Two Quantum Effects in the Theory of Gravitation
by
Sean Patrick Robinson
S.B., Physics, Massachusetts Institute of Technology (1999)
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy in Physics
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2005
© Sean Patrick Robinson, MMV. All rights reserved.
The author hereby grants to MIT permission to reproduce and
distribute publicly paper and electronic copies of this thesis document
in whole or in part.
MASSCHUSETSSE
OF TECHNOLOGY
JUN
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Auth or .............................
0 7 2005
LIBRARIES
...............
Department of Physics
May 19, 2005
Certified by
C ertifi ed by.........
-/
..............
-...
'I
.............................
Frank Wilczek
Herman Feshbach Professor of Physics
Thesis Supervisor
A ccepted by
~~~~~ ~~~~/
...........................
.
Professr Thom/Greytak
Associate Department Head fo
ducation
Two Quantum Effects in the Theory of Gravitation
by
Sean Patrick Robinson
Submitted to the Department of Physics
on May 19, 2005, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy in Physics
Abstract
We will discuss two methods by which the formalism of quantum field theory can be
included in calculating the physical effects of gravitation. In the first of these, the
consequences of treating general relativity as an effective quantum field theory will
be examined. The primary result will be the calculation of the first-order quantum
gravity corrections to the /3 functions of arbitrary Yang-Mills theories. These corrections will effect the high-energy phenomenology of such theories, including the details
of coupling constant unification. Following this, we will address the question of how
to form effective quantum field theories in classical gravitational backgrounds. We
follow the prescription that effective theories should provide a description of experimentally accessible degrees of freedom with all other degrees of freedom integrated
out of the theory. We will show that this prescription appears to fail for a scalar field
in a black hole background because of an anomaly generated in general covariance
at the black hole horizon. This anomaly is repaired and the effective field theory
is saved, however, by the inevitable presence of Hawking radiation in the quantum
theory.
Thesis Supervisor: Frank Wilczek
Title: Herman Feshbach Professor of Physics
3
4
__I__
Acknowledgments
The following body of work has benefited from the input of many individuals. Obvious
among these is my thesis advisor, Frank Wilczek. I would like to thank the Nobel
Foundation for making this an interesting year to write a thesis. I also need to
recognize the other members of my thesis committee, Eddie Farhi and Roman Jackiw.
Others who made significant contributions to the development of this work, but are
not specifically cited within, include Brett Altschul, Ted Baltz, Serkan Cabi, Qudsia
Ejaz, Ian Ellwood, Michael Forbes, Brian Fore, Vishesh Khemani, Joydip Kundu,
Vivek Mohta, Brain Patt, Dru Renner, Jessie Shelton, and Ari Turner. I especially
thank Michael Forbes for reading an early draft of this thesis. Finally, I would like to
acknowledge the exceptional support and motivation provided by my wife, daughter,
and parents, the importance of which cannot be overstated.
In celebration of the completion of this thesis, I compose the following cautionary
limerick 1 :
When working with quantum gravity,
There's not much that is easy to see.
The math's so opaque
that it's easy to make
an occasionalerror, or three.
This work is supported in part by funds provided by the U.S. Department of
Energy (D.O.E.) under cooperative research agreement DE-FC02-94ER40818.
'Incidentally,
Speak in limerick? Well, maybe I did.
But if I did, it was only to kid.
Speaking in limerick
is sort of a gimmick
behind which real intentions are hid.
5
6
_··^I_________
Contents
Overture
13
1.1 Quantum General Relativity and Yang-MillsTheory. .
14
1.2
Black Holes and Effective Field Theory .........
17
2 Gravitational Corrections to Yang-Mills /3 Functions
23
1
2.1
Introduction
2.1.1
. . ..
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
One-loop Divergences .......................
2.1.2 Asymptotic Safety .......................
2.2
2.3
26
..........
.
27
Technical Preliminaries .........................
..........
2.2.1 Background Field Theory ...................
.........
2.2.2 Definition of Newton's Constant .
.
....
Setup ..............................................
.
.
28
28
29
31
.
32
2.4 Expanding the Action .........................
..........
2.4.1
2.4.2
2.5
2.6
2.7
2.8
23
Expanding the Non-Polynomial Terms .
.
...
Expanding the Einstein-Hilbert Action .............
2.4.2.1 Curvature with Background Derivatives .
....
2.4.2.2 Some Useful Definitions and Identities ....... ..
2.4.2.3 Expansion of Curvature . ..............
2.4.3 Expanding the Yang-Mills Action .
.
......
Gauge-Fixing ...............................
Combining the Pieces ...........................
Compiling the Superfield .........................
Renormalization ...........................................
2.8.1 Computation of Functional Determinants ........... .....
2.8.2 Extracting the
Function .
.
49
.............
.
.
. . . . . . . . . . . . . . . . . ............
. . . . . . . . . . . . . . . . . . ............
3 Black Hole Effective Field Theory
3.1 Introduction ......................
3.1.1
3.1.2
3.1.3
.
......
2.9 Enlarging the Matter Sector and the Gauge Group
2.10 Coupling Constant Unification ...............................
2.11 Phenomenology
2.12 Commentary
.
50
53
54
57
59
. . . . . . . . . .
Hawking Radiation ..............
. . . . . . . . . .
Anomalies and Anomaly Driven Currents . . . . . . . . . . . .
Hawking Radiation and the Conformal Anomaly
7
32
33
33
34
34
36
36
38
41
43
46
59
59
60
61
3.2
3.1.4 Effective Field Theory Framework .
Spacetime Prel iminaries .........................
3.2.1 Spherica al Static Metrics .....................
3.2.1.1
3.2.2
3.2.3
62
63
63
65
67
Einstein's Equation ...................
Horizon Structure . .................
3.2.1.2
Kruskal Extension
69
................................
3.2.2.1
The Quantum Vacua .................
3.2.2.2
Euclidean Section ............................
.......
75
75
Wave E quation .................................
3.2.3.1
Spherical Harmonics .................
.......
3.2.3.2 Radial Wave Equation . ...............
3.2.3.3 Near-Horizon Action . ................
3.3 Thermal Radia tion ..................................
.......
3.3.1 Hyperci ubic Blackbody Cavity .................
3.3.2 Flux Versus Energy Density ...................
3.3.3 Spherica al Blackbody Cavity ...................
.......
3.3.3.1
Radial Mode Density .................
3.3.3.2 Spectral Densities ...................
3.4 Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Commentary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
3.6 Blackbody Spe ctrum from an Enhanced Symmetry? ......... ...
4 Finale
4.1
4.2
4.3
74
77
79
80
82
83
86
88
90
92
95
98
99
101
Summary .......................
Open Possibilities ...................
Conclusion.
......................
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
8
101
101
102
List of Figures
2-1 Feynman diagrams for two typical processes contributing to the renormalization of a Yang-Mills coupling at one-loop ...........
25
2-2 The schematic Feynman diagram represented by the functional trace
-TI[Mh]. A momentum p circulates in a virtual graviton loop coupled
to external gluons of momentum k ....................
46
2-3 The schematic Feynman diagram represented by the functional trace
-Tr[N].
A momentum p circulates in a virtual gluon loop coupled to
external gluons of momentum k ......................
47
2-4 The schematic Feynman diagram represented by the functional trace
1Tr[O+O_]. A momentum p circulates in a virtual gluon-graviton loop
coupled to external gluons of momentum k ..............
......
.
48
2-5 In Figure 2-5(a), the three Yang-Mills couplings of an MSSM-like theory evolve as straight lines in a plot of r- ' =_ 47r/g2 versus log1 0 (E)
when gravitation is ignored. The initial values at Mzo - 100 GeV
are set so that the lines approximately intersect at 1016 GeV. When
gravity is included at one-loop, the three lines curve towards weaker
coupling at high energy, but remain unified near 1016 GeV. In Figure
2-5(b), g is plotted for the same theory. All three couplings rapidly go
to zero near Mp, rendering the theory approximately free above this
scale................................................
55
3-1 Part of the causal diagram of a black hole spacetime, with inset detail
of a region near the horizon ................................
.
3-2 Sketches of three integrated mass functions and their associated h(r).
63
In 3-2(a) the matter distribution is relatively smooth and vanishes at
the origin, as in a normal star. In 3-2(b) the matter has a density
singularity at the origin, but is otherwise well behaved. In 3-2(c) a
potentially difficult-to-analyze situation is sketched ...........
70
3-3 In 3-3(a) typical profiles for the functions h(r) and f(r) are sketched for
an asymptotically flat black hole spacetime. The horizon occurs where
the functions vanish at r
In 3-3(b) the corresponding profile
= rh.
of r is sketched along with the line r = r. Note that r diverges
logarithmically at rh and approaches r at large r ............
9
71
3-4 A sketch of a typical effective radial scattering potential. The potential
for any metric qualitatively similar to the one sketched in Figure 3-3
will be qualitatively similar to the one sketched here for > 0 and
d > 3. The potential falls off exponentially for negative r and as is
typically dominated by the centrifugal term at large r,.
2
off as 1/r .
r, which falls
. . . . . . . . . . . . . . . . . . . . . . . .........
3-5 The thermal integral I(a) defined in Equation (3.129) is plotted for
the cases = +1 and ~ = 0. All three lines seem to converge towards
e 1 -a for a >> 1...........................................
10
81
94
List of Tables
3.1
3.2
A few physically interesting metrics that obey p = -P .........
The angular state degeneracies for total angular quantum number 1, as
determined by Equation (3.64), in a few chosen dimensions .....
.
11
67
79
12
_·
Chapter
1
Overture
In this thesis we shall describe two logically independent lines of research that represent small steps away from ordinary quantum field theory in flat, nondynamical
spacetimes and towards quantum gravity, the as-yet-undiscovered fundamental theory of quantum spacetime dynamics. The first strategy we investigate is that of perturbatively quantizing the small field fluctuations of general relativity, the first theory
of spacetime dynamics historically. This approach is famously limited in power and
much-maligned, but we will show by a specific example that useful physical predictions can nevertheless be obtained in this formalism. The second strategy is to not
attempt to give quantum dynamics to spacetime at all, but to instead only use quantum theory where it has already proven so successful: in the nongravitational aspects
of matter. In this approach, spacetime is described in a nonquantum way, either as
a nondynamical, curved background or using the classical dynamics of general relativity. This formalism is usually called semiclassical gravity. Like quantized general
relativity, semiclassical gravity is rather restricted in scope and cannot be considered
as more than a limited, but useful, model for quantum gravity. We will use semiclassical gravity to describe the behavior of a quantum field theory in the region outside
of a black hole.
The second strategy is often considered more respectable than the first, perhaps
because it never attempts to be more than a model, and thus its points of failure
are both more understandable and educational. We believe, on the other hand, that
both formalisms are useful as model theories for quantum gravity as long as they are
applied within their respective regimes of validity. By studying the conditions under
which a model theory begins to fail, we can learn which aspects of the true theory the
model theory lacks. Since semiclassical gravity and quantized general relativity have
different regimes of validity and different failure modes, they are complimentary tools
in the investigation of the properties of quantum gravity. They also have a broad
overlap region of validity, which is the domain of ordinary Minkowski space quantum
field theory. This provides for these models an anchor to known physics, which is
often claimed to be well understood.
In Section 1.1 we will describe the effects of including quantized general relativity
in the calculation the
function of a non-Abelian gauge theory. The calculation
of this quantity in the absence of gravitation [1, 2] is considered to be one of the
13
most important calculations of Minkowski space quantum field theory [3]. The full
calculation and discussion appears in Chapter 2. In Section 1.2 we will describe our
attempts to import the highly successful concepts of Minkowski space effective field
theory into the semiclassical description of quantum fields in a black hole spacetime.
This is discussed more fully in Chapter 3.
1.1
Quantum General Relativity and Yang-Mills
Theory
In Chapter 2, we will calculate to one-loop order in perturbation theory the f function
of the Yang-Mills coupling constant in an arbitrary non-Abelian gauge theory coupled
to quantum gravity. The core calculation of this chapter is based upon the work of
[4]. Here, quantum gravity is modeled by its low-energy effective field theory, which
is just quantized general relativity. This effective field theory should be an accurate
description for the quantum dynamics of spacetime at energy scales below the theory's
cutoff scale, the Planck mass, given in four dimensions as Mp -GN 1/ 2 1.1 X 1019
GeV, where GN is Newton's constant of universal gravitation.
Before the appreciation for the proper role of effective field theories in physics became widespread, common lore held that general relativity and quantum mechanics
are incompatible in terms of describing the physical phenomenon of gravitation. This
was primarily because quantum general relativity was finally proven to be perturbatively nonrenormalizable [5, 6, 7, 8, 9] shortly after the time when renormalizability
had become understood as an essential ingredient in quantum field theories of fundamental interactions. About ten years later, the discovery of quantum theories that
appear to describe gravitation
in terms of excited strings [10, 11, 12, 13, 14, for
introductions], rather than local fields, helped to cement the common lore.
Over time, however, appreciation has grown for the fact that even the best quantum field theories of reality (that is, the standard model) are, at best, effective theories
containing infinite numbers of nonrenormalizable interactions. The ideas of Wilsonian effective field theory [15, 16, 17] have taken deeper root in the intuition and
made possible the use of nonrenormalizable phenomenological models such as chiral perturbation theory [18, 19, 20]. Finally, about 20 years after the proofs of the
nonrenormalizability of general relativity, Donoghue [21] made compelling arguments
in favor of taking seriously calculations made with quantized general relativity and
treating the results of these calculations as genuine low-energy predictions of quantum gravity. It is in this spirit that we perform our calculation of the Yang-Mills
function.
The effective value of a renormalized coupling constant g in a quantum field theory
depends on the energy scale E at which it is probed via a universal function of the
theory known as the Callan-Symanzik /3 function [22, 23]:
E ag- =
14
E).
(1.1)
The remarkable discovery [1, 2] for four-dimensional non-Abelian Yang-Mills theories
was that these theories obey
3= (-g3l)2
11
C2(G)- 4 nfC(r
-
I)
bo
(4)23YM
(1.2)
for a gauge group G with nf fermions in representation r. This /3 is negative as long
as nf is not too large. Equation (1.2) integrates to give a running coupling of
1
gy (E)2
1
gy
___
b
(M) 2
In
(47r)2
/1
M2)
(1.3)
(1.3)
which demonstrates that the negative value of the /3function implies asymptotic freedom: gYM(E) -- 0 as E - oo, as long as b0 is positive. The only known asymptotically
free theories in four spacetimes dimensions are the non-Abelian gauge theories. Thus
a universe with laws of physics governed by non-Abelian gauge theories - as our
universe approximately appears to be - becomes simpler and simpler as it is probed
at more fundamental scales, as long as the matter content is simple enough.
We now want to augment this classic calculation with quantum general relativity.
The calculations will be done using the methods of background field theory, which
we will sketch briefly in Section 2.2.1. We will let the spacetime background and
dimension be arbitrary for as long calculationally feasible. This will require adopting
a definition for Newton's constant in d dimensions. We choose a definition, described
in Section 2.2.2, which preserves the interpretation of the nonrelativistic gravitational
force law as describing the areal density of diverging, but conserved, gravitational flux
lines. Then, in Sections 2.3 through 2.7, we perform the detailed expansion of the
coupled Einstein-Yang-Mills action in terms of quadratic fluctuations about nontrivial
gauge field and spacetime backgrounds. In particular, in Section 2.5 we gauge-fix the
theory using the Faddeev-Popov [24] procedure and calculate the ghost and gaugefixing Lagrangians. The gauge chosen to fix general covariance is reminiscent of
the R~ gauge [25], except that the original RC gauge was for a gauge field in a scalar
background and the current case is that of a gravitational field in a vector background.
In Section 2.8, we finally come to the central result of Chapter 2 by evaluating the
background effective action and extracting the ,Bfunction. In Section 2.9, the result
is generalized to arbitrary gauge groups and matter content. We find that to one-loop
accuracy, the /,3function is equal to the value calculated in the absence of gravity such as that given in Equation (1.2) - plus a new term lA]grav that is independent
of the gauge and matter content. In four spacetime dimensions, this term is given by
A/3grav(gy,E)
=-gYM
3 E
-2
7r
2
(1.4)
Note that this term is always negative. It will dominate the running of the coupling
when the energy is close to the Planck scale and the coupling constant is perturbatively small. Thus, it appears that the inclusion of quantum gravity effects renders
all non-Abelian gauge theories asymptotically free. The integrated running coupling
15
coming from the combination of Equations (1.2) and (1.4) is
1gy
_
1
r3E2 -M 2
g~ (E)2gy (M) 2 ex~p r
Me2
2 bo fEdk
+ 2~(47r)2,M
k e
r3E2 -k 2 }
M2
(1.5)
The logarithmic running of Equation (1.3) becomes modulated by a exponential in
E 2 . This has little effect at low energies, where the exponential is approximately
equal to one. As E approaches Mp, however, the exponential turns on very quickly
in comparison to the logarithm, and the coupling gets driven rapidly to zero. This
phenomenology comes with the caveat that the interesting physics is occurring very
close to the cutoff scale of the theory. However, taken at face value, this result seems
to indicate that Yang-Mills theories become approximately free at the Planck scale.
In Section 2.10, we explore the implications of Equation (1.5) for coupling constant
unification. That is, we consider a Yang-Mills theory with a simple gauge group
that is spontaneously broken at some high energy scale such that the theory at low
energies appears to be a Yang-Mills theory of some product gauge group with several
independent coupling constants, each with its own function. Without the context
of the unified theory, the low-energy values of these couplings could be taken to have
arbitrary independent values. However, since all the couplings secretly derive from
a unified theory at high energy with only a single coupling, the low-energy values
must conspire with the 3 functions in such a way that all the couplings evolve to the
same unified value at the breaking scale. The experimentally measured values of the
SU(3) x SU(2) x U(1) couplings of the standard model with minimally supersymmetric
matter content are consistent with such a unification in the real world with a breaking
scale of MGUT - 1016 GeV [26]. No matter how many couplings are in the low-energy
theory, only two of them may be chosen independently. The rest are then fixed by
the condition of unification.
If the field content of the low-energy theory is changed such that the functions
change without a corresponding change in the values of the low-energy couplings, the
unification will generically be spoiled. If the addition of gravitation spoiled unification in this way, it would indicate that the observed unification of standard model
couplings is a spurious coincidence. Fortunately, as we show in Section 2.10, this
is not the case for four dimensional gauge theories. Although the 3 functions are
changed in a non-trivial way given by Equation (1.4), we find that theories which
exhibit exact coupling constant unification in the absence of gravity continue to do
so with the same values of the low-energy couplings when Equation (1.4) is taken
into account. The values of the unified coupling and the breaking scale are slightly
altered. For a standard-model-like situation where the measurement scale M and the
putative breaking scale M0 obey a hierarchy of the form M < M0 < Mp, we find
that the new breaking scale Mu is given by
FinallyMinuection2.11
Mesomebriefremarksegardingthe(1.6)
we
Finly ° S 27r
rMPa
]
(
Finally, in Section 2.11, we make some brief remarks regarding the phenomenology
16
__·_
and possibleexperimental signatures of the calculated gravitational correction to the
running of coupling constants.
1.2
Black Holes and Effective Field Theory
The core result of Chapter 3 is based primarily on the work of [27]. In the context
of semiclassical gravity, we attempt to formulate an effective field theory for a scalar
field that lives in a black hole background. Our prescription for constructing this
theory ultimately results in a breakdown at the quantum level of the underlying gauge
symmetry of gravitation, general covariance. Demanding that general covariance
holds in the effective theory, as it does in the fundamental theory, forces each partial
wave of the scalar field to be in a state with a net energy-momentum flux (I given by
487r'
(1.7)
where r, is the surface gravity of the black hole event horizon. If each partial wave
mode is occupied with a blackbody frequency spectrum, then Equation (1.7) implies
a temperature of
TH =2-
(1.8)
which is exactly the Hawking temperature of the black hole.
The result (1.8) for the temperature of a black hole was originally found by Hawking [28, 29] and subsequently rederived by many other methods. Hawking radiation
is now understood as a kinematic effect resulting from the lack of an unambiguous
global definition for a particle number basis of Fock space when spacetime is not
globally flat.
Our construction can be thought of as arising from the presumption that the
physics observed by a given experimenter should be describable in terms of the effective degrees of freedom accessible to that experimenter. In the case of ordinary
Minkowski space quantum field theory, one can apply this presumption to an experimenter with limited energy available to probe highly excited states. In that case, the
effective physics observed by the experimenter is described by a theory in which states
above the high-energy cutoff have been integrated out, resulting in the standard story
of Wilsonian effective field theory [15, 16, 17]. The parameters and degrees of freedom
of the low-energy theory may be different from those that appear in the fundamental
theory.
We wish to consider an experimenter who lives outside of a static, spherically symmetric black hole. Such spacetimes have a global Killing vector (spacetime symmetry
generator) that appears locally like a time translation, but it is only timelike in the
region outside the black hole event horizon. Thus, the conserved quantity associated
with this symmetry can not be used as an energy outside of this region. Since the
observer cannot see beyond the event horizon of the black hole, however, this Killing
vector should be a perfectly reasonable choice with which to define the energy of
quantum states in an effective theory that only describes observable physics. Unfor17
tunately, the "vacuum state"' obtained with this definition is exactly the one considered by Boulware [30]. The Boulware vacuum has a divergent energy-momentum
tensor due to a pile up at the horizon of would-be outgoing modes (the UP modes
in the language of [31]), which take arbitrarily long amounts of coordinate time to
escape the near-horizon region.
Our approach differs from most previous work on Hawking radiation in that we
recognize the divergent energy of the horizon-skimming modes as an indicator that
the experimenter who observes these modes will not be able to probe them with finite
energy. Thus, the proper description of the observed physics is an effective theory
with these modes integrated out. In other words, we choose to take the lessons of
effective field theory seriously.
The effective theory thus formed no longer has observable divergences, but it now
suffers from an even worse problem; it contains an anomaly in general covariance.
As shown in [32], a two dimensional scalar field theory will violate general covariance
at the quantum level if the number of right-moving and left-moving modes are not
identical - that is, if the theory is chiral. The breakdown of general covariance means
that the energy-momentum tensor Tb of the scalar field is not conserved. In the case
of a single chiral scalar field, the anomaly takes the form
VaT: =
9
9o7rVf-g
dcdd
adaFr-c
(1.9)
where the a7c are the Christoffel symbols of the background spacetime.
We show in Section 3.2.3 that in the near-horizon limit, each partial wave behaves
like an independent two dimensional free massless scalar field. In our case, we have
eliminated the horizon-skimming part of each partial wave of the scalar field. So, this
effective theory is chiral and each partial wave exhibits an anomaly given by Equation
(1.9), but only in the near-horizon region. However, the fundamental theory contains
all the modes, so it has no anomaly. Some new physics must be introduced into the
chiral theory to carry out the job of anomaly cancellation that was formerly performed
by the degrees of freedom which were integrated out in the process of forming the
effective theory2 . Indeed, we find that demanding general covariance to hold in the
'The word "vacuum" is used here in the Fock space sense, meaning the state in which all momentum modes have zero occupation number. It does not mean that the state has minimal energy;
the energy of a state can not be unambiguously defined in a curved spacetime.
2
The ability to form a gauge invariant effective theory for a fundamental theory which cancels
anomalies between modes of very different energy seems to run contrary to decoupling theorems
which state that the only effect the high-energy modes can have in the low-energy theory is in the
renormalized value of low-energy coupling constants [33]. If the effective theory is at an energy
scale where some, but not all, of the modes involved in anomaly cancellation have been integrated
out, then decoupling should guarantee that the high-energy modes cannot cancel the remaining
anomalies in the effective theory. In theories like the electroweak standard model, which have
potentially anomalous chiral gauge couplings to fermions that gain a wide spectrum of masses via
Yukawa couplings to a Higgs field, the problem has been partially solved [34, 35] by the discovery of
a Wess-Zumino term in the low-energy effective action, but work continues on these models [36, 37,
for example]. We believe the present problem, in which gravitational anomaly cancellation occurs
between ingoing states of finite energy and outgoing states of divergent energy, may be another
18
effective theory places constraints of the energy-momentum tensor of the scalar field
in the form of a boundary condition that must be obeyed by each partial wave at the
black hole horizon. The boundary condition can then be used to solve the covariant
conservation equation for the energy-momentum tensor over all of spacetime. The
result is that the energy-momentum tensor must describe a flux of the form given in
Equation (1.7) in each partial wave.
