter2.WecanfurthercombinetherelationsinEquations 5{1 and 5{2 toeliminateanyreferencetothepotentials.TherststepistoactonEquation 5{2 with4=32,whichgivesus 2n44=320434Vo:(5{3) Commutingtheeightderivativesontherstterm(usingGHPtools,ofcourse)yieldstheusefulidentity whichwewillhaveoccasiontoexploitagain.CommutingthederivativesinthesecondtermofEquation 5{3 posesnoproblembecauseVcommuteswitheverything.NowitisasimplemattertoidentifytheresultingexpressionwiththetermsinEquations 5{1 and 5{2 toarriveatthefollowing 2inseveralearlierpapers.Stewart[ 21 ]andChrzanowski[ 18 ]havethesefactorscorrect,thelatterwithdierentsignconventions. 79

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65 ]andOrtigoza[ 66 ],whereexplicitcoordinateexpressionswereused,Equations 5{5 and 5{6 bothappeartobetrue.Thisfactappearstobecoincidentalsinceitisunclearhowitfollowsingeneralfromthefundamentalequationsofperturbationtheory.Theremainderoftheidentitieswewillpresenthavenotappearedintheliteratureinthisformandwecanonlyclaimtheyholdfornon-acceleratingtypeDspacetimes. Beforewecontinue,we'lltakealookatthecontentofEquations 5{5 and 5{6 inthecontextoftheKerrspacetime.Ifwewrite0R+2(r)S+2(;)and4=324R2(r)S2(;)andunderstandthetimedependenceofeachtobegivenbyei!t,thenEquation 5{5 tellsus:(1)theresultoffourradialderivativesonR+2isproportionaltoR2and(2)theresultoffourangularderivativesonS2isproportionaltoS+2.ThesameistrueofEquation 5{6 withthe+'sand'sswapped.NotethatEquations 5{1 and 5{2 (andtheirprimesintheORG)sayessentiallythesamethingwiththesubtledierencethattheangularandradialfunctionsarenotobviouslysolutionstothesameperturbation.NosuchambiguityarisesinEquations 5{5 and 5{6 Remarkably,wecanactuallytakethingsastepfurtherandarriveatexpressionsfor0and4independently.Webeginbyacting044=32onEquation 5{5 : Byrecallingthathasthesametypeas4=324(2carriesnoweight),wecansimplytaketheprimeandconjugateofEquation 5{4 ,anduseittocommutethederivativeson 80

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wherewemadeuseofEquation 5{6 inthesecondlineandcommutedeverythingthroughVinthethirdline.ThesecondterminEquation 5{7 becomes 3044=32V0=3V044=320=3V(044=324+3V4)=3V044=324+9VV4; wherewemadeuseofthecomplexconjugateofEquation 5{6 .Combiningtheseresultsgivesus wherewetooktheprimeoftherstequationtogetthesecondone\forfree."ThesearethesecondformoftheTeukolsky-Starobinskyidentities.Wenoteinpassingthatinthecontextoftheseparatedsolutionsof0and4=324,therelationsaboveallowforthedeterminationofthemagnitudeoftheproportionalityconstantrelatingR+2andR2[ 29 ]. Surprisingly,thisisn'ttheendofthestory.RecallthatinatypeDspacetimewealsohaveatourdisposaltheoutgoingradiationgaugewhere 2n40+34=32V0o; 2040; 81

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5{1 and 5{2 .NotethatwhereassatisestheTeukolskyequationfor4=324,0satisestheadjointequation|theTeukolskyequationfor0.FromthecomplexconjugateoftheprecedingequationsandtheirIRGcounterparts,wegetthefollowing: therstformoftheTeukolsky-Starobinskyrelationshipsforpotentials.NotethedierencebetweentheaboveandEquations 5{5 and 5{6 ,particularlythemissingfactorsof4=32andthefactthatVappears.AswithEquations 5{5 and 5{6 ,wecanobtainrelationsforeachpotentialindividuallybyacting044=32onEquation 5{14 andfurtherexploiting(theprimedconjugateof)Equation 5{4 .Theresultisthat Wecansummarizethislastidentitybywriting [044=324044=324+9VV4=32]f4=324;g=0; [44=320444=3204+9VV24=3]f4=320;0g=0: BardeenhasrecentlypointedoutanissueinthestandardtreatmentoftheTeukolsky-Starobinskyidentities[ 67 ].Inparticular,hendsthat,intheSchwarzschildbackground,thereisahithertounnoticedrelativesigndierencebetweentheodd-andeven-parityinthetermproportionalto@t(alternatively!whentimeseparationisperformed),whichbycontinuitypresumablypersistsintheKerrbackground.Bardeenarguesusingstandardtechniquesthatdon'tmakeclearthedierencebetweenthe'sandtheircomplexconjugatesontheright-hand-sidesofEquations 5{5 and 5{6 .However,recallingourdiscussionofparityinChapter3,aglanceattheseequationsrevealsthat 82

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83

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Inthischapterwewilladdresstheissueofthenon-radiatedmultipolesalludedtoinChapter1.TheissueisthatthemetricconstructedfromaHertzpotentialisincompleteinthesensethatitsmultipoledecompositionnecessarilybeginsat`=2becausetheangulardependenceofthepotentialisthatofaspin-weight2angularfunction.Toseethisexplicitly,wefocusourattentionontheIRGmetricperturbation(Equation 3{35 )intheSchwarzschildspacetime,wherethepotential,,canbedecomposedintosomeradialfunction,R(r),withexponentialtimedependence,ei!t,andaspin-weight2sphericalharmonic,2Y`m(;)(seeAppendixD,fordetailsaboutthespin-weightedsphericalharmonics).Ignoringtheradialandtimedependence,weseethatthecomponentsofthemetricperturbationhaveangulardependencegivenby andsimilarlyforhlmandhmm.Becausethespin-weightedsphericalharmonicsareundenedforjsj>`,theaboveexpressionsmakeitclearthatthemetricperturbationinthisgaugehasno`=0;1piecesandthereforeprovidesanincompletedescriptionofthephysicalspacetime.Bycontinuity,thesituationpersistsintheKerrspacetime.Howincompleteisthisdescription? Forthemajorityofthiswork,wehavefocusedourattentionongravitationalradiationintypeDspacetimes.Thisinformationiscontainedintheperturbationofeither0or4,aresultestablishedbyWald[ 16 ].Inparticular,Waldwasabletoshowthatwell-behavedperturbationsof0and4determineeachotherandfurthermorethateitheronecharacterizestheentireperturbationofthespacetimeupto\trivial"perturbationsinmassandangularmomentum.With0and4determinedbytheHertzpotential 84

