Developments in the Perturbation Theory of Algebraically Special Spacetimes

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Developments in the Perturbation Theory of Algebraically Special Spacetimes
Price, Lawrence Ray, Jr
Place of Publication:
[Gainesville, Fla.]
University of Florida
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Physical Description:
1 online resource (180 p.)

Thesis/Dissertation Information

Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Committee Chair:
Whiting, Bernard F.
Committee Members:
Klauder, John R.
Muller, Guido
Detweiler, Steven L.
Groisser, David J.
Graduation Date:


Subjects / Keywords:
Angular momentum ( jstor )
Distance functions ( jstor )
Einstein equations ( jstor )
Killing ( jstor )
Mass ( jstor )
Mathematical vectors ( jstor )
Scalars ( jstor )
Sine function ( jstor )
Spacetime ( jstor )
Tensors ( jstor )
Physics -- Dissertations, Academic -- UF
black, newman, spin
bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Physics thesis, Ph.D.


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. ( en )
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In the series University of Florida Digital Collections.
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Thesis (Ph.D.)--University of Florida, 2007.
Adviser: Whiting, Bernard F.
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by Lawrence Ray Price.

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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,


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 =


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,


a*cos(theta))))^(1/2)/B(t, r, theta, phi)]), Idn = vector([1/2*B(t, r,


phi)*2^(1/2)*((r ^2-2*M*r+a^2)/((r+I*a tcstea)*rIacsoht))

^(1/2), -1/2*B(t, r, theta,



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

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

[26] S. C'I .1..1 .s-ekhar, Proc. Roy. Soc. Lond. A392, 1 (1984).

[27] M. Campanelli, W. K~rivan, and C. O. Lousto, Phys. Rev. D58, 024016 (1998).

[28] C. O. Lousto, Class.Quant.Gray. 22, S569 (2005).

[29] S. C'I .1..11~ I-ekhar, The M~athematical Theory of Black Holes, Oxford University Press,

New York, 1983.

[30] A. I. Janis and E. T. N. x.--n! ll. J. Math. Phys. 6, 902 (1965).

[31] A. Petrov, Sci. Nat. State University of K~azan 114, 55 (1954).

[32] J. Goldberg and R. Sachs, Acta Phys. Polonica, Supp. 13 22, 13 (1962).

[33] W. K~innersley, J. Math. Phys. 10, 1195 (1969).

[34] M. Walker and R. Penrose, Commun. Math. Phys. 18, 265 (1970).

[35] L. P. Hughston, R. Penrose, P. Sommers, and M. Walker, Commun. Math. Phys. 27,

303 (1972).

[36] P. Sommers, Killing tensors and type {2, } spacetimes, Ph.D. dissertation, University

of Texas at Austin, 1973.

[37] M. Demaidiski and M. Francaviglia, J. Phys. A 14, 173 (1981).

[38] R. Floyd, The LI;,n of Kerr fields, Ph.D. dissertation, University of London,

[39] R. Penrose and W. Rindler, Sp~inors and Sp~acetime. Volume 2, Cambridge University

Press, New York, 1986.

[40] B. Carter, Commun. Math. Phys. 10, 280 (1968).

[41] B. Carter, Phys. Rev. 174, 1559 (1968).

[42] C. D. Collinosn and P. N. Smith, Comm. Math. Phys. 56, 277 (1977).

[43] B. P. Jeffryes, Class. Quant. Gray. 4, L17 (1987).

[44] H. R. Beyer and I. Craciun, gr-qc/0607070 (2006).

[45] A. Held, Comm. Math. Phys. 37, 311 (1974).

[46] J. M. Stewart and M. Walker, Proc. Roy. Soc. Lond. A341, 49 (1974).

[47] R. Sachs, in R.~ Il.:i. .I;, Group~s and T 'r.~ J.~ -i;, 1964.

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 .