Equations (1.7) and (1.8) are derived primarily in Section 3.4. The calculation
there is relatively brief and painless. However, a great deal of formalism needs to
be built-up to support those calculations. This comprises the bulk of Chapter 3.
In developing this formalism, we find a number of noteworthy intermediate results,
which we summarize below. We are also led to some interesting observations that
are not directly relevant to the core calculation of Section 3.4 and mostly appear as
footnotes in the main text. These are also summarized below.
In Section 3.2.1, we study the properties of the general static, spherically symmetric spacetime in d spacetime dimensions. We compute the components of the Ricci
tensor and scalar curvatures, as well as the Christoffel symbols, in a natural coordinate
system. This allows us, in Section 3.2.1.1, to construct and solve the d-dimensional
Einstein's equations for the most general background matter distributions allowed
by the symmetries. The d-dimensional versions of a few simple, well-known fourdimensional spacetimes are listed in Table 3.1. In Section 3.2.1.2 we examine the
conditions for the existence of a horizon such that the spacetime describes a black
hole. We distinguish between event horizons and Killing horizons, but argue that
given some rather general physical conditions, Einstein's equations imply that if an
event horizon exists at some constant-radius surface of the spacetime, then a Killing
horizon must also exist at the same location. We define and compute the surface
gravity of the horizon. Although much of the core analysis of Chapter 3 will not
depend in any way on the spacetime in question being a solution to Einstein's equations, we will find it necessary at several points to invoke a coincidence requirement
for Killing horizons and event horizons. Thus, it is reassuring that this requirement
is well-motivated by real physics. We also find in this section that the same physical
conditions that lead to the coincidence requirement also imply that the radial pressure
of the background matter seen by a static observer at the black hole horizon must be
equal to the negative of the observed energy density.
In Section 3.2.2, we construct the analog for this spacetime of the Kruskal extension [38] of the four dimensional Schwarzschild black hole. Unlike the Schwarzschild
case, the resulting metric for the general case is not obviously non-singular at the horizon, but we prove by construction that it is. In the process, we show that the choice
of the Kruskal coordinates is quite constrained. As in the original Kruskal extension,
the time translation symmetry of the original coordinates now manifests as a boost
symmetry. Further, we observe that translations in the Kruskal U and V coordinates
become spacetime symmetries at the past and future event horizons, respectively. The
existence of the Kruskal extension requires the coincidence of Killing and event horizons discussed in Section 3.2.1.2. As a bonus, we construct the Painlev-Gullstrand
example of the same generic phenomenon in field theory.
19
[39, 40] coordinates and tortoise coordinates for this spacetime. In Section 3.2.2.1, we
use our new Kruskal extension to reproduce the famous arguments originally made
by Unruh [41] for the four dimensional Schwarzschild black hole, thus proving that
Hawking radiation can be understood for this spacetime according to standard arguments. To further push this point, in Section 3.2.2.2 we follow [42]in constructing the
near-horizon coordinates for the Euclideanized version of the black hole and show that
the Euclideanized horizon exhibits conical singularity which is resolved by setting the
period of Euclidean time to be 27r/n. This agrees with Equation (1.8) by standard
arguments that associate the period of Euclidean time with inverse temperature. The
existence of the near-horizon coordinates also requires the coincidence of Killing and
event horizons.
In Section 3.2.3, we consider the partial wave decomposition of an arbitrary scalar
field theory in this d-dimensional black hole spacetime and solve the scalar wave equation by separation of variables. In Section 3.2.3.1, we derive the (d- 2)-spherical harmonics, which are an alternating product of Legendre P and Q functions. Demanding
regular solutions of the wave equation quantizes the arguments of the Legendre functions and therefore sets the spectrum of angular momentum quantum numbers. We
also find the degeneracy of angular momentum states with total angular momentum
quantum number 1 in d dimensions to be:
Dd(l) = (21+ d - 3)(1+ d-4)!
(1.10)
1!(d - 3)!
In Section 3.2.3.2, we tackle the radial equation, which is not solvable in general. In
the flat space limit, it is solved by Bessel functions; we define the (d- 2)-spherical
Bessel functions. In the near-horizon limit, the radial equation in tortoise coordinates
becomes the 1 + 1 dimensional massless free wave equation for each partial wave.
Finally, in Section 3.2.3.3, we apply the partial wave decomposition to the action
of a scalar field with arbitrary self-interactions. We find that whether or not an
interaction is important in the far-field limit is controlled by whether it would be
power-counting renormalizable in a quantum field theory. We also find that near the
horizon, the vanishing of the metric functions implies the vanishing of all terms in the
action except for the tortoise d'Alembertian term. Thus, the near-horizon action is
that of an infinite collection of free, massless, 1 + 1 dimensional scalar fields in 1 + 1
dimensional Minkowski space.
In Section 3.3, we study the spectrum of blackbody radiation. In Section 3.3.1,
we derive the blackbody spectrum and energy density in d dimensions for a multicomponent massive scalar field with either Bose-Einstein or Fermi-Dirac statistics.
In Section 3.3.2, we find that the relationship between the energy density p and
in d spacetime dimensions at
blackbody flux 1 for p massless fields of statistics
temperature T is given by
Vol (]3d-2)
(1.11)
Vol(§ - 2)
'
where I' is the unit n-ball and Sn is the unit n-sphere. The d-dimensional Stefan20
.____
_·
Boltzmann law is given by
2
I)= p 1 -- Pd)
2
(1-)/2
(d)(d- 1)!
Td
_T
2d-2(d- 2)F (-2)7rd/2
(1.12)
(1.12)
In Section 3.3.3, we engage in the unexpectedly challenging calculation of the blackbody spectrum of angular momentum modes. No closed form expression is ultimately
found, but various limits indicate that each partial wave behaves nearly like a 1 + 1
dimensional blackbody, including the correct statistics. We also remark on the difference between canonical momenta of angular variables and actual angular momenta,
which is understood using the Cartan subalgebra of SO(d- 1). This forms a pleasing
connection between group theory and mechanics.
Finally, in Section 3.6, we speculate on the existence of an enhanced spacetime symmetry giving rise to a thermal spectrum whose temperature is fixed by the
anomaly cancellation mechanism of our core calculation. We also make a speculative
remark about the possible anyon-like behavior of Euclidean black holes.
21
22
_I___
__
Chapter 2
Gravitational Corrections to
Yang-Mills Functions
In this chapter, we will calculate the one-loop contributions of virtual gravitons to
the running of Yang-Mills coupling constants using background field methods in
Minkowski space. We find that this renders all Yang-Mills couplings asymptotically
free, independent of any additional matter content. We also find that that the addition of gravity to a theory which previously displayed coupling constant unification
at a high energy MGUT does not upset this unification. Rather, the unification energy
is shifted up by approximately 23-MGUT/MP, which corresponds to a slightly weaker
coupling. For realistic grand unified theories, this shift corresponds to about 1010
GeV.
2.1
Introduction
Despite its problems with perturbative renormalizability [5], naively quantized general relativity can be taken as a low-energy effective theory for the true theory of
quantum gravity, just as the nonrenormalizable chiral Lagrangian of mesons is a lowenergy effective theory of the strong interactions. In this sense, quantum general
relativity cannot be taken as a fundamental theory and its predictions should not be
trusted above the built-in scale of the theory, MP- GN1/ 2 _ 1019GeV, just as chiral
perturbation theory should not be believed above its scale, f ~ 130 MeV. An interesting open possibility is that general relativity is nonperturbatively renormalizable.
This will be briefly discussed in Section 2.1.2.
In the case of chiral perturbation theory, we know that a new theory, namely
QCD, takes over as the appropriate description of the world above the cutoff scale.
Moreover, because this new theory is asymptotically free [1, 2], it is well-defined even
at arbitrarily high energy scales, and thus it can be taken as a fundamental theory.
Through its inverse effect, infrared slavery, asymptotic freedom also helps to provide
an explanation as to why the appropriate description at low energy does not look
anything like the fundamental theory, through the mechanisms of confinement [43]
and chiral symmetry breaking [44, 45].
23
While the existence - and precise form - of an ultraviolet completion for the
the chiral Lagrangian may be necessary for understanding its role as a description of
strong interactions, knowledge of the ultraviolet completion is usually not necessary if
one wants to use chiral perturbation theory simply to calculate the low-energy behavior of mesons (where "low-energy" here means below f). However, these predicted
behaviors should still be taken as genuine low-energy predictions of QCD. Likewise, as
compellingly argued by Donoghue [21], the predictions of quantum general relativity
at scales below Mp should be taken as unambiguous predictions of whatever the true
theory of quantum gravity may be.
Armed with this attitude, several authors have used quantum general relativity to
calculate the one-loop corrections to the nonrelativistic Newtonian potential, mostly
in the Born approximation [21, 46], but some by other methods [47, 48]. By reading
the numerator of the results of such calculations, one can interpret an "effective
Newton's constant," altered from its bare value by short-range quantum effects. Of
course, the short range in question is very close to the scale at which the theory
should be cutoff, and even then the effects are weak. So, unlike the running of the
QED and QCD couplings, the predicted form of gravitational running is not expected
to be experimentally falsifiable anytime soon.
On the other hand, allowing for the virtual production of any new species of
particle will change the rate at which couplings run. That is, the form of the CallanSymanzik 3 function [22, 23] - the logarithmic derivative of a coupling constant
with respect to the renormalization scale - should be altered with each new field
added to the theory. The addition of gravitons to the Standard Model should be
no exception. Again, it is not anticipated that this effect is of a directly measurable
magnitude for laboratory experiments, but it may disturb certain high-energy predictions of the theory. For example, the experimentally measured values of the standard
model coupling constants seem to conspire together with the theoretically calculated
coefficients of the Yang-Mills 3 functions (augmented with minimal supersymmetry)
to give a unification near MGUT - 10-3 MP [26]. This unification is highly sensitive
to the input parameters, as it is equivalent to getting three lines to meet at a point,
up to experimental uncertainties. Even a small perturbation of the function coefficients could push unification out of the experimentally measured range. If virtual
gravitons were to upset unification in this way, it would be somewhat disturbing,
as the unification of standard model gauge couplings is a necessary prediction of all
realistic grand unified theories.
With this in mind, we set about calculating the the scale-dependence of a nonsupersymmetric pure Yang-Mills theory coupled to quantum general relativity in 3+1
dimensions, to one-loop order. We will use effective action background field methods,
since they are quite natural for gravity and are known to be useful for gauge fields.
In principle, an arbitrary background spacetime could be used, in which case the
renormalization of Newton's constant and the cosmological constant could be studied,
too. Since renormalization depends only on ultraviolet physics, however, and all
spacetimes are locally Minkowski, we will restrict our attention here to the case of
Minkowski spacetime and, thus, zero cosmological constant. However, in the interest
of generality, calculations will be carried out in an arbitrary background with non-zero
24
9YMI
Figure 2-1: Feynman diagrams for two typical processes contributing to the renormalization of a Yang-Mills coupling at one-loop. Curly lines represent gluons. Double
lines represent gravitons. The three-gluon vertex * is proportional to 9 YM, while the
gluon-graviton vertex is proportional to E/Mp.
cosmological constant for as long as is feasible.
The form of the gravitational contribution to the function can be guessed without calculation, since all the new one-loop Feynman diagrams of interest are essentially
a three-gluon vertex with two legs connected by a graviton (See Figure 2-1). Since
the gluon vertex has strength 9 YM and gravitons couple to energy-momentum, one
expects' the inclusion of gravitons to add a term to the 3 function like
A/3 gra (gyM, E) =
aogyME 2 /MP
(2.2)
at energy scale E. Sections 2.3 through 2.8 of this thesis consist of the calculation of
the unknown coefficient a0 .
Once the calculational method is presented for the case of a single gauge field, it
can be extended to the case of multiple gauge fields with interacting matter almost
by inspection. This allows for discussion of realistic theories like the standard model.
It also allows for examination of any theory that exhibits high-energy coupling constant unification. Section 2.9 discusses the implications of gravitational corrections
in such theories. One hindrance of applying the results to interesting supersymmetric
theories, such as the minimally supersymmetric standard model, lies in the fact that
we are using only bosonic gravity in our analysis. Such an application only makes
'The guess of Equation (2.2) is for the case when gym is dimensionless, which demands d = 4.
Generically, each term in the ,3 function would be multiplied by an additional E d - 4 to account for
the units of gyM. So, in the absence of gravity, the / function takes the form
- Ed_4,
3(gYME) = bogM Vol
2(S(Sd ))d-4
(2.1)
where b is a number determined by the theory and the unitless, d-dependent numerical factor has
been arbitrarily chosen for convenience.
25
sense in a model where supersymmetry is treated as breaking above the Planck scale
in the gravitational sector while remaining unbroken in the rest of the theory.
2.1.1
One-loop Divergences
Finding a proper quantum treatment of gravitation is a long standing problem of
theoretical physics. The attempts to overcome the many technical difficulties of a relativistic local quantum field theory of gravity are legion. None have been completely
successful, but the effort spent on this problem has not been wasted, as several techniques originally developed to cope with gravity have ultimately proven essential elsewhere in field theory (for example, the Faddeev-Popov method [24]). In [5], 't Hooft
and Veltman applied the techniques they had previously used to prove the renormalizability of spontaneously broken gauge theories [25] to the problem of a scalar field
coupled to quantum general relativity. They showed that the one-loop divergences in
this theory can only be cancelled by counterterms that don't appear in the original
action. In other words, the theory is nonrenormalizable. To cancel the divergences
to all orders in renormalized perturbation theory would require an infinite number of
counterterms whose coefficients would require an infinite number of experiments to
determine, thus spoiling the scientific predictability of the theory.
To some degree, part of the nonrenormalizability of the theory can be attributed
to the matter to which it is coupled. This is because pure general relativity, without
matter, is actually renormalizable at the one-loop level; new counter terms only arise
at the two-loop level. One might hope, therefore, that some special combination of
matter fields and general relativity might be proven renormalizable, in effect choosing
a specific theory or family of theories as unique in this status. This is unfortunately
not the case. Dirac fields [7], Maxwell fields [6], and Yang-Mills fields [8, 9] all yield
one-loop divergences like the scalar field.
So, the quantum theory we plan to compute with has unavoidable one-loop divergences. Since we wish to calculate the Yang-Mills 3 function to one-loop order,
we should start with a one-loop renormalized action for general relativity. In four
dimensions this can be written as the "curvature-squared" action:
=
SG
J
dxV1 t{
1
6R
+ aR 2 + a 2 R,,vR'}X
(2.3)
where A is a cosmological constant, g is the determinant of the metric, R is the
Ricci scalar curvature, and Rab is the Ricci tensor curvature. The allowed "Riemannsquared" term has here been eliminated by rewriting it as a sum of the other two
curvature-squared terms and an unwritten topological density. The case of a, =
a2 = 0 is the standard Einstein-Hilbert action for general relativity. The coefficients
aeihave units of action, and can thus be combined with GN and the speed of light c to
3 - £pV/-/h,
where £p _ 1.6 x 10 - 3 5 m is the Planck length.
form a length V /GNj/C
The primary physical effect of the curvature-squared terms is in Yukawa corrections
to the non-relativistic Newton's law with characteristic lengths given by epv/i [49],
where we have returned to h = 1 units. A conservative modern limit on the length
26
I_·_I
scale of such forces is approximately 1 mm [50], which is much larger than ep.The
limit ai < 1064is thus extremely weak.
Since the curvature-squared corrections to the quantum action have such a small
effect on observable physics, we will ignore them throughout the rest of this thesis.
Assuming ai < 1, these terms will only become important at energies very close to
and above Mp.
2.1.2
Asymptotic Safety
An oft overlooked open possibility is that despite all the negativity of Section 2.1.1,
quantum general relativity may still be a perfectly well defined and predictive theory. This is because all that the nonrenormalizability proofs [5, 6, 7, 8, 9] really
show is that perturbation theory in terms of small fluctuations around a free field
theory fails for quantum general relativity. The theory may still be sensible non-
perturbatively if it has the property that Weinberg has dubbed "asymptotic safety"
[51, 52]. This is the case where the renormalization group flow of the theory exhibits
an ultraviolet fixed point with only a finite number ultraviolet attractive directions.
This condition ensures that there exists a finite-dimensional critical subspace in the
infinite-dimensional space of all allowed coupling constants such that renormalization group trajectories along this subspace are confined to it. The theory on the
critical surface is then parameterized by a finite number of couplings and has a well
defined continuum limit given by the ultraviolet fixed point. Theories for which the
ultraviolet fixed point corresponds to free field theory, such as in Yang-Mills theory,
are called asymptotically free. These have the advantage that they can be calcu-
lated with standard perturbation theory and are thus perturbatively renormalizable.
Generic asymptotically safe theories lack perturbatively renormalizability, but are no
less suited as nonperturbative fundamental theories because of this fact.
An example nontrivial asymptotically safe theory is that of a scalar field in
five dimensions obeying the symmetry
- [51]. This theory has no allowed
renormalizable interactions beyond a mass term, but it does exhibit an interacting
Wilson-Fisher ultraviolet fixed point. This fixed point has only two attractive direc-
tions, and is thus asymptotically safe. Another theory with a nontrivial ultraviolet
fixed point is that of fermions interacting via a four-fermion term in less than four
dimensions [53].
There is now evidence that four dimensional quantum general relativity may be
asymptotically safe with two attractive directions given approximately by Newton's
constant and the cosmologicalconstant [54]. Most of the study of this fixed point
has been carried out with the exact renormalization group equations for the so-called
effective average action [48] of Reuter, but the fixed point has also been found with
other methods, such as the proper time renormalization group [55]. These flow equations are infinite-dimensional nonlinearly coupled first-order partial differential equations. In practice, the equations must be approximated by truncating to some finitedimensional subspace. The observed fixed point seems to be stable against changes of
the truncation and acceptably insensitiveto the choice of regulator and gauge. It also
appears to persist for realistic matter content [56] and spacetime dimensions ranging
27
from 2 + E up to perhaps six [57].
In this chapter, we will be treating quantum general relativity coupled to matter
as a perturbatively nonrenormalizable effective field theory with a cutoff. The arguments of this section show that perhaps this theory is in fact valid to arbitrarily high
scales due to asymptotic safety. If it is also true that only Newton's constant and
the cosmologicalconstant are essential couplings at the fixed point, then the approximation of dropping the curvature-squared terms made in Section 2.1.1 is justified
even at high energy scales. Also, if the theory is asymptotically safe, then many of
the caveats that would need to be stated regarding calculating and interpreting the
results of the effective theory near its cutoff can be relaxed.
2.2
Technical Preliminaries
2.2.1
Background Field Theory
The background field method is especially well suited to the calculation we are going
to attempt. The application of the method to the calculation of one-loop Yang-Mills
/ functions without gravity is textbook fare. Indeed, our use of the method will follow
very closely to [58, Section 16.6]. A background expansion is always necessary at some
level for perturbative gravitational calculations, since these involve metric excitations
which are small fluctuations about Minkowski spacetime, or some other spacetime,
and perturbing about the singular state with vanishing metric would prove a poor
approach. The background method is thus a convenient choice, since it accommodates
this expansion naturally.
While ultimately equivalent to calculations that could be done with Feynman diagrams, the background method arranges the calculation differently. Whereas Feynman diagrams compute results process-by-process, the background method computes
them species-by-species. This is again convenient for us, since we want to examine the
effect of adding one new particle species (the graviton) to an established calculation.
One further advantage of this method is that a one-loop calculation corresponds to
evaluating a simple Gaussian functional determinant. If we were to attempt calcufunction to higher-loop accuracy, this method would lose much of its
lating the
advantage over Feynman diagram techniques.
The recipe for the background field method as we will be using it (to extract 3
functions) is as follows:
1. Write down the classical action. Identify the operators whose coefficients are
the renormalizable parameters of interest.
2. Expand each field that contributes to the operators of interest as a quantum
fluctuation about a classical background. Leave all other fields as they are,
effectively choosing zero background for them.
3. Identify the gauge freedom of the quantum fields and gauge-fix them. This will
introduce Fedeev-Popov ghosts [24].
28
-----
-
_ _ _
_
4. Ignore all terms that are higher than quadratic order in quantum fields. If desired, use the classical equations of motion for the background fields to eliminate
some terms. The action is now a background-dependent Gaussian functional of
quantum fields.
5. Using the generating functional, functionally integrate out all of the quantum
fields into functional determinates. What remains should be a gauge-invariant
functional of the classical background fields. This is the exponential of the
one-loop effective action.
6. Evaluate the functional determinates as the exponential of a polynomial series
in the background fields. As usual, interpret the divergent integrals encountered with a convenient scheme, such as minimal subtraction, that introduces
dependence on a mass scale E. The only terms in the series that need to be
retained are those that correspond to the original operators of interest.
7. Interpret the mass-scale dependent coefficients of the effective action as the
running couplings in the limit that the mass-scale is differentially close to the
renormalization scale.
8. Solve for the 3 functions using dg
/3dE/E.
9. Integrate the 3 functions to find the running couplings.
2.2.2
Definition of Newton's Constant
We will use a set of units for Newton's gravitational constant in d spacetime dimensions which reflects the physical interpretation of the nonrelativistic gravitational
force law as describing the density of diverging, conserved field lines, commensurate
with its origin in a Gauss law. These units are slightly different from certain other
conventions
[14, for example].
Start with the simplest generally covariant actions for a dynamical spacetime
metric, the Einstein-Hilbert action:
S
dd
=
Jd xv/R.
(2.4)
The overall coefficient is defined as - 2 so that perturbative metric excitations have
a canonically normalized kinetic energy operator when a factor of X is absorbed into
the field definition. When coupled to matter, this action yields the Einstein equation:
xK2
9gabR
=
2
2~2Tab,
(2.5a)
2 (Tab -d
2T)
(2.5b)
Rab -
or equivalently
Rab =
29
The energy-momentum tensor Tab is defined as
Tab = 2
6
matter
Jgab
(2.6)
The factor of 2 in the numerator ensures that tensor so defined - that is, by variation
with respect to the metric, of which there is usually one power in a kinetic matter
Lagrangian- actually represents the canonically normalized energy-momentum that
is, by variations with respect
is derived as the Noether current of translations-that
powers
in
a kinetic matter Lagrangian.
to matter fields, of which there are usually two
Consider an approximately Newtonian spacetime as seen by a nonrelativistic observer with timelike velocity ua. Contracting Equation (2.5b) with uaUband identifying the Newtonian potential o as the field whose gradient gives the acceleration of
free-falling particles, we get
X2(d-3)
2
V
p=_2(d-
where V2 is the
by the observer.
indices, UaUa=
limit. Equation
2) p '(2.7)
(2.7)
Euclidean Laplacian and p = -To is the physical energy density seen
We have used Roo = gijRoioj = -V 2 W, where i and j run over spatial
-1, and Ta - -p, since pressures are negligible in the Newtonian
(2.6) gives a force law between two point masses of
m1m2
X 2 (d- 3)
IF(r) = Vol (Sd-2)rd - 2 2(d - 2)
(2.8)
1 2
)/
Of course, we have Vol(Sd - 2 ) = 27r(d-
FP((d-1)/2)'
Since the physical essence of the 1/rd - 2 law is that there is a total flux generated
at some small distance which is conserved as it spreads out over the area of a sphere
at a larger distance r, we take the force law to represent the ratio of the areas of the
2(d-3) should be thought of as
spheres at these different distances. Thus, the factor 2(d-2)
the area of a sphere of some special unit radius, 2(d-23) Vol (Sd- 2 )ed- 2 . We then
-2
Md
as the basic unit for counting square areas. Thus,
define GN =
= 2d - 2 Vol (d-2)GN.
d- 3
(2.9)
of X gives a nonrelativistic Newton's
So, this choice for the d and GN dependence
2
Vol (Sd- )
law in d dimensions of F(r) = mlm 2 vol(S2 r,)
of radius r.
where Vol ()
30
_·_
_·____
is the volume of an $n
2.3
Setup
With the exception of the overall normalization of the gravitational field, we follow the
definitions and conventions of [59] for gravitational quantities as closely as possible.
