GRAVITATIONAL MODELS IN 2+1 DIMENSIONS
WITH TOPOLOGICAL TERMS AND
THERMOFIELD 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 thermofield 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 PowerCounting Renormalizability and Gravity
3.3 Topologically Massive Gravity .
3.4 Nonlocal Regularization .
3.5 Nonlocal Feynman Rules .
3.6 Renormalizability. .
THERMOFIELD 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
THERMOFIELD 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 ChernSimons
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 gaugeinvariant 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 thermofield 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 spacetime 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 energymomentum 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 antide Sitter.
Although local
curvature in sourcefree 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 conicalhelical 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 AharanovBohm 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 selfdual 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 selfduality
of the
YangMills field strength leads
to instanton solutions
similar
condition imposed on the gravitational connection leads to the EguchiHanson
gravitational instantons [7].
In Sect. 2.2 we review Wevl theory in d+ 1 snacetime 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 selfduality 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 conicalhelical 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^ niiinrnfm 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 powercounting
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 auantumstatistical
7
Here we follow Israel's approach, and first we rederive the wellknown 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 thermofield 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 multiblack 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, selfduality of YangMills gauge fields leads
to the classical instanton solutions. A similar condition imposed on the gravi
national connection leads to the EguchiHanson gravitational instantons. Mo
tivated by the recent interest in theories of lesser number of dimensions, we
investigate analogs of selfduality in the classical solutions of some theories in
2+1 dimensions.
In the case of YangMills gauge fields, selfduality 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
241 gravitational field.
The fluid is a rotating perfect fluid with velocity corre
duality condition and a massive KleinGordon equation.
When specializing to
axisymmetric solutions, we recover either the 2+1 dimensional G6del solution
or the boundary of 4 dimensional TaubNUT spacetime 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 7n... 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 pyAa),
/ 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
d1
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
0w
(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 d1
(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 igRy(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
ChernSimons 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 9RgV)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 KleinGordon 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 energymomentum 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 4t % 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 energymomentum 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 ChernSimons theory on Minkowski spacetime in refs. 20 and 21.
In this gauge the energymomentum tensor of Eq. (2.2.29) simplifies to
S
4A A0 )gv +
~04
(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
731
= 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 LeviCivita 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 + 22(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
energymomentum 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
{8P3
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 KleinGordon 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 spacetime 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 spacetime 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 = 1iN
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 22
(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
N2
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,
spacetime 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 spacetime 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 TaubNUT
solution for a fixed radial coordinate [28
Because we can
choose r
* o00,
we can think of our solution as the boundary of the TaubNUT
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
spacetime 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 "ChernSimons"
fluid.
As we have shown, this
solution is causal only if p
the TaubNUT
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 wellknown 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 PowerCounting Renormalizability and Gravity
which has not
Most gravity theories are not
powercounting 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 Lloop diagrams in d spacetime 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
npoint 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,Aqt1pv
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 tracefree 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), antide 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 ChernSimons action is conformally invariant, it is unaltered by this
rescaling.
Gaugefixing is performed by setting h"V
, 1 hP"
this neces
states the introduction of a Lagrange multiplier B1, and ghosts b6, ci
. The
resulting gaugefixed action is
SGF = S
+ S+
where
B1 1, h V
(3.3.8)
is the gaugefixing 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/ihn1) 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 superrenormalizable coupling.
expansion parameter p i
The true
s dimensionless, providing the first indication that the
theory may b
e renormalizable.
To prove the naive powercounting argument
we determine the highest degree of divergence D of any Lloop 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)n1
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 ChernSimons term of dimension 0 and possibly a cosmological term
of dimension
thus the theory is powercounting 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 AdlerBardeen
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 spacetime 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
Sexp 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
Kl1 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 gaugefixed in order that
we may
solve for W [];
then
symmetry T
represents the BRS symmetry of the gaugefixed 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 welldefined 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 gaugefixing 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 spacetime 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 onshell 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 welldefined 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~rmTT7~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 spacetime 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 npoint 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 ii 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
lPI .. 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 powerlaw 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
THERMOFIELD 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 thermofield 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 thermofield 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 multiblack 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 + 2MlnI2 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 TC
, 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 71
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:
Ht/+
= 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(ww ')
(4.2.12a)
(H'I
) = 6jj,'S(w w
(4.2.12b)
with respect to the KleinGordon 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 wellseparated 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
. eiw(tr) 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(tr)
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 (
1iwj 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 
ni 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(
=O1 G1
=_ 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)) = 01G1
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
1r
r sin 9
(1 2 1
(4.4.1)
where
and the indices a, /3 mean local frame indices and /i, v are spacetime 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,
ue) = (,
,) = (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(+)
FaI
)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 thermofield 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 wellseparated 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 thermofield 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 helicalconical 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
TaubNUT type solutions.
Interestingly, the
"matter part"
was described by
a rotating ChernSimons 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 thermofield 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 wellseparated 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 multiblack 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 thermofield 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 LeviCiviti 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}
1p4r(ap"
4p4
+ va p1l3
16i ppqv ,
Sq + 2(p2
71
The vertices from the ChernSimons term contain only h fields and three deriva
tives. 'I
q
The lowest order vertices coming from this term are
 : X/7jE7r( pyckKqp?7frpaa?77Tv
+ 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(fk
* 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) ,
sp
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 ,_... ,
UNIVERSITY OF FLORIDA
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3 1262 08556 8342