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



end if;


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,


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 +



+(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


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 +



+(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


result:=result w + times((conjugate(delta) +

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


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


return map(ETHP, temp);

elif op(0,f) = 'conjugate' then

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

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


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
1m=~ ;~ 1 c" o
m mo_ o l o P1
p p n~ i~~ I7:0 il(y&

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:

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


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


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

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).

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

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

vector([-1/2*Itatsin(theta)*2^(1/2)/(r-Itacstea) 0,


1/2*I*(r^2+a^2)*sin(theta)*2^(1/2)/(r-I* acstea)) ndn=


1/2*2^ (1/2)* ((r+I*atcos(theta))*(r-Itatcotha)/r22Mr^)^(

/2), 0,


a*cos(theta))))^(1/2)]), Idn=



1/2), 0,



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,


^(1/2)]), nup = vector([1/2*(r^2+a^2)*2^(1/2)/(B(t, r, theta,


-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,


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

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


~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

BI3c := proc()


end proc;

BI3pc:= proc()

return(GHPconj (GHPprime (BI3())));

end proc;

BI4 := proc()




end proc;

BI4p := proc ()

return (GHPprime (BI4 ())) ;

end proc;

BI4c := proc ()

return(GHPconj (BI4()));

end proc;

BI4pc:= proc()

return (GHPconj (GHPprime (BI4 ())));

end proc;

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


elif has (lo, conjugate (eta2)) then


elif (has (lo, xil) and not (has (lo, conjugate (xii)))) then


elif has (lo, conjugate (xii)) then


elif (has (lo, xi2) and not (has (lo, conjugate (xi2)))) then


elif has (lo, conjugate (xi2)) then


elif has(lo,h) then



elif has(lo,hl) then



elif has(lo,hn) then



end if ;


end proc;

GHPcomm := proc (whichcom, solvef or, whichvar)

local a,b;

(a,b) :=getpq(whichvar);


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


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

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

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)),


tetdnK := \{mdn =

vector([1/2*Itatsin(theta)*2^(1/2)/(r+I*acstea) 0,


-1/2*I*(r^2+a^2)*sin(theta)*2^(1/2)/(r+I~acstea)) ndn=


1/2, 0,


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,


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



elif has(10,conjugate(rho)) then



elif (has(lo,rhol) and not(has(10,conjugate(rhol)))) then



elif has(10,conjugate(rhol)) then



elif (has(lo,kappa) and not(has(10,conjugate(kappa)))) then



elif has(10,conjugate(kappa)) then



elif (has(lo,kappal) and not(has(10,conjugate(kappal)))) then



elif has(10,conjugate(kappal)) then



elif (has(lo,tau) and not(has(10,conjugate(tau)))) then



elif has(lo,conjugate(tau)) then

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

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

Thus our spacetime metric is

9ab = gb a Lb,



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


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


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:


yABd A XB d2 Sin2 d2


end proc;

GHP2p := proc ()


end proc;

GHP2c := proc ()

return(DGHP (GHPconj (GE2)));

end proc;

GHP2pc:= proc()

return(DGHP (GHPconj (GHPprime (GE2))));

end proc;

GHP3 := proc ()


end proc;

GHP3p := proc ()


end proc;

GHP3c := proc()

return(DGHP (GHPconj (GE3)));

end proc;


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

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

(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

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
+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

> 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:

*(times((delta p*beta +


elif(op(0,w)='eth') then


result:=result w + times((delta p*beta +


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 +


*(times((conjugate(delta) +p*betal -


elif(op(0,w)='ethp') then


result:=result w + times((conjugate(delta) +

p*betal q*conjugate(beta)),op(1,w));


result :=result;

end if;

end do;


end proc;




(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,

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),

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

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 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).

