Gravitational models in 2+1 dimensions with topological terms and thermo-field dynamics of black holes

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Gravitational models in 2+1 dimensions with topological terms and thermo-field dynamics of black holes
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GRAVITATIONAL MODELS IN 2+1 DIMENSIONS
WITH TOPOLOGICAL TERMS AND
THERMO-FIELD DYNAMICS OF BLACK HOLES







By

BETTINA E. KESZTHELYI


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



UNIVERSITY OF FLORIDA












ACKNOWLEDGEMENTS


would like


to acknowledge all


the professors I have worked


with


over


the years, especially my present advisor, Professor Pierre Ramond.


He was


responsible for introducing me to the study of dual solutions in 2+1 dimensions.

He also made valuable suggestions to me on the topic of thermo-field dynamics.

I would like to thank Professor Stanley Deser for his help and hospitality during


my stay at Brandeis University.

on topologically massive gravity.


I owe him special thanks for his suggestions

I would also like to thank Gary Kleppe for a


productive collaboration and for sharing his insights on nonlocal regularization.

Finally, I must thank Alexios Polychronakos for his invaluable advice early in

my career.




































Digitized by the Internet Archive
in 2011 with funding from
University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation












TABLE OF CONTENTS


ACKNOWLEDGEMENTS


ABSTRACT


I I I I I I I S a 11


. . I S S S S V


. I . a S 5 5 5 S I I 1


DUAL SOLUTIONS IN 2+1 DIMENSIONS


.1 Introduction


.2 Weyl Theory in 2+1 Dimensions .
.3 Stationary Solutions . .
.4 Solutions with Magnetic Field .
.5 Solutions with no Electromagnetic Field:


2.6 Stationary


Axial Symmetric Solutions


RENORMALIZABILITY


= 3 TMG


8


15
S 18
ity . 21
. 23


Einstein Grav


32


3.1 Introduction


3.2 Power-Counting Renormalizability and Gravity
3.3 Topologically Massive Gravity .
3.4 Nonlocal Regularization . .
3.5 Nonlocal Feynman Rules . .
3.6 Renormalizability. .

THERMO-FIELD DYNAMICS OF BLACK HOL


* I I .
* S f S .
* a *
* S S

ES .


33
. 34
38
. 41
. 43


. 48


4.1 Introduction


4.2 Massless Scalar Particles on the


Schwarzschild Background


4.3 Many Black Holes .
4.4 Neutrinos on the Schwarzschild Background


CONCLUSIONS


51
. .S55


S . . *66


APPENDIX A


APPENDIX B


FEYNMAN RULES


MEA


FOR TMG


69


SURE FACTOR FEYNMAN RULE


FOR TMG


REFERENCES


* I I 5 . 5 5 5 5 S S 5 *I7 6


BIOGRAPHICAL SKETCH


S S S S S S I 78


INTRODUCTION


PaRe








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

GRAVITATIONAL MODELS IN 2+1 DIMENSIONS
WITH TOPOLOGICAL TERMS AND
THERMO-FIELD DYNAMICS OF BLACK HOLES

By


BETTINA E.


KESZTHELYI


December 1993


Chairman: Pierre I
Major Department:


lamond
Physics


We consider two extensions of Einstein gravity in 2+1


dimensions.


First


we study some consequences of duality in three dimensions. In the case of pure


gravity, a dual Ansatz is shown to lead to pure gauge configurations,


and in


a Weyl invariant gravitational theory, duality arises as an equation of motion.

Its solutions obey the Liouville equation and describe a rotating Chern-Simons

fluid in a gravitational field.

Next we analyze the theory of topologically massive gravity in three space-

time dimensions, conjectured to be renormalizable by Deser and Yang, using


the nonlocal regularization.


The validity of this technique, however, depends on


the existence of a gauge-invariant measure for the nonlocal theory.


that such a measure exists,


Assuming


we show that the possible obstacle to renormaliz-


ability found by Deser and Yang does not appear.

Finally thermo-field dynamics is used to derive the Hawking radiation of


black holes emitting massless scalar particles or spin one half fermions.


also show


how to


generalize this method


to particle creation in spacetimes


- 1 I I


*, 1,


1


1 1













CHAPTER 1
INTRODUCTION


work we study two 2+1


dimensional gravitational models,


Weyl


gravity and topologically massive gravity.


We study the consequences of duality


the first case and renormalizability in the second.


We also consider the


Hawking radiation of black holes in 3+1 dimensions.

Field theories in lower space-time dimensions have recently attracted at-

tention as tractable toy models for realistic 3+1 dimensional systems and also


in their own right for their unusual topological properties.


For example, one


can hope to find 2+1 dimensional models involving gravity which have better


short distance behavior than those in 3+1 dimensions.


The latter are known to


be nonrenormalizable due to the presence of the gravitational coupling which

has negative mass dimension. Indeed, one may be able to discover 2+1 dimen-

sional theories which are renormalizable and may help one to better understand


the quantum behavior of the higher dimensional theory.


these theories are also interesting.


Classical solutions of


These can be used to study the nature of


gravitational singularities or can represent physical axially symmetric solutions

such as cosmic strings.


We consider two models addressing the above questions.


Since they both


involve gravity,


we review


the main features of 2+1


dimensional


pure


Einstein gravity.


These features are unique, but restrictive.


This is why it is


I I .I I *1 S -


mi n 1 j_ .-.- --





f9

be derived from the Einstein equation which is given by


G y, + Ag = TWV


(1.1)


where Gpy =


Rl v- jRgwv is the Einstein tensor.


Expressed in terms of the


Ricci tensor we have


R.V = 2Agt, + s(Tyv gtVT) ,


(1.2)


where A is the cosmological constant and the coupling t has mass dimension


The curvature tensor and


Ricci


tensor


both have


independent


components and one can write the curvature in terms of the Ricci tensor,


gIXRup + g pRX


- g9pRyx gvARx p


(1.3)


3R(g\Agvp gYpgYA)


From (1.2) and (1.3) one can see that the curvature tensor is completely de-


fined by the energy-momentum tensor Tay and A.


When Ty, = 0, the scalar


curvature is R = 6A so that any curvature effects produced by matter do not

propagate through spacetime and there are no dynamical degrees of freedom.


This could simply be seen by counting degrees of freedom.


the sign of A, the spacetime is either locally flat,


Depending upon


de Sitter or anti-de Sitter.


Although local


curvature in source-free regions is unaffected


by matter,


it can still produce nontrivial global effects.


Even in the simplest cases such


global effects are present. For example, for a point mass and A = 0 spacetime


is flat except along the world line of the particle. In the static


case


coordinates


can be chosen so that the constant time surfaces are conical [1].


This conical


spacetime is obtained by removing a wedge from Minkowski space and identi-


Ap =




3

source is also conical, but in this case, the points that are identified across the

deleted wedge differ in their time coordinate value by an amount proportional


to the angular momentum of the source.


Thus in the presence of a spinning


source the spacetime has a


"helical structure,"


a rotation about the source is


accompanied by a shift in time [2].


This conical-helical geometry characterizes


the spacetime outside of more general compact matter distributions, because

matter cannot affect the local curvature in source free regions.

Despite being locally flat except along the particle's world line, these space-


times have interesting global geometric properties.


For example,


there is an


analogue of the Aharanov-Bohm effect in


that a


vector parallel transported


around a loop surrounding the source experiences a nontrivial rotation, even


though the loop lies entirely within flat regions of the spacetime [3


Similarly,


two geodesics


passing of opposite sides of the source may intersect twice.


This


effect also arises in 3+1 dimensional gravity in the context of cosmic strings [4].

Here we will consider two extensions of the Einstein theory in 2+1 dimen-


sions,


Weyl gravity and topologically massive gravity (TMG).


The outline of


this work is as follows:


Chapter


we show that


Weyl


gravity


has self-dual solutions, and in


certain cases exact solutions can be

solutions, as described in Sect. 2.1 [


Our motivation for seeking such


5], is the important role they play in 3+1


dimensional theories.


For example in four


Euclidian dimensions self-duality


of the


Yang-Mills field strength leads


to instanton solutions


similar


condition imposed on the gravitational connection leads to the Eguchi-Hanson

gravitational instantons [7].


In Sect. 2.2 we review Wevl theory in d+ 1 snace-time dimensions.


Then we






gravity, a


Weyl gauge field and a real scalar field.


We find some interesting


features of the field equations, obtained from the Lagrangian of this theory.


These are useful to predict some of the general features of the solutions.


example we show that our solutions correspond to a special type of rotating


fluid immersed in a 2+1


dimensional gravitational field.


This indicates that


such solutions might have interesting applications in fluid mechanics.

back to this analogy throughout the following subsections of Chapter 2.


We refer

We also


show that the field equation of the Weyl gauge field in a specific gauge imposes

a self-duality condition between the field strength and the gauge potential.


In Sect. 2.3 we discuss stationary solutions in this gauge.


We show that


they can be classified by the nonvanishing elements of the field strength tensor.


We find that the case with no electric or magnetic field (Ei = 0,


to Einstein gravity in flat or in de Sitter space [2]


B = 0) reduces


The purely magnetic case


0) and the case with nonzero electric and magnetic field (Ei


B f 0) are more interesting. In the former case we find all the static solutions,

but not in the latter.


From here we proceed by solving the field equations for the E, = 0, B


case in Sect 2.4, and for the Ei


tinue with studying our solutions.


0 case in Sect 2.5. In Sect. 2.6 we con-


We find that in the axisymmetric case they


correspond to known 3+1


dimensional spacetimes.


It is also interesting that


in the proper coordinates our solutions have the same conical-helical geometry

characteristic to a large class of known solutions of 2+1 gravity as mentioned

above.


Weyl


gravity is one of the natural generalizations of Einstein gravity if


nne Wi;hss tn Qtiim1 t"h^ n-iiinrnfm bnTrr hon^ancn ;+ b.r0 hno ++nr 0br.4 rla e.a n


(Ei = 0,






behavior arising from conformal invariance [8]Q1)


In 2+1


dimensions there is


another possible choice, topologically massive gravity [9].


Chapter 3 we study the renormalizability of TMG


by using nonlocal


regularization


In Sect.


we describe why it is a good candidate for


a renormalizable theory with


the symmetries of


gravity.


There are several


arguments [9]


including ours [10],


indicating that TMG is renormalizable, but


none of them can be considered a strict proof.


The importance of the question


that if these arguments are proven


to be


true,


TMG


would be


the only


known theory with such properties.

In Sect. 3.2 we use general power counting arguments to show why certain

gravity theories are not renormalizable.

In Sect. 3.3 we review the main features of TMG, and we show that it is


power counting renormalizable.


However, in order to conclude that the the-


ory is renormalizable, one also has to show that the gauge invariance may be


maintained in


the regulated version of the theory without giving up the de-


sirable power counting behavior.


In other words,


without the use of a gauge


invariant regulator, additional


terms might


be required to cancel


the gauge


transformation of the effective action, and these might spoil power-counting


renormalizability.


For these reasons we use gauge invariant nonlocal regular-


ization.

In Sects. 3.4 and 3.5 we review the method of nonlocal regularization and

the nonlocal Feynman rules, respectively [11].


In Sect. 3.6 we apply these rules for


TMG.


Unfortunately, to find a proper


gauge invariant measure factor for the nonlocalized theory, or at least prove its

(1) It is necessary to consider a generalized theory, because Einstein theory in






existence, is an extremely difficult problem.


Only a perturbative method exists


which allows us to calculate it to any desired order in the fields.


In theories


without gauge anomalies (like


TMG),


it is reasonable to assume it


s existence.


We show that if this holds, the only anomaly term, found in ref. 9,


does not


appear, and the theory is renormalizable.

In Chapter 4 we consider another interesting problem of general relativity,

namely particle creation from the vacuum in spacetimes with causally discon-


nected regions.


It is known that the quantization of fields on such spacetimes


leads to particle creation from the vacuum as a consequence of the information


loss associated


with the presence of the event horizon(s)


In stationary


spacetimes with a simply connected event horizon (such as a stationary black

hole, an accelerated observer in Minkowski spacetime, or de Sitter type cos-


mologies), the emitted particles have a thermal spectrum [13].


This result has


been first obtained by Hawking


14] for black holes.


He has shown, that the


effective temperature of this radiation is TH = K, the Hawking temperature,


where i is the surface gravity of the black hole (units


are chosen throughout


such that k


= 1).