Boldface lowercase roman indices are gauge group indices, while normally faced
lowercase roman indices are spacetime indices.
The dynamics for a non-Abelian gauge field coupled to gravity in d spacetime
dimensions is given by the sum of the Einstein-Hilbert and Yang-Mills Lagrangians:
L=L
+Lm =
~~~~~~~~YM
X~-- 12 ac
bdFa (2.10)
2.x/--g[R - 2A]
3C
---
~~4gy~m
gg g
ab
2cd,
where X 2 is defined in Section 2.2.2, A is a cosmological constant, g = det gab, R is
the Ricci scalar, gYMis the gauge coupling,
Y~b
-A~4~--/\bAA
bJ4~+
Fa -/Aa
+ fbc
fA AaAbC~4
(2.11)
is the field strength, Aaais the gauge field, fabc are the structure constants of the
non-Abelian gauge group G, and /a is the derivative operator obeying Aagbc = 0,
i.e. the covariant derivative. Since Fabis antisymmetric under a -+b, the Christoffel
connections arising from the derivatives in Equation (2.11) will cancel against each
other 2 . Thus, the covariant derivatives here could be safely replaced with partial
derivatives or any other torsion-free derivatives.
The Lagrangian (2.10) is non-polynomial in gab, and the configuration gab = 0
is unachievable, so we expand gab about an arbitrary classical background gabwith
quantum fluctuations hab:
gab
=
gab + hab.
(2.12)
Indices are now raised and lowered with the background metric. We need to re-express
L in terms of hab and gab, up to quadratic order in hab. Higher order terms in hab will
only contribute to higher-loop processes. Once this is done, hab will look like a tensor
quantum field that lives in a classical curved spacetime. We also need to expand Aa
around a classical background configuration a a:
a:
Aa- = a + Aa ,
(2.13)
where a a obeys the classical equation of motion
DaFaab = 0.
(2.14)
Fabis the appropriately named function of classical fields only, and Da = Va - iatr a.
Here t a is a gauge group generator for the representation r, and V is the torsion-free
derivative operator obeying
Vagbc = 0.
(2.15)
2
We assume that gravity is torsion-free.
31
Of course, aaa is in the adjoint representation, so that in Equation (2.14),
[tciab = -ifabc.
(2.16)
Performing the expansions (2.12) and (2.13) on the Lagrangian (2.10) is an extended calculation. It is presented in detail for the case of an arbitrary background
spacetime in Section 2.4.
2.4 Expanding the Action
In this section, we show the tedious details involved in a particular method for expanding both the Einstein-Hilbert action and the Yang-Mills action in terms of per-
turbations about arbitrary backgrounds. The results for the gravitational portion
follow closely to results presented in [5], although the calculational method differs
significantly.
Expanding the Non-Polynomial Terms
2.4.1
Given the metric gab, we take the expansion as in Equation (2.12), with
an arbitrary background metric. We will now expand the inverse metric,
the math fact that given a matrix M = 11+ A, where
°
0
9ab
gab
being
using
is the identity, then M - 1 =
[-A] ' . This gives
gab= gab_ hab+ hhCb + 0 (h3).
(2.17)
We will also need the expansion of x/I--g:
/-g = f7=exp { 1Trln[5b+ ha]}.
We now Taylor series the logarithm, evaluate the trace, and Taylor series the exponential:
(2.18)
\/= = -g[1 + lh+ 1(h2 - 2habhab)] + 0 (h3)
where h h. To sum up, we define
g 9
--g
a gab +
_/i-[1
ab
(2.19)
+ D],
(2.20)
where
Iab =
D =
o
(h 3 ),
-hab + hahcb +
h+ 1(h2 -2habhab)+ 0
(2.21)
(h3 ).
(2.22)
D and Iab are infinite order polynomials (that is, nonpolynomials) in hab. In fact,
they are the source of all the non-polynomial graviton interactions. In practice we
will truncate these polynomials at second order or less in h, but by leaving them as
32
___
__
D and Iab for now, we will be able to both keep the calculation more organized and
keep track of the influence of the nonpolynomial terms.
2.4.2
Expanding the Einstein-Hilbert Action
In this section, we expand the Einstein-Hilbert action in terms of a quantum perturbation about an arbitrary background to second order in the quantum fields using
the method of background derivatives.
2.4.2.1
Curvature with Background Derivatives
We make the definitions, as in Wald [59]:
[Aa, Ab]Vc
-
RabcdVd,
(2.23a)
Rab
Racbc,
(2.23b)
R
Rabga
-
,
(2.23c)
where va is an arbitrary vector and Aa is the derivative operator commensurate
with the metric gab, that is Aagbc = 0. This derivative is unique (see, for example,
THEOREM 3.1.1 of [59]) and is usually expressed in terms of the partial derivative &a
as
b C
Aavb =b Ob b + rvc
AaVb =
/AaVbc =
(2.24a)
OaVb- rabvc,
aVbc- rabVdc - acVbd,
(2.24b)
(2.24c)
and so forth for arbitrary tensors v, where
rc
ab =
1g
(agbd
gcd
+ Obgad - '9dgab).
(.5
(2.25)
Inserting this form of Aa into Equation (2.23a), we get the standard result
d
d
e
d
e
d
Rabc = dbracDarbc+
racreb-rbcrea
(2.26)
However, we can also use the slightly non-standard expression
Aavb = VaVb+ C vc,
(2.27)
which relates Aa to some other derivative Va. Manipulations similar to those that
led to Equations (2.25) and (2.26) now lead to
Cab =
g
(Vagbd + Vbgad-
Vdgab)
(2.28)
and
d =Racd
Rabc
VbC Rabd ++VbCa
Rabc
VaCbc
CcC+ CacCedb-CbecCea,
C b(2.29)
33
(2.29)
where
[Va, Vb]vc
(2.30)
Rabc.Vd.
If there is an invertible symmetric tensor gab, perhaps numerically close gab, for
which the derivative satisfying Vagbc = 0 is a known quantity 3 , then the above constructions can become very helpful. We formally define a tensor hab such that
gab= gab+ hab.
(2.31)
We then have
Cab =
9 (Vahbd + Vbhad - Vdhab)
(2.32)
Of course, we want to choose the arbitrary symmetric tensors gaband hab defined in
Equation (2.31) to be the background metric and fluctuation, respectively, as defined
in Equation (2.12).
2.4.2.2 Some Useful Definitions and Identities
We define
Hab - (Vahb + Vbhc - Vchab).
Then
c _ Vahac
gHl-l
Vh
_ab
- 2ab),Vh
V
Cc
(h =gabh)
(2.33)
(2.34)
where Cc is the harmonic gauge factor.
Contracting Equation (2.33) over an upper and lower index gives
H
=
Vah.
(2.35)
Another useful relation is
_j__gabrdcrrd =abc
l h
v 2aC bd-2VcVagbd)hcd -
Vc(2habVahbc
-
ab).
(2.36)
2.4.2.3 Expansion of Curvature
Combining Equations
(2.19), (2.32), and (2.33) gives
(2.37)
Cab = Hab + Idb
'The proof that the derivative operator which annihilates the metric exists and is unique can be
extended easily to the derivative operator that annihilates any given symmetric, invertible tensor.
That is, given any symmetric, invertible tensor Vab, there is a unique derivative operator Va such
that VaVbc = 0. Thus, we can unambiguously take the Va in the above equations to be the operator
that satisfies Vagbc = 0.
34
· __··
Equation (2.29) then becomes
Rabc
=
Rabcd + V bH d-
V abd
H Hda+ O (h 3 )
(2.38)
+Vc(ICHab) - Va(IeCHceb)
+ HabHcee
- HecbHa+ 0 (h 3 ),
(2.39)
+Vb(IdHe)
- Va(Id Hbg) + Ha H
-db-
Applying Equation (2.23b) to Equation (2.38) gives
Rab =
Rab + VcHab-VaHCb
where Rab is the appropriately named function of gab. Using Equations (2.23c), (2.19),
(2.20), and (2.39) we find
/gR
= \/-(l
=
+ D)(ga + I )Rab
{R + [DR+ IabRab
c ]
b
[ab74-.lc
c + Vc(gabHacb-gcdHed)
=FfnFT-,n~abD
+ [DIabRab + DgabVcHa +IabVCHb
DVbHccb- Ia VaHcb
H(HabHce-gbHcea)
+ Vc(gabJcH-gacIbe)]
+0 (h3) }.
By using Equations (2.21), (2.22), (2.34), (2.35), and (2.36) and pulling the total
divergencesto the outside, this becomes
/--gR =
7
{R - Gabhab- hab [( 4 Gab + gabR)gcd-_ (4G ac + gaCR)gbd
gacgbdV 2 + gabgcdV2 + 2gacVdvb _ 2gcdvavb] hcd}
+ total divergence+ 0 (h3 ), (2.40)
where we've made the standard definition for the Einstein tensor:
Gab
Rab-
1R
g29abR.
General spacetimes allow for a cosmological constant.
(2.41)
We include this in the
gravitational Lagrangian by adding to Equation (2.40) the term
-2v/--gA =
-- {-2A-gabAhab
-4hab[(4gabA- 2gabA)gcd- (4gacA- 2gacA)gbd]hcd} + 0 (h3).
(2.42)
The final line of Equation (2.42) has been arranged in a form that makes obvious how
it should be added to Equation (2.40). The final result is
2£G = /--g [R-2A] =
KX
/g { [R - 2A]- [Gab+ gabA]hab
lhab [1 (4[Gab + gabA] + gab[R - 2A]) gcd _ (4[Gac + gacA] + gac[R - 2A]) gbd
-gacgbdV2 + gabgcdV2 + 2gacVdvb - 2gcdvavb] hcd}
+ total divergence+ 0 (h3). (2.43)
35
2.4.3
Expanding the Yang-Mills Action
The gauge field part of Equation (2.10) can be expanded as follows. First note that
applying the variation of Equation (2.13) to the field strength given in Equation (2.11)
results in
(2.44)
ab Fab+ DaAa - DbAa + fabcAbAc
where F
Va -iaat
is the appropriately named function of classical fields only and Da =
Using only the symmetry of the metric factors and applying Equation
a.
(2.44), we find
ac
ggbb
bd, ~a
a
bdA-a)g = gc [FabFcd
+ 2Fab(DcA- - DdA
)
__
_
O
~~~~~abc
ja
+(DaA - D Aa)(D Ad - DdA-) + 2fabFbAA
c
(A 3 A
Ab
+
(2.45)
However, using the antisymmetry of the field strengths, applying Equations (2.19)
through (2.22), and inserting Equation (2.45) gives
'a
pgac bdFa
vgg/-g
~abgcd
=
_~~~~~a
a ab
a ab
{ [FabFa
+ 4FabDaAab+ 2DaAb(DaAab
b aa
fabc~
a
DbAaa) +2 abcFa AbAcb
aaa\1
aad b 1
[2(F aadh
_ Fa Faabh) - 4(hbd- hg )Fab(D A d - DdA)]
aadcd1a2ab2
b cd --]
F aadhchd
(Fa
~-ab
ab
(FabFaadhhd _IF4abF b2h
FaFaabhcdhcd)
+ [2(Fab
} + 0 (fields 3 )
(2.46)
-1 FabFaab(h2 2hCdhcd) + FabFcadhachbd]
This can be simplified by completing total derivatives and using
Tab =-
J£vmo _
YMO
gab
-
2
2
_/::g~
Sgab
1
[
2
gYM
a
ac
ac b
11
Lacd
a\
49abFFacdFa)
lcd,
(2.47)
where LYMOis the appropriately named function of classical fields only. Equation
(2.46) becomes
a
-gacgbdca
4gyM~v~ =
-'
ab"cd
=~~~~~~~~~~~~
2[AaD2Aab _ AabDaDbAaa
faFbA
{[aFaab]
-4[DaFaAab]
-faFabA
-
aeb
aa - 2gym [Ta hab]2- 4ab[FaCb(DcAaa
a aa)_
DaA
)
+hab
[2 _ Tacgbd -mgymI abcd g 9'ef'
aFaef
+hab
[gM
F
(gab
y gcd
gabra
~1
rcad~
abFaDcAad]
hab
gacgb)d
- 22ag
b]
A]
+ F+a~bcd}
aeF abd] hcd}
+ total divergence+ 0 (fields3).
(2.48)
Before writing the fully expanded action, the contributions to the action due to gaugefixing need to be considered.
2.5
Gauge-Fixing
We regard 9ab as fixed with respect to diffeomorphisms and a a as fixed with respect to
the gauge group G. We attribute the variations of gab and 4aato transformations of hab
36
and Aa, respectively. Thus, we take the induced infinitesimal gauge transformations
due to diffeomorphisms to be
Jhab
VaTlb + Vb??a + VanJ/hcb + VbT7Chca+ 7lcVchab,
=
(2.49a)
(2.49b)
~~a,
JAa = AaVa?7 + cVcAa,
Jaa = ac Va' + rcVcaa.
(2.49c)
Acting with G, we get
JAa = a DaOa a + fabcAbac
(2.50a)
Jhab =
(2.50b)
0.
We need to fix the gauges on hab and Aa . We take the background-covariant
gauge-fixing conditions
-Ca(h)
Ca(h,A)
Ga(A)
where
Ca
2
2
FaabA= O,
(2.51)
- DaAaa = 0,
h)=
C (h)-Vbh
_ h~a
(2.52)
(h
-Vh
a).
ha).
(2.53)
By using the Faddeev-Popov [24] method and choosing Feynman-'t Hooft gauge factors, these each add a term to the Lagrangian
2K2 /CaC
AIgf:h =-
___
V/
~~~aab
2
-
249x
FaaC abF
2
(CaCa
2:X2
_
A29f:A
-
ca+ 2 X4 Fa bba
2b_FbAabba
a F
gyma~
12 ,/-gGcG
c
bc)
(2.54)
(2.55)
2gYM
respectively. Equation (2.51) is similar to an R~ gauge [25], which is here engineered
to cancel unpleasant graviton-gluon cross-terms that will appear later. We will eventually find that a convenient choice of gauge is ~ = 1, whereas ~ = 0 reproduces the
traditional harmonic gauge.
Equation (2.54) can be expanded using
-C Ca
2
a
=
h[-
4 42
abgcdV2 + gcdvavb + gabvcvd - 2gacvbvd]hh
+ total divergence. (2.56)
Likewise, Equation (2.55) becomes
-2G 1a
G
a1r
a Aab
2 - 2DaAaaDbAab
=--
1AaaDa'bAab
+
Dzx
ta.~bA-t
37
total
divergence.
(2.57)
Of course, these gauge-fixing terms appear along with the Faddeev-Popov ghost
Lagrangians through the application of Equations (2.49) to Equation (2.51) and Equations (2.50) to Equation (2.52). The form of the ghost Lagrangian in a non-Abelian
gauge theory is a standard result, and it is only slightly altered in a curved background
spacetime. We get
(A + 6A)ab
Serb
&a
j- [6abD2 _ fabc (DaAc + ADa)] ab
A/LFP:A =_/-3L-ga6G
_
-f(OD
(2.58)
(2.8
~
Ghosts also arise from the gauge-fixing of general covariance. Since both a and A'
transform as vectors, so does the combination FaaAab. That is,
6(Fa~~~~~~~'
ab" 'aa o
(F Aab) = bAaCVa
+ 7 Vb(FaAac )
(2.59)
The ghost Lagrangian is then given by
--/= a Ca(h
ALFP:h
+ h, A + A) b
[ gabV - [Va, Vb] + habv + Cbva + VbCa - h[V,
-4-z,
(FAVa
+2g9adHCVC-g_
Vb]
+ Vb(FaCAac))]
?b.
(2.60)
77b,
(2.61)
To second order in quantum fields, we have:
2.6
ALFP:h =
A/gq
[-gab V 2 + [Va, Vb] + 0 (h) + O (A)]
ALFP:A =
/'a
[_-abD2 + 0 (A)] ab.
(2.62)
Combining the Pieces
The classical fields gaband aa are governed by the classical version of Equation (2.10),
L =
GO
+ LYMO
=
1
a
4
A
-22/A][R-
\/FaabFaab
4gy2 M
(2.63)
By varying this action with respect to these fields, we get the Euler-Lagrange equa-
tions that govern their dynamics:
Gab+ gabA =
9C2
6L£wo
Vf/W 3gab
_
-
-2
9 2
2 Tab,
(2.64)
and
DaFaab= 0.
We will enforce Equation
(2.65) on a,
but we will not enforce Equation
(2.65)
(2.64). In
this way we can study the behavior of an arbitrary gauge field in any spacetime back38
ground, but we ignore the warping of spacetime by the background gauge field. That
is, we're making a "test field" approximation.
In contrast,
if we were to enforce both
Equations (2.64) and (2.65) and then restrict attention to, for example, Minkowski
space, we would be forced to consider only gauge field configurations with vanishing stress-energy. The most satisfying approach would be to force ab and a a to be
arbitrary simultaneous solutions of Equations (2.64) and (2.65), but the necessary
calculational techniques required for analysis subsequent to this point have proven
elusive.
By combining Equations (2.43) and (2.48), applying the equation of motion (2.65)
for the background field, and dropping total divergences, we can write
= 0 + iO(h)+ JO(h2)+-CO(A2)
+ LO(Ah)+ ·· ,
(2.66)
where
-g[R
X2
2
-
2A]
aab
1 1~---.1Fa
(2.67a)
Y=FabF
~~4gym
4g
=1 \=-g [~ab
2Tb] hab,
( (h)£O~~h)
=
[Gab + gab
abA - AXTab1
£O(h2)=-
(1
44JX2
-ghab [ X 2 L
ghab
1 ab cd
2
+ 2(GabgabA -
(2.67b)
9 acgbd)
9
2
c Faac
2+2
F F abd
g-g
2Tab)gCd - 4(G
-g acgbdv 2 + gabgcdv2 + 2gacVdVb
CO(A2)
-
2g/2[
=~~~~~a2a
2 Aa
a
[AD
+
AD2
A+
C + gacA -2T
Tc)gbd
2gcdvaVb] hcd, (2.67c)
b a a +Aafcabclcab
A aa DbDaA
AfabcFcabA
DD b +a.~
b
.b
,
(2.67d)
9YM
g/-F: [-gabDcAad + 2gad(DcAab- DbAac)] hb.
L0(hA)
=-'lOh)2g2M
(2.67e)
YM
Note that
CO(A)
was eliminated by the equation of motion (2.65).
Equation (2.67e) can be brought into a more symmetric form. By integrating by
parts on the first and third terms of Equation (2.67e), enforcing Equation (2.65), and
dropping total divergences, we get
-Mhab [9bFaadDd + DaFabC] A
gYM
(2.68)
/CFaabA
gYM
Repeating the procedure on the first term of Equation (2.68) gives the similar form,
LhA =
1
-2--/gA
a[cbaad
a [--g F
_ aac
d - LIFJ
gYM
1
Lab- 2
gYM
39
-gCaF
aaba
Ab.
(2.69)
We now symmetrize Equations (2.68) and (2.69) to get
{hab [D F
2--V
£hA =
2gYm
+g F
Dd] AC
+Aa [-DaFacb
-
gcbFaadDd]hab+ 2CaFaabAa} . (2.70)
Equation
(2.54) will give contributions to CO(h2), O(hA), and LO(A2), whereas
Equation (2.55) will only contribute to £O(A2). Adding the gauge fixing contributions
to Equation (2.67c) using Equation (2.56) we get
1
O(h2) + ALo(h2)
0(h 2 )
I2 ~-ghb [(gacgbd 1 ab cd)aV2
)V 2 _ gX2
gh ab[(gacg
--4X 2
-
___
+2(Gab+ gabA-
2
4) -
2gd[V ,Vc] ++
cd - 4(Gac+ gacA2T ab)g
aa
X2FaacFabd
d9
M
(2.71)
2T ac)gbd hd,
2
where we've used the fact that
hab(gabVcvd - gcdVavb) hcd = Va (hVbhab - habVbh) - VhVdhcd + VahabVbh
=
Va (hVbhab
(2.72)
habVbh)
-
is a total divergence, which we've dropped. Adding the gauge fixing contributions to
Equation (2.67d) using Equation (2.57) we get
4
+Af[gf:h
O(A2)+ A/gf:A + .. O(A2)
-
/A~
1
aaFbbc] A2
A b,2
[4-gabjab[D2]- 6ab[Va,Vb] + 2 fabcFcab+
2gym
gYM
1
(2.73)
where we've used Equation (2.16) and
[Da, Db] = [Va, Vb]
-
iFabtra.
(2.74)
Finally we can add the gauge fixing contributions to Equation (2.70).
L(hA)+
CO(hA)
+ A'gf:h
""''O(Ah)
-
2
{hab [DaFabc + gbcFaadDd] A + A [DaFacb
bd A +aa A c
= ~~V--2gy-----[D F {
C F
--2YM
+2(1
-
_ gcbFaadD]d hab
(2.75)
)CaFaabAa}.
Bringing together Equations
we get
(2.67a), (2.67b), (2.71), (2.75), (2.73), (2.58), and (2.60)
£ I: L + h + £h + hA+ A2+ L,-2+ £O,2
+ . . .,
40
_ __
(2.76)
where
=h
Lo =
V/-g[R2A]--X2
~4gyXm
= -
-
Lh 2
2
Gab + gabA -
-- --
12x/L-ha
4X -----
=
b
C2
- 2gbd[Va,
-acFabd
Vc]
9
2F
o)
YM
2Tab)gcd - 4(Gac + gaCA--2Tac)gbd]
v--'g{hab [DaFabc+ gbcFaadD]
2
(2.77b)
C2Tab] hab,
lab-
+ 2 (Gab + gabA -
(2.77a)
Ia
[(gaCgbd _l gab9cd)(_V2 _
+
hA
aab b,
\/=Aa [gabab[D
2]
(2.77c)
Ac
+Aa [-DaFacb _ gcbFaadDd] hab + 2(1 LA2 = i
hcd
)CaFaabA}
(2.77d)
FaacFbbc]Ab ,
(2.77e)
_ 6ab[Va vb]
+ 2 fabcFcab +
2 X2
YM
La2
=
g_/Z-a
[_6abD 2] ab,
/,2
=
g-7a [-gabV
2
(2.77f)
+ [Va, Vb]] qb*
7
(2.77g)
,
These expressions can be evaluated for Minkowski space by taking g = -1, V
and Rab = R = A = 0. The ellipses in Equation (2.76) indicate terms of higher
than quadratic order in quantum fields. Equation (2.77g) will only contribute only
to the renormalization of X (or simply to an infinite constant in the effective action,
in the Minkowski space limit), so it will be ignored from here in. Physically, the
terms in Equations (2.77) that are proportional to background field strengths represent magnetic-moment-type interactions of the dynamical fields with the background
fields.
2.7
Compiling the Superfield
The term LhA deserves special attention. It is equivalent to
6
6
gab
[L-"'
6ac
A
I
hb
=
aac9
[L
6
L gb
hab]
J
A =
2 { 69ab
+
[jA j1 h,
h
6a
[
L6 gab
9
hab A
J
}
.
(2.78)
If Equation (2.77d) were in this more symmetric form, we could rewrite the quadratic
terms in L by using the superoperator V4o, 1
41
,
where
(g,a)
is a classical
superfield. That is, we wish to write
h2 + LhA +
where I-
£p2 =
A2
(2.79)
(It X5L (I),
is the superfield of quantum fluctuations.
h, A)
Equation (2.77d) is the sum of two parts. One of these has differential operators
acting only on hab while the other part has differential operators acting only on A .