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+



(p+1)*beta +(q+1)*conjugate(betal)),times((DD -



elif (op(0,w)='eth' and op(0,0p(1,w))='th') then


result:=result w + times((delta (p+1)*beta

+ (q+1)*conjugate(betal)),times((DD p*epsilon -


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 +


(op(1,0p(1,0p(nops(w),w))))) *(times((delta -

(p-1)*beta +(q-1)*conjugate(betal)),times((Delta +



elif (op(0,w)='eth' and op(0,0p(1,w))='thp') then


result:=result w + times((delta (p-1)*beta

+ (q-1)*conjugate(betal)),times((Delta + p*epsiloni



elif (dl='thp') and (d2='ethp') then


elif (dl='ethp') and (d2='thp') then


elif (dl='thp') and (d2='eth') then


elif (dl='eth') and (d2='thp') then


elif (dl='eth') and (d2='ethp') then


elif (dl='ethp') and (d2='eth') then


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

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 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:


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)

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-

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);


error "routine not built to handle that

function: X1", op(0,f);

end if;


error "routine not built to handle that type: %1",


end if;

end proc;

DGHP := proc(expr)

local 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)

comparatively simple derivation of the same result. This is the approach we will follow


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

elif (has(lo,phil) and not(has(10,conjugate(phil)))) then



elif has(10,conjugate(phil)) then



elif (has(lo,chil) and not(has(10,conjugate(chil)))) then


elif has(10,conjugate(chil)) then


elif (has(10,chi2) and not(has(10,conjugate(chi2)))) then


elif has(10,conjugate(chi2)) then


elif (has(lo,omegal) and not(has(10,conjugate(omegal)))) then


elif has(10,conjugate(omegal)) then


elif (has(10,omega2) and not(has(10,conjugate(omega2)))) then


elif has(10,conjugate(omega2)) then


elif (has(lo,etal) and not(has(10,conjugate(etal)))) then


elif has(10,conjugate(etal)) then


elif (has(10,eta2) and not(has(10,conjugate(eta2)))) then

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

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.

+-{(( 9 p')(' r'' + F'a) +2pa m
+- { (- + p )8 p) ) (8 1 + 2-' -r) + 2f '(D 2p)h,

- 8 7 7' p 2p'r knr,B4

{ (-a(' p' r) s + a'( + R'e' } ha(r'--r)
+- { (9 p + ) (8 2) p(P' 27' )p + 2'(P p) }h~a- 2z
2('-3rB+-'21-- +4r)--(1-2r),

+-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

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

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,

(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 resulting proposal for a generalized RW gaugfe is

hmm = 0,

hmm = 0,
(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
a'hlm c7le.. = 0

B'hm cll, = 0.

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


(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

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.



elif (has(lo,taul) and not(has(10,conjugate(taul)))) then



elif has(10,conjugate(taul)) then



elif (has(lo,sigma) and not(has(10,conjugate(sigma)))) then



elif has(10,conjugate(sigma)) then



elif (has(lo,sigmal) and not(has(10,conjugate(sigmal)))) then



elif has(10,conjugate(sigmal)) then



elif (has(10,Psi0) and not(has(10,conjugate(Psi0)))) then


elif has(10,conjugate(Psi0)) then


elif (has(lo,Psil) and not(has(10,conjugate(Psii)))) then


elif has(lo,conjugate(Psii)) then


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
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.

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


elif type( f, function ) then

if op(0,f) = 'th' then


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);


error "routine not built to handle that

function: %1", op(0,f);

end if;

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

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.

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 +


(op(1,0p(1,0p(nops(w),w)))))*(times((DD -(p+1)*epsilon

-(q+1)*conjugate(epsilon)),times((DD p*epsilon -


elif (op(0,w)='th' and op(0,0p(1,w))='th') then


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 +


(op(1,0p(1,0p(nops(w),w)))))*(times((Delt +




elif (op(0,w)='thp' and op(0,0p(1,w))='thp') then


result:=result w + times((Delta +


+ptepsiloni + q*conjugate(epsiloni)),0p(1,0p(1,w))));

elif (op(0,w)='*' and op(1,0p(0,0p(nops(w),w))) ='th'


[1] Y. Minor, M. Sasaki, and T. Tanaka, Phys. Rev. D55, 3457 (1997).