These results have been confirmed and de-


rived in a number of ways, and several attempts have been made to gain a

better understanding of the features of the Hawking process and the physical


role of the event horizon [14,15,16].


A particularly interesting approach into


this direction is the one used


Israel [16],


who considered the problem of


particle creation on the Schwarzschild background.


The idea is to quantize the


fields in the full analytically extended Schwarzschild spacetime (known as the

Kruskal extension) in order to keep track of particle states on the hidden side


of the horizon as well.


The same idea allows us to annlv a auantum-statistical




7

Here we follow Israel's approach, and first we rederive the well-known re-


suits of Hawking radiation of a black hole.


dard results.


This method reproduces the stan-


We look for possible extensions to the case of many black holes.


The organization of this chapter is as follows:


In Sect.


4.1 we briefly review


the canonical formulation of thermo-field dynamics for free fields [17] and show

how it can be generalized to describe particle creation in general relativity [16].


In Sect.


4.2 we review Israel's paper [16]


. The quantization of a massless


scalar field on the Schwarzschild background is considered by using thermo-


field dynamics. We conclude that the results obtained this way are equivalent

to the earlier ones. In the rest of the paper we describe the possible generaliza-


tions of this approach.


In Sect.


4.3 an approximate multi-black hole solution is


considered as an example to demonstrate how to extend the method to space-


times with many causally disconnected regions.


Finally in Sect.


4.4 we derive


the Hawking radiation of a black hole emitting neutrinos and antineutrinos.













CHAPTER 2
DUAL SOLUTIONS IN 2+1 DIMENSIONS



2.1 Introduction


In four Euclidean dimensions, self-duality of Yang-Mills gauge fields leads

to the classical instanton solutions. A similar condition imposed on the gravi-


national connection leads to the Eguchi-Hanson gravitational instantons. Mo-

tivated by the recent interest in theories of lesser number of dimensions, we

investigate analogs of self-duality in the classical solutions of some theories in

2+1 dimensions.

In the case of Yang-Mills gauge fields, self-duality cannot be imposed on

the field strengths, as in four dimensions, but only between the gauge poten-


tial and


the field strength.


In order to find such a solution,


we investigate


a simple theory involving gravity and an Abelian gauge potential, linked by


the requirement of Weyl invariance.

duality (in a special gauge). We stu


The classical equations of motion demand


idy stationary solutions of these equations.


We first solve the equations of motion in the pure gauge case, when our theory


reduces to Einstein gravity with nonzero cosmological constant.


Then we find


the general solution when


the magnetic field does not


vanish


for these the


conformal factor of the two dimensional space satisfies a Liouville equation.

The physical situation corresponds to a special type of fluid immersed in a


24-1 gravitational field.


The fluid is a rotating perfect fluid with velocity corre-






duality condition and a massive Klein-Gordon equation.


When specializing to


axisymmetric solutions, we recover either the 2+1 dimensional G6del solution

or the boundary of 4 dimensional Taub-NUT space-time solutions.


Weyl Theory in 2+1 Dimensions


order to


consider


gauge fields,


we use


as a principle


Weyl's


original


theory which links gravity to electromagnetism through a


"gauge"


principle


For the reader who may be unfamiliar with this type of theory,


we start


with a brief review.

In the second order formalism of general relativity, the fundamental fields


are the metric


field


addition


to the usual


diffeomorphism of general relativity,


Weyl requires invariance under conformal


rescaling or "gauge" transformation


gpv(x) Q

g111(2) -+ 4


x)gpv(x


2(x)g{"(x


(2.2.1)


(2.2.2)


x) --


In Einstein


Ap(x) + 2, In l (xa


theory the spacetime is described


(2.2.3)


Riemannian geometry,


where one has a metric connection,


the covariant derivative of the


metric is zero,


gv = 0 .


(2.2.4)


This implies that lengths and angles are preserved under parallel transport. In

order to get a locally scale invariant theory Weyl weakened this condition, and

required only that


( 9 f\


7 7-n... 4 \ n....






where A4 is a vector field.


That is, in


Weyl's theory only the angles, but not


the lengths, are preserved under parallel transport.

If we assume that the torsion is zero, then Eq. (2.2.5) can be solved for the


connection in terms of gpy and A


The result is


= { -2A +6aA -p-yAa),
/ zl


(2.2.6)


where { i}


is the usual Einstein connection.


Notice that Viv


a is gauge in-


variant and symmetric. From Fyw


a we can construct the conformally invariant


curvature tensor:


Rpva


=- Dpa


- Ov1 pa


+ va


-- Fpa


7v7Y


(2.2.7)


The conformally invariant Ricci tensor and the scalar curvature are given by


contracting (2.2.7) with the metric.


In d + 1 dimensions the Ricci tensor is


1
<(d DD1 A, D A, + gp D AC )


RV = Rpvy


(2.2.8)


- 1
(A A, A- gy AaA )


and the scalar curvature is


-g= VRV = RE


+ d DAC


Aa A0


(2.2.9)


where RE and RE


are the usual Einstein Ricci tensor and scalar curvature,


respectively, and Dp is the covariant derivative with respect to { a, }.


Notice


that the Ricci tensor is not symmetric, since it contains the antisymmetric field

strength tensor F0e. From Eq. (2.2.8) its antisymmetric part is given by


d-1
F---
2 rL


(2.2.10)


nC 1'1 1 l


7Fy


SRE
- Ul


Rp R,?


nnrl 7? .. -e c ol oA-





11

because the connection is symmetric.

The covariant derivative with respect to the connection given by Eq. (2.2.6)

acts on a scalar field y and on a vector field Vp of Weyl weight w in the following

manner


, (2.2.2.12)


w
v, 8pV, -I' Va + -A V,
^v -ir/ ^ v'1" 1^r


(2.2.13)


A field 2 is of (Weyl) weight w if it transforms under conformal rescaling as


0--w


(2.2.14)


Weyl invariance limits the form of the gravitational interaction.


construct a purely gravitational action,


One cannot


first order in the curvature,


and reproduce Einstein's theory (except in the "trivial" 1+1 dimensional


when it is a surface term).


case,


Mathematically this means that, under a conformal


rescaling


-+4 d+1


VHT


(2.2.15)


where g = det gp,, and


(2.2.16)


the Einstein action is not invariant:


16W7G


d+lx -


R 167
16irG


dd+lx gR Q d-1


(2.2.17)


where G is Newton's constant.


The action IE is invariant only if


that is


in 1+1 dimensions where it is the Euler characteristic. In 3+1 dimensions the

Einstein action is not gauge invariant (an invariant gravitational action can be


fnrmedrl bit i hno n nhr minrlrntirin 7?


(n *hP nthhor nnrt IVlnvxM^ll' c


nrfi;nn


VkIf


up = 9P +


Apup


R --






Weyl invariant.


This is not surprising, because electromagnetism does not


require any dimensional constant.


In order to reproduce Einstein's


theory, many authors [8,19] have intro-


duced a scalar field of unit weight, that is


(2.2.19)


corresponding to its canonical dimension.


Then the action


d4x i-gRy(2


is indeed invariant.


(2.2.20)


9 acquires a nonzero value o0 in the vacuum, it yields


the Einstein action with 16r = 2.


Weyl invariance in 3+1 dimensions allows


for a kinetic term for 9 as well as a potential term, thus making it a dynamical


field.


However, it is not clear how to generate such a vacuum value in a theory


that is not only without a scale but also not renormalizable.


In 2+1


dimensions one can also


introduce a scalar field with


Weyl weight


1 (not its canonical dimension) in order to construct a gauge invariant action,

given by


d1 (gRy+cpFA,
d3x( ll -


+)A


(2.2.21)


in this


case,


Weyl invariance allows us to write a cubic potential


term


for t,


but not a kinetic term.


Thus we do not expect 2 to correspond to a


dynamical degree of freedom.


We further note that the action contains only a


Chern-Simons term for the vector potential and no Maxwell term.

The equations of motion obtained from varying the action with respect to

the fields and the metric are the following:


_? -.+ -lcp







(2.2.23)


1
(RV 9-RgV)9 + {p ** v} = (V, VV gpvV!Va)
2


+ ~ gpv + {I


<-+ v}


(2.2.24)


It is interesting to note that, even

derivatives of the scalar field, p still


though our Lagrangian does not contain


1 obeys a Klein-Gordon equation, with the


source


as the Chern-


imons density


1 eD FapA A,
- i1
EC/?9fl


(2.2.25)


using Eq.


2.2.23


) and assuming the


Bianchi identity


on Fy,


(absence of a


magnetic monopole).


Actually


9 is a derived field which obeys a first order


differential equation of the form


"9 =


(2.2.26)


where Vuj is a vector field chosen such that


V- Fa pfalY


=0 .


(2.2.27)


This means that


9 is covariantly constant along a direction locally determined


by the electric and magnetic fields.


Before looking for solutions of these equa-


tions,


we make the


following


simplifying observations.


First, not all


of the


above three equations are independent.


The trace of Eq.


(2.2.24) is equivalent


to Eqs. (2.2.22) and (2.2.23).


This allows us to use only the last two equations


when seeking solutions.


Second,


Weyl vector terms can


be incorporated


into the energy-momentum tensor of the matter fields. As a result,


Eq. (2.2.24)


n r' V-.,,,: 1.,,n a stct I 1 /' ,P.. 1 r* lvi 4-t % r .64 4 n ri C, rc t t ti.


----a F, = gTVp
2g


Vg

y^ ) nr /-yv r* v f n r% f~^r r^ //






where the energy-momentum tensor Tpn is given by


-1 (Dcov- gDa,) + Avpp + ApaO


gpv A a


+A 2
+-(-
'6


- AaAa)gsv +
4


(2.2.29)


To proceed,


we use the Weyl invariance to go into a gauge where


(2.2.30)


which is allowed


as long


as y


does not vanish anywhere.


There, Eq. (


2.2.23


rewritten


1 P Fap = 0 A7
1


(2.2.2.31)


from which it follows that DoAn


=0.


From Eq. (2.2.31) one also finds that


A, satisfies


D"'DaA Av + Aa R =
4


the field equation of a massive vector field.


(2.2.32)


A similar result has been found for


the Abelian Chern-Simons theory on Minkowski spacetime in refs. 20 and 21.

In this gauge the energy-momentum tensor of Eq. (2.2.29) simplifies to


S-
4A A0 )gv +
~0-4


(2.2.33)


a form reminiscent of a fluid.


It is not quite perfect since the pressure p and


the density


p depend on AaAC


as well as


on 2)0.


Furthermore the fluid has


a velocity vector proportional to Ap

will study the properties of this flui(


which obeys the constraint (2.2.31).


d in greater detail in the context of exact


solutions.


If the electromagnetic field is zero, i.e.


E = B = 0, we find from Eq. (2.2.23)


-. I


A
6


Tv=


AUA,


c = Q ,


AA ,


/~ L II 11





15

and =the Einstein equation reduces to

and the Einstein equation reduces to


(2.2.35)


Clearly,


A2
+2).


when the Weyl invariance is gauge fixed to a constant


(2.2.36)


we obtain


Gpy = gplV6


(2.2.37)


reducing the space to an Einstein space with cosmological constant.


following,


In the


we discuss stationary solutions.


Stationary Solutions


the stationary


case


the most general form of the line element in


component notation is


= -N2(dt + Kidxz)2


+ yijdxzdx3


(2.3.1)


where N


, Ki and Yij depend only on the spatial coordinates, x and i,


That is, the metric components are


, goi =


, gij = 7ij N2KiK


(2.3.2)


with inverse components


1
-N2


+ "y Ki ij


vg= =N


, 7 = det 7ij


Sgi
, g1


>0.


-=7-
-7


(2.3.3)


(2.3.4)


The remaining reparametrization


auge freedom, t --* t + A(r), can be fixed by


setting DiKz


= 0,


where D; is the two dimensional covariant derivative with


T I I 1 1 r~ 1 S .


1 1 A
G, = 4(A A, + D A, + D A ) + ggy(2Da A


goo00 = -N


7-31
= -ys^.Ki


J J






1
-2N


1 -(D
= N2KikKik -


Dj(N3Kij) ,


(2.3.6)


1- k
_ 2NyirKkJlKkt


(2.3.7)


where Kij


= DiK1


= OiKj jKi, and 2R is the two dimensional


curvature.


We choose spatial conformal coordinates.