If we now choose ~ = 1, then these two parts are conjugate to each other. This is
exactly what we want for the "off-diagonal" terms in Ve. With this choice of gauge,
then, we have
42~vtv hab~
=r
'
[A' d(2.80)
(Vabcd Vabbf
A '
V
where
Vabcd=
-abcd(72O)
h~~~~~~~~~
4X2 T
LkV
--
2 (FaacF
+ X2YM(
2g2,-oF abd+ FaadFabc
I (gbd[Va,Vc] + gbc[va, Vd] + gad[vb, Vc] + gac[vb, Vd])
2
(G aC + gacA - _2TaC)gbd
2
(Gad+ gadA -
-
(Gbc +
(G
+ gAA- -
'2Tad)gbc_ (Gbd+ g bdA-
+(Gab + gabA- -X2Tab)cd + (G + dAVabab=
21
1
V = +
T
)gac
2T )ga ],
(2.81a)
FaacFbbc] (281b)
[-gabab[D2] _ jab[va, vb] + 2 fabcFcab + ,2
2g2YM
Vabac =
2Tbc )gad
2-
M
[gbCFaadD+ gacFabdDd + DaFabc+ DbFaac],
(2.81c)
[gcbFaadDd+ gcaFabdDd+ DaFacb + DbFaca].
(2.81d)
4gym
We have defined the tensor
Tabcd
(2.82)
1 (gacgbd + gadgbc _ gabgcd)
We will later need the inverse of T in d dimensions, which can be determined from
Tabef (Tefcd + 2 d-gefg
where ifabcd is the identity on the
(2.83)
).abcd,
=
d(d + 1) dimensional space of symmetric 2-tensors4 .
2~~~~~~~~~~~~
4
The projector onto the space of traceless tensors in four dimensions is PT = (I + T), while the
the
space
onto
of
"pure trace"
projector onto projector
the space
of "pure
trace" tensors
tensors is
is Pss =-](lI - T).
T). In
In aa spacetime
spacetime of
of dimension
dimension d,
d,
42
_
_
I·__·I
_I__
_
Finally, we can rewrite the source term (2.77b) as
= [ ·-2 (
£h= L4
-I.
habV
{Tl/Aa)_--
Thus we have
(2.85)
L £Lo + L + L2 + L2.
2.8
(2.84)
Renormalization
Beginning from this section, we specialize all calculations to Minkowski space.
We define the generating functional Z as
-
Z
DgDAeif dsC[g,A]
J
-
DhDAeis[haA]
=
=
J DgDAeiS[gsA]
eiw[g9,a]
(2.86)
where, in the last line, we have defined the effective action W as a functional of the
classical fields. To evaluate W, we first gauge fix and then manipulate terms as in
Sections 2.3 through 2.7:
eiW[g,a]=
f
=
) e{a i
DhDADr
S[
g h a A]+A S gf :h+As gf: A+ / SF P:h+ As FP:A }
J
}
DhDADaei{So[,a]+Sh+Sh2+shA+sA2+s2
J
DIDacei{S[9ai+s++s*2+sc2}
We can eliminate the S. term by writing
~ ~~
3
+
+4,2
gti)-~-g~2~
=
[()J+ v J] vl'
-v- 1J]
-e [(I +--~~~
Then we shift = I+ V 1iJ, which has no effect on D
Renaming 4Yto 4, this leaves
eiW[,a]
D
i{So[9a]- fddXJtv1J+Sp
4
In this expression, both eS ° and the
- 4.()J(~V()1JtV-l1J. (2.87)
(that is, D( = DV4).
2+S% }
-term are constants and the Gaussian
the T appearing in these projector equations is given by Tabcd = abcd - 2gabgcd. The T appearing
in Equation (2.81a), however, is given properly by Equation (2.82) in any spacetime dimension. The
identification of these objects for d = 4 is a coincidence.
43
integrals over
and a can be evaluated as functional determinants:
ezW[9a] ~ exp iSo -
det[V]-2 det[V ]+1
ddxJtVl
J
= exp {iSo -
ddxJV
lJ
j-
½Trln[V,] +Tr ln[VQ]
,
(2.88)
where Equation (2.77f)tells us that V, = -D 2 . Thus,
iW[g, a] , iSo[g, a] -
J
ddxJtV
J -
Tr ln[V,] + Tr ln[VQ].
(2.89)
Note that each term in Equation (2.76) is invariant under both diffeomorphisms and
a formal gauge transformation of the background gauge field where Aa is treated as
adjoint matter. Since each term of Equation (2.76) is a gauge invariant functional of
the classical field, so must be each term of W. Thus each term must be proportional
to
f ddx/
FaabFaab,
at least to first order.
We begin evaluating Equation (2.89) by turning our attention to the V, term.
First we extract out an overall normalization constant from the superoperator in the
following way:
Vabbf
Vabcd
V1 =
Vaecd
g
oabef
ab
0
2
0
0
1)
1
2 [ghcd + Mhcd
Ohbf(
292M
t _ oaecd
M b ef + Nabef
{ _ 3abgef + M
~2gyM
(2.90)
where
1 [FaecFafl
d-4 abgf) (-a2 )-l[FaecF
=~~~~~X
(-b~ + j~
abcd(_2+)-'Lo
=
Mabcd
-
2X2
Nabab =
9
(2ab
+ 1jd-gabg f) (02)-i
[Tecgfd
4Tefg - 4T 9g|, (2.91a)
~YMYM
(_2
[Faa Fbbc],
gabsab (1 + (-0 2 )-1 D 2 ) + 2fabc(_a 2 )-Fcab
+a
2=(--O2)-l
X2
2)
a a bDeFaf]l
_ca b ,
abjab
((T abgfcFaedD
1+
(--2)-1
-' 2fabc(-T ef
NMbab=
b
=
[Faa
cF bbc]_)-F
MbAb=
YM
-
-
ef
-
TCdf],
g2 M
oabac
2
+2-3C
2 2'--l1fLaedl"
- -
Ige
eIf ar ] a
(Tabef
,d
tD--{DeF
Oac" =(--a2)-l[-g fcFadDd - DeF acf]la°bi
d-4 gab
-2-'ge(91)
(2.91c)
(2.91b)
(2.91a)
(2.91c)
(2.91e)
\29d
(2.91e)
44
__
___·
__
Observe that each term in Mh and N is quadratic in Fb. So,
-21Trln[V~] = -
1Tr
n[N] - !T In [
M
oh
[ O
'"'2T
1 [
2
+
+NMAln[]1
O_
MA +N
M,2+O+O_
MhO++O+(MA+N)
OMh+(MA+ N)O_
(MA+ N) 2 + 0_0+
+ ...
(2.92)
We can drop the Tr ln[N] term, as it simply represents an infinite constant. Working
to the order of the fields that appear in the the classical action, we can also drop the
Mh,,MAN, N2 , MAO, and NO terms. The ellipses in Equation (2.92) indicate that
all the excluded terms are also of higher order in the classical fields. Thus we have,
1
- -2T
1in[V]
2Tr(
o0
+
O_ MA+ N
] 1( °( 0
]]
M2 +
+)
- 1Tr[Mh- 1O+O_]- 1Tr[MA- 1MA+ N - 1O-O+]
-Tt[Mh - O+O_1- Tr[MA- MA + N],
(2.93)
where the last line follows from trace cyclicity.
The MA and MA terms from the second trace, along with the S0 and V terms in
Equation (2.89), are exactly what would be found for a Yang-Mills theory evaluated
to the one-loop level in the absence of gravitation. Evaluating these terms for d = 4
and inserting the form of So therefore gives
iW[g, a]
- Tr[Mh- O+O_]- 1Tr[N]4
ddxtV
--2 + (47r)22 n
] FabM,
(2.94)
where bo = C2 (G), E is the background energy scale, and M is a renormalization
scale at which gyM(E)IE=M = 9YM is imposed. Ignoring gravity, this would lead to
~(gYM)
-(47r)2
yM
We now return to evaluating the remaining terms in the effective action. We'll
start with the J-term. Expanding VDas in Equation (2.90), we get
(O_- MA+N
JddxJtV1J - 4ddxJt[[
4
+f
Mh-O+O_
MhO++O+(MA+N)
O_Mh
+(MA+N)O_ (MA+N)2++00+
-1
[ -4C2T-1(_02)
+"'.
0
_292Y (_,02)-1J
t~~
45
J.
(2.95)
p--+
Figure 2-2: The schematic Feynman diagram represented by the functional trace
-Tr[Mh]. A momentum p circulates in a virtual graviton loop coupled to external
gluons of momentum
k.
Since J = (Tab o) , we can see that all terms in Equation (2.95) have four or
(
t
more powers of Fab. Since we are only working to the order of fields appearing in Lo,
none of these terms make any contribution to the effective action.
The final terms, - 1Tr[Mh- O+O_]- Tr[N], are each evaluated in turn in Section
2.8.1, using an ultraviolet cutoff Auv in d dimensions.
2.8.1
Computation of Functional Determinants
The term - Tr[Mh] in Equation (2.94) represents quantum contributions to the gauge
field from integrating out a single graviton loop, as in Figure 2-2. It can be evaluated
as follows:
-Tr[Mh]
J
ddX]abcd?2{
2- (-O2)
(tb labcdLco 9 1 ( a ef++
4gabf)
2 )-l[FaecFafd]
d-2_ (Q g)
}
ld-(ab\
-2 (Tabef
+ --gabgef)(- 2)-l[Tecgfd-1Tefg~cd-Tcdge]}
2
_ -1
whee
X
w4gw
M 2
i~~~~
d-2
(ld(d +1)-4 +2~ - (d-4)(d+ 1))/
ddx{a(-a )- [FaFa ] }
2 tf
(2.96)
where we've used the fact that
g9abT=
-
1
(d - 4)Faab
9YM
46
= (d - 4)£o.
(2.97)
P
4
k-*+
k-*
2
gYM
Figure 2-3: The schematic Feynman diagram represented by the functional trace
-1Tr[N].
A momentum p circulates in a virtual gluon loop coupled to external
gluons of momentum k.
We now evaluate the integral as
I
ddx{(_2)-1[Fab
Faab ] =
ddk FaFaab
I(27r)d ab
f (2ir)
Iddk
b
fa
-
J (27r)d
-
r[Mh =
-.
i
t4
[
aab
aab
(2-r)d
=(-7r)
F
-2i
1
J(27r)dp2
pEd
d
d-
dP
Jtab
d ab '
So, finally
b
ddp
E1-
PEj(27r)d p2
i
2
/ d 22) rEppddPrEP
b (27r)da~d
(Ad-2 _ Ed-2 )
!(d(d+ 1)- 4 - (d- 4)(d + 1
92
YM
(47x)d/2r(d/2)
J
ddFa
Faab (2.98)
d . ab.P
d-
U
-2
Ed-2
d-2
ddXFaFaab. (2.99)
The term - Tr[N] in Equation (2.94) is a gauge-fixing contribution from Equation
(2.51). It is an integration over a gluon loop, as shown in Figure 2-3. We evaluate
47
p-4
9
gYM
YM
Figure 2-4: The schematic Feynman diagram represented by the functional trace
'Tr[O+O_]. A momentum p circulates in a virtual gluon-graviton loop coupled to
external gluons of momentum k.
this, using Equation (2.98), as
292Jd x ab
-2T[N]
1
-49
Ad-2
2
g(4r)d/2r(d/2
= -i
- 1 [Faa Fbbc] }
{(-2)
-
Ed- 2'
d-2
)
aab
JXFa
(2.100)
The trace Tr[O+O_] represents a process where the external gauge field emits
and reabsorbs a virtual graviton, as in Figure 2-4. This becomes
12o+oI
)
=1
ddX OabOca°-cab}
| 2 IdX+
=
2
2
dx
(
gYM
}
(_2) -1 [gfcFaedDd+DeFafc (ab
[gI'
{
+
L]
X (_092)-1[-gicFhagg-
f
X2 /dkdp{-
2g-
()d
(
{
(z
+
+
d-4 abg )
g g
j
DhFc]l[abhi
a[geFacf(k)(-p- k)f + kCFade(k)]
d + d-9ab gcd) (
)2[gbeFa(-k)p 9 + kaF(eab-k)]}
In passing to momentum space, we dropped the gauge field part of the covariant
derivatives, as these will only produce terms which are of high powers in the classical
field. To the order of fields in which we are working, the classical equation of motion
48
___II
for aaa, Equation (2.14), becomes kaFaab(k) = 0. Applying this, we get
2
I
ddk
ddp{
1
+ k)2
(27r)d(27r)d Faa(k)Fcd(-k)p2(p
x [(d+1d2
2
) gacpbpd
+k2gacgbd]}.
The k 2 term produces a higher derivative term in the effective action whose properties
we are not interested in. Ignoring it, we get
2
5111[0+0_1
__ -i1 4
4 - (d -
Ad- 2 _ Ed - 2
4)d(d2))
g2 (4)d/ 2r(d/2)
ddXFaFaab.
d-2
(2.101)
2.8.2
Extracting the 3 Function
Combining Equations (2.99), (2.100), and (2.101) of Section 2.8.1 together gives
- Tr[Mh- OO] - Tr[N] -i
2
2
AdEd- 2 IIddXFa
UV
(47r)d/2(/2) d- 2 J
2~gY~m(4rd2Pd
ab'
4
aab
(2.102)
where
C(d) = 2 d(d + 1) - 4 -
(d -4)(d
d)-dd)----
(2.103)
- 1)(d + 2)
is a function that is positive-valued for all integers 1 < d < 8 and negative-valued
for d = 1 and integers d > 8. Note that C(4) = 6, which is the maximum for real
positive d.
The ultraviolet divergence in Equation (2.102) is regulated by counter termswhich we've not been writing - whose values are determined at energy scale E by
a renormalization condition at scale M. Subtracting divergences by including these
terms gives:
_X 2 C(d)
F
~-_I.1
~4
- Th[Mh-O0+O-]- T4[N]
2
Md-
I
2r(d/2)
[g92 (47w)d/
2 -
Ed-2] 1
d-2
Ad
a raab
]jJ - -ab'
(2.104)
Evaluating this for d = 4 gives
a]- -4 Jd4xg.
W[g,
1
92
3
bo
2
+ g2 (4-)2 (E2 - M2) + (n
2
Thus, when E is differentially close to M -
gyM(E)
_
2
1
gyM(M)
2
+
X
2
gym (M)
-
and only then -
running coupling constant at scale E by
1
Fab
--
(4) M 2
gYM
~2
3
M! (E
(47r) 2
49
2
2 _
2
) +
(2.105)
we define the one-loop
bo
(47r) 2
in
E
2
M 2 '
(2.106)
Under these conditions we find
d
(gym
-( ( 4 )2(2EdE) + bo2 2dE
g--y-(-7
2
(
47r) E
2
(2.107)
and thus,
/3(gymE)
dgym
~
(Y
dlnE
d( /gd(1/gbo
~gyP3d(1/m)
=2Y
dE
=g
_3
-
Eb2
(4 X)2YM
3
(4 7r)22YM
(2.108)
Using the definition X 2 = 167r/Mp, we find that the unknown coefficient of Equation
(2.2) is now determined
to be a =-3
7r
0.95.
The form of the running coupling can now be found by integrating the 3 function.
x3 E 2
Equation (2.107) can be integrated using the integrating factor exp {
This yields
I1
2
gyM(E)
gym
(E) 2
gy
(M)>
exp {3C47
()2
1o
2 (E2_
M2)} +2 (4 bo
17r)2 J
Ed
dk
keXP
S
2
(4r)2kLI
}
(2.109)
Notice that for E > M,
9YM
(E) will always be less than the value it would have had
in the absence of gravitation. Indeed, as E -- o, gyM(E) -- 0 independent of the
values of b0 and gYM(M). That is to say, the addition of gravity to a pure Yang-Mills
theory renders its coupling asymptotically free, even if it were not so before adding
gravity. In fact, near Me the coupling turns off very rapidly in comparison to the
usual logarithmic running. Remember, however, that any discussion of the theory at
or above Me is dubious at best.
It is interesting that quantum gravity perturbations can cause gauge couplings to
run even in theories that exhibited exact conformal invariance, and thus vanishing
/ function, before gravity was added. Two notable examples in four dimensions are
= 4 super-Yang-Mills [60, 61, 62]. For these
pure U(1) electromagnetism and
theories, the exponential integral in Equation (2.109) has zero coefficient, so we are
left with
a(E)
gyM(E) a(0) exp{ -3 () 2E2}.
(2.110)
The couplings in such theories run down from their infrared values as Gaussians of
width /76Mp, corresponding to a 10% reduction in a at about 0.1Mp.
2.9
Enlarging the Matter Sector and the Gauge
Group
We will now examine what happens to the preceding analysis if we include scalars,
spinors, or additional gauge fields. Since we are only interested in the renormalization
of gauge couplings, we do not expand our new scalar and spinor fields as quantum
50
___._
-____
-....
perturbations about a classical background. Therefore, every term in the Lagrangian
containing these fields is of quadratic order or more in quantum fields. Thus, adding
these fields doesn't change any of the terms already calculated in Equation (2.76).
Each new field simply supplies one new term which is quadratic in the matter field
and zeroth order in all the other fields. These terms have no effect on the functional
integration over hab and Aa as described in Section 2.8. That is, adding matter to our
previous discussion will not change the terms already present in Equation (2.108),
but will simply add to them. Indeed, these terms will contribute in exactly the same
way as they did in the absence of gravity. Namely, they add to the logarithmic
running. So, adding matter to our previous discussion only changes the coefficient b0
in the function and has no effect on the gravitational term. Moreover, the matter
contribution is exactly what it was in the absence of gravity.
The addition of new gauge fields is slightly more complicated. Each new field
contributes a term to LO and corresponding terms to Lh (via Tab) and Lh2. Likewise,
the th gauge field brings in a LCA term and an LhA. cross term. Of course, gaugefixing each new symmetry brings in a new gauge-fixing term and a corresponding
ghost field. We also need to augment the gauge condition of Equation (2.51) with
an additional term for each new gauge field. This adds additional couplings of the
gravitational ghosts to the new gauge fields, but these terms will drop out to second
order in the quantum fields. Each LA2 and LhA, receives a similar contribution from
the cross terms generated by gauge-fixing. Gauge-fixing in this way also generates
new cross terms between gauge fields. With so many graviton-gluon and gluon-gluon
cross terms, we need to enlarge the definition of the superfield to include the graviton
and each of the new gauge fields.
The net effect of all this is that to second order in the fields, we get
N
L ;: L +
+ 2 + EL,,
+..,
(2.111)
i=1
where N is the number of gauge symmetries,
N
Lo =
-1 4g 2 (i)abF'(i)
a
aab
1Tab
L4=f=,
bV2=
0
A%
a
0
A(2a
abcd
h
d
A(5)e
LN~ Al)f
a_(iaLa2
Vab
O=
b
habt
V
_~b2 D
(2.112a)
e
Va
2~
(2.112b)
..
Vbbf
b
abbbf
l~ab
Faber
bcd
ia
a~~i
'
51
f
(2.112c)
Ab
(2.112d)
and
r
1
Vabcd
h
_
[abcd(_a2
L T:K2
4X2
(o°)
N
2
-
0)
+
+
-
2 (FaacFabd
( i ) +FaadFabc)
'(i)
~~~~~~i=l
gi
(i)
+ 2(Tacgbd + Tadgbc- Tabg cd + Tbc gad + Tbdgac
Vbab
1 [-gab6ab[D2] + 2f(a) c()
c ()
iCF)
dbdD()d +D +DbF]
va[
__
2gi2
vabab =
+ f()
a
)
Tcdgab)]
(2.113a)
(2.113b)
(2.113c)
(2.113e)
Fbcb
Equations (2.112) and (2.113) are written for a Minkowski space background. The
energy-momentum tensor, Tab, now refers to the total energy-momentum tensor of
the background fields,
N
cr(i)
ac (b (i) + aF(nacd
a)
Jab
1(i)c
4d
-abg2
d [r(i
) 1 r()+b
a.ab
1
-
i
(2.114)
i
The covariant derivatives appearing in the above equations include a term for each
gauge symmetry. However, the ghosts and gauge fields that they act on are all in
adjoint representations of their respective symmetries and singlet representations of
every other symmetry. Thus, these derivatives don't produce any unexpected cross
terms.
We now need to evaluate the effective action. The a term can again be shifted
away, as in Equation (2.87), producing a constant shift in the effective action. We
again find, as in Equation (2.95), that this shift makes no contribution to lowest order
in the field strengths. After extracting an overall normalization from the remaining
quadratic terms and performing functional integration, we need to expand in powers
of the fields. We again find that all of the terms of V, and most of the terms in
V~are of higher order in the classical fields than appear in the classical Lagrangian.
This includes every term involving a power of Vij. The result is that every surviving
term of the operator whose trace we are evaluating is of exactly the same form as
those we evaluated in Section 2.8. One copy is produced for each gauge symmetry,
and each is multiplied by its own coupling constant. The exception to this is the
graviton contribution, which is instead equal to a sum over multiple copies of the form
presented in Section 2.8. Thus we only need to evaluate N copies of the integrals
already shown in Section 2.8.1. Once this is done and terms are recombined to give
the N running gauge couplings, we find that each one get renormalized separately
and in exactly same way as in our analysis of a single gauge field.
52
II
Combining these observations about adding matter and extending the gauge symmetry, we can now see how the analysis applies to a theory with arbitrary gauge and
matter content. The d function for each gauge coupling is equal to what it would be
in the absence of gravitation plus the new term, which is identical for each coupling.
The only possible exception to this would be if the original theory is supersymmetric. In this case one might include gravitinos, which we have not included in this
calculation, since including only gravitons breaks supersymmetry.
We noted at the end of Section 2.8 that the addition of gravity to a pure YangMills theory rendered the coupling asymptotically free. This, in itself, is not so
impressive, since all pure Yang-Mills theories are already asymptotically free. We can
now, however, make a much stronger statement about the gauge theory coupled to
matter. Since the running is of the exact same form in the presence of matter as
without it
only the value of b is changed - we can now see that the addition
of gravity renders all gauge couplings asymptotically free even in the presence of
arbitrary numbers and types of matter fields. Of course, the meaning and utility
of asymptotic freedom is somewhat ambiguous here, since none of these calculations
should be extrapolated too near to or beyond Mp.
2.10
Coupling Constant Unification
Now consider a theory in the absence of gravity with N gauge symmetries and matter
content such that at some high energy all couplings take on the same value. Defining
the symbol yi = 1/g2 for the ith gauge coupling, each one runs as
yi(E) = y7+
E2
In - .
(2.115)
The condition for unification is that there exists an energy Eo such that for any pair
{i,j} we have
(4-7r)2
y~
_ y~
_-(Ir)(.o
° =inI
2 !b;i-/20
=
Eo
.
(2.116)
n'
When gravity is added to the theory, the form of the running follows Equation (2.109):
y\5(E)=
f~x2tE2_M2)}+2
(E2-M2)+
Y {3
0i ~_
IT4-,--y-exp
boE
dk
(4~~~~~
3()
_2
1
2(4 7r)JM
k~ ex
3
(47r)2
(E 2 - k2)} . (2.117)
The condition for unification at some energy Eu now becomes that for any pair {i, j},
we have
22 byep
b-i9
exp
{3
?2 (E2 _ M2)} =J
7r2kex
exp{3(
(7
2 (E
2-k2)}.
(2.118)
But, since we assumed the theory exhibited unification before gravity was added, the
ratio (Ayo/Abo)ij in Equation (2.118) has already been determined to be a constant
53
for all {i, j}. Equation (2.118) is therefore independent of the values i and j. Thus,
any value Eu that solves the equation
in Eo
E--fEdkuexp 3(dkdk
) (M - k2)}
4
(2.119)
2
is an intersection point for all N gauge couplings. Note that since the exponential
function is always positive, the integral in Equation (2.119) is a monotonically increasing function of Eu. Thus, Equation (2.119) has exactly one solution and the
gauge couplings intersect at exactly one point.