[2] T. C. Quinn and R. M. Wald, Phys. Rev. D56, 3381 (1997).

[3] S. Detweiler and B. F. Whiting, Phys. Rev. D67, 024025 (2003).

[4] B. F. Whiting and L. R. Price, Class. Quant. Gray. 22, S589 (2005).

[5] T. R;--- and J. A. Wheeler, Phys. Rev. D108, 1063 (1957).

[6] F. Zerilli, Phys. Rev. Lett. 24, 737 (1970).

[7] F. J. Zerilli, Phys. Rev. D2, 2141 (1970).

[8] K(. Thorne, Rev. Mod. Phys. 52, 299 (1980).

[9] R. Geroch, A. Held, and R. Penrose, J. Math. Phys. 14, 874 (1973).

[10] S. A. Teukolsky, Astrophys. J. 185, 635 (1973).

[11] W. Press and S. Teukolsky, A-r ~1inph--s. J. 185, 649 (1973).

[12] S. Teukolsky and W. Press, Astrophys. J. 193, 443 (1974).

[13] M. P. Ryan, Phys. Rev. D10, 1736 (1974).

[14] B. F. Whiting, J. Math. Phys. 30, 1301 (1989).

[15] J. N. Goldberg, A. J. MacFarlane, E. T. N. i.--us! lIs, F. Rohrlich, and E. C. G.

Sudarshan, J. Math. Phys. 8, 2155 (1967).

[16] R. M. Wald, J. Math. Phys. 14, 1453 (1973).

[17] S. Detweiler and E. Poisson, Phys. Rev. D69, 084019 (2004).

[18] P. L. Cl!!. .1,....---1:!, Phys. Rev. D11, 2042 (1975).

[19] J. M. Cohen and L. S. K~egeles, Phys. Rev. D10, 1070 (1974).

[20] L. S. K~egeles and J. M. Cohen, Phys. Rev. D19, 1641 (1979).

[21] J. M. Stewart, Proc. Roy. Soc. 367, 527 (1979).

[22] E. Newman and R. Penrose, J. Math. Phys. 3, 566 (1962).

[23] A. Ori, Phys. Rev. D67, 124010 (2003).

[24] L. Barack and A. Ori, Phys. Rev. D64, 124003 (2001).

[25] C. O. Lousto and B. F. Whiting, Phys. Rev. D66, 024026 (2002).

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)

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.

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, k")kcV bPbc

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.

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


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,

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

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)

1 1 1l .1

and our task is complete.

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:

(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 +


*(times((DD p*epsilon -q*conjugate(epsilon)),

op(1,0p(nops(w),w)) ));

elif(op(0,w)='th') then


result:=result w + times((DD p*epsilon -


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 +


*(times((Delta + p*epsiloni+


elif(op(0,w)='thp') then


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')


(p,q):=getpq(op(1,0p(nops(w),w)) );

result:=result w +


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

S2007 Larry R. Price

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


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


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.

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

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

PK = (I<'_j )-1/3(p +p), S =-<_ -/
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 '

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

which represents the largest class of gaugfe transformations consistent with form

invariance. This requirement also restricts

S(t, 4) = pt + S(4),


while stationarity again necessitates

P(t) = a~t.


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





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

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

GHP5c := proc ()

return(DGHP (GHPconj (GE5)));

end proc;

GHP5pc := proc ()

return(DGHP (GHPconj (GHPprime (GE5))));

end proc;

GHP6 := proc ()


end proc;

GHP~p := proc()


end proc;

GHP~c := proc()

return(DGHP (GHPconj (GE6)));

end proc;

GHP~pc := proc ()

return(DGHP (GHPconj (GHPprime (GE6))));

end proc;

COM1 := proc ()

GHP3pc := proc ()

return(DGHP (GHPconj (GHPprime (GE3))));

end proc;

GHP4 := proc ()


end proc;

GHP4p := proc ()


end proc;