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5{1 and 5{2 )thisbegsthequestionofwhyweshouldconcernourselveswithsuchtrivialities. Theansweris,inpart,thatthesetrivialperturbationsrepresentthelargestcontributiontotheself-force,asshownbyDetweilerandPoisson[ 17 ].Althoughitisunclearifsuchcontributionspersistinallgaugeinvariantquantitiesofinterest,suchascertaincharacterizationsoftheorbitalmotionoftheparticle[ 68 ],thereisinfactamorecompellingreasontobeconcernedwiththenon-radiatedmultipoles.Inrecentwork,Keidl,FriedmanandWiseman[ 69 ]havelookedattheproblemofcomputingtheself-forceinaradiationgaugeinthecontextofastaticparticleintheSchwarzschildspacetime.Intheircalculation,theyfoundtheperturbationsofmassandangularmomentumarisingintheconstructionofaHertzpotential.Thus,althoughtheHertzpotentialcannotbeusedtodeterminetheseperturbations,itmuststill\know"aboutthemandtheymustbedeterminedbysomeothermeans. Inthischapterwewillpresentageneralprescriptionforcomputingthenon-radiatedmultipoles.Morespecically,wewillconsidertheproblemofcomputingtheshiftsinmassandangularmomentumduetoapointsourceinacircular(geodesic),equatorialorbitaroundablackhole.SpecicallyweareafterexpressionsforManda,theshiftsinmassandangularmomentum,intermsoftheorbitingparticle'smass,,andorbitalparameters.Theideaisquitesimple:matchaninteriorspacetime,(gab;M),toanexteriorspacetime,(g+ab;M+),dieringonlyinmassandangularmomentum,onahypersurface(ofcodimension1),p,containingtheperturbation.Thebasicconditionsforagoodmatchingare(1)thatthemetriciscontinuousacrosspand(2)therstderivativesofthemetricarecontinuousexceptwherethesourceisinnite.TheseconditionsarecompatiblewithIsrael'squitegeneraljunctionconditions[ 70 ]. Beforewecandoanymatching,wemustrstdeterminethegeometryofp.Insphericallysymmetricspacetimes,theobviouschoiceisthesimplest|the(round)2-sphere,aswe'llseebelowinourcalculationinSchwarzschild.FortheKerrspacetime, 85

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Oncewe'veagreedonap,fulllingourrstmatchingconditionrequiresustosimplyequatethecomponentsofthemetric(onp).Inotherwords, [gab]g+abjpgabjp=0;(6{4) wherejpindicatestherestrictiontop.Theonly(slight)complicationthatariseshereisensuringthatthereisenoughfreedominthemetricperturbationtoperformthematching.Thiswillgenerallyrequireperformingagaugetransformationontheinteriorandexteriorspacetimes.Althoughthisintroducessomegaugedependenceintotheproblem,theendresult-Mora-isinfactgaugeinvariant,aswewillseebelow. Imposingthesecondconditionisabitmoreinvolvedbecauseofthepresenceofthesource.Bychoosingagoodmatchingsurface,p,wecaneectively\smearout"theangulardependenceofthesource.If,forexample,pisa2-sphere,wecanusethecompletenessrelationstowritetheangulardeltafunctionaccordingto Similarrelationsholdforcompletesetsoffunctionsondierentclosed2-surfaces.Thesourcenowconsistssolelyofaradialdeltafunction.Tohandlethis,weimposetheperturbedEinsteinequationsas,forexample, lim!0Zr0+r0Eabdr=Zr0+r08Tabdr!;(6{6) whereEabdenotestheperturbedEinsteintensorandTabdenotesthestress-energytensorofthesourceandr0isthelocationofpasseenfrombothsides.ForadeltafunctionsourcedueaparticleofmassinacircularequatorialorbitoftheKerrspacetime, g(rr0)(cos)(t);(6{7) 86

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d;0;0;d d)isthefour-velocityoftheparticleparameterizedbypropertime(),r0istheradiusoftheorbitand=d dt.Forcircularequatorialgeodesics d=(r20+a2)T d=~La~E+aT with where~E=E=and~L=L=aretheenergyandangularmomentumperunitmass,respectively.WecanrecoverthecorrespondingresultfortheSchwarzschildspacetimebysimplytakinga!0.BecausetheintegrationinEquation 6{6 ispurelyradial,itisclearthattheonlytermsthatactuallyparticipateintheintegralontheleftsidearethoseinvolvingtworadialderivatives.ThisiswhereourformoftheperturbedEinsteinequationscomesin.WhileitisgenerallyquitetediousandimpracticaltocomputetheperturbedEinsteintensorforabackgroundmoregeneralthanSchwarzschildandpickoutthetermsinvolvingtwoderivatives,itisaquitetrivialtaskfortheEinsteinequationsinGHPform.Allweneedtodoispickoutthepiecesinvolvingtwoofand0(amindlesstaskwiththeaidofGHPtools),pluginourfavoritetetradandvola!Notethattheseconditionsonthesecondderivativesaregenerallyinvariantwithrespecttochoiceoftetrad.Becauseofthis,wewillwritethejumpconditionsoutinthesymmetrictetrad,whichisobtainablefromtheKinnersleytetradbyasimplespin-boost(Equation 2{16 ) 87