E 0 . Assuming this is the case,
If E0 < Mp, then it should be true that Eu
then Equation (2.119) can be approximated by
2
InEo
Eo dkEexp 1 3
-o
Mn ;f:odk
k
L (4r)2 (M - k2) + Ex Eo
lnM
]
0
3E
2 (M 2
()2
(rexp
F20).
2
(2.120)
Thus,
Eu
Eo + Eo / J
k2)}] exp{3
[1-exp{ 3 '(2 (M
2
(M2E
(2.121)
Now explicitly using Mp > Eo > M, we get
Eu
P_
Eo [1 +
(M
(2.122)
Examining Equation (2.122) for the numerical values E0 = MGUT ~ 1016 GeV and
Mp ; 10m GeV reveals that the new intersection point is shifted from E0 by less
than a part per million. The one-loop flows of a - 1 - 4r/g 2 for a theory with running
quantitatively similar to the minimally supersymmetric standard model with gravity
are shown in Figure 2-5(a). The dramatic switching-off of the Yang-Mills interaction
near Mp is made more apparent when the couplings themselves are plotted, rather
than cr- 1 , as in Figure 2-5(b).
2.11
Phenomenology
In this section, we will briefly comment on a few possible experimental implications
of the preceding analysis. First, however, we will discuss the physical content of the
result.
As apparent from Figure 2-5(b), the Yang-Mills couplings turn off so quickly near
Mp that free field theory (zeroth-order perturbation theory) should become an excellent description of the gluon dynamics. To within the accuracy of this calculation,
high-energy gluons do not couple to anything but gravity. For a theory with only
Yang-Mills couplings (that is, no Yukawa couplings, masses, or scalar self-couplings),
this implies that all physics near Mp is well described by free fields coupled to grav54
___
___1__
____1____11
ru
-
7
fUO
6050
.
40'~"'.
I
"'-.
~....
30
X~~~~~~~~~~
~~....................................................
20
-- -- ---- -- - - - - - - - - - - - - -'"~.-'-. --
10
2
0
4
8
6
(a)
1
1
1
4
'
16
8
20
14
16
18
20
loglO(E/GeV)
1.4
1.2
1-
-
0.8
....
....... ...............
0.6
.....
.
-------------0.4
0.2
2 4
(b)
6
8
10
12
log 1 0(E/GeV)
Figure 2-5: In Figure 2-5(a), the three Yang-Mills couplings of an MSSM-like theory
evolve as straight lines in a plot of a - _ 47r/g2 versus log1 0 (E) when gravitation
is ignored (dashed lines). The initial values at Mzo
100 GeV are set so that
the lines approximately intersect at 1016 GeV. When gravity is included at one-loop
(solid lines), the three lines curve towards weaker coupling at high energy, but remain
unified near 1016 GeV. In Figure 2-5(b), g is plotted for the same theory. All three
couplings rapidly go to zero near Mp, rendering the theory approximately free above
this scale.
55
ity. On the other hand, gravitation itself may be extremely complicated at this scale.
Nevertheless, it is somewhat reassuring that all of the most complicated and potentially divergent high-energy dynamics is sequestered in the gravitational sector. It is
difficult to say how seriously this discussion should be taken, however, since all of the
most interesting statements are being made near or above the theory's cutoff. These
statements can only be strictly meaningful if the renormalization group for the quantum effective action flows to a nonperturbative ultraviolet fixed point, as discussed
in Section 2.1.2.
We saw in Equation (2.122) that the unification scale of a grand unified theory
(GUT), as predicted from low-energy data, is slightly raised by the inclusion of gravity.
One might hope that such a raising of the GUT scale is significant enough to effect
predictions of GUT phenomenology, such as the proton lifetime or other "undesirable"
consequences of GUT theories. We believe, however, that all effects are likely to
be in the parts per million range or smaller, which is an accuracy that GUT scale
experiments are not likely to reach for some time.
We should consider the case where the measured values of low-energy couplings
have experimental uncertainties. In this case, the hypothesis of coupling constant
unification can be tested against some accepted goodness-of-fit parameter, such as
x2. Accordingly, the best-fit values of the unification scale and coupling will have associated fitting uncertainties and potential cross-correlations. When gravity is taken
into account in the 3 function, we expect the best-fit unification values to change according to the discussion of Section 2.10. An important open question whose answer
is not obvious at this point is how the fitted uncertainties and 2 will be affected
by the inclusion of gravity. Can gravitation make non-unifying models (such as the
non-supersymmetric standard model) unify to within experimental error, or can it
push generic almost-unified models (such as the minimally supersymmetric standard
model) out of a statistically acceptable range? The first of these possibilities would
suggest that coupling constant unification is not evidence for low-energy supersymmetry. The second possibility would either rule out many supersymmetric GUTs (if
the coupling constants are no longer unified) or make the conditions for unification so
strict that if unification persisted with the addition of gravity, the case for low-energy
supersymmetry would be significantly strengthened.
There has been some speculation in the literature, mostly inspired by the possibility of building large extra dimensions in string theory, that the fundamental scale
of quantum gravity may in the multi-TeV range rather than at 1019GeV. It has been
noted that among the predictions of such scenarios is the creation of microscopic black
holes in high-energy colliders and cosmic rays experiments. The present calculation
suggests that such experiments should also observe a reduced strength of standard
model interactions, presumably disturbing the properties of extensive air showers and
hadronic jets in some way. Working out the details of these experimental signatures,
however, is beyond the scope of this thesis.
A rather different probe of the TeV-scale gravity scenarios is suggested by Equation (2.110). There, we saw that the fine structure constant of a theory with vanishing
,3 function in the absence of gravity runs down from its zero-energy value as a Gaussian of width V/76Mp. Quantum electrodynamics is such a theory at energies below
56
_ .........·
the threshold for electron-positron creation, approximately 1 MeV. Without gravity,
the fine structure constant should be a genuine constant at these scales. Logarithmic
running should only set in well above threshold, where the electron mass can be ignored. With gravity included, the fine structure constant should run as a Gaussian
at low energies, with possible effects on atomic spectra and electromagnetic nuclear
decays. However, even with a speculatively low Planck scale of 1 TeV, the shift in the
fine structure constant at threshold would be a part in 1012, which does not appear
to be within experimental reach.
2.12
Commentary
We have found that for a very generic class of four dimensional gauge theories, the
addition of gravity adds a term to the one-loop 3 function of the form
3 E2
A3grav(gyM, E) = -gYM; - 2
(2.123)
The new term renders all Yang-Mills couplings asymptotically free. In fact, the couplings turn off from the usual logarithmic running quite quickly near the Planck scale.
This correction to the running might be observable if the world is governed by certain
unorthodox theories of particle physics. For a theory whose gauge couplings exhibited
unification before gravity was added, the unique unification point is maintained, but
shifted to a slightly higher energy (Eu/E 0 , 1 + 3 E/Mp) and weaker coupling.
The following are some important caveats about the preceding calculation:
* We have only used an effective theory for gravitation, and are trying to interpret
it somewhat close to its cutoff. This is the strongest critique of the calculation. If
quantized general relativity has an ultraviolet fixed point in its renormalization
group flow as discussed in Section 2.1.2, however, the results given here may, in
fact, be reliable to higher energies.
* Quantized general relativity should properly include terms of order R2 at oneloop, as discussed in Section 2.1.1. These terms usually have little effect on
observable physics, but they may begin to become important near Mp. Also,
the coefficients of these operators should be allowed to run, as should X and
A. The running of these coefficients could become important in about the
same range as where the gravitational contribution to the Yang-Mills function
become important. Again, however, an ultraviolet fixed point may mitigate
these problems.
* Restricting the background spacetime configuration raises the problem of back
reaction. The "test field" approximation may not be valid at very high energies.
That is, if the background gauge field carries a very large energy density, it
should be allowed to warp the background spacetime.
The caveats just given are issues of principle. There are also several warnings to be
made of a less fundamental nature with regard to model building:
57
* We live in an approximately Robertson-Walker spacetime, not Minkowski, which
is well modeled as having a small positive cosmological constant. We do not
expect large scale structure to have dramatic effects on the ultraviolet physics
discussed here, however, so this is a minor point.
* We have not considered extra dimensions as they are often implemented in
realistic models of particle physics, whether they be compactified, orbifolded,
deconstructed, warped, or otherwise of popular interest. We have only considered the case where all matter fields, gauge fields, and gravity propagate in
the same bulk space and none of the fields are confined to lower-dimensional
defects.
* We have made an inadequate treatment of supersymmetry in the gravitational
sector by not including gravitinos. If we imagine that below Mp supersymmetry
is broken only in the gravitational sector, while remaining valid in the gauge and
matter sector, then the analysis given here is intact. However, this restriction
may be too unnatural for serious application to some models.
58
-
K
Chapter 3
Black Hole Effective Field Theory
In this chapter, we will show that in order to avoid a breakdown of general covariance
at the quantum level, the total flux in each outgoing partial wave of a quantum field
in a black hole background must be equal to that of a 1 + 1 dimensional blackbody at
the Hawking temperature. Specifically, we attempt to formulate an effective quantum
field theory only in the region outside of a black hole and discover that this theory is
anomalous unless the stated radiation flux is included.
In Section 3.2, we will turn our attention to the physics of d-dimensional black hole
spacetimes in classical general relativity and the behavior of scalar wave equations in
such spacetimes. Then, in Section 3.3, we will derive some useful but nonstandard
results regarding the properties of blackbody radiation in spherical coordinates and
in d spacetime dimensions. Finally, in Section 3.4, we will attempt to formulate an
effective field theory of scalar field modes that live only outside a black hole event
horizon.
3.1
Introduction
Hawking radiation from black holes is one of the most striking effects that is known,
or at least widely agreed, to arise from the combination of quantum mechanics and
general relativity. On the other hand, potential sources of conflict between the central
principle of general relativity, general covariance, and quantum theory may exist in
certain situations in the form of gravitational anomalies. Both the anomaly and
Hawking radiation result from, in a certain sense, ambiguities of the quantization
process in curved spacetimes and both vanish in the absence of spacetime curvature.
3.1.1 Hawking Radiation
Hawking radiation originates upon quantization of matter in a background spacetime
that contains an event horizon - for example, a black hole. One finds that the occupation number spectrum of quantum field modes in the vacuum state is that of a
blackbody at a fixed temperature given by the surface gravity of the horizon. The
literature contains several derivations of Hawking radiation, each with strengths and
59
weaknesses. Hawking's original derivation [28, 29] is very direct and physical, but
it relies on hypothetical properties of modes that undergo extreme blue shifts, and
specifically assumes that their interactions with matter can be ignored. Derivations
based on Euclidean quantum gravity are quick and elegant, but the formalism lacks
a secure microscopic foundation [42]. Derivations based on string theory have a logically consistent foundation, but they only apply to special solutions in unrealistic
world-models, and they do not explain the simplicity and generality of the results
inferred from the other methods [63, 64]. In all these approaches, the Hawking radiation appears as a rather special and isolated phenomenon. Here we discuss another
approach, which ties its existence to the cancellation of gravitational anomalies.
3.1.2
Anomalies and Anomaly Driven Currents
An anomaly in a quantum field theory is a conflict between a symmetry of the classical
action and the procedure of quantization (see [65] for a review). Anomalies in global
symmetries can signal new and interesting physics, as in the original application to
neutral pion decay r ° - -y7 [66, 67] and in 't Hooft's resolution of the U(1) problem
of QCD [68, 69]. Anomalies in gauge symmetries, however, represent a theoretical
inconsistency,leading to difficulties with the probability interpretation of quantum
mechanics due to a loss of positivity'. Cancellation of gauge anomalies gives powerful
constraints on the charge spectrum of the standard model, which were important
historically
[70, 71].
A gravitational anomaly [32]is a gauge anomaly in general covariance, taking the
form of non-conservation of the energy-momentum tensor. A gravitational anomaly
can only occur in theories with chiral matter coupled to gravity in spacetimes of
dimension 4k + 2, for integer k. The chiral matter could be either a chiral fermion or
a 2k-form with an (anti-)self-dual field strength. An important case is the self-dual
scalar field in 1 + 1 dimensions. This is a scalar field constrained to obey
0a4 = EabO('0.
(3.1)
That is, it has only right-moving modes and is thus chiral. In this simplest case,
which will be crucial for us, the anomaly then reads [32, 65, 72]
~VaT
= 96 l
1
+/
c
da r c
(3.2)
The energy-momentum tensor of the chiral scalar is therefore not conserved in curved
spacetimes
[32].
There are several cases in physics where anomalies have been connected to the
existence of current flows. Pair creation in an electric field has been related to a
chiral anomaly [73]. The existence of exotic charges on solitons, with or without the
1'Athird kind of anomaly is a "conformal anomaly", which is a quantum contribution to the trace
of the energy-momentum tensor Taa . This is the source of scale-dependent renormalization effects in
otherwise scale-free theories.
60
__
existence of zero modes, has been related to anomalous charge flows that arise in
building up the soliton adiabatically [74, 75].
Especially closely related to our problem is the connection between anomaly cancellation and the existence of chiral edge states in the quantum Hall effect [76, for
example]. The effective action modeling the electromagnetic field in this state involves
the Chern-Simons Lagrangian density confined to the sample area:
S oc d3 xeA(x)abCFab(x)Ac(x),
(3.3)
where the EA function is 1 within the sample region A and 0 outside. The gauge
transformation Aa - Aa + O~adoes not leave this action invariant, because after
integration by parts a term appears from the derivative acting on eA. The extra
term is confined to the boundary and is proportional to a there. This variation
takes the same form as the anomaly of a 1 + 1 dimensional massless, charged, chiral
fermion field on the boundary. Since the theory we are modeling is gauge invariant, an
adequate effective theory must cancel the boundary gauge variations. This motivates
one to expect the existence of massless chiral edge states, whose anomaly cancels
the boundary gauge variation. Such states do in fact arise, as can be proved from
more microscopic considerations. We will demonstrate a similar phenomenon for
gravitational anomaly cancellation, with Hawking radiation playing a role analogous
to the edge current.
3.1.3
Hawking Radiation and the Conformal Anomaly
Many years ago Christiansen and Fulling [77] showed that it is sometimes possible to
use an anomaly in conformal symmetry to derive important constraints on the energymomentum tensors of quantum fields in a black hole background 2 . This anomaly
appears as a contribution to the trace Ta of the energy momentum tensor in a theory
where it vanishes classically. By requiring finiteness of the energy-momentum tensor
of massless fields as seen by a freely falling observer at the horizon of a 1 + 1 dimensional Schwarzschild background metric and imposing the anomalous trace equation
everywhere, one finds an outgoing flux given by GN
Nmmr, Ta whra
(r), i where
2 f2;G
h m is the
black hole mass, which is in quantitative agreement with Hawking's result. This is a
beautiful observation, but it is quite special, and might be regarded an isolated curiosity. Specifically, the limitation to massless fields is quite essential to the analysis,
as is the limitation to 1 + 1 dimensions. Indeed, only the absence of back-scattering
for massless particles in 1 + 1 dimensions allows one to relate flux at the horizon
- which is the simple, universal aspect of Hawking radiation to an integral over
the exterior. Also, as a conceptual matter, the central role ascribed to conformal
symmetry seems rather artificial in this context.
2
For a recent application, see [78]
61
3.1.4
Effective Field Theory Framework
Our goal is to formulate an effective theory for the behavior of fields in the region
outside the horizon. The relevant dynamics of the interior (that is, the part that
effects the exterior) is assumed to be captured by an account of the horizon, regarded
as a dynamical system. At the classical level, there is a very useful effective membrane
theory of the horizon, which can be derived in a fairly straightforward way from the
classical action [31, 79].
A delicate issue arises, however, when one moves to the quantum theory. To
identify the ground state of a quantum field (say, for definiteness, a free field), one
normally associates positive energy with occupation of modes of positive frequency.
But in defining positive frequency, one must refer to a specific definition of time.
In the exterior region, where the effective theory is formulated, there is a natural
definition of time, for which translation t - t + to leaves the metric invariant. This
time coordinate becomes mathematically ill-defined at the horizon, and the "ground
state" associated with its use (the Boulware state [30]) is physically problematic,
since in it a freely-falling observer would, upon passing through the horizon, feel
a singular flux of energy-momentum. The singular contribution arises from modes
that propagate nearly along the horizon at high frequency. In the Boulware state,
these modes have non-trivial occupation. The Unruh vacuum [41], which is nonsingular, is defined instead by associating positive energy to these modes, so they are
unoccupied. Mathematically, it is implemented by associating positive energy with
occupation of modes that are positive frequency with respect to translation of the
Kruskal coordinate U, which will be discussed in detail in Section 3.2.2.
Our proposal arises from elevating this state-choice to the level of theory-choice.
That is, we suppose that the quantum field theory just below the membrane, to
which we should join, does not contain the offending modes: in effect, that they can
be integrated out.
There is an apparent difficulty with this, however. Having excluded propagation
along one light-like direction, the effective near-horizon quantum field theory becomes
chiral. But chiral theories contain gravitational anomalies, as discussed above. In our
context the original underlying theory is generally covariant, so failure of the effective
theory to reflect this symmetry is a glaring deficiency. Analogy to the quantum Hall
effect, as in Section 3.1.2, suggests that one might relieve the problem by introducing
a compensating real energy-momentum flux whose divergence cancels the anomaly
at the horizon. We will show that the energy-momentum associated with Hawking
radiation originating at the horizon does the job. One can extend the discussion to
construct an effective theory for the interior as well as the exterior bulk, separated by
a chiral bilayer membrane near the horizon. The primary features of our framework
are sketched in Figure 3-1. In this context, the horizon acts as a sort of hot plate,
radiating both in to and out of the black hole, similar to pair-creation in a constant
electric field.
62
_·
_
Figure 3-1: Part of the causal diagram of a black hole spacetime, with inset detail
of a region near the horizon. Dashed arrows indicate unoccupied modes, while solid
arrows indicate occupied modes. The white region is the infinitesimal slab near the
horizon where outgoing modes are eliminated.
3.2
Spacetime Preliminaries
We will be studying the behavior of scalar fields in static, spherically symmetric
background space times. It will be useful to have certain facts on hand about these
spacetimes, where they come from, and properties of their wave equations.
3.2.1
Spherical Static Metrics
We wish to construct a metric ds2 = gabdxadxb for the general static, spherically
symmetric spacetime in d dimensions in a suitably intuitive coordinate system. We
choose coordinates (t, r, Q) such that Q are the d- 2 angular coordinates 9 i on the
unit sphere S d - 2 with metric d 2 - sijdOidOj . Spherical symmetry allows setting
gri and gti to zero and to declaring the remaining t-r components of the metric to
be angle-independent. The static condition allows setting tr to zero and makes the
metric components t-independent. We can further choose r to be the coordinate that
measures the areas of spheres in the sense that a spatial surface of constant coordinate
r and t has an area3 Vol(Sd-2 )rd- 2 . The general metric is now given by
ds2 =-f(r)dt
2
+ h(r)-ldr2 + r2dQ2
3
(3.4)
The word "area" is used here only because d-2 is a boundary of a spatial region. When
appropriate, we refer to the volume of such a spatial boundary as an "area" or "surface area" in
order to distinguish it from the volume of the spatial region enclosed by the boundary.
63
where f(r) and h(r) are arbitrary functions of the coordinate r. The angular part of
the metric is diagonal and given by
d-2
IH sin2
j
.
(3.5)
j=i+l
i=l
Thus,
1 is the azimuthal angle ranging over [0,27r] and the rest are polar angles
ranging
over [0, 7r]4 .
The determinant of the metric (3.4) is given by
g =-f
h
r2(d- 2) det
Q.
(3.6)
Note that the metric may not be invertible at points where f or h vanish, unless they
vanish simultaneously. This condition will be explored further in Section 3.2.1.2. The
Christoffel symbols are
f'
rttr = rtrt
f'h
=
tit
(3.7a)
(3.7b)
=
rr
rr
P
.
ir
jr-
rj
r~]
jk
(3.7c)
= -rhsij,
(3.7d)
1e
(3.7e)
r
= same as unit Sd-2,
(3.7f)
with all other components vanishing. The primes in Equation (3.7) indicate differentiation with respect to r. The Ricci tensor is given by
+ rcdab
r - adIcb.
Rab= acrab- aarcb
(3.8)
All the off-diagonal components of Rab vanish. The rest are given by
1
2
Rrr
Rij
4
f f'h'
24!!
2fh 1(f ) 2 fd]
f
f)
fj) h'd
h -r2J'
1
92f
f'h'
1
2
f
2fh
2
f
2
= sij[(d-3)(1-h)- rhrln '/].
Note that this ordering of angles is reversed from some conventions.
64
-------
'
2
(3.9a)
-2
(3.9b)
(3.9c)
For completeness, we list the Ricci scalar
R =-h f[
f f f'frIn
f
2)4,ln
ln(d- + 2(dV- +
r2
h
2)(d- 3)(l -h) (3.10)
and Einstein tensor
1
(3.11)
Gab-= Rab- -gabR,
2
which has components
Gtt = ~
dd-2f
-2
Grr
=_d
r
[rd 3(1 - h)]
2 r In
-
h
fh'
Gij =sijr2 h [- - Orin
2f
Since Rab and
gab
(3.12a)
'-
(3.12b)
rd 4
rn
+ -a nj
h r
d4G]
.
d -2 fh J
(3.12c)
are both diagonal, Equation (3.11) implies that Gab is, too.
3.2.1.1 Einstein's Equation
While not a strict requirement for the core analysis of Section 3.4, it will be interesting
to consider spacetimes that are solutions of the d-dimensional Einstein equations for
some given background matter source energy-momentum tensor Tab:
5(C2
2 Tab,
(3.13a)
2 (Tab d- 2 9abTc)*
(3.13b)
Gab =
2
or equivalently,
Rab=
Recall that X2 was defined in Section 2.2.2 as 2 = 2- Vol(Sd-2 )GN. Since Gab
is diagonal, Equation (3.13a) implies that Tb is, too. Equations (3.13a) and (3.12c)
together imply that Tij is equal to sij times a scalar function of r only. Thus, we can
write Tab as
Ttt=
p(r)f(r),
Tr = P(r)/h(r),
Tij = S(r)r2sij.
(3.14a)
(3.14b)
(3.14c)
The physical energy density of background matter measured by a static observer in
these coordinates is given by p. Likewise, P gives the radial pressure measured by
this observer. For many matter models, the tangential pressure S is not independent
of p and P. For example, a perfect, static, isotropic fluid obeys S = P, but this fact
will be neither relevant here nor true in general. The trace of the energy-momentum
65
tensor is
(3.15)
T = -p+ P+ (d-2)S.
The tt-component of Equation (3.13a) can be integrated immediately, using Equations (3.12a) and (3.14a), to give
h=1-
[
C3
X2
(d- 2)rd-3 jx
rd-
d 2 p(x)dx,
p()dx
(3.16)
where C is constant of integration parameterizing physically different solutions with
the same given matter distribution. A non-zero C is equivalent to placing an additional point mass at the origin. The explicit appearance of C could be absorbed into
the lower limit of the integration. This expression can be simplified by introducing
the "integrated mass" function
m(r)
Vol(S d- 2 )
xd- 2 p(x)dx,
(3.17)
/0
where we allow for a possible point-mass singularity in p at the origin. This function
has the intuitive -
but, in general, incorrect5
-
interpretation as the total mass
inside a sphere of radius r. Equation (3.16) then simplifies to
_
2GNm(r)
h(r)= 1- (d N3) r().
(3.18)
Given the solution for h(r), the rr-component of Equation (3.13a) can also be
integrated, using Equations (3.12b) and (3.14b), to give
___
f(r) = Ah(r) exp
-{
I
X[p(X)
+PWx]
(x)
dx
(3.19)
The integration constant A allows an arbitrary rescaling of t. It can be set to any value
without loss of generality, or otherwise absorbed into the lower limit of integration.