GHP4c := proc ()

return(DGHP (GHPconj (GE4)));

end proc;

GHP4pc := proc ()

return(DGHP (GHPconj (GHPprime (GE4))));

end proc;

GHP5 := proc ()


end proc;

GHP5p := proc ()


end proc;

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(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


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)

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 +


(op(1,0p(1,0p(nops(w),w)))))*(times((DD -(p-1)*epsilon

-(q-1)*conjugate(epsilon)),times((Delta + p*epsiloni +


elif (op(0,w)='th' and op(0,0p(1,w))='thp') then


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



*(times((delta (p+1)*beta +

(q-1)*conjugate(betal)),times((delta p*beta +


elif (op(0,w)='eth' and op(0,0p(1,w))='eth') then


result:=result w + times((delta (p+1)*beta

+ (q-1)*conjugate(betal)),times((delta p*beta +


elif (op(0,w)='*' and op(1,0p(0,0p(nops(w),w))) ='eth'

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.

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));


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}

+ -((' p') ( pr) + (B -r) r') -r(8' + p) -r( ) + p)2 mm

+{((P-2p)a' +(r + ') + (r -')2 non, (B-6)

{D'(pD p' p') + 2p'p' 'r-7) 'h-7) ,U}i

--{D(B pr + p)r + 2pp')hl,z
- { (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


- 2f'p' 2r'( p)( r')- 3}him a'h,

-{-(2P 2p(P' 27')p + '( 2p~ 2p)P 2pf + 47' p'(h+ pm






end proc;

COMlp := proc()


end proc;

COMic := proc()


end proc;

COMlpc := proc()


end proc;

COM2 := proc()





end proc;

COM2p := proc()


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


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")
1 1
m" = (xa iya), m" (Xa iya),

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

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

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

~ 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


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].

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.



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


-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, a']

- ee' + 011 II)


The GHP equations are:






~ 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,

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

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,


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,

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.


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

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)

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



- a irmb T-mb) na Emb mb)

+ma(@mb ; "IiI.) + Eii, mb)

- a Klb + ~b) na ~b nb)

+m,(p'lb ; "';,.) m('b +Ob)



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)


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)

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


error "routine not built to handle that type: \%1",


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


elif type( f, function ) then

if op(0,f) = 'th' then

return apply(thp, f);

elif op(0,f) = 'thp' then


return map(THORNP, temp);

elif op(0,f) = 'eth' then

return apply(thp, f);

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)
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

Full Text






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


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'sconstructionforKerr .................. 18 ......... 19 ............ 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


........................... 72 4.4ImposingtheIRGintypeD .......................... 73 4.5Discussion .................................... 76 5THETEUKOLSKY-STAROBINSKYIDENTITIES ................ 78 6THENON-RADIATEDMULTIPOLES ...................... 84 6.1Schwarzschild .................................. 91 6.1.1Massperturbations ........................... 91 6.1.2Angularmomentumperturbations ................... 96 ....... 97 .............. 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


Thedetectionofgravitationalwavesisthemostexcitingprospectforexperimentalrelativitytoday.WithgroundbasedinterferometerssuchasLIGO,VIRGOandGEOonlineandthespacebasedLISAprojectinpreparation,theexperimentalapparatusnecessaryforsuchworkissteadilytakingshape.Yet,howevercapabletheseexperimentsareoftakingdata,theactualdetectionofgravitationalwavesreliesinasignicantwayonmakingsenseofthecollectedsignals.Someofthedataanalysistechniquesalreadyinplaceuseknowledgeofexpectedwaveformstoaidthesearch.Thisismanifestedintemplatebaseddataanalysistechniques.Forthesetechniquestobesuccessful,potentialsourcesofgravitationalradiationmustbeidentiedandthecorrespondingwaveformsforthosesourcesmustbecomputed.Itisinthiscontextthatblackholeperturbationtheoryhasitsmostimmediateconsequences. ThisdissertationpresentsanewframeworkforblackholeperturbationtheorybasedonthespincoecientformalismofGeroch,HeldandPenrose.ThetwomaincomponentsofthisframeworkareanewformfortheperturbedEinsteinequationsandaMaplepackage,GHPtools,forperformingthenecessarysymboliccomputation.ThisframeworkprovidesapowerfultoolforperforminganalysesgenerallyapplicabletotheentireclassofPetrovtypeDsolutions,whichincludetheKerrandSchwarzschildspacetimes. Severalexamplesofthepowerandexibilityoftheframeworkareexplored.TheyincludeaproofoftheexistenceoftheradiationgaugesofChrzanowskiinPetrovtype 6