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2~2;0;a 2~2;0;a Withthistetradchoice,theradialjumpconditionsare: wheretheomittedequationsfollowbytakingtheprimeand/orcomplexconjugateofthoselisted(thefactorsofand~2remainunchanged;afeatureofthesymmetrictetrad),anditisunderstoodthatequalityonlyholdsinthesenseofEquation 6{6 .AtaglanceEquations 6{14 { 6{18 mayappearinconsistent,withthesameleft-hand-sidebeingequatedtodierentright-hand-sides.Infact,thecirculargeodesicnatureofuaensuresthatthisisnotthecase. Whatwehavenotyetaddressedisthequestionofwhat,precisely,wemeanbymassandangularmomentum.SuitabledenitionsarisefromtheHamiltoniantreatmentofGeneralRelativityinitiatedbyArnowitt,DeserandMisner[ 71 ].ThegeneralideaisthatbecauseMinkowskispaceprovidesanunambiguousnotionofenergyandangularmomentumthroughtimetranslationsandrotations,respectively,wecanadaptthesenotionstocurvedspacesifthemetricbecomesMinkowskianatspacelikeinnity.ThustheADMdenitionsrequireustorestrictourattentiontoasymptoticallyatspacetimes, 88

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72 ],butforourpurposesitwillsucetosimplyconsidertheasymptoticfalloofthecomponentsofthemetric.Moreprecisely,forasetofcoordinates(x;y;z)inametric,gab,andr=p TheseconditionsaresatisedbytheSchwarzschildandKerrspacetimeswewishtoconsider,butwemustbecarefultochooseanappropriategaugeforthemetricperturbationtoensurethatEquations 6{19 aresatised.Assuminganasymptoticallyatspacetime,theADMmassisdenedby 16limS!1IS(DbabDa)radS;(6{20) wherethesymbolsneedabitofexplanation:wedenotethehypersurfaceofconstanttbytanditsboundarybyS.Thethree-metricontisab.Thenab=ab0ab,with0abbeingthemetricofatspacetime(inthesamecoordinatesasab)and=ab(0)ab.Additionally,Daisthecovariantderivativecompatiblewith0ab,raistheunitnormaltoS,anddSisthesurfaceelementonS.Foranarbitrarymetricperturbation,hab,thisevaluatesto 16limr!1Z20Z02rsinhrrdd;(6{21) 89

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8limS!1IS(KabKab)arbdS;(6{22) wherewehaveintroducedtheextrinsiccurvature,Kab,oftandtherotationalKillingvectora.ForagenericmetricperturbationoftheKerrspacetime,wehave 8limr!1Z20Z0rsinht1 2r2sin@rhtdd:(6{23) Thoughthesedenitionsprovidethemostgeneralprescriptionforcomputingthemassandangularmomentum,forstationaryandaxiallysymmetricspacetimes(thosecontainingbothtimelikeandaxialKillingvectors),theKomarformulae[ 73 ]evaluatedatinnityallowustocomputethevalueoftheperturbations 2Tgab)natbp 2Tgab)nabp whereisspacelikehypersurfacethatextendstoinnity,naistheunitnormaltoit,taandaarethetimelikeandaxialKillingvectorsandp 6{7 { 6{12 ,theKomarformulaegive(fortheKerrspacetime) 90

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wheref=12M=r.AccordingtoBirkho'stheorem,theonlystatic,sphericallysymmetricsolutiontotheEinsteinequationsistheSchwarzschildsolution.Thus,weareassuredfromtheoutsetthatperturbingthemasswillsimplyleadustoanotherSchwarzschildspacetimewithamassM+M.Thenonzerocomponentsofthecorrespondingmetricperturbationaregivenby rhrr=2M rf2;(6{29) whichiseasilyobtainedbylinearizingamassperturbationofEquation 6{28 .Inordertocharacterizemassperturbationsmoregenerally,wewillintroducemorefreedomby 91

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wherewe'vetakenacuefromRegge&Wheeleranddecomposedthegaugevectorintosphericalharmonics.Notetheabsenceofandcomponentsinourgaugevector.Wehavedeliberatelyomittedthesecomponentsonthegroundsthattheyinterferewiththeforminvarianceofthemetric.InordertodeterminethefunctionsP(t;r)andQ(t;r)aswellastheappropriate`andm,we'lllookattheircontributiontothemetricperturbation.Ourgaugetransformation,ab=$gab,hastheform where\sym"meanssymmetricandwe'vedroppedthefunctionaldependenciesforsimplicity.First,we'llfurtherspecializethegaugetransformationbyinsistingonpreservingtheformofEquation 6{28 .Aconsequenceofthisisthathtr=(f@rP(t;r)+f1@tQ(t;r))=0: 92

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6{19 requireQ(r)=O1 r; r2f; WecangiveaninterpretationtobyconsideringEquation 6{33 withM=Q=0,inwhichcaseitisclearthatisjustarescalingofthetimecoordinate. Inordertoperformthematching,weneedtoadaptourgenericperturbationtotheinteriorandexteriorspacetimesandchooseaparticulargaugetoperformthematching.Wewillbeginwiththedescriptionofthemetricontheinterior,gab.HereM=0,sotheperturbationispuregauge.Furthermore,ontheinteriorthereisnoneedtoimposeasymptoticatness.Instead,wewillchooseQ(r)sothattheinteriormetricisregularonthehorizonandleavetheformofP(t)untouched.Asuitablechoiceis r02M!i;(6{37) wherer=r0isthelocationofpandisaconstantinsertedfordimensionalreasonsandi>0.Thevaluesofandwillbedeterminedfromthejumpconditions.Proceedingto 93