This can be done locally without


loss of generality in a two dimensional space:


dij = dij

dl2 = (dx


>0),


(2.3.8)


+ dy


In these coordinates the DZ


Ki = 0 condition reduces to OiKi = 0.


This allows


us to write Ki = eij V.


with 12 =


where eij is the two dimensional Levi-Civita tensor


-E21 =


With


this form


of the metric,


the field equations (2.2.22),


(2.2.23)


(2.2.24) in the p = p0 gauge become


.ij iAj -


- N0
2


N2 + Kj(Aj


- AoKj))


(2.3.9)


eijQjAo = N--(A
2J


- AoKi) ,


(2.3.10)


A


1 3N2


2-


AaAa


)_212
2N2


(2.3.11)


N3
Oj(-- eijAV) = Ao(Ai


(2.3.12)


1-
1N(DiON -
A 2
=-( -


- 6jAN) --ij(iV)2
23 43


- AaA a)gij -


(2.3.13)


AiAj,
2-


where A is the two dimensional flat Laplacian.


solutions can


be characterized


by the nonvanishing


components of


- iJD2)N


- DjlKi


, (


- AoKi) ?







magnetic field,


Ei = Foi.


B = -2-%-e~Efij, and the space components of the electric field,


We have the following cases.


a) B


- 0,


E;=


We have already seen that this case reduces to Einstein


gravity in flat or in de Sitter space [2].


b) B


This means that A0 is constant and from Eqs. (2.3.10)


we have Ai =


AKi. .


Further, since B is


different from zero, A0 itself cannot


vanish.


Thus Ai = AoeijjV


allows us to rewrite Eq. (2.3.9) as


(2.3.14)


2N


which by comparing with Eq


(2.3.12) leads to


= NO = constant.


(2.3.15)


Note that our gauge condition (2.2.30) still allows us to make constant gauge


transformations to rescale N0


(and 0).o


We fix it


choosing NO


= 1.


this purely magnetic


case


we were able to find all


the solutions of the field


equations.


c) E;


, B = anything. In this


case,


when the electric field is not zero,


we did not


find any static solutions.


We can say


however that


if only


component of the electric field is nonzero,


say El


then the solutions depend


on only one spatial coordinate x1.

In the next two sections we solve the field equations in the purely magnetic


case with B


0 (Sect.


2.4) and with B


= 0 (Sect.


2.5).


In the latter case,


when our theory reduces to the Einstein case,


the equations of motion.


we find the general solution of


In Sect. 2.6 we continue with studying our solutions,


obtained for the pure magnetic case.


We show that in the axisymmetric case


one


E;=






2.4 Solutions with Magnetic Field


In this section


we solve Eqs. (2.3.9)-(2.3.13) in the purely magnetic


case


At the end of the previous section we showed that in this


case


Ag = constant, Ai


= AgKi and NO = constant.


Using these results we


find that Eq. (2.3.10) is trivially solved and the remaining equations are


PoAo


1
--Alnb +
a


(2.4.1)


3 AV
2 (


(2.4.2)


3


AV 0,
8a = 0 ,


(2.4.3)


( )


where we have set NO = 1.


2 + A,


(2.4.4)


From the above equations one finds that 4 satisfies


a Liouville equation


In 4 = /3'


where


(2.4.5)


A. All the other quantities can be expressed in terms of the


solutions of this equation and the constants 0 and A as follows


A2 =
0n-


1
4


(2.4.6)


3 ) ,


(2.4.7)


(2.4.8)


Note that because AO is nonnegative only the A


g values are allowed.


This


leads to 3


< that is, it can be either negative or positive.


From the above equations one finds that the magnetic field,


AO
B = V


= _A0


(2.4.9)


0, Ei = 0).


1
= 2


Ai = Aoeij jV


e3ijiA --







the length of the Weyl gauge field,


SA0A


- -A2,
- "0'


(2.4.10)


and the Einstein scalar curvature,


2R + 4KijKij


-AIln + 2-2(AV)2


2
S8


(2.4.11)


16A
3 )'
3


are constant.


We note that in this case, the perfect fluid analogy mentioned


in Sect. 2.2 is complete, because as follows from Eqs. (2.2.30) and (2.4.10) the

energy-momentum tensor is just


-=16 2 gA AA


(2.4.12 )


The normalized velocity is then


Up =


(2.4.13)


Ao
IAO


that is,


uP = 1 and ui = -ij jV


The equation of state relating the density


{8P-3

p =O0


where the p = 0 case corresponds to dust.


(2.4.14)


p to the pressure p is given by


(2.4.15)


otherwise,


We note that the (weak and domi-


nant) energy conditions [23], stemming from demanding causality, lead to the


condition p


which gives further restriction on /: either ,3


< 0 and p


or 0


< / 3


In addition to this equation of state,


the velocity vector up


obevs the further equation


> Ip




20

which is indicative of rotation. In particular this means that the velocity obeys

a massive Klein-Gordon equation, and that the fluid is incompressible.

Depending on how we choose the value of A, R can be positive or nega-


tive.


Positive R corresponds to a compact space-time manifold (i. e.


"closed


universe").


In this case the solutions can be characterized by topological in-


variants of the manifold.


For negative R,


that is, for noncompact manifolds


(or "open universe"


), one can define the energy and the angular momentum of


the solution.


We also find that our solution is


conformally flat.


three dimensional


space the Weyl tensor is always zero in the absence of matter.


However there


is another tensor, the Cotton tensor [24],


=(1 EaflVfRi


+ eaflV^R ) ,


(2.4.17)


which plays the same role as the Weyl tensor in higher dimensional spaces. It

is symmetric, covariantly conserved and vanishes if and only if the spacetime


is conformally flat.


For example all the vacuum solutions as well as the point


particle and rotating solutions of Deser et al.


[2] are conformally flat.


In our


case C~" is vanishing; that is, our solution is also conformally flat.


general solution


Liouville equation is


given


terms of


complex functions


z) and g(z),


_2
S(f(


z ff g(z)


(2.4.18)


where f(


z) and g(


) are such that they give real positive values to 4.


Explicit


forms of f and g, that satisfy this requirement, are known [25]


case.


As the simplest


we will consider axial symmetric solutions But first we tirli R the R =


-g(z))2







2.5 Solutions with no Electromagnetic Field:


Einstein Gravity


Although we have already shown that this case reduces to Einstein gravity

with a cosmological term, it is instructive to elaborate on the form of the static


solutions.


The equations become


1
-Aln4 1
4)


3N2
2


(AV2
4,


A2
3=


(2.5.1)


di (N3%V)=


(2.5.2)


1 (D N
NA(D lN


- siAN) 4 (6


Sij =
si-?


(2.5.3)


To solve these equations,


we have to consider the N = constant and


$ constant


cases


separately.


a) N


= constant:


Without loss of generality


we can set N


= 1, then from Eqs. (2.5.1) and


(2.5.2) we obtain


cp = /3


(2.5.4)


^AV)
84,)-


(2.5.5)


(A')


(2.5.6)


Notice that these equations are similar to the ones we obtained for the pure


magnetic


case.


The field


4 satisfies a Liouville equation


(Eq.


(2.5.4)), and,


from Eqs. (2.5.5) and (2.5.6),


can be obtained in terms of the solutions of


that equation.


However there are differences


between


the two cases.


The above equa-


tions have solutions only for nonnegative A as follows from Eq. (2.5.6) (in the


mnornetic caspe we have A


This imnlies (PFn


(21..4V that,


13 has to he


A ln =
(a


'06ifJ






depending on the sign of ft.


Here we note that because the B = 0 case corre-


spends to flat space-time solutions, i.e., Aln$ = 0 and AV


= 0,


we do not


consider it here.


Instead,


we discuss the more interesting /3


> 0 and 0


cases.


We can say that because 3 > 0 in this case,


solutions while in the magnetic


case


we obtain only one class of


we have both classes.


# constant:


To solve our equations in the N


constant


case


we follow similar tech-


niques used in ref. 2 to obtain multiparticle solutions for the Einstein equations

with nonzero cosmological constant. Note that our solutions are more general,


since the solutions of ref.

case in our notation), wl


We start with separating Eq


2 correspond to nonrotating sources (a = N3 = 0

while ours describe rotating sources as well.


(2.5.3) into the spatial trace


--AN + -N a2
NQ 22N4


A 2
3'


(2.5.7)


and into the traceless part


j kMk = 0 ,


2.5.8


where Mi = 1-iN


a =


NA = constant,


(2.5.9)


as follows from Eq. (


Note that if we define a complex function M


MI +iM2 =

for M, whici


'aN


then Eq


. (2.5

= M(


I is solved by


.8) becomes the Cauchy Riemann equation

z). Equations (2.5.1) and (2.5.7) become


SA 2 4
5^ 2N4


0 ,2
3


(2.5.10)


b) N


9iM + ijM






After multiplication by


9tN


and integration with respect


to Z,


(2.5.11)


becomes


z)zN ( N2
6 b


+ 2 -2) e(z)
4 2


(2.5.12)


where e(


z) is an arbitrary integration


"constant"


If we introduce a real pa-


rameter


z
1/
2 \


dw
M(w) +


(2.5.13)


then Eq. (2.5.12) becomes


8 N


= N32
(A o0


(2.5.14)


This


is a first order ordinary differential equation for N(()


the solutions are


real only if


e is a real constant, and they are given by standard integrals through


N(()


N(c~o)


- CO


(2.5.15)


The solution for the spatial conformal factor 4 is given in terms of the solutions


of Eq. (2.5.15),


z) and the constant parameters A, (o0, a and


( 2N2


2M(


as one can


z)M(


see from Eq. (2.5.1


Thus in the N


2
a 2-2


(2.5.16)


and the definition of M(


constant case, the solution is given by Eqs. (2.5.9),


(2.5.15)


and (2.5.16) in terms of an arbitrary holomorphic function, M(


z). Once M(


is specified,


the explicit forms of N(x),


V(x) and


x) are obtained by the


above equations.


2.6 Stationary


Axial


Symmetric Solutions


SN-


N-2


Maw)
M(w})


N+2


2+e


+ E






given by Eq. (2.3.1


with N


= 1 and 7ij = bnij and the problem reduces to


the solution of a Liouville equation (Eqs. (2.4.5) and (2.5.4)) for the spatial


conformal factor 4


. This means that


the spatial part of the spacetime is of


constant curvature (negative if /3


> 0 and positive if Q


<0).


In order to sim-


plify our discussions,


we consider only axial symmetric solutions.


The most


general such solutions are given in terms of two real parameters, a and v


the radial coordinate


r = q/(x2


+ y2)


as follows [25]


2 2v
a


+ a2V)2


8v2a2Y
, =8
22v


- a2)2


<0,


(2.6.1)


>0,


(2.6.2)


case


would correspond to


the flat solution, A4


= AV


= 0.


The parameter a can


be absorbed into r


by introducing (


The other


parameter, v, has to be nonzero; otherwise


b would be zero.


Because of the


invariance of 5 under inversion, r


- 1
r


, it is enough to consider the v


case.


the magnetic


case,


space-time components Ki = eij jV


of the metric


are given by Eqs. (2.4.4) and (2.4.6):


f\2 T -,- ^^


VI -


2vr2V
r2v a2v


(2.6.3)


In the


B=0


case,


as one can see from Eq.


(2.5.6), Ki is given by the same


expression,


constant b in Eq.


set b=


(v 1).


with different numerical factors.


We choose the integration


(2.6.3) such that IKi is nonsingular at the origin; that is,

Then, the line element in spherical coordinates reads


n


Ei =


1.


|I --







Note


the metric is singular at


- 00


because 4


vanishes there.


the f3 > 0 case it is also singular at r = a.


The latter is very much like the


case of solutions of Einstein's equations that describe a rotating fluid.

also that because the diffeomorphism invariant quantities such as th


"""" ~""~~r"~"` "~"" r"~*'1


Notice


curvature and the length of the Weyl vector are nonsingular everywhere (they

are constants) these are only coordinate singularities.

Since we have an explicit solution, we can complete our perfect fluid analogy

discussed previously by calculating the normalized velocity. From Eq. (2.4.13)

we find that in spherical coordinates


U0 =


ur = 0 and


U, =-


-79


(2.6.5)


a2v


where we have used that AgKi =


Ai in the purely magnetic


case.