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


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


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


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


1{4 ),wenowconsiderthetwosectorsofthemetricperturbationindependently,writing and ThenalstepbeforeappealingtotheEinsteinequationsconsistsofchoosingagauge.Equation 1{3 isinvariantunderthetransformation whereaisanarbitraryvectorand$istheLiederivative.Takingtheodd-paritysectorasanexample,theRegge-Wheelergaugevectortakestheform whereisafunctionchosensothattheoddparitypartofthemetricperturbation 1{10 takestheform Similarsimplicationsariseintheeven-paritysector. 12


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


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



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


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


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


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


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


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


InChapter2,wewilldeveloptheformalismnecessaryforbuildingourframework.Additionally,theframeworkwillbepresented,whichincludesanewformfortheperturbedEinsteinequationsaswellasaMaplepackagethataidsnotonlyintheirapplication,butanycomputationintheformalismofGeroch,HeldandPenrose.Chapter3thenprovidesafurtherdiscussionofboththeRWandTeukolskyformalisms,phrasedinourframework.InChapter4,thenecessaryandsucientconditionsfortheexistenceoftheIRG(inalargerclassofspacetimesthanweconsiderelsewhere)aredeterminedwiththeaidofourformoftheEinsteinequations.Chapter5thenusestheIRGmetricperturbationtoderivesomeimportantrelationshipsbetweenthecurvatureperturbationsrepresentedby0and4,whichareofimportancefortheinversionproblemdescribedinthischapter.Furthermore,thisapplicationshowcasessomeofourMaplepackage'smostusefulfeatures.InChapter6wethenpresentaverydierentapplicationofourframeworkinconjunctionwithmorestandardtechniquestoaddresstheissueofthenon-radiatedmultipoles. 22


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


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


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


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


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


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


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


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


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


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;


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


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


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


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


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


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


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


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


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


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


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


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


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


Thefp;qgtypeofanyquantitymaybeobtainedbytheuseofthegetpqfunction,whichreturnspandq,inthatorder:> 1=0; =0; =0; 1=0;


TherealusefulnessofGHPtoolscomesnotfromitsbookkeepingabilities,butratheritsabilitytoperformsymboliccomputationswithintheGHPformalism.TheseabilitiesbeginwiththeDGHP()procedure,whichexpandsderivativesofobjectsoccurringinanexpressioninaccordancewiththerulesofderivations.Forexample> 1)>


1))> Todate,DGHP()canhandlepowersandlogarithms(theonlyfunctionsthisauthorhasencounteredintheGHPformalism),buttheprocedurecanbeeasilymodiedtoaccommodatejustaboutanyfunction.BuildingcomplicatedexpressionsinvolvinglinearcombinationsofderivativeandmultiplicativeoperatorsiseasilyachievedwiththehelpoftheGHPmult()procedure.TheseexpressionscanthenbeexpandedwithDGHP().Asanexample,considertheexpression()4:> Simplifyingsuchexpressionsis,inthecontextoftypeDspacetimewithoutacceleration,handledbythetdsimp()procedurethatsubstitutestheknownvaluesofthederivativesofthespincoecients(storedinthegloballyavailablelisttdspec;suchaprocedurecanbeeasilygeneralizedtoencompassanyspacetime,shouldtheneedarise)intoitsargument.Thusourpreviousexamplesimpliesconsiderably:>