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r2M!j;(6{38) where,inanticipationofthematching,we'vechosenthesamedimensionalconstant,,thatweusedinthedescriptionoftheinteriorspacetimeandj2.Withbothmetricsspeciedwenowturnourattentiontomatchingthespacetimes. Becausebothbackgroundmetricsarethesame,itwillsucetomatchtheperturbationsonly.Byimposing[hab]=0,wearriveatthreeuniqueconditions: r0+f0[]Y00+M r20[Q]Y00=0; dr#Y00M[Q]Y00r0M=0; [Q]=0; whereweusedf0=f(r0).OurchoicesforQ+andQ( 6{38 6{37 )ensurethatthethirdconditionissatised.WecansolveEquations 6{39 and 6{40 togetequationsfor[]andM: []="dQ dr#=(i+j) dr#Y00=(i+j)Y00; wherewe'vemadeuseofEquations 6{38 and 6{37 .Nextwewillusethejumpconditionstosolvefor. Applicationofthejumpconditions(Equations 6{14 { 6{18 )issimpliedbythefactthatourmetricperturbationispurespin-0.Thusweonlyneedconsiderthejumpconditionsforthespin-0componentsofthemetricperturbation(hll,hln,hnnandhmm).ForsimplicitywewillworkwithEquation 6{15 ,thoughitcanbedirectlyveriedthatthe 94

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6{7 ,wehaveforthetetradcomponentsoftherelevantobjects: rr2M r02M!iY00 rr02M r2M!jY00 16~2 r20f0(rr0)(cos)(t); withalltheMdependencereplacedaccordingtoEquation 6{43 .ImposingEquation 6{6 thenleadsto"@hmm r20f0(cos)(t); r20f01X`=0`Xm=`Y`m(=2;t)Y`m(;);(6{47) wherewe'vedecomposedtheangulardeltafunctionsaccordingtoEquation 6{5 .WecaneliminatethesumontherightsideofEquation 6{48 bymultiplyingbothsidesbyY00(;),integratingoverthesphereandexploitingtheorthogonalityofthesphericalharmonics.Theresultisthat i+j;(6{48) wherewe'veusedY00(;)=Y00(;)=(4)1=2.Finally,wehave []=(4)1=2~E r02M Theseequationscompleteourconstructionofthematchedspacetime.Notethattheaboveonlyrestrictsthedierencebetweenontheinteriorandexterior.IfwerecallEquation 6{41 ,weseethatthesameisgenerallytrueofQ(r)aswellifwedroptherequirementsofregularityintheinteriorandasymptoticatnessintheexterior. 95

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6{23 ,itisclearthatourmetricperturbationwillacquireanhtcomponent.RealizingthisasaperturbationtowardstheKerrspacetime,wewillwriteitas r;(6{51) whichisjustthelinearizationabouta=J=Mofthecorrespondingcomponentofthe(background)Kerrmetric.Becauseofthis,therewillbenonzerocontributionstohlm,hnmandtheircomplexconjugateswhichmeansthatwemustnowtakeparityintoconsideration.Tothatendwewillintroduceagaugevectorwithcomponents (6{52) (6{53) 2(+0)+S(t;r)i sin wherewe'vedenedY+`m=1 2(+0)Y`m=1 2(1Y`m+1Y`m)andY`m=i 6{51 has.Forananswer,welooktothesourceterms.AquickcomputationrevealsthatTlm=Tnm=Tlm=Tnm,fromwhichitfollows 96

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wherethe\-"onhab,referringtotheinteriorspacetime,istobedistinguishedfromthe\-"onY`m,whichreferstoacombinationofspin-weight1sphericalharmonics.Inthissituation,wemustmodifyourrequirementofforminvariance(whichisalreadybrokenbytheperturbation)totherequirementthatonlyhtremainsnonzero,whichpreservestheminimumfreedomtomatchtotheexterior.Firstwesetht=0,whichimpliesY`m=0or1Y`m=1Y`m.Thiscanonlyholdifm=0,whichmeanstheperturbationisaxiallysymmetric.Movingon,weturnourattentiontoeliminatingh.ThisentailscosY+`0sin@Y+`0=0; 21Y`0+1Y`0sin: 97

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4sin:(6{57) Finally,itiseasytosethr@rS2r1S=0; r; withallothercomponentsvanishing. Continuityofthemetricperturbation([hab]=0)requires wherewe'veusedtheequalityof1Y10toexpandY+10.Asbefore,theradialjumpconditionswilldetermine.Inthiscasewe'llusetheodd-parity(imaginary)partofEquation 6{17 .Therelevanttetradcomponentsaregivenby: 16~2 r30f1=20(rr0)(cos)(t): 98

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f1=201Y10=i16~L r30f1=20(cos)(t)=i16~L r30f1=201Xl=0`Xm=`1Y`m(=2;t)1Y`m(;): Equation 6{57 weobtain r30p ItthenfollowsfromEquation 6{60 that whichispreciselywhatistobeexpected|alloftheangularmomentuminthe(otherwisenon-rotating)spacetimecomesfromtheangularmomentumoftheparticle.OnceagainwecanverifydirectlyfromEquation 6{23 thatwehavecorrectlyidentiedtheangularmomentumofthespacetime. Incontrasttothesituationwithmassandangularmomentumperturbations,whereitwaseasytowritedownthegeneralformoftheperturbations,wehavenogeneralformforthemetricperturbation.Withoutpriorknowledgeoftheperturbation,wemustresorttosolvingtheEinsteinequationstodeterminetheperturbation.ThishasbeencarriedoutbybothZerilli[ 7 ]andDetweiler&Poisson[ 17 ].Theresultisametricperturbationthatcan 99

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Mr20sinsin(t)(rr0);hrr=2~E Mr20 Mr20coscos(t)(rr0);hr=~E Mr20sinsin(t)(rr0): 7 ]thatthegaugetransformationleadingtothisdescriptioncanbeinterpretedasatransformationfromanon-inertialframetetheredtothecentralblackholetothecenterofmassreferenceframe. Theprimaryissuewithtreatingthenon-radiatedmultipolesinthecontextofmatchedspacetimesisthechoiceofthematchingsurface,p.Mostofourdiscussionwillbefocusedonthisissue. 16 ],describedearlier,assuringusthatinnitesimalmassperturbationsoftheKerrsolutionleadtootherKerrsolutions(withinnitesimallydierentmasses,ofcourse)becausesuchperturbationsdonotcontributetheperturbationsof0or4(whichwewillverifyshortly).Thuswehavethe 100