Generally, it is chosen such that f = 1 at some special value or r, typically r = cc.
Equation (3.19) illustrates that if the matter source obeys p = -P, as it does for
vacuum, pure cosmological constant, or electrostatic field, we get the simple result
f(r) = h(r), everywhere. A few simple metrics of physical interest that obey p = -P
are listed in Table 3.1.
The exact forms of p(r) and P(r) will depend on the dynamics or equation of
state of the matter source in question, which we will not be concerned with here. It is
often useful for the intuition to considerthe case of f(r) = h(r), although the general
case will be studied throughout. However, as will be discussed in Section 3.2.1.2
- and as is already apparent from Equation (3.19) - Einstein's equations imply
5In general relativity, there is no global notion of "total energy" or "total mass" within a region.
However, in asymptoticly flat spacetimes an unambiguous notion of the the total energy in the
spacetime, the ADM mass [80], does exist. Also in such spacetimes, a Newtonian limit can be taken
and the expression (3.17) does indeed agree with the Newtonian mass.
66
__
Name
Schwarzschild
Reissner-Nordstrom
1-
de Sitter
de Sitter
Schwarzschild-de Sitter
f (r) = h(r)
P= -P
1-(d-3) r
0
2GN rd-3
m
+ (d-3) 2 GN
(d-3)
Vol(Sd-2) r2 (d- 3 )
1
1-
1
-(d-2)(d2A
1) 2
d-2)(d-
(d-
r
2
2V (
r (
(d-2)
222
2A
Table 3.1: A few physically interesting metrics that obey p = -P.
that if a surface of constant r exists such that one of the functions f or h vanishes
there, then the other must vanish there, too. We will always use this condition when
needed in further sections. Interest in the special case f(r) = h(r) would be well
motivated, however, by the study of matter distributions that could possibly give rise
to such solutions in classical general relativity. As will be shown in Section 3.2.1.2,
such spacetimes saturate the null energy condition and include several physically
interesting cases (again, see Table 3.1).
3.2.1.2
Horizon Structure
Two types of horizons will be encountered when studying black holes: event horizons
and Killing horizons. With the coordinates chosen as they are for the static metric
(3.4), the condition for an event horizon (a null hypersurface that separates spatial
regions where timelike trajectories may escape to timelike infinity from regions where
they may not) to exist for some constant r hypersurface is that the one-form normal
to the hypersurface has vanishing norm: gabOrbr = h(r) = 0. A Killing horizon for
the global Killing vector = 9t (an infinite redshift surface) will exist when the norm
of vanishes: g9b7/ar/b= -f(r) = 0. We will consider the conditions under which
these horizons might exist.
Since the integrand in Equation (3.19) is real, the exponential is manifestly nonnegative. Thus, the sign of f(r) and h(r) must always match. In particular, the zeros
of f(r) and h(r) coincide. The only possible exception to this might occur when the
integral in Equation (3.19) has a positive pole. Assuming the background matter is
nonsingular near a given event horizon, Equation (3.19) reads
XJ2 rh[p(rh)+P(rh)]
f(r) - (r- rh) +
(d-2)h'(rh)
(3.20)
for r very close to rh, where rh is defined by h(rh) = 0. As long as this exponent
is non-negative, f(rh) will vanish and the zeros of f(r) and h(r) will match. The
exponent can only be negative if h(r) approaches 0 with a slope of sign opposite to
that of p + P, and even then the exponent may still be positive.
As will be discussed below, most spacetimes that describe stars or black holes will
be asymptotically flat and obey the null energy condition. This means that near the
outermost event horizon both h and p + P are non-negative. Thus, it is the case
that for sufficiently well behaved matter sources, at least the outermost event horizon
67
will indeed obey f(rh) = 0. Since this is the only event horizon that can be observed
from infinity, it is sufficient and unambiguous for physics outside of rh to refer to
this hypersurface as the event horizon6 . The event horizons of static, spherically
symmetric spacetimes are therefore also Killing horizons. We can then simply talk
about "horizons" without confusion.
The fact that the event horizon is also a Killing horizon means, among other
things, that in the static coordinates (3.4), a patch of spacetime near the horizon
looks locally like Minkowski spacetime in Rindler coordinates. If this were not the
case, analysis of the behavior of wave modes near the horizon would be much more
difficult (see Section 3.2.3). Arguably, the universality of black hole thermodynamics
depends, at least at the kinematic level (Hawking radiation), on the ubiquitousness
of Killing horizons.
From the form of the metric (3.4), it can be seen that f(r) must remain positive
for t to remain a timelike coordinate. If f(r) does change sign, as it does at a
Killing horizon, then the global isometry t - t + to can no longer be referred to as
a "time translation," and this spacetime cannot be considered truly static. Rather,
such a spacetime is merely stationary. This loss of staticness allows time-irreversible
processes to occur. As such, spacetimes of this sort may contain spatial regions that
can be entered but cannot be escaped along timelike trajectories. Also, at some level,
irreversibility may ascribe an entropy to such spacetimes [82].
The quantity often called the surface gravity of a Killing horizon, 3, is defined by
/aVaqb = -lb evaluated at the horizon, with tl normalized to 1 at a conventional
point, typically infinity. For the metric (3.4) this gives = f' . The surface gravity
rh
is so named because in a static, asymptotically flat spacetime it typically measures
the magnitude of the acceleration of a static particle at the horizon, redshifted to an
observer at infinity. However, in our spacetime, this acceleration is actually given by7
K;
fL=l
(3.21)
rh
If we define the corresponding quantity for the event horizon, namely
--
h'
then we find
/c
A.(3.22)
~~~~~~=
We will find that the relevant quantity appearing in the equations of black hole
thermodynamics is in fact nc,not a.
One of the classical energy conditions often imposed on the choice of energy6
Nevertheless, there do exist spacetimes for which some of the zeros of f(r) and h(r) do not
coincide. See, for example, the GHS black hole 81] and similar dilaton black holes. These may
occur because the matter sources become singular - in which case the hypersurface is a singularity,
not a horizon - or because a zero occurs inside of another horizon where h(r) has changed sign.
As just discussed, such "inner horizons" are always hidden from observers at infinity by horizons
obeying f(rh) = h(rh) = O.
7It may be worth noting that the acceleration as redshifted to infinity of a static point at a
constant coordinate r that is not a horizon is given by f'v/h7,
which limits to Equation (3.21) as
r - rh.
68
____·II _Y
momentum tensor is the so-called null energy condition (NEC) (Tabnanb > 0 for any
null vector na), which demands p+ P > 0. This condition appears to be necessary for
the classical stability of many systems, including perfect fluids [83]. From Equation
(3.19), we see that saturating this bound yields f(r) = h(r). From the same equation,
we see that the NEC also demands that for h(r) positive everywhere, as in a normal
star, the integrand is always positive. Thus, the exponential is greater than 1 and
f(r) > h(r). If regions of both positive and negative h(r) exist, the situation is more
complicated and If(r) may be less than h(r) (but always the same sign). There is
no physical significanceto an intersection f = h other than f = h = 0, since such
intersections can be moved around arbitrarily by rescaling t.
The null radial component of Equation (3.13), Gabnanb
d- 2 h(I h
r
f
h
)=
2 (p +
=
2Tabnan b,
~~~~2
).
reads
(3.23)
Interestingly, examining Equation (3.23) near the horizon using f(rh) = h(rh) = 0
we find simply
(p + P)l = 0.
rh
(3.24)
This shows that Einstein's equations force any form of matter to saturate the NEC
at a horizon in a static, spherically symmetric spacetime.
Another physically reasonable condition to impose on the background matter is
the weak energy condition (WEC) (TabUaub > 0 for any timelike vector Ua), which
reads p > 0 here. Applying the WEC, Equation (3.17) tells us that m(r) is a positive,
monotonically increasing function of r. Even with this restriction, however, h(r) may
have an arbitrary number of zeros, as shown in Figure 3-2. Thus, the WEC says little
about the horizon structure of these spacetimes.
3.2.2
Kruskal Extension
In order to properly understand the global structure of the spacetime under consideration and the nature of its horizon, we want to define a Kruskal-like extension of
this metric in analogy to the known extension of the d = 4 Schwarzschild metric
[38]. Primarily, we wish to find coordinates in which the the metric is manifestly
nonsingular at the horizon. But we would also like the new coordinates, in analogy
to the d = 4 Schwarzschildcase, to show that the global time translation isometry
of the old coordinates becomes a boost symmetry in the new coordinates. Further,
we would like the new coordinates to maintain the stationarity of the metric and to
exhibit an explicit on-horizon translation symmetry, which should intuitively exist
given the close similarity of this spacetime to d = 4 Schwarzschild space8 .
8
We can also construct a Painlev6-Gullstrand [39, 40] coordinate system for this spacetime. By
defining r = t + f dr S1-, the metric (3.4) becomes
ds2 =-f(r)dr
2
+ 2f(1 -h)
h
69
drdr + dr2 + r2dQ2.
(3.25)
m,h'
r
(a)
r
(b)
r
(c)
Figure 3-2: Sketches of three integrated mass functions and their associated h(r).
Solid lines indicate the integrated mass function m(r), while dotted lines indicate the
metric function h(r). Vertical dashed lines indicate the position of the outermost event
horizon, if it exists. In 3-2(a) the matter distribution is relatively smooth and vanishes
at the origin, as in a normal star. In 3-2(b) the matter has a density singularity at
the origin, but is otherwise well behaved. In 3-2(c) a potentially difficult-to-analyze
situation is sketched.
70
fh
r,
I
I
I
I
I
I
:
:.
I
I
I
r
rh
r
rh :
I
I
I
I
(a)
(b)
Figure 3-3: In 3-3(a) typical profiles for the functions h(r) (solid line) and f(r) (dotted
line) are sketched for an asymptotically flat black hole spacetime. The horizon occurs
where the functions vanish at r = rh. In 3-3(b) the corresponding profile of r is
sketched along with the dotted line r = r. Note that r diverges logarithmically at
rh and approaches r at large r.
The strategy will be to first transform from (t, r) to a set of coordinates (u, v) that
parameterize ingoing and outgoing null geodesics. Like t and r, these coordinates will
be poorly behaved at the horizon, but it will be possible to construct a rescaled set
of null coordinates (U(u), V(v)) that can be extended past the horizon. We we then
attempt to impose the condition that a translation in t appears as a Lorentz boost in
the Minkowski-like coordinates associated with the null coordinates (U, V). This will
greatly restrict the possible form of the new coordinates, leaving only three arbitrary
real constants to be fixed in the coordinate transformation. We will then check
whether the metric is regular at the horizon in these coordinates. The regularity
condition will fix one of the free constants, while the other two are simply global
rescalings of U and V that can be chosen by convenience. We will then show that
translations in U or V are on-horizon isometries.
The first step is to introduce the analog of the tortoise coordinate r defined by
cr
/ 1
/fh
0-~ =
'
(3.26)
The generic form of r is sketched in Figure 3-3. With this coordinate the metric
becomes
ds2 = (r)(-dt
2
+ dr 2) + r2dQ2
(3.27)
where r is now thought of as an implicit function of r.
These are the locally
Minkowski coordinates of a free-falling observer. Radial null geodesics (dQ2 = ds2 =
In these coordinates the spacetime appears stationary, but not static. It also has the nice feature of
having spatially flat equal time slices. Finally, they demonstrate that the event horizon at f = h = 0
is a non-singular surface. The case of h(r) = f(r) = 1 2GNm is the original Painlev6-Gullstrand
case, which has recently been applied widely [79, 84, 85, 86].
71
0) in this spacetime obey
~(at0r~)
1f~~
h(3.28)
r =fh'
2
which can be combined with Equation (3.26) to show that t ± r, is a constant along
such paths, with the plus sign for ingoing geodesics and the minus sign for outgoing
geodesics. We thus define the ingoing and outgoing null coordinates
u = t+r,
(3.29a)
v = t-r,,
(3.29b)
which are constants along the respectively named trajectories.
Obviously, Equations (3.26) and (3.28) exhibit a singular behavior at a horizon
where either f(r) or h(r) vanishes. This is the coordinate singularity we hope to
eliminate with a coordinate transformation. The singularity is made much worse if
f(r) and h(r) do not have the same sign everywhere. We thus impose the physical
choice of spacetimes discussed in Section 3.2.1. That is, we demand that the zeros of
f (r) and h(r) coincide.
As outlined above, we wish to construct new null coordinates U(u) and V(v) such
that the Minkowski-like coordinates (T, R) defined by
U = T+R,
V = T-R
(3.30a)
(3.30b)
obey
T
-yT
-y3(to)R,
(3.31a)
R
- yR- y3(to)T
(3.31b)
when t - t + to, for some function f3(to). Of course, 1/7
1- -/. This demands
that
U
V
- y(1-3)U,
- y(l + )V,
(3.32a)
(3.32b)
or equivalently
UV -UV,
V
U
(1 +
1-
(3.33a)
V
U
(3.33b)
Equation (3.33a) means that the product UV - A(r) is a time independent function
72
___ _____. _
of r only. Equation (3.33b) means that V/U depends exponentially 9 on t as
V
B(r) e' t
(3.34)
with B(r) and k arbitrary. That k must be an r-independent constant can be seen
from Equation (3.33b), which can be solved to give
l(to) = tanh kto.
by construction,
Since 3 is r-independent
U2
(3.35)
so is k.
- 2 kt
= (A/B)e
and V2 = ABe 2 kt. For U to be a function of u
We now have
only and V to be a function of v only, while maintaining simultaneously that A and
B are functions of r only, it must be the case that A = cucve2kr* and B = cv/cu, for
arbitrary constants cu, cv. The (U, V) coordinates now take the form
U = cue
(3.36a)
(3.36b)
ku,
V = cvekv,
and metric in the now reads
f(r)e- 2 krdUdV + r2dQ2 = fk
ds2= k2c
dU
(3.37)
To examine the metric near the horizon, we Taylor expand the metric functions
h(r) and f(r) around r = rh using the notation of Section 3.2.1.2:
f(r) - 2(r - rh)+
f((rh)(r- rh,
(3.38a)
() (r- rh.
(3.38b)
n=2
h(r)
= 2(r - rh)+E
En=2
n=2
This yields
(r-rh +Sjf"(rh) A+
1
h"(rh)
ro-rh
r-r
h
near the horizon, where r is an integration constant.
rh + 1/(2j).) The metric is approximately
(r -ro) +
(3.39)
(A sensible choice is ro =
k
ds 2 r
9
- (rh
r-rh) K+O ((r-
rh)
K)
dUdV + r 2dQ2.
(3.40)
For a sufficiently well-behaved real function a(x) to obey a(x+y) = b(y)a(x) for a given function
b(y), also sufficientlywell-behaved, it must be true that a(x) = Aek z and b(y) = eky . This can be
proven by writing a(x + y) = e . a(x) and Taylor expanding about y = 0.
73
This metric will only have a non-vanishing determinate at r = rh if k = . As
promised, regularity has fixed the value of the free constant k. We fix the scale
factors cu and cv by aesthetics, choosing cv = -cu = 1/r. This choice gives as close
as meaningfully possible to a unit determinate at the horizon, such that the (T, R)
coordinates are a close as possible to Minkowski coordinates.
The final expressions for the Kruskal extension of the metric (3.4) are given by
U=
_ 1-e-`(t-r.)
K(tr*)
(3.41a)
V= l
e (t+r*)
(3.41b)
and
2,
ds2 = f(r)e-2xrdUdV + r2dfQ2 = f(r) dUdV+ r2dQ
K2UV
(3.42)
with r given by Equation (3.26). Neither form of the metric given in Equation (3.42)
is obviously nonsingular at the horizon, but we have shown by the constructions of
this section that they are. The horizon at r = rh is now seen to occur at both U = 0
and V = 0. The horizon is a bifurcate Killing horizon with bifurcation surface given
by the S d - 2 at U = V = 0 (t = 0, r = rh). The U = 0 surface also corresponds to
t = +oo and is therefore referred to as the future horizon, while the V = 0 surface
corresponds to t = -oo and is referred to as the past horizon. Since all of the metric
components are t-independent, they depend on U and V only through the product
UV. The metric is thus trivially invariant under translation of one coordinate along
the hypersurface where the other coordinate is zero. This shows that translation
along the horizon is a valid local isometry.
3.2.2.1
The Quantum Vacua
We have constructed a coordinate system for an arbitrary static, spherically symmetric black hole in d dimensions that has all the familiar properties of the Kruskal
extension of d = 4 Schwarzschild spacetime. In particular, the fact that the Lorentz
boosts of the local Minkowski-like coordinates near the horizon generate t-translations
in the original coordinates of Equation (3.4) shows that near the horizon, the original
coordinates are more akin to Rindler coordinates than Minkowski ones. Equation
(3.39) shows that the logarithmic divergence or r, which will ultimately translate
into a logarithmic phase singularity in wave modes defined with respect to the global
Killing vector 77,is a universal feature of horizons in this family of spacetimes. Furthermore, just as for Schwarzschild spacetime, the final form of the coordinates (3.41)
shows that U --* U + Uois an isometry along the past horizon V = 0, and thus
O-u
is a Killing vector there.
Thus, it is meaningful to follow the logic of Unruh [41] and define a quantum
vacuum state, the so-called Unruh state or -vacuum, for this spacetime l0 by defining
' 0 The quantum vacuum for a star dynamically collapsing to a form black hole, such that the
metric of this collapsing spacetime settles down at late times to a stationary black hole metric of
the form (3.4), is thought to be well represented at times long after the collapse by the -vacuum of
74
__
positive energy states as those that have positive frequency with respect to E on the
past horizon, and letting this vacuum propagate outward and forward in time. The
vacuum state defined in this way differs from the one obtained by defining positive
energy with respect to the global Killing vector 7, which is called the Boulware state
or r1-vacuum [30]. In fact, Unruh found that the -vacuum is a thermal ensemble
of i-frequency states at temperature TH
, in exact agreement with the earlier
results of Hawking [29, 28]. The 7-vacuum has divergences in its energy-momentum
tensor arising from horizon-skimming modes, despite appearing empty" to static observers. This divergence renders the Boulware state an unphysical candidate vacuum.
The Unruh state, on the other hand, is well behaved despite its nonintuitive mode
occupation spectrum.
Unruh's arguments go thorough essentially unchanged using the spacetime (3.4)
and its Kruskal extension (3.41). Thus, we find a temperature for the -vacuum of
this spacetime given by TH. Furthermore, the 7-vacuumof this spacetime also has
divergences that render it unphysical.
3.2.2.2
Euclidean Section
A relatively easy route to determining the thermal nature of a spacetime is to consider
the Euclidean section of the complexified coordinates, following the method of [42].
This can be effectively accomplished by letting t -- i. Using the Taylor expansions
of equation (3.38) and defining x 2 = (2/Y)(r- rh), the near-horizon metric becomes
ds2
dx 2 + K22x2 dr 2 + r2dQ 2 .
(3.43)
This is a metric on the manifold Cone2 ® Sd-2, with the conical singularity occurring
at the horizon. The singularity is removed if the coordinate Kir is periodic in 27r.
The period of T- must then be given by 21. Since the inverse period of Euclidean
time is interpretable as the temperature of a system, we see that the temperature
of the black hole is given by TH. Note that without the condition f(rh) = h(rh) =
0, no coordinate x exists which gives a near-horizon conical metric like Equation
(3.43). This demonstrates that, in general, the stated condition is necessary for the
interpretation of a Euclidean black hole as a non-singular thermal system.
3.2.3
Wave Equation
Consider an interacting classical scalar field 0(x) living in a spacetime with the metric
(3.4). This field has an action given by:
S = | ddx+{
_-V2 X-,
_ZAn(
the eternal black holes we have been studying.
"Here, "empty" means all modes have zero occupation number.
75
}.
(3.44)
The A,, are a set of arbitrary coupling constants. In particular,
mass of a weakly coupled excitation of this field. We will expand
of the free, classical wave equation such that
(x) =
V2 0
m2 gives the
X in the eigenmodes
2
dpapp(X),
(3.45)
m2op.
=
(3.46)
A sufficiently large set of quantum numbers p label the eigenbasis. The abstract formal expression dtp simply represents an appropriate measure over the modes under
whichf dpp(x)p(y)
-
d(x - y) and f ddx /ip()q(x)
= pq. Ofcourse,the
full theory will not necessarily obey Equation (3.46), but the expansion (3.45) is nevertheless a complete functional basis that may be applied to any field configuration,
regardless of the field equations it solves. In a second-quantized quantum theory, the
coefficients ap would be promoted to field operators.
We will exploit the high degree of symmetry in the background spacetime and
use separation of variables to examine the normal modes of this field. As remarked
in Section 3.2.2, the wave equation arising from the action (3.44) will take a much
simpler form in the tortoise coordinates described by Equations (3.26) and (3.27). For
the sake of generality and to illustrate the nature of the useful cancellations inherent
in certain coordinate systems, we now examine the wave modes in a coordinate system
with metric
ds2 =-f (r)dt2 +j(r)-ldr2 + r2dQ2,
(3.47)
where r = (r) is an arbitrary radial coordinate. The metric (3.4) will be recovered
with the choice r = r, j = h, while the tortoise coordinates will be recovered from
r= r, j = l/f.
We factor a given basis function as
r
p _2
1
R(t, r)Y(f).
(3.48)
With this factorization, the eigenvalue equation becomes
V
mp
[m
1_tfj2_
fi
2 Rfln[-])
2 (r d-2(fj)1/4)
((fj)1/4)
2r~~~
1 1
+y2i
d-2
l[Iji
[ar]2
R
9i [(sin i)
+lsin
aiY]
j] (sin Oi)
1
(3.49)
Clearly, all the variables are separable. We first turn our attention to the angular
part. The radial part will be approached in Section 3.2.3.2.
76
______
·__
-- ...
3.2.3.1
Spherical Harmonics
The angular part of Equation (3.49) is again separable, so we write
Y(Q) = E1 (01 )E2(0 2 ) ...
(3.50)
Ed-2 (Od-2).
We then get
d-2
Y-E
i=l
i [(sin ')'-'i Y]
[rdl-2+sin
2 ji]
9i) i-1
(sin
d-2
0i [(sin
i=1
[--~jl
d-2+
i)'-'
i]
(3.51)
sin
2
O] (sin
i)'-1
By standard separation of variables methods, the above must be equal to a constant,
which we name -1(1 + d - 3). Further, the angular separation clearly requires the
following equations to hold:
Ol2e= - n2 ,
a2[(sin 0 2 )0 2 6 2]
sn 2
sin 02
03 [(sin 0 3 ) 2 0 3 e 3 ]
n2O~' e2
sin2
sin 2 03
sin2 03-03
sini-l
[(sin
(3.52b)
(3.52c)
(3.52d)
sin 2 0i
Oi
d-3)d-4ad-30 d-3]
sind-4 d-3
ad-2 [(sin Od-2 )d-3ad_2 ed_2 ]
sind-3 d-2
82
2
ni-i
i [(sin i)i-laiei]
ad-3
=- n
(3.52a)
n2_
nd-4
sin2 d3
d-3 -
sin 2 Od2
d-2
=
- n2
-n_3(
d-3,
(3.52e)
- 1(1 + d - 3)EOd-2,
(3.52f)
where the ni are constants. Defining x = cos 9 i and
F(x) = (sin Oi) 2 oi
=
(1
X
)
j
i 4i,
(3.53)
we find each function obeys
(1-x 2 )0xF-2x0F +
where i = 1 .. d-2,
no - 0, and n_
F(x) = L
2(
)+ ni
(
2
l(
1 - x2'
(3.54)
+ d -3). This is solved by
(i 1) 2 /4+ n
(V)(i-2,/4+n-;
1/21 ,
(3.55)
where L is a Legendre function of the indicated order.
In principle, Equation (3.54) is solved by any linear combination of the Legendre
77
P and Legendre Q functions. We will now show that the physical requirements of
single-valuedness and finiteness of the scalar wavefunction restrict both the choice
of Legendre function and the allowed values for the ni. In general, the Legendre
functions have singularities unless they are P functions with integer parameters or Q
functions with half-integer parameters. Looking at i = 1, we see that
(3.56)
E = eimlo,
where ml = n is restricted to be any integer by the condition of single-valuedness.