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


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


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


Asarstapplicationofourframework,wewillprovideamoredetaileddiscussionoftheRegge-WheelerandTeukolskyequations.ThisleadsquitenaturallytoadiscussionofthemetricperturbationgeneratedfromaHertzpotential,whichwillplayamajorroleinsubsequentchapters.Ourstartingpointisageneraldiscussionofparitythatdoesnotassumeeithersphericalsymmetryorangularseparationfromtheoutset. 50 ]. Ourrstassumptionisthatourspacetimemanifold,M,admitsaspacelike,closed2-surface,S,topologicallya2-sphere,withpositiveGaussiancurvatureandapositivedenitemetricgivenby wheremaandmaaretwomembersofanulltetrad.Forasphericallysymmetricbackgroundabisproportionaltothemetricofthe(round)2-sphereandmaandmacanbedirectlyassociatedwiththebackgroundmetric.Moregenerally,weallowforthe 52


Forexampleconsidersomevector,va,denedinthespacetime:va=vlna+vnlavmmavmma: Theseareallthetoolsnecessaryforwhatfollows. WebeginbyconsideringtheprojectionofvectorsdenedinthespacetimeontoS.ToidentifytheoddandevenparitypieceswestartbydecomposingageneralvectoronS (3{4) 53


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


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


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


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


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


(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


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


TheZerilliequation[ 7 ]describingeven-parityperturbationsoftheSchwarzschildspacetimehassofareludedadirectdescriptionintermsofgaugeinvariantperturbationsoftheWeylscalars.However,theinformationcontainedwithintheZerilliequationcanbeobtainedthroughthemetricperturbationthatfollowsfromtheTeukolskyequation,whichisthefocusoftheremainderofthischapter. 10 { 12 ]camedirectlyfromconsideringperturbationsoftheWeylscalars.We,however,areinterestedinobtainingitdirectlyfromtheEinsteinequation.UsingTeukolsky'sexpressionsforthesourcesof0and4,wecanobtainthisdirectly.ThesourcesoftheTeukolskyequationaregivenby whereT0andT4arethesourcesfor0and4,respectively.MakingthereplacementTab=1 8Eabintheequationsaboveleads(afterproperlyrearrangingthederivativeswith 61


[(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


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


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


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


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


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


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


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


[;~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


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


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


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


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


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


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


63 ]hasarguedthatmassandangularmomentumperturbationsarenotgivenbyanysolutiontotheTeukolskyequations,andStewart[ 21 ]hasshownthatthesecannotberepresentedinaradiationgaugeintermsofapotential.Whatwehavedoneisidentifythegaugefreedomwhichremainsinthefullysatisedradiationgauges,neitherinterferingwiththeradiationgaugeprescriptionnorrulingoutthepossibilityofmassandangularmomentumperturbations.ByrealizingtheexplicitconstructionoftheradiationgaugesfortypeIIbackgroundspacetimesandbyidentifyingtheremaininggaugefreedomwhichtheyallow,wehave,inasense,completedataskinitiallyembarkeduponbyStewart[ 21 ],thoughinthedierentcontextofHertzpotentials. 77


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


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


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


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


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




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


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


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


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


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


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


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


wheref=12M=r.AccordingtoBirkho'stheorem,theonlystatic,sphericallysymmetricsolutiontotheEinsteinequationsistheSchwarzschildsolution.Thus,weareassuredfromtheoutsetthatperturbingthemasswillsimplyleadustoanotherSchwarzschildspacetimewithamassM+M.Thenonzerocomponentsofthecorrespondingmetricperturbationaregivenby rhrr=2M rf2;(6{29) whichiseasilyobtainedbylinearizingamassperturbationofEquation 6{28 .Inordertocharacterizemassperturbationsmoregenerally,wewillintroducemorefreedomby 91


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


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


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


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


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


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


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


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,[ 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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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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