Thus our


solution corresponds to circular flow with vorticity, va = --.-ap 72vy:
Wl~l '


2
= -50
2


that is


Be:


, vi=O ,


, only the time component is nonzero, and it is constant.

fore we proceed with the discussion of the space-time structure of our


solution,


we note that the v


case always can be brought into the form of


the v =


case


by rescaling the radial coordinate.


As we shall see,


the only


difference


that the range of the angular coordinate will change.


To see this


let us define a new radial coordinate,


(= a


In terms of


the line element becomes


e scalar






where the new angular coordinate 4"


= v4 ranges from 0 to 2ruv (if 0


27r); that is, the points with Ib


and 4/


+ 2rrv are identified.


In the following we consider the positive and negative 3S cases separately.


We consider the


v =


1 case to simplify our discussion.


a) >0:


The solution of the Liouville equation is given by Eq. (2.6.2).


The metric


in this case is given by


(2.6.5) with


the lower sign, and it is singular at


=1.


ce, we have to consider the (


< 1 and C


cases separately.


< 1, the change of the radial coordinate


- C2) gives the following


line element


= -(dt -


2
2 d 2dg2)


(2.6.7)


where


- 00


27 and 0


A final,


"hyperbolic,"


formula is obtained by defining F = sinha,


= -(dt --(cosh
/3


a -


1)dw)


2 2
/ (


+ sinh2 ad 2) ,


(2.6.8)


and 0


Similarly for E


> 1, the change of the radial coordinate


- 1) gives


+ 1 + 1)d, )


db2) ,


(2.6.9)


2 d(2
+ {(- +
3 i2+ 1


where -oo < t


oo and 0


. The final


" hyperbolic "


formula is obtained by defining ( = sinh a:


= -(dt + (cosh + 1)d0})
/3


+ (d2
-


+ sinh2 ad4'


(2.6.10)


where 0 < a < oo.


We note that this is the metric of thpe flel univprsp [9Rl


-0
= -(dt + /


nI


< J>


( = (


2(/(1





27

We have already mentioned that our solution is conformally flat (the Cotton


tensor is vanishing).


Here


we show that one can find a set of coordinates, in


terms of which

time. In the v


, the form of the metric reduces to the flat solution, with periodic


= 1 case, the metric is given by


= -(dt + Kidxi)2


+ $dxdxzi


(2.6.11)


where Ki and 4 are given by Eqs. (2.6.3) and (2.6.1)-(2.6.2), respectively.

us introduce new coordinates,


= r2


and denote the corresponding angular coordinate X.

of these coordinates is


(2.6.12)


The line element in terms


= -(dt +


+ (dp


+ p2 dx


(2.6.13)


where one has the lower sign if r < a, and the upper sign if r


Let us make


one more coordinate transformation:


(2.6.14)


Then the metric reduces to the familiar form


= -(dr z dx) + a-(d2
'IP


- o a(dpi2
+y(^


+ p'2dx


(2.6.15)


+ p2dx2)


where we have introduced a new periodic time coordinate


(2.6.16)


-a 2 ,


= -dt'2






< oo, but the coordinates cover only the 0


r < a or the a < r


part of the spacetime.


Let us make the following coordinate transformation:


sin2 2
2 (2+1


t
(2.6.17)


,0 <


< 00


- 00


< 00 .


(2.6.18)


In these coordinates the line element has the form


= -4 (dT


_+ (dO2


sin2


+ sin 2d2) .


(2.6.19)


As we have already mentioned the spatial part of the metric is a two dimen-


sional sphere.


And because the curvature of this


sphere is 21R =


-Aln,/4 =


IiI, the factor in front of the spatial part, 2/|/ 1, is the square of the radius.

The Euler characteristic, j f d 2xJ 2R = 2, is that of the sphere.


The metric is regular everywhere, except at 0 = it,


string type singularity.


where it has a Dirac


One can remove this singularity by introducing a new


time coordinate


1
+ ^-r
W-~C


(2.6.20)


The metric then becomes


-- cos
La'


+ ~ (dB2


+ sin2 Od 2).


(2.6.21)


This is regular at 0 = x, but not at 0 = 0.


0, 4) to cover the northern hemisphere (0

9, ', ) at the southern hemisphere (2 < 0


One can therefore use the coordinates


_< ), and the coordinates

. Because i is an angular


b) 0





29

on the sphere and the curvature (or radius) of the sphere. Namely, in order for


the field to be regular, single valued with time dependence eiwT


- 2~ integer,


, the equality,


(2.6.22)


has to hold.


It is interesting to notice that the form of our metric is


the 3+1


dimen-


sional Taub-NUT


solution for a fixed radial coordinate [28


Because we can


choose r


-* o00,


we can think of our solution as the boundary of the Taub-NUT


solution.


The topology of the boundary (and of any r = constant surface) is


locally


, but globally


t is that of a deformed sphere in the following sense.


Killing vector field


where the


defines


is parametrized by 6 and 4


a nontrivial Hopf fibration:


and the fibres are circles.


- S2


Thus the


topolgy is a


"twisted product"


xS2


. Thus the solution can be characterized


with the Hopf invariant of the mapping from the compact three dimensional

space-time manifold to the two dimensional spatial part, and with the Euler

characteristic of the latter.

As in the positive If case the solution is not only conformally flat, but also


can be brought into flat form,


In the v = 1


with unconventional range of the coordinates.


case the metric is given by


= (dt + Kidx


+ 4dx'dxt


(2.6.23)


where Ki and 4 are given by Eqs. (2.6.3) and (2.6.1)-(2.6.2), respectively.

us introduce new coordinates,


(2.6.24)


= r2


+a2







and denote the corresponding angular coordinate X.


The line element in terms


of these coordinates is


= -(dt + 2od)2
(4 1 __


on (dp2


+ P2dx


(2.6.25)


Let us make one more coordinate transformation


(2.6.26)


The metric then reduces to the familiar form


2= -(dt + ) + (dp2


+ p'2dx


(2.6.27)


where we have introduced a new periodic time coordinate,


=*+ 20x


(2.6.28)


The metric given by Eq. (2.6.20) is flat, but again the range of the coordinates


is unusual, t and t + integer


x o are identified, and 0


p' < 1/a.


Notice


both


cases,


metric can


transformed into


Minkowski form, if we introduce a new periodic time coordinate.


this feature,


Because of


we suspect that our theory is equivalent to a finite temperature


one.


We have discussed our stationary solutions in


the axial symmetric


case.


We found that in the positive and negative / cases the solutions have different


properties.


In the Einstein case (


correspond to 2+1


Ei =


B = 0) one has only the solutions that


dimensional Godel universes, because /3


In the case of


nonzero magnetic field (B


, Ei = 0) however, one can have the solutions


= -dt'2


Ott (dp12
+ T^


+ p'2dx2) ,





31

We have also observed in Sect. 2.4, that in the latter case our solution is


similar to that for a rotating "Chern-Simons"


fluid.


As we have shown, this


solution is causal only if p


the Taub-NUT


3
KTm-


This means that the fluid analogy holds for


case, and for the GSdel case with /


- -, but in the latter the


pressure is negative.













CHAPTER 3
RENORMALIZABILITY OF


D = 3 TMG


3.1 Introduction


By now it is well-known that perturbative quantum gravity in four space-


time dimensions suffers from the problem of nonrenormalizability.


This may


be cured by


going to lower dimensions, but in


this case the theory is much


less interesting,


because gravity in D


the absence of matter


has no


dynamical degrees of freedom.


topologically massive gravity


Recently Deser and Yang [9] have shown that


three dimensions has the possibility


being renormalizable. Because this theory is massive, it does possess dynamics


even in three dimensions.


Although such a three dimensional theory clearly


does not describe the universe in which we live, it would be of great theoretical

interest to find such a renormalizable theory with the symmetries of gravity.

Deser and Yang have shown, by using an unusual parametrization of the


metric,


that


TMG


has power counting behavior consistent with renormaliz-


ability.


This by itself does not establish the result, because one needs to show


that the theory may be regulated in such a way to preserve both the theory's


gauge invariance and the desirable power counting behavior.

the newly discovered nonlocal regularization [11] to this theory.


We will apply

We will show


that using this regulator, the possible obstacle to renormalizability discussed


Deser and Yang does not appear, and that if this technique is valid,





33

technique depends on the existence of a functional integration measure which


is invariant under the nonlocally generalized gauge symmetry,


at this time been proven.


3.2 Power-Counting Renormalizability and Gravity


which has not


Most gravity theories are not


power-counting renormalizable due in part


to the presence of a coupling with negative mass dimension.


To determine


whether any theory of gravity has the hope of being renormalizable we look at

the generic ultraviolet behavior of L-loop diagrams in d space-time dimensions.


First,


we note


that


n all


geometrical gravity


theories,


the propagator and


vertex have reciprocal power behavior.


For example, in Einstein theory the


propagator A p


-2 and the vertex V


~ p2 in any dimension. Higher derivative


terms suc

different


as R2 and R3


behavior.


can be


added to the Einstein action with somewhat


Adding an R2 term introduces p


dependence into the


propagator which improves the


properties of the theory, however, such a


theory is either not unitary or not causal or both. Adding higher powers of

R does not affect the propagator but worsens the UV divergences because the


vertices contain higher powers of moment.


Assuming this generic reciprocal


behavior


, the divergence of a one loop


n-point function is proportional to


ddp(AV)"


, Ad


Because of the topo-


logical relation


= N Nv +1,


(3.2.1)


where NI and NV are the number of internal lines and vertices,

higher loops have one more power of propagator than vertex. Each


respectively,


loop also has


I i i I i A I,- 1 d-






number of counterterms we must have d r


Because unitarity forbids


r > 4 and there are no propagating degrees of freedom in a pure gravity theory


in d=


the only possibility is d = r = 3.


We will


see that


TMG has this


property.


3.3 Topologically Massive Gravity


The action for TMG is given by SE +


where the Einstein and Chern-


Simons terms are respectively


d3 xv/ R ,


(3.3.1a)


Scs =


L
-


d3x ePI ,(P 8(, pF,
) t,-Aq-t1pv


ir )r vp)


(3.3.1b)


The field equations are third order in derivatives of the metric, and they are

given by


- 1 Cpv


=0,


(3.3.2)


where G^P and CI"'


are the Einstein and Cotton tensors, respectively. Eq (3.3.2)


can be split into a trace


R = 6A ,


(3.3.3)


and a trace-free part,


1 R)


(3.3.4)


Just in


case


of Einstein gravity the solutions of Eqs (3.3.3) and


(3.3.4)


are spaces with constant curvature, that is, de Sitter (A


> 0), anti-de Sitter


< 0), or flat (A = 0). But unlike three dimensional Einstein gravity,


TMG


has a single dynamical mode, a graviton with mass m = UK.


If, as usual, one expands the metric about the flat background.


-2


--g


SE = K


= -(RI1







one finds that the h propagator has both p


and p


components and one


cannot apply the simple power counting arguments discussed in the previous


section.


Instead


, following Deser and Yang 1) we parametrize the metric ac-


cording to


(3.3.6)


gYu = + + hy1,) = 4 g$ y .

where h satisfies h2 = 0!2) The action becomes


d3 x x/-


[8g9VQ9^ + 2R()) + (


(3.3.7)


Since the Chern-Simons action is conformally invariant, it is unaltered by this


rescaling.


Gauge-fixing is performed by setting h"V


,- 1 hP"


this neces-


states the introduction of a Lagrange multiplier B1, and ghosts b6, ci


. The


resulting gauge-fixed action is


SGF = S


+ S+


where


B1 1, h V


(3.3.8)


is the gauge-fixing term and


SG =-


+ D!"c


- 39' (qa^g D(c )]


(3.3.9)


is the ghost action.


Here D is the covariant derivative with respect to g.


total action is then invariant under the BRST transformation


S = #(c"


cAy)


- c


A,A


(3.3.10a)


Sht"


+D"


= CV,"I


+ cC,"


A + hV"c


A h"Ac


(3.3.10b)


- 3
3


+ hYi)(c


,A + hafa,a)


2 la Dc


Ir-


SF =


-1'Scs(g) .


,M + h "


=D^ c"


+ h c






= c c1


Sba =


(3.3.10c)


(3.3.10d)


Bo = 0 .


(3.3.10e)


The propagators are


1
4"


< h Vhaf


-4pT(PeYa pZV


+ evf pm) ,


(3.3.11a)


(3.3.11b)


-2 1
p (?77pv P


-2pppv)


(3.3.11c)


where


= ?r7v


pjppV


Thus


with


parametrization


(3.3.6),


propagator


has the desired p


-3 behavior.