We now have, for i = 2,
(3.57)
02).
e2 = Lm
1 (COS
In order for this function to be finite over all values of 02 for integer ml, L must be
a P function and n 2 must obey n = m2 (m 2 + 1) for m2 a positive integer obeying
m2 >
ml . This yields
(3.58)
02 = P2(cos 02).
For i = 3 we get
3 = (sin 3) 1/2Lm2
2l/
(cos 03 ),
(3.59)
which requires that L must be a Q function and n3 must obey n2 = m3 (m 3 + 2) for
m 3 a positive integer obeying m3 > m2 . This yields
e3
(3.60)
= (sin 03)-1/2Qm2+1/2(COS93).
This pattern continues, giving for angle i
t ) 2 _/)E = (zs+in
Lm)
(sin~
miL(i-2)/2
(3.61)
(o
where L = P for i even, L = Q for i odd, and mi a positive integer obeying mi > mi- 1 .
Finally, for i = d- 2 we find
Oi (sin
i= (sin
"-4 rMd-3s+(d-4)/2 (COS d - 2 )
d-2)2)
Ll+(d-4)/2 (cos
)
(.2
(3.62)
where L = P for d even, L = Q for d odd, and a positive integer obeying > md-3.
This, of course, motivated the choice of form for the constant in Equation (3.52f).
So, the angular distribution can be parameterized by an eigenbasis labeled by the
quantum numbers {, m}, where m = {m1,. . , md-3}. Thus, we write Y = Y/m(Q).
Only the "highest" angular quantum number, , will appear in the radial equation,
so it will be useful to calculate the degeneracy, Dd(l), of angular states with a given
value of 1. The construction outlined above makes clear that
l
Dd(l) = E Dd-l(n).
(3.63)
n=O
Starting from D4 (1) = 21 + 1, Equation (3.63) can be iterated to find the proper
78
__._
_·
d
Dd(l)
= 0, 2 otherwise
3
1 for
4
21+1
5 ( +1) 2
6
(21+ 3)(1+ 2)(1+ 1)
7
~f(21 + 4)( + 3)( + 2)(1 + 1)
8
1-(2
9
+ 5)( + 4)( + 3)( + 2)( + 1)
9(21+ 6)(1+ 5)(1+4)(
+ 2)(1 + 1)
+3)(
Table 3.2: The angular state degeneracies for total angular quantum number 1, as
determined by Equation (3.64), in a few chosen dimensions.
expression for higher dimensions1 2 . The exact expression1 3 for arbitrary d and
Dd(1) = (21+ d - 3)( + d- 4)!
l!(d - 3)!
is
(3.64)
Some values of Dd(l) are tabulated in Table 3.2.
We chose the normalization of the Ylm such that
J
dd
3.2.3.2
2 QYlm()Yk(Q)
=
6
(3.65)
ilkcmn
Radial Wave Equation
Separating off the time dependence by writing R(t, r) = eiwtR(r), the eigenvalue
equation (3.49) now simplifies to
R2
f[
2=
1 1
2
fj&
fjo9-
-
~fJ
j
2(
f2
Oft
2r ([r]
2)ar
Or
[j)4
ar 2
(r(fj)/4)
d-(fj)l1/4
/)
n (r -2 (j
(fj)l/4)
d-3
fl(l( +d-3)]R.
(3.66)
There is no general solution to this equation, but one can be found for certain special
cases, and approximate solutions can be found in others.
o2 0f course, one could also start with D3 (1) =
{1
for
for
1=0
1ll> 0
l3We are aware of three different original proofs of this expression, each developed independently
when the problem was presented by the author as a puzzle to a group of fellowstudents. The method
of Guido Festuccia and Antonello Scardicchio counts the number of lattice points in a certain discrete
(d - 3)-simplex [87]. The method of Ian Ellwood formulates the problem as a combinatoric "ballsand-buckets" problem [88]. The method of the author uses inspired guess work checked against
Equation (3.63).
79
In Minkowskispace (f = j = 1,
=
r), Equation (3.66) becomes
- (I + (d - 4)/2)(1+ (d - 2)/2) + k2] R(r) =
2
r
~~~r
where k2
(2)
d-3
2
=
m2.
2 jl+(d-3)/2(z)
(367)
This is solved by R(r)/r(d- 2 )/ 2 = k(d-2 )/2 J (kr), where J(z) =
hVh
is the 1th (d- 2)-spherical Bessel function, and j(z) is the uth
ordinary, cylindrical Bessel function. This Bessel function will also be the solution
to the full radial wave equations in regions of spacetime that are approximately flat,
such as near spatial infinity in a black hole spacetime.
Another situation of interest is the wave equation in tortoise coordinates (j = 1/f,
= r, Or/ar, = \/f-h).The radial wave equation becomes
[ar* +
w2 -
fV(r*)] R = 0,
(3.68)
with
1 [(I+ d-4)
r*)
2
(1 + d-2)-2
r(f h)+
(d -2)(d -4) (-h)+(d-2)
2
f
44
(3.69)
Using the fact that h and f both vanish when f does, we see that all the terms in
V are nonsingular at a horizon, f = h = 0. Thus, the overall factor of f causes
this effective radial scattering potential to vanish near the horizon. In fact, using
f
2(r - rH) and h 2(r -rH), wefind
fV(r*) me2 ~r* (1(I+d-3)+ (d-2)rH5 +
r}
2)
(370)
which vanishes exponentially fast as the horizon at r - -oc is approached. A sketch
of a typical effective radial scattering potential is given in Figure 3-4.
Thus, the dynamics of the radial wave function near a horizon is identical to that
of 1+ 1-dimensional flat spacetime with coordinates t and r,. The solutions are simply
plane waves of the given frequency. Using the coordinates of Section 3.2.2, these can
be written as
/
(3.71)
+ IVI- i/.
eu + eiwv , Uiiw
R(t, r*)
The identification of the near-horizon dynamics of this spacetime with that of standard Schwarzschild-type black hole physics is now essentially complete. Essentially
all qualitative results for the case of f(r) = h(r) = 1 - 2GNm have a direct analog in
this spacetime as long as the Killing horizons and event horizons are coincident.
3.2.3.3
Near-Horizon Action
Now that the dynamical modes of a scalar field in this spacetime are understood, we
can execute a partial wave decomposition of the action (3.44). The field is expanded
80
__11
__·_·_____
fV(r,)
r
(r.=-oo)
r=rh
a few rh
r,
Figure 3-4: A sketch of a typical effective radial scattering potential. The potential
for any metric qualitatively similar to the one sketched in Figure 3-3 will be qualitatively similar to the one sketched here for I > 0 and d > 3. The potential falls off
exponentially for negative r and as is typically dominated by the centrifugal term at
large r, . r, which falls off as 1/r 2.
as
1
0
ERi tr)Y ()
(3.72)
The action becomes
S = Jdtdrdd -2Q
- E
{
JRimYim
[
fjl
+n( fj,91
-r
d-2
r (fj)/4
2~~~~
2,k,m,n
2
E_
ar ln (
( [ar]
n fli
(
rd-2 (f j)l/4) _ fk(k
r
(fj)114
ar 2
2
+d 3)] RknY
7,2
-Rim
- 2)(f
1/4-rj)l/4
-f
r(n-2)(d-2)/2(fj)n/4
E E
[RimYm
...
RinMn~nn ] }
11 I...nml...mn
R
1 _E
dtdr
2- 2 ([-ar
[ar
-00
=-
n=2
-
+fjO9- d i
d-2
11/...
..
In (rd-2
(fj)/4)
,Or In (r
An Vf/
r(n-2)(d-2)/ 2 (fj)n/4
2 (rd2(fj)1/4)[r]
_
E
1n Ml
81
fl1(l+rd3)
2
Rlm
nC~m'R
{} l1ml
'· .. · RlnM
Mn
(3.73)
where nis the set of group theoretic constants obtained by integrating the local
product of n properly normalized d-dimensional spherical harmonics over Sd- 2 . In
particular, Equation (3.65) tells us that 2C{I1,M12} = 611 26mm2
The radial wavefunction, Rim, in unitless, just like a canonically normalized 1 + 1
dimensional scalar field. All the units of the original d-dimensional field are carried
by the prefactor 1/r(d - 2 )/ 2 in Equation (3.72). The mass units of A are [An] =
2- (n- 2)(d- 2)/2. Thus, we can write the unit-carrying factors in the interaction
series of Equation (3.73) as
An
r(n-2)(d-2)/2 A ,,r2-[A,,]
(3.74)
'
This rewriting emphasizes that interactions which are not perturbatively renormalizable- and are therefore governed by coefficients with negative mass dimension fall off at large distances faster than 1/r 2 in a partial wave decomposition. Superrenormalizable interactions fall off slower than 1/r 2 and marginal interactions fall off
as 1/r 2 . The scaling relative to 1/r 2 is important because the large-distance scaling
of the centrifugal term in the kinetic part of the action is given by 1/r 2 .
If we now evaluate the action (3.73) for the tortoise coordinates as in Section
3.2.3.2, we get
S
J
dtdr, {-i
REm[--t +
a
--
r2
- fV(r.)] RIm
lm
An
,2-[~n
EZ
E
n=3
1l... ln
ml
nC/{
im}
Rlnmn
(3.75)
...IMn
where V is defined as in Equation (3.69) with m2 = A2. Again, f(r) vanishes exponentially fast near the horizon in tortoise coordinates. So, near the horizon, the
action becomes simply
S
d2XE -RimLmRim,
(3.76)
I,m
d2 x
where
= dtdr, and D = -_t + a2,. This is the action for an infinite collection of
free scalar fields in 1 + 1 dimensions. This description of the dynamics becomes exact
at the event horizon, however the notation and coordinates used in this section fail
there.
3.3 Thermal Radiation
The theoretical derivation of the blackbody spectrum is standard fare for 3 + 1 dimensions with wave modes labeled by Cartesian coordinates, but the derivation for
d-dimensional spherical coordinates is somewhat less well known. We will illustrate
this derivation in this section. For purposes of comparison and introduction of some
necessary mathematics, we will first study the spectrum of blackbody radiation in d
82
Cartesian coordinates. Some subtleties of the spherical case will be illuminated by
comparison with the Cartesian case. Minkowski space will be used throughout this
section.
3.3.1
Hypercubic Blackbody Cavity
I
Consider a real scalar field '(x,
t), where I is some kind of p-dimensional polarization
index representing p internal degrees of freedom14 . Further assume that the field is
sufficiently weakly coupled that each polarization component can be treated as an
independent field obeying an action similar to Equation (3.44) with all interaction
coefficients higher than A2 M2 set equal to zero. Then each field component obeys
the classical equation of motion
V2 qI(x) = M2 q'(x).
(3.77)
The solutions to Equation (3.77) may be expressed as a sum over modes labeled by
by a wave vector ka obeying k2 = _M2:
0)k
= A, sin (kaXa) + B/ cos (kaXa),
(3.78)
for arbitrary real coefficients A, and BI. We take the state to be labeled by the d -1
spatial components of ka and fix the frequency of each mode by wk02= M 2 + kiki .
We now confine the field to live in a cubic box of side length L by demanding
Dirichlet boundary conditions at x i = {0, L} for i = 1... d- 1. This demands B = 0
and
ki=
7r
mi,
(3.79)
where mi is a spatial vector whose components are non-negative integers. The full
space of modes available to the system is given by the tensor product of the space
of all vectors of integers with the state space of the polarization index. The theory
is then effectively quantized by stating that the energy in each such mode is a nonnegative integer multiple of Wk. We denote the integer as nml. The total energy in
the hypercube is then given by
p
U= E
00
00
... E
E
Wk
I=1 ml=O
where we understand that
wk
Wknm,
(3.80)
md-1=O
is given by
2
2d-1
+ rM Z
2m
.
(3.81)
14 Sufficiently simple external degrees of freedom will also be captured by the following discussion.
For example, in a well-chosen gauge, the transverse polarization of an Abelian vector field behaves
essentially like an internal index on a scalar field with p = d - 2.
83
The set of integers nmI defining the quantum state must obey the appropriate statistics 15 for the field I. That is, nmI is unrestricted if the field obeys Bose-Einstein
statistics, but can only take the values 0 or 1 if the field obeys Fermi-Dirac statistics.
So far, we have described a pure quantum state of the theory. At a finite temperature T and zero chemical potential, the system will be in a mixed state governed by
the partition function
Q=
e
k
(3.82)
where fi- 1/T. This can be evaluated to give
00
00
lnQ=-p. E .
E ln(1-e-k),
ml =0
(3.83)
md-1l2°
where = 1 for bosons and ~ = -1 for fermions. The overall factor of p occurs because
the energy is independent of p, so each polarization mode contributes equally. The
average occupation number of a given momentum mode in the thermal state is then
given by
I=
p
p
1
/3e9 Wk
I=1
(3.84)
efk-
The total energy in the hypercube can now be found by combining the expressions
(3.80) and (3.84), or by
(U) = --
&a
00
00
lnQ=
-
00
E
ml =O
Wk(nm)
ml =0
md-1=
oo
..miZd
Md-1=0
°
PWk_
(3.85)
-~
In principle, Equation (3.85) can be evaluated and the summand can be interpreted as
the spectral energy density over the quantum numbers mi. Makingthis interpretation
is problematic, however, in the L - o limit. The problem is illustrated by looking
at the M = 0 case, such that Wk = Iml, and examining the scaling with L:
00
(U) =
E
ml=0
L-O plr
L
00
ml =0
oo
..
5E
md _1-0
Iml
e7r31ml/L _
00
r
1
'~
md
1=0Iml
$r
1-
Iml(
(3.86)
-)2L
which is nonsensical for several reasons. The total energy does not scale with the
volume of the hypercube, L d -l. In fact, the total energy appears to either diverge
or vanish as L - 2 , depending on the statistics. The expansion itself is not even well
150f course, the spin-statistics theorem says that if OI is a scalar field, then it obeys Bose-Einstein
statistics. However, we want to allow for the case where -bz is a single real component of larger
multiplet, which may obey Fermi-Dirac statistics. In this case, the polarization index I would also
run over spin.
84
------
_____I··_
__I
defined, since it is an expansion in the ratio of a divergent quantity to a infinitely
large quantity.
The major mistake in Equation (3.86) is that in the L - o limit, the mode
density of mi states diverges, as can be seen from Equation (3.79). That is, the
modes cease to be countable and discrete. As the limit is taken, we should pass from
a state labeling in terms of quantum numbers mi to a labeling in terms of physical
momenta ki, with a mode density determined by the differential limit of Equation
(3.79). The sums over mi then become integrals over ki as
L-+o d- 1
(U) L-p
Ldli
d
PWk
k
(27r)d-I ek
(387)
(3.87)
-
where now Wk is understood as VM 2 + kik i. The factors of 2 in the denominator of
the measure arise because the integrals over the ki run over both positive and negative
values, whereas the mi were only summed over non-negative values. Equation (3.87)
scales properly with the volume, so that even in the infinite volume limit we can
define the spectral energy density over ki modes.
For the current case of Cartesian coordinates, the failure of Equation (3.86) and
its resolution by passing from quantum numbers to physical momenta are obvious
and the discussion has been overly pedantic. Similar failures will be encountered
when using spherical coordinates to study the spectral density of angular momenta
in blackbody radiation. In that case, however, the nature of the problem and its
solution will not be as obvious.
Using the spherical symmetry of the infinite volume limit and defining k =
/kii = ki, Equation (3.87) becomes
(U)
=p Vol (Sd- 2 )
V
kk-
2 VM 2
+
(3.88)
J-
(27r)d-1
This definesthe spectral energy density over the magnitude of the spatial momentum,
via p f dkuk(k), as
Uk(k)
= Vol (Sd - 2 ) kd - 2 M 2 + k 2
)=
(23.)
1
(3.89)
eg3vM+-k3
-
Similarly, we can define the spectral energy density over the frequency 16 as
Vol (Sd2) W(W
- 2 _ M2)(d-3)/2
u'(w) = p (2 7r)d-1
e,
(3.90)
where w runs over [M, oo].
The total energy density can now be evaluated using either Equation (3.89) or
Equation (3.90). Simple analytic results can be found for the case M = 0, which will
also apply when T
M. In this case, w = k and Equations (3.89) and (3.90) match.
6
Note, dk =
wdw 2
2
V1W --M
85
They give
Vol(Sd- 2) x
d-1
= pTd (2rd-(27r~d-1edx. x_
(3.91)
The integral can be evaluated by pulling the exponential into the numerator, performing a Taylor series in e- x, doing the integral, and resumming the Taylor series.
The result for the general definite integral is given by
b ex
X
e
e
( ~(d)(--xdog-1
-
=
( d-_e-b
-_ _
dd-1 [d(aa
_
L
~
-(=-)/
01
/
l
d a ll - a ea-e°b
z
a
1
a=d)
ane-a
ana- bneb
'I-~~~~)(d)(d-1)!
2-j
(i
~n=O
(3.92)
where ((s) is the famous Riemann zeta function, which can be defined for real s > 1
as
00
=(s1
E(3.93)
n=1
This series arises in the evaluation of the integral with the ~ = 1, for bosons. For
fermions, = -1, the corresponding series is
00
=E n----1~~
(_l)n-l·
(3.94)
n=1
This series can be evaluated using
C(s)-NW=
(s) 2 +
+
+ .
2.C.=
(s),
(3.95)
so that
N(s) =
1-
-)
(s),
(3.96)
which is the origin of this factor in Equation (3.92). So, Equation (3.91) becomes
p=p
3.3.2
-
2d_
(397)
2r (-1) 7r(di)/2T
Flux Versus Energy Density
In the previous section, we considered the energy density p of a massless scalar field
in a infinitely large hypercubic cavity at temperature T. We now want to calculate
the energy flux (power per unit area) ·P emitted from a blackbody with this same
temperature. Any number of textbook arguments can be followed to derive an expression for 4D. The key insight in all of them is that thermal equilibrium requires
the condition of detailed balance on every mode of the system, which is a far more
restrictive condition than simply total energy conservation.
86
___·_·
At the end of the day, the calculation can be cast as finding the flux through a
chosen spatial boundary. We choose a set of (d- 1)-dimensional spherical coordinates
for the spatial momenta at some given point of the boundary using the same set of
angular conventions introduced in Section 3.2.1. We orient the coordinates such that
the vector normal to the boundary at the chosen point lies at the "north pole" of the
coordinates. Then the calculation of the flux proceeds exactly as the calculation of
the energy density, with two key differences. First, the polar angle 0 d-2 only ranges
over [0,7/2] instead of [0, 7r]because we are interested in the flux passing in only one
direction through the surface. Second, there will be an additional factor of cos (0 d - 2)
multiplying the phase space measure to account for the scalar product between the
flux vector and the area element of the boundary.
The only change to the mathematics of Section 3.3.1 is in the passing from Equation (3.87) to Equation (3.88). We used there dd-lk = dd-2Qkd-2dk and f dd-2Q =
Vol (d-2).
For the flux calculation, instead of Vol (S d - 2) we will encounter
dd-2 Q cos (od- 2 )e(7/2
r2r
-
O'
j2dj
r7r
7r
d2...
27r( d - 2 ) / 2
(d
(d ---2)2)
3
7r/2
dod- 2 cos (
J
0~~~~~
d-
d-3
2)
H (sin (i))
}
i=2
1
(-2)
= 1
dod-
dd-2
d- 2 Vol(S 3) = Vol(Bd 2)
(d2)
V/7(d - 2)
- Od-2)
F(
(-2
~
(3.98)
) Vol (Sd-2),
)
where (x) is the step function and B is the n-dimensional unit ball1 7: the compact
subspace of IR bounded by Sn- 1 . The fact that the expression for the energy density
becomes that for the flux when Vol (Sd-2) is replaced by Vol (Id-2) makes physical
sense, since Bn is the projection
of Sn onto iRn.
We are left with the relationship of flux to energy density as
Vol (B d - 2 )
(-1
v(d-
2)r
(d
)P
(d
F(~j-l)
Vol (d
2 )P.
Vol
(3.99)
2 )P.(
Thus, the d-dimensional Stefan-Boltzmann law is given by
(
-2
1- -)
(1- )/2
(
1
2d-2(d- 2)F (d-2)
d/2
Td
(3.100)
Thermodynamic relations like Equations (3.97) and (3.100) are independent of the
mode labeling that was used to compute them. So, when attempting to rederive
(3.100) using an angular mode labeling instead of the Cartesian one used in Section
17
For example, lB2 is the unit disc of volume 7r bounded by the circle S1 of area 2r. Also,
the unit 3-ball of volume 47r/3 bounded by the sphere S2 of area 47r.
87
3
is
3.3.1, it will be sufficient to rederive Equation (3.97) and then apply Equation (3.99).
The case of d = 2 will be important to the core analysis of Section 3.4. Since
Equations (3.99) and (3.100) are somewhat ambiguous for d = 2, we explicitly list
the results for that case here:
b1
3.3.3
2
PT2
(3.101)
12'
(12 ) (1-E,)/2
PT
311
Spherical Blackbody Cavity
We will now repeat the calculation of Section 3.3.1 in a spherical cavity of radius R
instead of a hypercubic cavity. We define spherical coordinates as in Section 3.2.1
with f = h = 1 and decompose q$I in partial waves as in Section 3.2.3. The field still
obeys Equation (3.77), but instead of expanding classical solutions in the plane wave
basis (3.78), we separate variables as in Section 3.2.3:
5'(x) =
(3.102)
eiwktRkl(r)Ylm(f1),
r
2
where the (d- 2)-spherical harmonics Y
part of the field obeys Equation (3.67):
were defined in Section 3.2.3.1, the radial
[oq (1+ (d - 4)/2)(1 + (d - 2)/2) + k2] Rkl(r) = 0,
(3.103)
/,2
and, again, k2 = w2- M 2 .
As discussed in Section 3.2.3.2, Equation (3.103) is solved by
Fk 2ir
d-3)/2
= k(d 2 )/2 J(kr)r(d 2 )/ 2 Rkl(r)
r~~~~~~~~~~~~~
(3.104)
j(d-3)/2(kr),
where j,(z) is the vth cylindrical Bessel function. The boundary condition at r = R
demands that J(kTZ) = 0. This quantizes the allowed values of k as
k =
A7n1
'
(3.105)
where the pure numbers A, are defined by J(Al) = 0.
The modes are now labeled by the quantum numbers {n, 1,m, I}. The quantization of the angular quantum numbers was discussed in Section 3.2.3.1. The
new radial quantum number, n, takes all positive integer values. Upon secondquantization of the q 1, the energy of a singly occupied mode is given by the frequency,
wnl = M 2 Anl/7Z2. Analogously to Equation (3.80), the total energy in the cavity
for a pure quantum state is given by
p 00
U=
E E
(3.106)
wnlnlnmI,
I=1 n=l l,m
88
____il
_
where the integers nlmI are the occupation numbers specifying the state.
Except for the detailed labeling of modes, the partition function for a thermal state
is computed exactly as it was in Equations (3.82) and (3.83). In fact, the expressions
are somewhat simpler because the energy only depends on the two quantum numbers
n and l:
00
lnQ= -p
00
EEDd(l)l n (1- e-w'n).
(3.107)
n=1 1=0
The angular momentum degeneracy Dd(l) is given by Equation (3.64). The average
occupation number of a given n, 1} mode in the thermal state is then given by
(nnl)=-
1 0
pDa(l)
aw
InQ = , ,
.
,3 19U~n1 eIcwnl
(3.108)
-
Likewise, the total energy in the cavity is given by
(U)=-l
(U
-~'9
a
00
pDd()wnl
n = E 2Q
E PD)Cl
n=l 1-O ef~wnl
-
(3.109)
This had better be proportional to the volume of the cavity Viz= Vol(Bd-l)7d-1.