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r2+a2cos2; r2+a2cos2; r2+a2cos2: BecausethecalculationsintheKerrspacetimearesignicantlymorecomplicated,wewilltakeashortcuttodeterminingtheangulardependenceoftheperturbationbylookingatthetetradcomponentsofthemetricperturbation,aresultwhichwewillinanycaseuseshortly.Inthesymmetrictetrad(Equations 6{13 )wehave withallothercomponentsvanishing.Becausebothhllandhnnarespin-weight0,theyhaveanaturaldecompositioninto`=0,m=0scalar(ordinary)sphericalharmonics.Furthermore,utilizingEquation 3{27 weseethat andsimilarlyfor4.Therefore,accordingtoWald'stheorem,weareensuredthatEquations 6{70 and 6{71 areaperturbationtowardsanotherKerrsolution. Withtheangulardependencedetermined,weareledtoconsideragaugevectoroftheform 101

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whilestationarityagainnecessitates Nextweturnourattentiontothematchingproblem. Inordertoclarifytheissuesinvolvedinthematchingproblem,we'lltakealookatthematchingconditionsthemselves.Supposewe'vechosensomep,buthaveyettospecifyitexplicitly.Thatis,wehavenotyetwritten(orimposed)r=something.Thefullsetofmatchingconditionsnowtaketheform d#2arM=0; dr#rM d#(a2cos2+r2(r2+a2)+2amr)=0; where=r22Mr+a2and~2=r2+a2cos2asbeforeandwehaveimposedtheconditioninEquation 6{79 intheothers.NotethatthisreducestotheSchwarzschildresultinEquations 6{39 { 6{41 bytakinga!0andsettingr=r0.Thissetofequations 102

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dr#; []=(r2+a2)2 dr#; []= (r2+a2)2[]; d#=a2sin2 whichisagaineasilyseentoreducetotheSchwarzschildresultintheappropriatelimit.Fromtheseequationswecanseeclearlytheissuesinvolvedinchoosingamatchingsurface.First,becausetheleftsidesofEquations 6{81 { 6{84 areallconstant,thismustbereectedintherightsidesaswell,whichcurrentlyexhibitdependenceonbothrand.Presumably,somechoiceofr=r()willenforcethis,thoughitiscurrentlyunclearwhatthatchoicemightbe.Notethatbecauseofthis,r=constantsurfacesdonotappeartobegoodformatching. WhatwehaveencounteredappearstobeaninstanceofalongstandingproblemwithmatchingtheKerrsolutiontoasource[ 74 75 ].Namely,thereisnoknownmattersolutionthatcorrectlyreproducesthemultipolestructureofthefullKerrgeometry.Inourproblem,we'retryingtoforcetheissuebyspecifyingboththemetricandthesource.Ontheotherhand,becausewe'renotmatchingtheentiresource,whichincludesquadrupoleandhighermoments,butonlythenon-radiatedmultipolesthatmerelytakeusfromoneKerrsolutiontothenext,itisnotclearthatthematching(inthisinstance)shouldfail.Thoughweareunabletoperformthematchinghere,wemaintainthatnothingforbidsit. Mostauthorsfacedwiththisissueturntothe\slowrotation"approximationandkeeponlytermslinearina.InthisapproximationtheKerrmetriccanbeviewedastherstorderperturbationoftheSchwarzschildsolutiontotheKerrsolution.Thatis,thebackgroundisgivenbySchwarzschildplusatermidenticaltothatinEquation 6{59 .It 103

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6{81 { 6{84 andallowforamatchingonr=constantsurfaces(whichareround2-spheresinthiscase).BecausethisapproachfailstoshednewlightonthesituationinthefullKerrspacetime,wewillnotfollowithere.Instead,wewillfocusonEquations 6{66 6{69 ,whichweknowtobecorrect. Let'sreviewthesituation.WehaveestablishedthatthemetricperturbationinEquations 6{66 { 6{69 isaperturbationtowardsanotherKerrsolutionwithdieringmass.Furthermore,wepreviouslyestablishedthatM=~E(Equation 6{26 ).Theproblemisthatwearecurrentlyunabletoperformthematching.Inpractice,therelevantportionofthespacetimeistheexteriorwheregravitationalradiationandthenon-radiatedmultipolesareobservedfarawayfromthesource.Becauseofthis,wecontendthatconsiderationsfromtheKomarformulaandWald'stheoremtogetherprovidethecorrectperturbationintheexteriorspacetime,independentlyofanymatchingconsiderations.ThusourresultislikelyusefulintheEMRIproblemeventhoughwelackthemetricperturbationeverywhereinthespacetime.Moreover,theperturbationisstillsimpletointerpretandasymptoticallyat,soitisamenabletosomeanalysis. Thisbeingthecase,weremarkthatmassperturbationsoftheKerrbackgroundremainconnedtothes=0sectoroftheperturbation.Itislikelythatthisistrueingeneral(atleastintypeD),butageneralproofofthisremainselusive.Furthermore,contrarytowhatonemightexpectintheKerrspacetime,themassperturbationdoesnotmixsphericalharmonic`-modes,butispurely`=0.Wenowturnourattentiontoangularmomentumperturbations. 104

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(r2+a2cos2)2; Thecorrespondingtetradcomponents(inthesymmetrictetrad)aregivenby ~2; ~2; wherewehaveomittedthecomplexconjugates.Thoughitisnotimmediatelyobvious,thisperturbationmakesnocontributionto0or4,ensuringthatthisisavalidangularmomentumperturbation. Inlightofrelativelystraightforwardresultsformassperturbations,thenontrivialformofEquations 6{90 { 6{94 comesasasurprise.Unlikemassperturbations,angularmomentumperturbationsarenotconnedtoasinglessector,whereasonemightexpectthemtobeexclusivelys=1,asintuitionfromworkingintheSchwarzschildbackgroundwouldleadustobelieve.Notethatalthoughtheperturbationappearsinthes=2sectorofthemetric,thevanishingofthes=2componentsoftheWeylcurvaturekeep 105