The vertices may


be found by


expanding the


action


desired


order


Appendix


they


clude terms of the form 4)


n >

fields


,3(Viph)n+l and bO2(c/ihn-1) where


2. Note that the vertices from the Einstein term contain exactly two 4

. while the other vertices contain none.


We now notice that negative powers of m (equivalently, positive powers of

n) may never appear in any Feynman diagram, since they do not appear in any


vertices or propagators, hence K is a super-renormalizable coupling.


expansion parameter p i


The true


s dimensionless, providing the first indication that the


theory may b


e renormalizable.


To prove the naive power-counting argument


we determine the highest degree of divergence D of any L-loop diagram in the


theory.


Let Nf


and Nt,


be the number of internal lines of species x and the


number of vertices of degree 9Y


, respectively.


The degree of divergence is then


+ vcY pIPfl


+ e37 Pva


>=z


> = z


-BQ ,


< $


< bpc,


8 (W)n-1







Using (3.2.1) in (3.3.12) one obtains


(3.3.13)


where Nuh
V/


and N? are the number of ghost and vertices respectively. Since


any vertex has at most two ghost or two $ fields, the terms in parentheses are


nonnegative and thus the degree of UV


divergence is always


These diver-


gences can be absorbed into the coefficients of the Einstein term of dimension

-1, the Chern-Simons term of dimension 0 and possibly a cosmological term


of dimension


thus the theory is power-counting renormalizable.


The loophole in the above argument is the assumption that gauge invari-

ance may be maintained in a regulated version of the theory without giving up


the desirable power counting behavior mentioned above.


Without the use of


such a gauge invariant regulator, there is the possibility that additional terms

might be required to cancel the gauge transformation of the effective action,


and that these terms might contain negative powers of m.


To see if quantum


corrections to the theory violate the BRST invariance through such a term, we


look for nontrivial solutions to the BRST


cohomology problem as follows. Let


A = QF be the possible violation of BRST


symmetry,


where Q is the BRST


transformation and P is the effective action to some loop order.


We consider


general solutions to the cohomology problem


QA=0,


=0


(3.3.14)


If the solution is trivial


A= Qr'


we can add Fi


as a counterterm to P


to cancel the anomaly.


If the solution is nontrivial and


is indeed generated


D = 3 (Ngh


- -(N$ -)





. I


I~






determine,


through such an analysis,


that


there is one such possibility:


term


,a= -


arises from


+ higher order terms,


the BRST transformation of the effective action,


terterm necessary to cancel


this term


(3.3.15)


then the coun-


will add negative powers of the mass


and hence ruin renormalizability.


Deser and Yang showed that to one loop


in dimensional regularization this term does not appear.


Unfortunately, since


the / function for p vanishes to one loop, one cannot apply the Adler-Bardeen


theorems [29,30,31] to conclude that it cannot occur at higher loops.


to determine whether or not this


Thus,


term arises, one must use a suitable gauge


invariant regularization.


3.4 Nonlocal Regularization


In this


section


we review the method of nonlocal regularization.


Details


may be found in ref. 11.


Consider a generic action in d space-time dimensions


which can be written as a free part plus an interacting part:


S[i] =


ddx iFijfij + I[f]


(3.4.1)


where qij


are fields of any type,


and Fij


course contains derivatives.


define the nonlocal smearing operator


S--exp I


where A is the regularization parameter.


(3.4.2)


The local limit is obtained by taking


-+ oo limit.


Our convention is


the derivatives


in an $2


act on


everything to the right, unless otherwise specified.







For each field qSi,


we introduce an auxiliary field 4'i of the same type, and


construct the regulated action


ddx (i


~j- i~iz51 +IMt+ 1


(3.4.3)


where


-1)


(3.4.4)


It is to be understood that i are auxiliary fields which are to be eliminated

using their equations of motion:


=0


(3.4.5)


Multiplying (3.4.5) by O


we obtain the unique solution for 4


as a functional


ti[]W = QOij


(3.4.6)


Equation


(3.4.6) can


be solved iteratively for t.i


The solution for 4i


has a


convenient graphical expression:


the unregulated theory,


0'i is given by


with a factor of 2


evaluating tree amplitudes of


- 1 on each propagator (see ref. 11


for details). Substituting this solution into (3.4.3) gives the nonlocalized action

for the 6 fields,


S([] = S[,()].


(3.4.7)


Suppose that S[] is invariant under any symmetry


Ti[ef]


(3.4.8)


Let T


consist of a linear part plus a nonlinear part, T


, then S[j, ]


as defined by (3.4.3) will be invariant under the new symmetry


(3.4.9a)


b6i =


= Tt


84;=


[, ] =


/ 3- -


bl[# + ^}]


+rTn


Tyl


+ E2T"i[ + 1],






where


K-l1 -1
. 3


(3.4.9c)


62M-~b


In order to obtain (3.4.9b ) one must use the equation of motion (3.4.6) for the


& field.


Note that the nonlocalized symmetry transformations can be chosen


such


linear part


is independent of


the auxiliary fields.


Generally,


must


be gauge-fixed in order that


we may


solve for W [];


then


symmetry T


represents the BRS symmetry of the gauge-fixed theory.


Classically, the nonlocal action Sf[] is equivalent to the original S[4]


former being obtainable from the latter by some field redefinition.


ence arises upon quantization.


The differ-


The old functional measure does not exist in


the new basis due to ultraviolet divergences.


To quantize the theory a new


measure must be constructed which is well-defined in the new basis, is analytic


in the moment, and obeys the symmetries of the theory.


The invariance of


the quantum theory under the nonlocalized symmetry requires the


invariance


of the functional integral


Di([Di] (o]Gauge fixing) exp (iS[])


Although


(3.4.10)


the full action including gauge-fixing terms is invariant under the


symmetry transformations, a measure factor


[])= eiSM[] ,


must be introduced to insure invariance of the functional measure:


([D4o][w]) =0 .


(3.4.11)


(3.4.12)


S[ ,]





41

The condition of Eq. (3.4.12) relates the variation of the measure factor to the

Jacobian of the transformation via


SSM [ -4


661 i
s & m J


(3.4.13)


Tr{ tJ + w] Ojj Kim + 4 t
IZJ 9kL3'L


where the second equality uses (3.4.9) and the trace is over space-time coor-


dinates.


We can use (3.4.13) to solve for the measure factor order by order,


resulting in a completely invariant theory.


In practice, this is difficult to do for


higher order terms, and it is hoped that further study of nonlocal theories will


reveal easier ways of generating the measure factor.


We must also note that it


has not yet been proven that it is always possible to construct an appropriate


measure factor to all orders.


If such a measure factor does not exist for the


theory then a local symmetry is potentially anomalous.


For our arguments


concerning TMG, we will be assuming that an appropriate measure does exist.


3.5 Nonlocal Feynman Rules


We have described how to obtain the nonlocal action (3.4.7) by solving the


auxiliary field equation of motion.


However the


Feynman rules for Green


functions


theory


are inconvenient for


calculations due


to all


interactions induced


when


the auxiliary field is eliminated.


Instead


we will


work with the Feynman rules derived from the action S[#, 4


which are closer


to those of the original theory and enforce the condition


that


satisfy its


classical field equation by requiring that


there are no closed loops consisting


of only 4 lines (4 must be on-shell in any diagram). Since one is interested in

amplitudes involving the physical field no auxiliary fields appear as external




42

The general Feynman rules in the theory in terms of S[4, 4'] are as follows.

The 6 and 4 propagators are


exp
ex(p (r A2


(3.5.1)


-io=


-_2


2 exp


F


(3.5.2)


respectively.


The field is indeed an auxiliary field which should not appear on


any external legs,


as its propagator, from Eq. (3.5.2), has no pole.


The vertices


are of the same form as in the local theory.


The higher induced vertices in


the S[#] theory are obtained graphically from the S[(4, ] theory: they are the

connected tree diagrams which follow from using the local interaction vertices


but with propagators replaced by -iO (i lines).


There are also vertices from


the measure factor which will be connected only to 4 lines.


reducibility of Feynman diagrams in the theory in terms of


Questions such as


[4i, f] are resolved


as in the S[ ] theory with the additional requirement that 4 lines cannot be

cut. Feynman rules for nonlocal TMG are collected in Appendix A.


We have shown that


there should exist an appropriate measure factor in


order to have a well-defined anomaly free quantum theory.


In most


cases


extremely difficult construct it, however it can be computed perturbatively to

any order in the coupling of the theory. If we expand Eq. (3.4.13) we can obtain

a set of Feynman rules for calculating the variation of the measure factor under


the nonlocal symmetry.


Writing


K = Oik (Skj Okl51
/ 62 1


= Oik(6kj + kl


(3.5.3)


+ Okl Ji m
0(f~i0(f,


621
Omn
C~rmTT-7~T


iS2
F+ie


f00


=-i


= -i





43

and inserting this in (3.4.13), one obtains


SSM ] =-


-Tr S% 6 r c []Sk


62I
+ Oki
kSl 16m

-. .)**[*>


(3.5.4)


+ Okl
6

np S pkm
0 bpm


We may read


diagrammatic rules from


this expression


by writing it in


momentum space.


Since there is only one trace over space-time coordinates,


we need only


look at one loop diagrams.


Each diagram has a single vertex


factor coming from


The remaining vertices are arbitrary in number and


are the same as those discussed above.


The first type of vertex always connects


to an internal line with


"propagator"


The other vertices connect either to


two internal lines with "propagator" 0 or to one internal line of each type.


external legs correspond to either or fields.


TMG are given in Appendix B.


These diagrammatic rules for


By computing all one loop n-point diagrams of


this type one obtains a perturbative expression for SSM which must be inverted

to get the measure factor SM.


3.6 Renormalizabilitv


We now apply this method to TMG. We associate auxiliary fields P


with the fields 4


, hlv,


cP respectively.


In this field basis each field is massless,


so the smearing operator is simply


= exp (2 /A2)


(3.6.1)


The gauge transformation laws for the fields then become nonlocalized accord-

ing to (3.4.9a ):


I/1 >T., [ \


IL f IL\PL'I.( ,\


kty,


EC- <2







bhP"


= cV,


+- CP ,


,a + 2 (h+k)(c


(3.6.2b)


iP(h + k) p(c + d)Q,

(h + k)"V(c + d) ,

(h + k)0"(h + k)ap(c + d)aP]


= E2(c + d),(c


(3.6.2c)


-Ba ,


(3.6.2d)


6Ba= 0 .


(3.6.2e)


We see that nonlocal regularization gives a regulated theory which is au-


tomatically


BRS invariant, but it must be checked that the desirable power-


counting behavior of the unregulated theory still persists.


Negative powers of


m could be generated either in the measure factor, or by the loop integration


themselves.


We will examine each of these possibilities.


We now


examine (3.4.13)


to determine which


terms could


possibly


a contribution of the form (3.3.15). Tr cannot contribute to this term.

This is because and b, c each couple only to h, so every tree graph which con-

tributes to 4' or dp must include at least one h. By the same token, each term


in k" includes either an h, a pair of ghosts, or a pair of b's with derivatives


on them, none of which is


what we are looking for.


So the 4 term does not


contribute to the possible anomaly.


for the same reasons.


c ghost term also does not contribute,


Since b and B do not contribute to the measure factor


at all, the only possible contribution is


from Tr


We find from (3.6.1b),


that this contribution comes from the diagram shown in Figure 1.


Since its


6b,=


+d)"


rnUca


- (c + d) (h + k)r"


+ (h + k)" (c + d)"


+ d)g ,







contracted into after the internal momentum has been integrated over.


is necessarily zero by Lorentz invariance.


This


Therefore there are no contributions


to the measure factor of the form (3.3.15).


C
a11111


Contribution to the measure factor of the form of Eq. (3.3.15).
Here a wavy line with a bar corresponds to the k prop-
agator where k is the auxiliary field field for h. A wavy
line with a dot corresponds to a "propagator" given by
the smearing operator 2 as described in Sect. 3.5.


the remaining we show that such powers do not


arise from the loop


integration themselves.


This situation


would


correspond


to divergences in


the limit where m


-*0,


A remains finite.


Since nonlocal regularization


regulates all int


egrals at p -*-


, unregulated divergences can only occur for


p -+ 0, it i.e.


in the infrared, which is not at all affected by this procedure.


contrast


to ultraviolet divergences,


which are determined


by the net


effect of all the propagators around an entire loop, infrared singularities are


determined by a single propagator, or any group of propagators,


whose mo-


mentum goes to zero.


If we let all the loop moment b


e independent and keep


r nm avr C art tn n r, nri r' n' A1 4. n C r,^ nn ,l, nfl .i a tar4: nnn an i-i an ..rn nr1a 4 Vt r 4. *


Figure 1.







ghost propagators actually help matters because they


go like .
P


The only


possible divergences of this type will be when one or more of the momentum-


conserving delta functions give 6 3(0).

going into the vertex go to zero. He


This can happen if all of the moment


iwever, each vertex contains derivatives,


which will in momentum space give powers of the moment which will neces-


sarily soften these singularities.


Specifically, each vertex either contains three


derivatives, which softens the singularity to at most logarithmic, or two deriva-


tives plus 4 or ghost propagators or positive powers of m, or both.

there is no possibility of a power law singularity for m -- 0.


In any case


In higher


oops,


there is the possibility that a 6(0) singularity might


generated not by one vertex but by a combination of two or more.


This can


happen when some subgraph is imbedded inside another graph.


For m


- 0


this graph may be singular when the momentum on the line connecting the


subgraph to the other graph is zero.


If the momentum factors associated with


the vertices lie on the subgraph then they will not cancel the singularity. In this

case we must argue by induction against the possibility of any problem being

caused. Suppose that the theory is proved to be renormalizable to N 1 loops,


and that this renormalization has been carried out.


diagram of the type described above.


Then consider an N loop


The subgraph of this graph has fewer than


N loops, so by assumption the theory must at this point contain counterterms

to make this subgraph finite. But by simple dimensional analysis, the subgraph


must have dimension 3 n/


where n is the number of 4 or ghost (not h)


lines coming out of the subgraph. Since the subgraph plus counterterms must

be finite, the graph cannot achieve this dimension by being proportional to the


l-PI .. 1 1 1- -., *


nn /- I r^ i- -. A







the singularity from the delta functions to be at most logarithmic.


Note that


logarithmic infrared singularities are not a problem as they merely indicate the

presence of In A/mn terms.

We therefore see that infrared power-law divergences do not arise as m -*


0, so that negative powers of m do not arise from loop integration.


Thus,


assuming that a gauge invariant measure exists for the nonlocal theory,


theory is indeed anomaly free and hence renormalizable.


Our result will still


hold if


this assumption is false, if the violation of


gauge invariance is such


that the noninvariant terms in the effective action vanish to all loop orders in


the local limit, A


-> 00.


Since


TMG


has no actual gauge anomalies, this is


a reasonable assumption, but is by no means a foregone conclusion.


Thus at


present we have discussed but one approach which gives strong support to the


conjecture of Deser and Yang.


Our result cannot be considered a proof until


the existence of the appropriate measure factor is established.













CHAPTER 4
THERMO-FIELD DYNAMICS OF BLACK HOLES



4.1 Introduction


It is known that the quantization of fields on spacetimes with causally dis-

connected regions leads to particle creation from the vacuum as a consequence


of the information loss


associated


with


the presence of the event horizon(s)


In stationary


as a stationary black hole,


spacetimes with a simply connected event horizon (such


an accelerated observer in Minkowski spacetime,


or de Sitter type cosmologies), the emitted particles have a thermal spectrum


This result has been first obtained by Hawking (14] for black holes.


has shown,


Hawking temperature,


the effective temperature of this radiation is


I=
_ K


where K is the surface gravity of the black hole (units


are chosen throughout such that k


=h= c=G=1


These results have been


confirmed and derived in a number of ways, and several attempts have been

made to gain a better understanding of the features of the Hawking process


and the physical role of the event horizon [14,15,16


A particul


approach into this direction is the one used by Israel [16], who

problem of particle creation on the Schwarzschild background.


early interesting

considered the

The idea is to


quantize the fields in the full analytically extended Schwarzschild spacetime

(known as the Kruskal extension) in order to keep track of particle states on


the hidden side of the horizon as well.


The same idea allows us to apply a





49

Here we briefly review the canonical formulation of thermo-field dynamics


for free fields.


This will provide the main ideas and all the technical tools we


need (for more general and detailed discussion see e.g., ref. 17).


The central idea


of thermo-field dynamics is to express the statistical average of any operator

0 as a single vacuum expectation value


(o) = (o()


0(/)>


(4.1.1)


where p is


the inverse temperature.


This can be achieved by augmenting the


physical Fock space F by a fictitious, dual Fock space F


That is, for each op-


erator O(a ,,aj ) and each state vector I


1 anflj
,- vnj


we introduce


a dual operator O(a tw


) and a dual state


vector n>2)


W= FWI
= v",. i ao


where a .


' ., a and a j
U)] LWIJ


are creation and annihilation operators of the


wj modes (j labels the degeneracy of the energy level w) with the usual com-


mutation (anticommutation) relations.


Namely, for bosons the only nonzero


commutators, and for fermions the only nonzero anticommutators are


a j,aj,] ]= [a j,aW,,j] = 6jj,8(w -w


(4.1.2)


{a~wj, ,aj,} = {ij,


wj } = 6j'(w w


(4.1.3)


respectively.


The states I0), and


0) are the vacuum states annihilated by awj


and a.j respectively.


the direct


product


Fock space


, spanned


the state


vectors


n,m) =I n


the temperature dependent vacuum state


0(p)) in (4.1.1)


is given by a Bogoliubov transformation of 10,0)
..- 4 I I -


n =


j ),







with the 0, parameters defined by


sinh2 8 =

sin2 = =


Ce 1
1


for bosons,


(4.1.5)


for fermions.


We also introduce the operators


) e


-iG t


such


they


satisfy


same


commutation


(anticommutation)


relations


(4.1.2) and (4.1.3).


The state


0(3)) is annihilated by the operators amj(i)


and a


and the entire Fock space can be constructed successively from


0(3)) using the creation operators a4j(j) and a4


Using the above construction of


0(3)) the


statistical average of any phys-


ical operator (a functional of a4j and atj only) can be expressed
*/ *i


expectation value of the form (4.1.1).


as a vacuum


In particular, as it is easily seen from


Eqs. (4.1.4) and (4.1.5), the average number of the wj modes are given by the


familiar Fermi and Bose


distributions.


The formalism described above can easily be generalized to black holes by


identifying the physical Fock sp


zon, and the tilde space


ace


with particle states outside


with particle states inside the horizon.


hori-


The above


decomposition of the whole Fock space into the direct product space O E

corresponds to the conventional definition of positive frequency modes (parti-


cle states).


It is known [13,14,15,16] that this definition with respect to the


Schwarzschild time coordinate t (which is defined by a timelike Killing vector

field 8/Ot everywhere outside the horizon) leads to a mode expansion of the


aj(p) = e


a eiG
wi


at i


"'*( 0 = e


j().





51

mode expansion. Note that the latter corresponds to positive frequency modes


with respect to


Kruskal


time coordinate,


which defines a Killing vector


field only on


the horizon.


This linear combination is generated by a Bogoli-


ubov transformation,


which is uniquely determined by the requirement that


the fields be analytic on the horizon [15,16


The formalism described above


can be applied to the problem of particle creation by black holes, and as we

shall see, far from the black hole it leads to a thermal distribution with the

Hawking temperature.


We begin by reviewing of Israel's paper [16


The quantization of a massless


scalar field on the Schwarzschild background is considered by using thermo-


field dynamics. We conclude that the results obtained this way are equivalent

to the earlier ones. In the rest of the paper we describe the possible generaliza-


tions of this


approach. In Sect.


4.3 an approximate multi-black hole solution is


considered as an example to demonstrate how to extend the method to space-


times with many causally disconnected regions.


Finally in Sect.


4.4 we derive


the Hawking radiation of a black hole emitting neutrinos and antineutrinos.


4.2 Massless Scalar Particles on the Schwarzschild Background

We first consider the creation of massless scalar particles in the gravita-


tional field of a black hole.


We consider the Schwarzschild metric


= (1- 2M)dt2
V


- (1 2M)- dr2
r2
-(i- -r~


- r2 d2


(4.2.1)


as an example, and look for solutions of the massless scalar field equation,


1(
J--( ',CL g (-g)- 2),y = 0 ,


(4.2.2)


1 .1 P 1 r r 1 Psi r*








fwl obeys the equation:


d
~dr*T
dr*


d
dr*


(4.2.4)


S2M ( + 1) (r)
-r r2 /


r*=r + 2MlnI-2 -1
p2


(4.2.5)


Let us consider the solutions of Eq. (4.2.4) which correspond to outgoing


modes at the past horizon I"-


Near the horizon these solutions behave like


fwlm e


-ZWtU


(4.2.6)


where w


> 0 and u = t r*


This is the usual definition of positive frequency


states with respect to the Schwarzschild time. Everywhere outside the horizon,

4 is a timelike Killing vector.


It is


also possible to define positive frequency modes with respect to the


Kruskal time coordinate U.


In Kruskal coordinates [15],


= -4Me


U
4M


= 4Me4M


(4.2.7)


S= t r*


the Schwarzschild metric has the form


= 2Mr


r-
2M


dUdV


- r2


(4.2.8)


On the past horizon T-C


, 9/OU is a null Killing vector.


Note that fwim defined


by Eq. (4.2.6),


fwlm e


-zwun


-iw lnlU


(4.2.9)


is also a complete set of positive frequency solutions (w

the KrniTkal time T_


> 0) with respect to


Here


v=t


+r*







Since 7-1-


divides spacetime into two causally disconnected regions, the one


outside the horizon (region I) and the other inside the horizon (region II), two

"Schwarzschild" modes can be associated with any given solution fwimn:


1 Ym9
2i riwI


- "-lnjU
A*, 'r


outside the horizon


inside the horizon,


(4.2.10a)


F


O

=< -7^^~


outside the horizon


,p)e+ln U


inside the horizon.


(4.2.10b)


Note however that


are singular because U


goes


to zero on


the future


horizon.


On the other hand, the following linear combinations:


H-t/+


= F+)cosh, + F(


sinh0u ,


(4.2.11a)


- F(+) sinh 8


are analytic in the lower half complex U


cosh 0,


plane if tanh 6, = e


(4.2.11b)


The modes


are positive (negative) frequency "Kruskal" modes. Both F )
Wj


and H(
Wi


are complete sets of modes, satisfying the orthonormality conditions


(F


F))=Sjji6(w-w ')


(4.2.12a)


(H'I


) = -6jj,'S(w w


(4.2.12b)


with respect to the Klein-Gordon scalar product,


( F,2)= i


(FrO* F2 F2&pFf) ndZ ,


(4.2.13)


where nP is a future directed unit vector orthogonal to the Cauchy surface

and dE is the volume element in E.


Ti rinn+;'7n rtb0 1 crnClnr- fn1A di ; +ormc F f /!/


nirP nYvnn~n


, p)e


FW+)


F


+ F,


-^n\







where a ,
Uwj


and a()


are creation and annihilation operators,


respectively,


obeying the usual commutation relations, that is the only nonzero commutators

are


,a, =t 6j6(w- w').


(4.2.15)


The alternative expansion of (c(x) in terms of H )
wj


(+t,)(t)W a


+a


t(K)HI


+h.c.)


(4.2.16)


The operators a(:t (,)


a .(c)
WI


also satisfy the commutation relations


.15), and are given by the Bogoliubov transformation:

a(() = exp(-iG)a exp(iG)


(4.2.17)


= cosh9 a()


- sinh9 a


where the hermitian operator G is defined by


i (+)t (-)t
^^J Wj


The physical vacuum state near the black hole, the


--a ( a
~wI w


(4.2.18)


"Kruskal"


vacuum, is de-


termined by requiring that freely falling observers encounter no singularities


as they pass through the horizon.

annihilated by the operators a),
wn


0) denotes the "Schwarzschild" vacuum


then the "Kruskal" vacuum:


0(K)) = exp(-iG) 10)


(4.2.19)


is annihilated by the operators a(t) (n).

Far from the black hole, at infinity, the observable quantities are the vac-


uum expectation values of the operators of the form O(a4


the Kruskal vacuum. In particular the average number of


a i) calculated in


wj modes at infinity,







given


a thermal


distribution


with


temperature equal


to the


Hawking


temperature TH.