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Anothersurprisingfeatureisthefactthattheperturbationiscomplexandthusexhibitsbothtypesof\parity".Althoughthestaticnatureoftheperturbationguaranteesspin-weightedsphericalharmonicangulardependence,wemustbecarefulnottospeakofparityintheSchwarzschildsense,butrathertherealandimaginarypartsoftheperturbation.Inanycasetheimplicationsofthisfactarepresentlyunclearandremaintobedeterminedinfuturework. Firsto,onemayspeculatethatourrequirementofforminvarianceisperhapstoostricttoallowforapropermatching.Thisdoesnotappeartobethecase.AresultofCarter[ 76 ]impliesthat,duetostationarityandaxialsymmetry,theKerrmetric(inBoyer-Lindquistcoordinates)haspreciselytheminimumnumberofnonzerocomponents.Havingestablishedindependentlythatthemassandangularmomentumperturbationspreservethesespropertiesofthebackground,Carter'sresultsuggeststhattheproblemlieselsewhere. Thisleadsustoconsiderwhethertheintroductionofaninnitesimallythinshellofmatter(whichiseectivelywhatpis),necessarilyintroducesnon-Kerrperturbations.Ashell(ofsomecurrentlyunspeciedshape)wouldpresumablybeadierentiallyrotatingobject.Itisunclearwhetherthisdisruptsthestationarityoraxialsymmetryoftheexteriorspacetimebytheintroductionofperturbationsthatwehaveneglected 106

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16 ]actuallyspeciestwoothertypesofperturbationsthat0and4cannotaccountfor:perturbationstowardstheacceleratingC-metricsandperturbationstowardtheNUTsolution.IntheworkofKeidl,et.al.[ 69 ],wheretheyconcernedthemselveswithastaticparticleintheSchwarzschildgeometry,itwasfoundthatthespacetimeontheinteriordiersfromthatontheexteriorbyaperturbationtowardstheC-metrics.ThismakesphysicalsensebecauseastaticparticleisnotonageodesicoftheSchwarzschildspacetimeandthusrequiresaccelerationtokeepitinplace.Thoughwehavenoobviousphysicalreasontoexpecttheseperturbationsforcircular,equatorialorbitsoftheKerrgeometryandevidencefromtheSchwarzschildcalculationsuggeststheyshouldnotcontribute,wehavenotyetprovenaresulteitherway. Finally,onequestionthatwehaveoverlookedentirelyisthequestionofthestabilityofathinshell.IntheSchwarzschildbackground,thisproblemhasbeensolvedbyBrady,LoukoandPoisson[ 77 ],whoshowedthatathinshellisstableandsatisesthedominantenergyconditionalmostallthewayuptothelocationofthecircularphotonorbit(locatedatr=3M).TherearenosuchresultstoreportonfortheKerrspacetime.TheclosestthingtoastepinthisdirectionistheworkofMusgraveandLake[ 78 ],whoconsiderthematchingoftwoKerrspacetimeswithdierentvaluesofmassandangularmomentum.Unfortunately,theseauthorswereforcedtoresorttotheslowrotationapproximationdiscussedearlier.Strictlyspeaking,withoutknowledgeoftheexistenceofastableshellofmattersucientlyclosetotheblackhole,wearelefttoquestionthevalidityofourprocedure.Thisisaproblemweleaveforfuturework. 107

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Firstandforemost,bytakingadvantageoftheGHPformalism,ourframeworkemphasizesandexploitsthosefeaturescommontoallblackholespacetimes|theirnullstructureasmanifestedintheirPetrovtype|which,sinceTeukolsky'sderivationoftheequationthatbearshisname,hasbeentheonlyprovenroadtoprogressinthisdicultsubject.SuchfeatureshavemadeanappearancethroughthesimplicationinthebackgroundGHPequationsdiscussedinChapter1.Thesehaveleadtousefulsimplicationsthroughout.Besidesthesefeatures,thebuilt-inconceptsofspin-andboost-weighthaveallowedussomeintuitiveinsightintothenatureofthefundamentalquantities,withoutresortingtoseparationofvariables. ThecreationofGHPtoolsistheonlyreasonanyofthisworkwasfeasibleintherstplace.Coordinate-independencecomesatthepriceofhavingtoperformmanynontrivialsymboliccomputations.GHPtoolshasnotonlyallowedustoperformsuchcomputations,butalsotopresenttheminafullysimpliedway,bringingsomeclarityeventopreviouslyknownresults.ThisisperhapsmostevidentinourtreatmentoftheTeukolsky-Starobinskyidentities,wheretheuseofGHPtoolsmaskedallofthehorrendouscomputationalcomplexityinvolvedintheirderivation,byprovidingsimpleandconciseresultsintheend. Furthermore,thecoordinate-freenatureofourframeworkhasfurtherallowedustoworkingreatgenerality.Thiswasseeninourtreatmentofthecommutingoperatorsof 108

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Finally,ourtreatmentofthenon-radiatedmultipolesdemonstratesthepowerofourframeworkwhencombinedwithexistingtechniques.OurresultsintheKerrspacetimerepresenttherstattemptattreatingthispartoftheperturbation.Thoughwewereunabletoobtainthedescriptionintermsofamatchedspacetime,weneverthelessprovidedaperturbationsuitableforuseinmetricreconstruction. Perhapsmostpressingisthegeneralizationofourresultforthenon-radiatedmultipolesintheKerrspacetimetoencompassmoregeneralorbits.Inparticular,orbitsnotlyingintheequatorialplaneareofparticularinterest.Suchorbitsnecessarilycontaino-axisangularmomentum,whichinturnarewidelythoughttoberelatedtoCarter'sconstant(associatedwiththeKillingtensor).ForsuchorbitstheKomarformulaefailtocompletelycharacterizetheseo-axisangularmomentumcomponents,soitisclearthatwemustlookelsewhereforasolution.OnepotentialavenueforprogressistheEinsteinequationsthemselves.Aswenotedinthepreviouschapter,massandangularmomentumperturbationsarebothstationaryperturbationswithangulardependencecharacterizedbythespin-weightedsphericalharmonics.ThesimplicationsthisbringsforworkingwiththeEinsteinequationsisimmenseandmayprovetomaketheproblemtractable,withoutrecoursetopurelynumericalmethods.Inanycase,itseemsclearthatourframework, 109