4.3 Many Black Holes


The method can be


extended to spacetimes with more than two causally


disconnected regions.


Because such a solution is not known


we consider the


idealized case of N well-separated black holes as an example.


Our assumptions


are as follows: (1) they can be considered static; (2) far from the black holes the

metric is approximately Minkowski; (3) the radiation emitted by the individual


black holes is uncorrelated.


scalar field equation


By approximation (1) and (2) the solution of the
. e-iw(t-r) and near the horizon of the i'th black


hole the metric is approximately Schwarzschild.


Thus, fwim,


~ e i where


i=1,2...N and ti are the surface gravities of the horizons.

With these assumptions a linear combination of the "Schwarzschild" modes


can be found which is analytic everywhere.


The creation, annihilation opera-


tors and the corresponding vacuum state are given by a Boguliubov transfor-


mation.


The new vacuum state will not appear to be empty for a stationary


observer far from the black holes.


The spectrum of the emitted particles (in this


idealized case) will be the sum of the individual black hole's thermal spectra.


We first consider the N=2 case.


The solution of the field equation is


such


fwim


e
C


:iw(t-r)
T hiU^ lt/'*


far from the black holes


(4.3.1)


at the horizon of the i'th black hole.


In order to define a complete set of modes we divide the space into two cells,


1 l l .. 1 1 .1 1 1 1_ 1


mi_.~ _- --- i--- -- A


fwmn







are such that F /)
zw3


are nonzero in the i'th cell only, and they are given by


0

X EIW7'm(
-\y Z7;i


1 r m(9


- tInlUil
K .


inside i'th horizon


(4.3.2)


elsewhere,


outside i' th horizon


(4.3.3)


elsewhere,


with the normalization,


F!)
(F.w
vtwa


,F!) = 6i,6j6(w w').


(4.3.4)


Analogous to the one black hole case


we consider another set of modes such that


they are given by a linear combination of the above defined


modes and are analytic everywhere.


"Schwarzschild"


These are given by


H(lw = F) cosh O1, cos F(+) cosh 02w sin ,j


(4.3.5a)


FI- sinh01, cos


H(+) (+) cosh 02w sin
2wj 2wj


sb -


F(-! sinh 02 sin
2w3


+ Fl) cosh 01,
+ lw~j


cos C,


+ F(-! sinh 02w cos ,w + Fl(- cosh 02w sin w

H(-{ = F(I) sinh 01 cos q, F) sinh 02 sin ~
-w 2w wj


- cosh 901 cos
.1


H( = Fj) sinh 02w

+ F2] cos


cos w


(4.3.5b)



(4.3.5c)


F(-) cosh 02w sin w ,
2wj


+ F(+) sinh 1,w sin ,w


(4.3.5d)


;h 01, cos 4w + F{W) cosh 01w sin ,


Note first


the solutions are matched on


the horizons of the individual


black holes independently by choosing the 0, parameters such that


HX) !(region I of the i'th black hole)


/,_t 1- 4 9


(+)


F. w
SW}


, )e


) -


.A 0, t'\





57

which is exactly the analyticity condition for the modes in the case of one black


hole.


The remaining freedom is used to match the solution between the black


holes, that is on the


"wall"


separating the two cells.


To do this we set the d,


parameters to be


tan 4w =-


cosh 81w
cosh 22w


1 -exp(
1 exp(


_27wU
_2xw
K1


(4.3.7)


) 1/2


H +) are also normalized according to
z 3

(H H ) It=66= -w


To quantize we expand in terms of F )
Tow


(4.3.8)


(+) (+) (- (-)t
F.iwj wa t +f Fa j


+ h.c.)


(4.3.9)


w,WJ

where a!+ satisfy the commutation relations
twj


,a i, ] = ,i'6jfj'6(w ) ',


the other commutators being zero.


(4.3.10)


The vacuum state is defined by


S ) =0o.


(4.3.11)


But we can also expand in terms of Hif:


(H+a) (+ ) +H-)a(n (
1-iwj tuij + JiwH zj uj


) + h.c)


(4.3.12)


where a+ () also satisfy the commutation relations,
ZWJ


! )-' ) !+*,)t .()] = tii'6 Sjj16(w -w') ,
a ,a


(4.3.13)


1 I 1 1 -


n-i i 1 1 1I




58

The operators ao (ti,) are given by the Bogoliubov transformation

.(+-) a(+) (+) cosh 0 sin g
lwj( ) = alwj Cosh 1o COSqw + a2w cosh2 sin


(4.3.15a)


- a() sinh 1i, sin a- sinh 2, sin


a2j.( )= -a2j cosh2w sin


> =


(+) cosh 01,
"t alw


cos w


+ a2. sinh 02 sin w a(lwj cosh 92w sin

-a(+). sinh 01, cos 4, a2 t sinh 02w sin
-awjs1`82 sn 3


(4.3.15b)


(4.3.15c)


+ ac
lwj cosh (i^ cos 4' +


- a(3 sinh 02w
a2wj


cos d,


(+)
2wj cosh01io


. cosh 02w sin


+ awj sinh w1, sin f,0


cos {w) a


() cosh 61w sin


If we introduce the hermitian matrices


~I
1,2
2=l,2


Sa 3- -Wj )


(4.3.16)


0- (taw+ia- t
[^I^W zo3 jo


0= exp(i&,


cos p
- sin C,


COS Aw


cos 'N
- sin d.,,


cos ~


(4.3.17)


where


E2=i


O
20 and
72


oa2


-1


,then Eqs. (4.3.15a)


- (4.3.15d)


can be written in a compact form:


(+)

( j )
a2w


<'(li)
to)f


2L(


=O-1 G-1
=_ GI 1


(+)
alwj 1


(+)
a2wj


a 3)


(4.3.18)


-I -* -r -.


(4.3.15d)


aI (


a2w3 (





59

one black hole only, such that the corresponding expansion functions Hiwj are


analytic on the horizon of that black hole.


The matrix 0 is a two dimensional


rotation between the operators near different black holes.


parameter,


It depends on one


to be chosen such that the modes be continuous in the region


between the two black holes.

The vacuum states are given by


0) = 0


(4.3.19)


0())= 0 ,


(4.3.20)


where


0(K)) = 0-1G-1


The expectation value of the number of wj modes is


, (o ) (+)t (+)
, -) (() a aw, j


(+) t (+)
+ a2wj a2wi


O(K))


(4.3.21)


= sinh2 G1 + sinh2 02 =


2jrw
K1


_ 2xrw
K2


Note that our result does not depend on t, only on the Oi,


other words


parameters.


, it depends only on how we match the solutions on the horizon.


Generalization to arbitrary N is straightforward.

N cells, each of them containing only one black hole.


We divide the space into

We define normal modes


which are nonvanishing in


one cell


only,


and are given


Eqs.


(4.3.2) and


(4.3.3) except now i goes from


to N.


A linear combination of these modes


can be found that is analytic everywhere,


by first matching the solutions on


the horizons of the individual black holes, then in the region between the black


holes, that is on the "walls"


of the cells. Again, the corresponding Bogoliubov


za3


S()






the components of a 2N


component vector.


G is a product of N


transforma-


tions of the form given in Eq. (4.3.19) with i=l...N.


The N parameters Q0 are


chosen such that the new modes Hiwj analytic on the horizon. Now O is an N

dimensional rotation, mixing the particle states near the horizons of different


black holes.


The N(N -


1)/2 parameters (4, in the N=2 case) are to be chosen


such that the solution is continuous in the region between the black holes.


expectation value of the wj modes is again unaffected by these rotations, and,


similar to Eq. (4.3.21),


we obtain


) -


sinh2 i =


(4.3.22)


S2rw
,i


The spectrum of the created particles is the sum of the thermal spectra of the

individual black holes.


4.4 Neutrinos on the Schwarzschild Background


Now we examine massless spin one half fermions, that is neutrinos.


the scalar case,


As in


we have to start with finding the normal mode expansion of


the Dirac equation near the horizon.


We will use the vierbein formalism.


shall see that, with a suitable choice of the vierbein fields, the Dirac equation is


separable [32]


The metric tensor in this formalism is related to the flat metric


through


vierbein


which satisfies orthonormality,


Vp V,


= 6pa
- CgN


and completeness,


VP(x)V f(x)1, a


-gpv


conditions.


In particular,


Schwarzschild case the latter is


g9, = V a(x) V1 19( =


f1 2M
1--r


-r sin 9


-(1 -2 -1


(4.4.1)


where








and the indices a, /3 mean local frame indices and /i, v are space-time indices.

We can choose the vierbein such that its nonzero components are


2M )1/2
r


=(1


2M -1/2
r


(4.4.2)


= r sin O9


rsin 0


2M 1/2
F


The massless Dirac equation in curved spacetime becomes


(4.4.3)


where tb is a Dirac spinor field with (1


- 75


The gamma matrices are


given by


=V a


(4.4.4)


and they are the curved space counterparts of the usual flat space Dirac ma-


trices,


a(l1
-Y


They clearly satisfy the anticommutation relations:


V} = 2g= l


(4.4.5)


which are the curved space generalization of the flat space anticommutation

relations,


(4.4.6)


,7 = 217


The spin connection, FT. is given by


rF(x


) AKh


,a]Vva(x


(4.4.7)


We are looking for solutions of the form


\, -- x


- r


T+) = 0o,


S


= (1


=(1-


V29


:-M_1/2


(1-


) = 0.


VV/(x


I





62

This leads to the following coupled first order differential equations for R1, R2,


S1, and


2M
1O-- -r i--


zwr


R1 = kR2 ,


1 2M
1 -- --


1- 2M
1---- BrR2


zUr


1 2M
1- r
r


O9 Sl +
sin


s2-


m
sin 9


R2 = kRi ,


S1 = kS2 ,


S2 =


-kS1


(4.4.9)


where the separation constant k is to be chosen such that Si(09) and 52(0) are

regular at i9 = 0 and 9 = Ir. We are interested in the solution of the radial

equations at the past horizon. We find that


R1 ~" exp(

R2 ~ exp(


near the horizon.


(4.4.10)


-iwu

-izwv


(4.4.11)


Thus, for neutrinos, the solution which corresponds to out-


going waves at the past horizon is


I ~e
V,


-t1m7


(1>


-iwu


S1(9)


o)0


(4.4.12)


For antineutrinos


the solution of


Dirac equation is given


charge


conjugation (C


= i,2


0
-2202


77*
'*
1 "'7


*
92

r/2
'4f


(4.4.13)


where tr4 ~ exp(iwu) represents outgoing negative frequency waves at the past


horizon.


Hence the solution corresponding to outgoing antineutrinos at the


past horizon is given by


n\


-T=C
Iltu = ~





63

To quantize we expand the field 4


+ h.c.)


(4.4.15)


where at and b, represent creation operators for neutrinos and antineutrinos


respectively,


while aw and bt are the corresponding annihilation operators


b 10)=0,


satisfying the following anticommutation relations:


at,} = {b,


(the other anticomnmutators are zero).


,} = 6(w w')


The spinors un and


(4.4.16)

form a complete


orthonormal set,


u-e) = (-,


,) = (v,


,v ) = (- ,


) = S( w')


(4.4.17)


(others are zero), and near the horizon they are given by


UwC= e


-zwu
e


-imtp


S1(0i)


1
O
1
0


~ e


- IlnlUII
K


1
0
1
0


(4.4.18)


5-,= eiwu
Vwd --


Sr(o)


-ln Uj


0
- 1
0


(4.4.19)


us consider only neutrinos and define a complete


set of positive fre-


quency modes on the whole extended Schwarzschild spacetime:


-_ lnlU
mx /


1
0
1
0


outside the horizon


(4.4.20)


inside the horizon,


(1


eimp


(a, u + bLvi







The field 4 can be expanded in terms of complete set of modes F)
wj


S(F(+ a(+)
F-aI


)a(-)t
WI


+ h.c.)


a(+)(t) + H()a(.)t(a) + h.c.
Uw 3 j


(4.4.22)


where the modes H(I
U)1


are defined by


H(+
w3

w3(~


= F() cosw 0 F(


= F+) sin + F


(4.4.23)


(4.4.24)


Both F(
w3


satisfy the orthonormality conditions


(F
Cd)


F(j). (()3
Cdj) ~ WJ


) = 6jjf6, (w w) .