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AnotheravenueworthpursuingisthecommutingoperatorassociatedwiththeKillingtensorduetoBeyer[ 44 ](cf.Chapter1).RecallthatBeyer'soperatorcommuteswiththescalarwaveequationinKerr.ItisverytemptingtothinkthatsuchanoperatorwouldexistfortheTeukolskyequationaswell.TheGHPformalism,andGHPtools(ofcourse),providetheidealenvironmentinwhichtostudysuchquestions.Furthermore,inthecontextofworkperformedbyJeryes[ 79 ]concerningtheimplicationsoftheexistenceoftheKillingspinor(whichincludesadiscussionoftheTeukolsky-Starobinksyidentities),itisnaturaltothinkthatsuchanoperatormayinfactshedsomenewlightontheTeukolsky-StarobinskyidentitiesintheformpresentedinChapter5.Additionally,theexistenceofageneralizationofBeyer'soperatorcarrieswithitthepossibilityofnewdecompositionoffunctionsintheKerrspacetime|justastheexistenceoftheKillingvectors@ @tand@ @leadtoseparationintandaccordingtoei!tandeim(respectively),theeigenfunctionsofageneralizedBeyeroperatormayprovideanewseparationofvariablesintheKerrspacetime.Thisiscertainlyapossibilityworthpursuing. Finally,bothGHPtoolsandourformoftheperturbedEinsteinequationsareentirelygeneralandreadyforusebyresearchersinterestedinmoregeneral(orevenmorespecialized)backgroundsthanPetrovtypeD.Inparticular,theclassoftypeIIspacetimesseemsalikelycandidateforfurtheranalysis,especiallywiththeaidoftheintegrationtechniqueofHeld.Wehaveonlybeguntoscratchthesurfaceofthewidevarietyofproblemsthesetoolscanhelpsolve. 110

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Inthisappendix,wegivetheGHPcommutators,eldequationsandBianchiidentities,aswellasthederivativesofthetetradvectors.Thefullsetofequationsisobtainedbyapplyingtothoselistedprime,complexconjugationorboth.Whenactingonaquantityoftypefp;qg,thecommutatorsare: [;0]=(0)+(0)0p(00+2+11)q(00+2+11); [;]=+000p(00+1)q(00+01); [;0]=(00)+()0+p(00+211)q(00+211): TheGHPequationsare: TheBianchiidentitiesaregivenby: 111

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Finally,thederivativesofthetetradvectorsaregivenby alb=la(mb+mb)na(mb+mb)+ma(mb+mb)+ma(mb+mb) (A{14) amb=la(0lb+nb)na(0lb+nb)+ma(0lb+nb)+ma(0lb+nb) (A{15) 112

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InthisappendixwewritethecomponentsoftheperturbedEinsteintensorforanarbitraryalgebraicallyspecial(PetrovtypeII)background.WehaveassumedthePNDisalignedwithlaandmadeuseoftheGoldberg-Sachstheorem.NotethattheequationsforElm,EnmandEmmarecomplex,soElm=Elmandsoon: 113

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2f0(00)+0(00)+(20)0+(020)0+200ghll+1 2f()+()ghnn+1 2f(0+0+)(0)(0+30+30)+2(+0)+(2)0+(020)0(+)(0+0)22ghln+1 2f(020)(00)+(0+0+0)0(00)(20)0(2)0+0(0)ghlm+1 2f(020)(0)+(0+0+0)0(00)(2)0(2)0+0(0)ghlm+1 2f(2)(0)+(0+)(+)2(00)2ghnm+1 2f(2)()+(0+)(+)2(0)2ghnm+1 2f(0)(00)+(0)0ghmm+1 2f()(0)+(0)0ghmm+1 2f(0+0)(+0)+(000+)(2+2)+(020)+(2)0+(3020)+0(32)20+20+20()ghmm; 2f(00)(0)+(20)0()0+(0+0)+0(00+)+3+0ghll+1 2f(+)(+0)(3+0)20ghln+1 2f(0+0)(2)+(0+2020)40+22+(0+)(20)(0+02)0(40)ghlm+1 2f(2)0(+24)20(0)ghlm+1 2f(2)+2()ghnm

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2f()(00+)+2ghmm+1 2f(+)(+0)+20(2)(0)+2ghmm; 2f(00)0+00+00ghll+1 2f(+)(0)(020+)+0()ghnn+1 2f((00+0)(0+0)(030+)0+(+0)020320ghln+f0(020)0(02)+1 24ghlm+1 2f(0(020)+0(2+20)+0(040+2)+20(00)+200ghlm+1 2f0(020)+0(2+2)2(0)ghnm+1 2f(00)(+2)+(020)+20()222+(03)+0(20+40)(02)ghnm+1 2f(00)000ghmm+1 2f(00)(+0)+2000(2+2)+0(0)0ghmm+1 2f(0+00)(00+)+2(020)(00)0+200+(0)0+003ghmm;

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2f0(000)+200+0(0)0(0)+200ghll+1 2f()+2ghnn+1 2f(0+00)(+)0(+)+(0+00)2+(0)(0)+0(20)0(2+0)2(0)+200+0ghln+1 2f(020)(02)+(0+2020)+2(0)0020020()3ghlm+1 2f(020)(2)+(0+202)+2(00)00020020()3ghlm+1 2f(2)(020)+0(22)2+40ghnm+1 2f(2)(20)+0(22)2+40ghnm+1 2f(0)0(00)(2)0ghmm+1 2f()0(0)(2)0ghmm+1 2f20(00)()0(00+0)0(+)(020)0+(0+2)0(0)+(+)0()22ghmm: 116

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WeprovidedetailsoftheintegrationthatleadtoEquation 4{17 and 4{25 .Asitturnsout,thetypeIIcalculationisactuallymuchsimplerthanthethetypeDcalculationbecauseitusesatetradinwhich=0=0.ThereforewewillworkoutthetypeDcalculationindetailandthetypeIIresultmostlyfollowsbysettingcertainquantitiestozero,asindicatedbelow. Wewillneedsomeresults(andtheircomplexconjugates)fromtheintegrationofthetypeDbackground: ~0= 22(~+1 2)2++2+1 21 2+1 21 (~+); 2=3: Asnotedinthetext,6=0leadstotheacceleratingC-metrics,whichweincludeforfullgenerality.HenceforththecorrespondingquantitiesintypeIIspacetimescanbeobtainedbysetting==)0and)2 117

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22(2+); 2=23; theequationfor~0notfollowingfromthelimitingprocessmentionedabove.Notethatthequantity~0isneverusedinanyoftheintegrationsweperforminthetypeIIbackgroundspacetime.Wewillalsoneedthedenitionsofthenewoperators: ~0=0~~0+(p )+1 2(p2 ~= ; ~0=0 wherepandqlabeltheGHPtypeofthequantitybeingactedon.Additionally,inSections 4.2 and 4.4 wemakeuseofthecommutator [~;~0]=00 1 221 +~ 221 +~0 o;(C{13) whichisvalidintypeDand(with=0)typeIIspacetimes. Wenowbeginwith whichintegratestriviallytogive Withthisinformationinhand,wecannowintegratetheequationgoverningm: (+)m+(+0)l=0:(C{16) 118

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C{11 withp=1leadsto 1 (m)+0l+~l l=0;(C{17) which,aftersubstitutingEquation C{3 ,thecomplexconjugateofEquation C{4 andEquation C{15 alongwithsomerearranging,yields Integrationthengivesus l1 andthesolutionformthenfollowsfromcomplexconjugation +l(~0+)l:(C{20) Finally,weareinapositiontodealwithn,bywriting intermsofHeld'soperators(Equations C{1 C{3 and C{4 )as l1 22 SubstitutingEquations C{3 C{4 C{5 C{15 C{19 and C{20 ,rearrangingtermsandlettingthedustsettleleadsto 2l2+1 2l2l1 +l(2+2)[2(~+)+2(~0+)]l[(~+)+(~0+)]l+2m1 +2m1 119

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2l+1 2l+l+1 2l1 2+h 21 ~0l[(~+)+(~0+)]l+m m1 2m1 m1 andourtaskiscomplete. 120

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Inthisappendix,wepresentthebasicsofthetheoryofspin-weightedsphericalharmonics[ 15 80 ].ThesefunctionshaveanaturalplaceintheGHPformalismandprovideasimplealternativetothemorecomplicatedtensorsphericalharmonics.Thediscussioninthissectiontakesplaceontheround2-sphere.Inthatcase,theactionofonsomequantity,,ofspin-weightsisgivenby @+icsc@ @#(sin)s;(D{1) andtheactionof0is @icsc@ @#(sin)s:(D{2) Thespin-weightedsphericalharmonics,sY`m(;),arethendenedintermsoftheordinarysphericalharmonicsby (`+s)!sY`m(;)0s`;q (`s)!(1)s(0)sY`m(;)`s0;(D{3) butareundenedforjsj>`.ThebasicpropertiesofthesY`mareeasilyseentobe Foreachvalueofs,thespin-weightedsphericalharmonicsarecomplete: 121

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122

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Larrywasbornin1978,inElPaso,Texas.Heistheeldestchildof(theelder)LarryPriceandPamelaVilla.Atlastcount,hehasapproximately6siblings.Fromtheagesofaboutvetotwelve,heattendedafunnysortofschoolwherethestudentswereallforcedtodressthesameandgatheronFridaystolistentoamaninadressreadfromabigbook.Hewastreatedwellthere,buthisentryintotheTexaspublicschoolsysteminthefthgradeprovedtobeagoodmove.Inmiddleschool,Larryrealizedheunderstoodalgebramuchbetterthanhisteacher(whohappenedtoalsobetheschool'sbasketballcoach),apointthathemadeclearinclassateveryopportunity.Itgoeswithoutsayingthathisinitialdesiretopubliclyhumiliatejockssubsequentlygrewintoamuchdeeperinterestinmathematicsandphysics.Theseinterestswerefurtheredinhighschool,whereLarryexploredotherareasaswell.Amongtheseisthetheater.FewpeopleareawarethatLarryhasperformedinleadingrolesinseveralmusicals,aswellasanoperetta.Upongraduatinghighschoolin1997,LarrydecidedthatitwouldbebesttogetasfarawayfromElPasoashecould.Tothisend,heattendedasmallliberalartsschoolnamedReedCollegeinPortland,Oregon,wherehespentsomeofthebestyearsofhislife.ReedprovidedavaluableopportunityforLarrytofurtherpursuethesciencesandreadsomereallygreatbooksatthesametime.Italsogavehimtheopportunitytointeractwithmanyinterestingpeoplefromwidelydierentbackgrounds.ItwastherethatLarrycameincontactwithNickWheeler,atrulyuniqueindividualwhoremainsatrustedmentor.Alas,allgoodthingsmustcometoanend,andsoLarrygraduatedfromReedwithaB.A.inphysicsin2001.Withhispathuncertainatthetime,LarrydecidedtostayinPortlandforthefollowingyear.ThereLarrytriedhishandasacomputationalchemistforSchrodinger,Inc.Thepeopletherewerefantasticandthepaychecksweren'tbad,butheneedmore 179

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180