(4.4.25)


As in the bosonic case


H


are positive (negative) frequency modes, which are


analytic on I--


if tan 9,


The corresponding creation and annihilation


operators a ( ) () are given by the Bogoliubov transformations:
w3r


o '(+) =
a .( )=


a t cos, + a-) sin 0
a I


- exp(


-G)a' exp(G) ,


(4.4.26)


(- (+) sin + (-)t cos
a w())= -a sm +a cOS


= exp


-G)a j exp(G) ,


(4.4.27)


where


=E


S,(+)ta (-)t
^^j aw


- a(+) a().


(4.4.28)


The vacuum annihilated by a (t) is given by
"J


I0( K(- = exn(-G10\


(4.4.29)


or H


+F^-
* F(?


= (


Ssin 0 ,


cos 6,





65

The number of the wj modes detected by an observer at infinity is given

as a vacuum exceptation value


(n"j)u = (O(la) a.)a 10())> = sin2 0a =

For antineutrinos we have the same result,


(4.4.30)


,j) = sin- 2 =
1)'/7 sin2 O^ -


(4.4.31)


This is, as expected, a thermal distribution of fermions with effective temper-

ature equal to the Hawking temperature.

The result obtained above can easily be extended to the case of many black


holes.


One should follow essentially the same steps as we have in Sect 4.3 for


the bosonic case,


and find similar results.


In particular,


one finds that


spectrum of the created fermions is the superposition of the thermal spectra

of fermions created independently by the individual black holes.


Summarizing our results,


we have found that, in agreement with previous


calculations, the thermo-field approach lead to particle creation in a spacetime


with


causally


disconnected


regions.


the case of


a single


black


hole


spectrum of the emitted particles is thermal with effective temperature equal


to the


Hawking temperature.


the case of well-separated black holes the


spectrum is the superposition of individual black hole spectra.


Furthermore our


approach suggests that these results mainly depend on the analytic behavior


of the fields on the horizon, but not on the statistics of the particles.


This


been demonstrated by quantizing both a massless boson and a fermion field.

Consequently, we hope that similar studies will lead to a better understanding













CHAPTER 5
CONCLUSIONS

We have investigated two extensions of Einstein gravity in 2+1 dimensions,


Weyl gravity and topologically massive gravity.


We have also considerated the


applications of thermo-field dynamics to particle creation by black holes.

In the case of Weyl gravity we considered the consequences of duality in


the context of Weyl theory in three dimensions.


We constructed a theory of


gravity with Weyl invariance and a noncanonical scalar auxiliary field, as a lab-

oratory to study duality between the gauge field and its field strength. There


it appears as an equation of motion.


We have studied the classical solutions,


and found that they can be classified by the nonvanishing components of the


field strength.


There are stationary solutions only if the electric field is van-


fishing.


If the magnetic field is vanishing as well,


., in the pure gauge case,


our theory reduces to Einstein gravity in flat or de Sitter space.


solution was found.


The general


In the case when only the magnetic field is nonvanishing,


the problem reduces to the solution of a Liouville equation.


We studied the


axial symmetric solutions in more detail.


the helical-conical structure,


they are 2+1


We found


characteristic to


dimensional analogs of the known 3+1


that the solutions have


dimensional gravity,


dimensional GSdel and


Taub-NUT type solutions.


Interestingly, the


"matter part"


was described by


a rotating Chern-Simons fluid with intriguing properties.


Consequently this


work mirht, have interest.inip annlicationns in fluid mechanics.







Next we studied the renormalizability of TMG by using nonlocal regulariza-


tion.


We found that the theory is renormalizable under a certain assumption,


namely when


the nonlocal measure factor exists.


Although


we cannot give


a general proof of its existence, it can be constructed perturbatively, and its


existence and gauge invariance can be checked to any order.


We showed that a


possible anomaly which could spoil its power counting renormalizability does

not occur. If our assumption is valid, topologically massive gravity is the only

known example of a renormalizable and dynamical theory of gravity.


Finally,


we have used thermo-field dynamics to study particle creation in


causally disconnected spacetimes.


We have chosen 3+1 dimensional black hole


spacetimes, because these are the best known examples with the above prop-


erty.


We have found that our results are consistent with


those obtained by


different methods. In particular the thermal character of the vacuum has been

derived for the emission of massless scalar particles and for the emission of


neutrinos.


We also discussed how to generalize the method


to space times


with many disconnected regions.


For definiteness we considered the example


of many well-separated black holes, and found that the spectrum is the super-


position of the individual black hole thermal spectra.


Approximations were


necessary in order to obtain the multi-black hole metric, but only because we

do not know any exact solutions of the Einstein equation with the above prop-


erty, not because of the failure of our formalism in a more accurate case.


the same time our example clearly shows the basic ideas.

This method not only provides a new technical tool to discuss the quanti-


zation in such spacetimes,


but it also helps us to understand the features of the




68

we have found that the analytic behavior of fields on event horizons is crucial


to the derivation of the spectrum of the created particles.


depend on the statistics of the particles.


The effect does not


This has been demonstrated explicitly


by quantizing a massless scalar field and a neutrino field on the Schwarzschild


background using thermo-field dynamics.


In both cases the spectrum of the


produced particles is thermal with effective temperature equal to the Hawking

temperature.


As in earlier works,


we also have found that the particle creation process


is due to the presence of the event horizon.


To see this one should note that


the physical observables are the (temperature dependent) vacuum expectation


values,


they


contain information only


about


particle states outside the


horizon(s) (only these states have nonzero contribution).


But we have learned


more than that by realizing that the spectrum of the radiation depends on the

properties of the event horizon, namely on the number of disconnected pieces,

and on the behavior of the fields near the horizon.












APPENDIX A
FEYNMAN RULES FOR TMG


The Feynman rules for TMG follow from the action described in Eqs. (3.3.1),


(3.3.7), (3.3.8) and (3.3.9) in the usual way.


The corresponding rules of the


nonlocalized theory can be obtained by applying the general rules of Sect. 3.5 to


TMG. Associated with each field of the local theory is an auxiliary field.


in the nonlocal theory auxiliary fields 4,


Thus


kny and dP are associated with the


original fields 4,


hpy and c


. The ghost field b1 does not require an auxiliary


field because its BRS variation is linear.


In the nonlocalized theory the form


of the vertices are unchanged except that now the lines represent both phys-


ical and auxiliary lines.


The propagators of the local theory are replaced by


smeared ones in the nonlocal theory.


The original propagators are multiplied


by 2


defined in (3.6.1),


while


the auxiliary field propagators are 1 E multi-


plied by the original ones.


In constructing Feynman diagrams in the nonlocal


theory, one does not include loops involving only auxiliary fields or diagrams


with auxiliary fields on external lines.


Details can be found


ects. 3.4 and


Below we list the Feynman rules for


theory are described above.


TMG where the rules for the nonlocal


An auxiliary field line is represented by putting


a bar on the corresponding physical field line.


First we list the propagators.


The $ propagator is


C






The h propagator is


fiv ap
,, t~txx~(-/xlx


+ TE lPpva


+ eLfr P/Ia)


Finally, the ghost propagator is


It


S-------


P( p2 Pv)


where rply = diag(-1, 1, 1), Epya is the 2+1 dimensional Levi-Civiti tensor and

the projection operator Pr" is


-=


p2


(A.1


Next


we give the vertices.


Since the vertices involve all orders in the h


field we shall only give vertices at most cubic


4 point vertices.


in the h field and no higher than


The vertices arising from the Einstein and kinetic terms of


(3.3.7) are quadratic in


the ) field and contain two derivatives.


The lowest


order vertices are


~kV
k


i/Cl{ap {tp (8k


e+ 3p


+q2))


- 16(k3fl + kpc v) 2pvqp}


1-p4r(ap"
4p4


+ va p1l3


16i ppqv ,


Sq + 2(p2




71

The vertices from the Chern-Simons term contain only h fields and three deriva-


tives. 'I







q


The lowest order vertices coming from this term are


- : X/7jE7r( -pyckKqp?7frpaa?77-Tv


+ PpkpkK1 va]7


+ pvkaktrlprlvprl


7*O~
Ta-
TO'-


pv k^ kp rtpaQ P 7a7ra

P kk Tkra'nprlglrra-


+ gp ka qprp Ea rirv


* qkpriaprt


7KUrTVr)


+ permutations,


(YaKrP(f-k


* qp, + k


Sqp)


qp(k


+ aq9(k 1(


Kp -


k1Y]p))


A7v (


-2k7


(7ap rloagKQep


-- rlpprcnqag/p)


1p.( taK -q I


q7 Er)


KZICo -


+ 4tina(r,


QK e -


+ 3y p~'ilva(o 'l


191


plra K


'J~v}


+ permutations.


The vertices arising from the ghost part of the action in (3.3.9) are


- AP1


2qan


p -


iep)


* ^


- e, r


p(Qa qarLu


ns(,a(17pWW^


.7,5, a6 kQ rla r1p 77pTa









-i (ppkvqrlap + pflk r1va


- ppqlrlav


q


a\


Ji^(puklaAT?, j+ PkpTJalAVI)


Here it is understood


that one must symmetrize each


pair of indices on an


external h line and take permutations of the various identical h fields arising

from a vertex as indicated in the figures.


2- 2pkp va 4pai, kv)













APPENDIX B
MEASURE FACTOR
FEYNMAN RULES FOR TMG


The Feynman rules for the nonlocal BRST


variation of the measure were


discussed in Sect. 3.5 and were derived from Eq. (3.5.4).


There are two types


of vertices,


those from


the nonlocal


theory as described in


Appendix A


the other type connects to internal lines, one of which has

Such internal lines are represented with a dot on their legs.


"propagator"


All the diagrams


contributing to the variation of the measure factor are one loop diagrams and


contain


only


one of


second


type of vertex


and an


arbitrary number of


vertices from the original theory.

cT in the vertices that follow. T


All dashed lines correspond to a ghost field


hose diagrams coming from the contribution


of the variation of the 4 field are


1
- Qv + pv


0- -


'P
' p


1
3 pv $-qv




74





-- -,V/pq + mp qi) ,


s-p


vJ4('7p/pPv + ?lvppj'})


- n (p+ 0vpp)


vertices


arising


from the contribution of the variation of h are


P


(i 1 p


+ Vrrp
771Qflp


- Urp7(pp +


P X


rl^r^p?)




75





j2 (api + eup7C.3


e .4

*'-p


,k p


2
3J~ft(Pfih;'crp


'lv
7l~7?A77 + pA 7Kp


Finally, the


vertex


coming from the contribution of the variation of the ghost


field


p
-__- ___


'P


-Pp'7ay + qy raB


p\


p \













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WTA (2 TTI Dr Pirb Pnr TR.0 1 10e (1f 72\











BIOGRAPHICAL SKETCH


Bettina E. Keszthelyi was born April, 8, 1961, in Budapest,


Hungary. She


received


the Diploma in Physics from EStvos


University in


1984.


She then


travelled to the United States to study in the graduate program in astrophysics


at the


University of Chicago from 1984 to


1986.


She decided to change her


research direction to the area of particle physics and subsequently went to the


University of Florida from 1987 to


1993.


She spent much of the 1992 to 1993


academic year on leave studying field theory at


Brandeis


University.


She is


currently an Honorary Fellow at the University of Wisconsin.






I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope
and quality, as a dissertation for the degree of Doctor of Philosophy.


Pierr .C Ramond, Chairman
Professor of Physics
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope
and quality, as a dissertation for the degree of Doctor of Philosophy.


Pierre Sikivie
Professor of Physics


I certify that I have read this study and that in my opinion it
acceptable standards of scholarly presentationI and is fully adequo
and quality, as a dissertation for the degree of1 Dqdctr df Philosop


^L, ,


forms to
in scope


I1
*-" <^
^ ^


Jame, R. Ipser
Professor of Physics


I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope
and quality, as a dissertation for the degree of Doctor of Philosophy.


sk N. Fry
ciate Professor of Physics


I certify that I have read th


acceptable
and quality


standards of scholarly


study and that in my opinion it conforms to
presentation and is fully adequate, in scope


as a dissertation for the degree of Doctor of Philosophy.


SrCri


Louis S. Block


Professor of Mathematics


This dissertation was


submit ted to the Graduate Faculty of the Department


of Physics in the College of Liberal Arts and Sciences and to the Graduate
School and was accepted as partial fulfillment of the requirements for the degree
of Doctor of Philosophy.


December


1993


Dean, Graduate School


1 ,_... -,





































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