As in the case of the hypercubic cavity, we cannot simply take the 7 - oo limit
of Equation (3.109), interpret the result as a spectral density unl (n, 1), and then sum
over n, l} to get the total energy. For the reasons discussed in Section 3.3.1, such a
procedure gives meaningless results. One's intuition from commonly interpreting the
I quantum number as the actual, physical angular momentum of a wave mode would
lead one to believe that in the large 7Rlimit, the n integral can be performed in some
properly regulated way, leaving a quantity that can be appropriately interpreted as
the angular momentum spectral density ul(l) which would obey
00
(U) = Eui(l).
(3.110)
1=0
Unfortunately, this is not the case. Instead, the quantum number must be regulated
due to a diverging density of states by passing to a new variable e = /R. This new e
has units of linear momentum 19 and actually would be the physical linear momentum
8 Care
should be taken not to confuse the symbols for occupation number and radial quantum
number. Likewise, the symbol m is being used differently in this section than it was used in Section
3.3.1.
19 The quantum numbers {1, m} were never really the angular momentum operator eigenvalues
to begin with. As defined in Section 3.2.3.1, they are more properly referred to as the canonical
momenta conjugate to the angular coordinates. This is why there are d-2 of these quantum numbers.
Angular momenta are the generators of the rotation group SO(d - 1), which is (d - 1)(d - 2)
dimensional. Of course, a wonderful point of connection between group theory and mechanics
occurs here in that the d- 2 conjugate momenta form a set consisting of the Casimir operator for
the representation and a Cartan subalgebra of SO(d- 1), which is why these quantum numbers are
all that is needed to label physical states.
89
if space were an Sd- 2 of radius R. Also, with this definition we get
2fd-3
(d - 3)!
Dd(l) -= d- s d_
(3.111)
We will approach the radial part of Equation (3.109) by eliminating the sum over
n in favor of a sum over physical radial momentum k, which is simply related to A
by Equation (3.105). This will require knowing the phase space density On/OA in the
limit of large 7R. We will now evaluate On/OA by two independent methods.
3.3.3.1
Radial Mode Density
One approach to finding On/OA is to find an approximate explicit expression for Ani
which is valid in the limit 1 - oo. We begin on this route with an asymptotic series
expression for jn(Z) of the Hankel type (large argument, fixed order) [89, Section
III.3.14.1, for example]:
2
-jn(Z) = cos
si(
-
7r(
~r(n - 1))[
2
+))
2
\ -E(-l)mL'(n+ + 2m)
mo(2m)!r(n
+ 1 - 2m) (2z)-2m
(-1)m F(n +
+ 2m + 1) (2z)2m1
- 2m
(2m
+ )!(n + -z)
- 2m
2m -1)
- 1)
.m--0 (2m + 1)!F(n +
(3.112)
2~~~~~
Thus, we have
(2>
(d-4)/2
R
(d-2)/2
[e
=o
J (Z) =
7r)
z
7r(21+d-2))
4
=wz
yjl+(d-3)/2(Z)
(-1)m r(l + (d- 2)/2 + 2m)
(2m)!r(l + (d- 2)/2- 2m)
(-1)m r(l + (d- 2)/2 + 2m+ 1)
- sin z -7r(21 + d-2)
4
[m=o
(2m+1)!r(l+ (d-2)/2-2m-1)
(
Z)-2m]
(2Z)-2m- 1
(3.113)
Since we will be writing
b is an integer, then
= RI for large 7?, we can apply the following math fact: if
F+1 - b)
(a+
r(a
+ 1-b)
- b) =(a+ b)(a+ b- 1)... (a + 2)(a+ 1)a(a- 1)... (a-b+ 2)(a-b+ 1)
=[a(a + 1)- (b- 1)b][a(a+ 1)- (b-2)(b-1)]
...
[a(a+ 1)-12][a(a + 1)]
z[a(a + 1)]b,
(3.114)
90
_ 1_
where the last line holds when a > b. Then we have
(2>
(d-4)//2
Rkl(z/k)
7
/
7r
cos
j
- sin z -
~d2
(-1)M ([I+ (d- 4)/2][ + (d- 2)/2])
Y- (2m)!
2z
M=0
+2
z
142
=COS Z (-
- sin
[I
d12)COS
[ + (d - 4)/2][ + (d - 2)/2])2m+1
(-l)m
0 (2m
+ 1)!
2z
2
T [Id2T-
(2
(
2m
00
I[1+ (d-4)/2][ + (d- 2)/2])
2])
+
((
]
2
[I+ 2]sin
([ + (d-4)/2][ + (d-2)/2]>
2z
[I+ (d-4)/2][1 + (d-2)/2]\
d
=cos~z~~ -~
2z
j
2z
(3.115)
So,
J (z) zoo 2 ( 7
A,,
_,
7 [, , d - 2
2 j
V2 [
(d-2)/2
{-·
2
N
'~'
[I
(d - 4)/2][I + (d - 2)/2] \
2z
I
'
(3.116)
This vanishes when z = Anlobeys
-
[I +
2
+
[I+ (d- 4)/2][1 + (d- 2)/2]
I
n- -
= '7r
2
1
(3.117)
such that
On _ 1 1
[ + (d-4)/2][1 + (d -2)/2]>
A 7-
2A2
J
[I + (d- 4)/2][1 + (d- 2)/2]
71<
A2
(3.118)
This last expression holds for large values of A, such that A21 >> 12, Thus, the mode
hierarchy implicit here is A > 1 > 1, or equivalently 7R= A/k >>R/k >>1/k. This
makes sense, since k and are physical momenta which should have finite values as
R-
o.
Equation (3.118) can also be derived by the method of Ari Turner [90], which is
to look at the WKB solution to (3.103). This is
() - /4 exp i r
jRnlV(7r)
Rnl(r) ';z V(r>- 1 / 4 exp i/
V(x)dx} ,
(3.119)
with
V(r) = k 2 - [I+ (d - 4)/2][1 + (d- 2)/2]
(r)-
2~~~_
-
91
(3.120)
Taking an appropriate real solution, this will vanish when the phase is rn. So,
[Anl
(3.121)
+ (d- 2)/2 d
[1+ (d- 4)/2][1
/
x2
kr
J
which again yields
1_I + (d-4)/2][1 + (d-2)/2]
On = 11
aA
3.3.3.2
r
A
(3.122)
2
Spectral Densities
We can now return to Equation (3.109). Using R -- oo, either of Equations (3.118)
and (3.122) become
On
I On I
£
an
1 an
,\
TRak
7r
1
2
k2
(3.123)
Note that the square root sets either a lower limit on k or an upper limit on
depending on the order of integration. We get
(U)
00dk
OdRan
2pd3Rd-37gk
=ek
(U)
-
dk]
(d- 3)!(e
d£ (p
=fo°°0ofk
r(d =VZ L
d-3
7r
-3)!d
k-
e,
i)
k 2 e,3wk -
3)!Vol(I d- 1)
k
£e2 d- 3Wk
k2 edk
dk p Vol (Sd-1 2 ) kd- 2wk
Jo (27r)d- efPWk- '(314
-
(3.124)
which is in perfect agreement20 with Equation (3.88). As in Section 3.3.1, this yields
a spectral energy density over the radial momentum given by Equation (3.89) which
can be exchanged for the spectral energy density over the frequency given in Equation
(3.90). Either of these spectra could then be integrated to give the energy density of
the thermal state. The methods of Section 3.3.2 could then be applied to give the
blackbody flux.
We still need to calculate the new physical quantity made available by analysis in
spherical coordinates: the angular momentum spectral energy density. Reversing the
20
The
e integral in Equation (3.124) was performed using fl dxx v-_
r (n-')=
r (n+2)
alsousedtheidentity
2
2
-
=
r[(+W)/21.
~2n
92
----------
We
order of integration in Equation (3.124), we find
(U)
V
_
d
_
_
_2p__
k.
1
J
P=w(d - 3)!Vol (Bd-1)
___________
dk
d - a
_d
k
00
-r(d- 3)!Vol(d-1) J
£d
'
d
,
-
efk
e
/
I)
2
Wk
M
-
3
2
W2 M2
2
e
2
-
.
(3125)
(3.125)
Some remarks are in order regarding the form of Equation (3.125). Note that the
d-dependence has dropped out of the k or w integrals. This is because these integrals
describe the thermodynamics of only the t- r section of spacetime, which is independent of dimension. In fact, for low partial wave modes ( < M) or large masses, the
frequency form of Equation (3.125) yields
ue(e)
pDd(e)
d-
Vol (
00
)
A
dw
L-
1I
M2 e
-
(3.126)
suggesting, via Equation (3.90), an infinite collection of massive 1 + 1 dimensional
modes carrying an internal quantum number with degeneracy vol
pDd()l),
each therdl)
mally occupied according to standard 1 + 1 dimensional physics. This degeneracy
factor is just the normal counting of angular momentum states, taken to the continuum limit. Thus Equation (3.126) is telling us that all the original {l, m} modes with
e << M are all uniformly occupied like 1 + 1 dimensional blackbodies.
Another important limit of Equation (3.125) is high temperature (T > M, £). In
this limit, the mass and angular momentum contribute to the w integral only through
the lower integration limit, so we can use Equation (3.92) to get
uepT2 £d- 3
3(d -3)! Vol (
3)
_
2
~V+M 2 /T (1T (1- ~)/2
d - l)
2
Equation (3.127) should be valid for both massive and massless fields near
(3.127)
=
0, as
long as T > M.
For M = 0, Equation (3.125) yields
U,=2p
()7r(d-
ue(e)
3)! Vol (d-1)T
T2d3~o
ed I(3e)
2
3
318
(3.128)
where
I(a) =
dx
_
-a
(3.129)
Note that I+(O) = 7r2 /6 and I_(0) = 7r2 /12. The function I(a) is plotted in Figure
3-5. For << T we again find Equation (3.127), evaluated at M = 0. Again factoring
out the angular degeneracy factor, Equation (3.127) implies that for <<T each of
93
1.8
IC3
a
Figure 3-5: The thermal integral Is(a) defined in Equation (3.129). The solid top
curve indicates the bosonic function I+(a). The dashed bottom curve indicates the
fermionic function I_(a). The dotted central curve indicates ~ = 0, corresponding to
Boltzmann statistics. All three lines seem to converge towards el-a for a
1.
the original {l, m} modes contributes an amount
-T/ e
(1)(2 -)/2
p.T2ee/T
6
(3.130)
to the energy density p. At = 0, this agrees with Equation (3.101), showing once
again that each partial wave in this regime behaves like a 1+ 1 dimensional blackbody.
As was noted in Section 3.2.3.3, near a black hole event horizon all effects of
masses, interactions, and angular momentum in the radial wave equation of a field
are exponentially suppressed, with the suppression becoming exact at the horizon. In
particular, this means that all modes of blackbody radiation near an event horizon
behave just like the = 0 mode, which makes no contribution to the effective radial
scattering potential. Thus, we expect that the energy density in every partial wave
of d-dimensional blackbody radiation at the event horizon of a d-dimensional black
hole is given exactly by the 1 + 1 dimensional result of Equation (3.101).
Figure 3-5 shows that, at least for large ( > T), we have
ue(£) (d-
-r(d-3)!
Vol (d
T2d3e/T.
1)
(3.131)
Equations (3.127) and (3.131) together should accurately reproduce the exact M = 0
spectrum given in Equation (3.128) except for e - T. It is interesting to see that
apparently all the effects of statistics are in the small e modes. Also, the Fermi-Dirac
94
...
I_·___
factor appearing at small f is , the factor for d = 2. This again shows, as was noted
with Equations (3.126) and (3.130), the special role of the 1+1 dimensional blackbody
in the partial wave spectrum. Of course, when summed up over all values of £, the
Fermi-Dirac factors must combine to give 1- 2/2 d , in order to properly reproduce
d-dimensional physics. It is not obvious from Equations (3.128) and (3.129) exactly
how this happens, but the constructions of this section guarantee that it does.
3.4
Calculation
We now return to formulating an effective field theory outside of a black hole. The
effective theory is formed by eliminating the dangerous horizon-skimming modes by
hand - to avoid singularities in the energy-momentum tensor near the horizon while simultaneously adding a compensating energy-momentum tensor - to avoid
anomalies.
Consider the partial wave decomposition of a scalar field in a static, spherically
symmetric background spacetime. In suitable coordinates, the metric of the spacetime
can be written as in Section 3.2:
2
ds= -f(r)dt
2
+ 1 ~~1
dr2 +r22d
h(r)
22.
(3.132)
The properties of this metric were discussed extensively in Section 3.2. In particular,
we adopt here the physically motivated assumption that the zeros of f(r) and h(r) are
coincident. In this scenario, we found that physics near the horizon can be described
using an infinite collection of 1 + 1 dimensional fields, each propagating in a spacetime
with a metric given by the "r-t" section of the full spacetime metric (3.132). We will
also adopt this simplification.
For the reasons discussed in Section 3.1.4, we impose the constraint that outgoing
(horizon-skimming) modes vanish near the horizon as a boundary condition. We take
this condition to be localized on a slab of width 2 straddling the horizon at r = rh
with - 0 ultimately (see Figure 3-1). The energy-momentum tensor in this region
then exhibits an anomaly of the form (3.2).
For a metric of the form (3.132), the anomaly is purely time-like and can be
written as
VaTa - Ab-
1
a
1=
aNia,
(3.133)
where the components of Na are
Nt = Nr = 0,
N
=
Nt =
1
(3.134a)
192 (f'h' + f"h),
(3.134b)
19 2in ( ah").
(3.134c)
192n-r
95
The parameter a is an arbitrary number with no effect on physics21 because Ab is
independent of N'.
The contribution to effective action for the metric gab due to matter fields that
interact with this metric is given by
-i in (
W[gab]
(3.135)
D[matter]eis[matter'9ab),
where S[matter, gab] is the classical action functional. Under general coordinate transformations the classical action S changes by A5S = - f ddxx/IgAbVaTb where T is
the energy-momentum tensor and A is the variational parameter.
General covariance of the full quantum theory requires 6xW -=0. We write this
as
-SA0vW=
J
d 2 X/7AbVa
J
{TbH + TaE@++ Tbe
2
= d2xAt {r (NtrH)+ (z-To
-
+ (/-Tit+
dxArv-g{(To
-
}
TXt + Nt) ae+
Tx[ + Nt) a@-}
T~) ae+
+ (orre-Tx;)
}
(3.136)
T rh- e) are scalar step functions and H = 1 +- e_ is a
scalar "top hat" function which is 1 in the region between rh ± E and zero elsewhere.
The anomalous chiral physics is described by Txb via Equation (3.133). The energymomentum tensors Tob and T b are the covariantly conserved energy-momentum tensors outside and inside the horizon, respectively. Constancy in time and Equation
(3.133) together restrict the form of the Tb up to an arbitrary function of r, which is
where E)+ = ( (r
the trace T, and two constants of integration, K and Q:
Ttt = -(K + Q)/f - B(r)/f - I(r)/f + Ta(r),
Trr = (K + Q)/f + B(r)/f +I(r)/f,
V/-LTr = - K +C (r -f-h-g
h Trt,
wherewehavedefinedC(r) - frrr /A(x)dx,-f B(r)
2
(3.137a)
(3.137b)
(3.137c)
f (x)Ar(x)dx,and I(r)
f rhTa (x)f (x)dx.
A few remarks regarding the evaluation of Equation (3.137) are in order. A trace
could arise from a number of physical sources, among them a conformal anomaly. We
assume, however, that I/f
= Tad is finite. Since we will be concerned with the
rh
rh
conditions imposed by canceling potential divergences, finite terms will play no role.
Moreover, the terms containing the components of Ab vanish at the horizon. Note
21
Interestingly, for ac = 1 the antisymmetric part of Nab is equal to -eabR/(192ir),
d = 2 Ricci scalar as given in Equation (3.10).
96
where R is the
that for the diagonal terms in Equation (3.137), the limit r -- rh depends on whether
rh is approached from above or below, since f flips signs as the horizon is crossed.
The limit on 1/f is exactly antisymmetric, so
lim -= ( r-rh-*0+f
lim -).
(3.138)
r-rh-*0-f
Finally, A/
is a finite number at the horizon, given by
Section 3.2.1.2.
/-5 in the notation of
We can now take the - 0 limit of Equation (3.136). The term Or9(N[trH)
vanishes
in this limit. Using the relation (3.138) to take all limits from above, and the small
e expansions,
aE_
=6r 1 _
0a4- = -- [+1-r
r±12_2
±- o-
(3.139)
...* ] 6(r - rh),
the variation (3.136) becomes
6AW =
d2xAt {[Ko - Ki 6(r - rh)
- [Ko+Ki- 2Kx-2Ntr]a(r - rh)+-.}
-J
{ [KO+Qo+Ki+Qi
d2xAr
2Kx 2Qx]
-_E [Ko+QofKi-Qi]
(r
O(r-
-rh)
rh) +..
..
(3.140)
The ellipses represent higher order terms in with higher derivatives of 6-functions;
the coefficients of these terms are simply repetitions of the ones given above. The
delta functions in Equation (3.140) indicate that only the on-horizon values of the
energy-momentum tensors will contribute to the possible loss of general covariance.
The finite trace terms make no contribution in comparison to the divergent K + Q
terms.
Since At and Ar are independent arbitrary variational parameters, each of the four
terms in square brackets in Equation (3.140) must vanish simultaneously, but only
need do so at r = rh. These four conditions can be solved to give
Ko= K =Kx + ,
(3.141a)
Q = Qi =Qx - P,
(3.141b)
where
= Ntr| =
rh
.
(3.142)
48ir'
These conditions fix the 4 of the 6 constants Q and K. The total energy momentum
tensor
b
-
T = To
becomes, in the limit
-
_ob
+
+ T
XbH
(3.143)
0,
Tb = TCb + Tb,
97
(3.144)
where Tc~ is the conserved energy-momentum tensor which the matter in this theory
would have without any quantum effects, and TD is a conserved tensor with K =
-Q = ), a pure flux.
As discussed in Section 3.3.2, a beam of massless blackbody radiation moving in
= T 2 . Thus
the positive r direction at a temperature T has a flux of the form
we see that the flux required to cancel the gravitational anomaly at the horizon has
a form equivalent to blackbody radiation with a temperature given by T = /(27r).
This is exactly the Hawking temperature for this spacetime, as discussed in Sections
3.2.2.1 and 3.2.2.2. Thus, the thermal flux required by black hole thermodynamics is
capable of canceling the anomaly. If we fill each partial wave of the full d-dimensional
theory so that each one behaves like a 1 + 1 dimensional blackbody source at the
Hawking temperature, then we reproduce the core of the standard calculation of
black hole emission. This is exactly what one requires such that when the partial
waves are propagated outwards from the black hole and undergo mode-dependent
scattering from effective potential due to spatial curvature outside the horizon, the
resultant occupation density at infinity is that of a d-dimensional blackbody at the
Hawking temperature, modulo greybody factors.
3.5
Commentary
In contrast to the preceding argument based on gravitational anomaly cancellation,
it appears difficult to generalize the conformal anomaly derivation [77] to arbitrary
dimensions using partial wave analysis. In that framework the connection between
the anomaly and the Hawking flux is made through an integral over all of space. In
our framework the connection between the anomaly and the Hawking flux is made
through a boundary condition at the horizon, which is accurately described using
1 + 1 dimensional physics, irrespective of the true dimension.
Comparing the fluxes for thermal radiation of massless bosons and fermions in
1 + 1 dimensions, we find, as in Equation (3.101), that the boson flux is twice that
of the fermion flux. This same factor of two appears in the relative values of the
conformal anomalies (central charge) and of the gravitational anomalies. There does
not appear to be any comparably simple correspondence in higher dimensions. However, when the 1 + 1 dimensional field in question is really a single partial wave of a
higher dimensional system, the blackbody angular momentum spectrum discussed in
Section 3.3.3.2 guarantees us that this factor of two arising from the 1 + 1 dimensional
anomaly adds up to an overall (1 - 2 / 2 d) - 1, which is the proper statistical factor in
d dimensions.
In the context of an eternal black hole one can find a role for thermal radiation
incoming to the black hole by imposing additional boundary conditions near the past
horizon (V = 0, in the language of Section 3.2.2) that are symmetric with the ones
we imposed above near the future horizon. This corresponds to the Hartle-Hawking
state [91].
98
___
3.6
Blackbody Spectrum from an Enhanced Sym-
metry?
While the arguments advanced here show a pleasing consistency between the existence
of Hawking radiation flux and gravitational anomaly cancellation, they do not in
themselves suffice to show that the spectrum of radiation is thermal. One might
hope to single out the thermal state by imposing an appropriate symmetry. Indeed,
thermal states support a form of time-translation symmetry that makes sense even
near the horizon, namely translation by discrete units 3 of imaginary time.
Specifically, in Section 3.2.2.2 we generalized the Euclidean method of [42] to
show that the Euclidean section of the black holes we are studying contains a conical
singularity at the horizon unless the period of Euclidean time is taken to be /3 = 1/TH.
If multivalued coordinates are allowed, however, the singularity can be removed by
taking the period to be any integer multiple22 of this factor, /3 = n/TH, giving a
discrete set of allowed temperatures for this spacetime, Tn = TH.
Perhaps less objectionable is consideration of the Kruskal coordinate U. We argued in Section 3.2.2.1 that respecting translation in U as a good spacetime symmetry
at the past horizon, V = 0, can be seen to give rise to Hawking radiation. We also
noted, by construction, that t-translation is a global Killing vector. From the definition (3.41a), we see that under a time translation t - t + a, U -+ e-aU. If we want
Hawking radiation modes to be properly analytic in the complex frequency plane, we
should demand that that U is invariant under this transformation when a is a pure
imaginary number. This demands that a = 2rin/n, for any integer n, which says
that the black hole must be invariant under discrete translations of imaginary time
of magnitude 3 = 2n/n.
Again, this gives a discrete set of allowed temperatures,
T. = 1TH
While neither of these arguments is quite satisfying, they do make plausible the
idea that an enhanced discrete translation symmetry can fix the thermal nature of the
black hole system, and that temperatures of different magnitude can be accommodated as different units for the periodicity by T = 1/,3. If we assume that a symmetry
of this form exists, then anomaly cancellation fixes the unit. One could certainly wish
for a less formal, more physically enlightening perspective, however.
22
With this integer factor n, one must travel in a circle around the origin a distance 2rn in
Euclidean time to return to one's starting point. This makes the black hole appear like an anyon
[92] of spin 1/n. Such particle states are indeed allowed in the two dimensional r-x plane of Equation
(3.43), but their physical significance is uncertain.
99
100
____II__
Chapter 4
Finale
4.1
Summary
In Chapter 2, we found that the addition of gravity to four-dimensional Yang-Mills
theories adds a term to the one-loop 3 functions of the form
Agrav (gym, E) =
3 E2
3-9
E2
,
(4.1)
which renders all Yang-Mills couplings asymptotically free. To some extent, this
result simplifies the physics of the the early universe because, for what it is worth,
gluon dynamics can be ignored and we only need to worry about quantum gravity.
We believe this gravitational correction is only directly observable if the true scale of
quantum gravity is unexpectedly low. We also found that the gravitational correction
does not spoil coupling constant unification in a theory whose Yang-Mills couplings
exhibited unification before gravity was added. The unification point remains unique
and is shifted slightly in energy (approximately one part in 106 for realistic theories).
We showed in Chapter 3 that the Hawking radiation from a rather generic class
of spherically symmetric black holes in arbitrary dimensions can be understood as
arising from a gravitational anomaly in the 1 + 1 dimensional effective quantum field
theory that governs physics close to the horizon. This helps to raise the connection of
anomalies to Hawking radiation above the level of an isolated curiosity. We did not
prove definitivelythat anomaly cancellation requires the radiation to be thermal, but
argued that the radiation has several features in common with blackbody radiation
at the Hawkingtemperature.
4.2
Open Possibilities
As mentioned in Section 2.11, there are several questions raised by the calculation of
the gravitational correction to the Yang-Mills function. For example, how is the
calculation implemented in TeV-scale gravity theories and what are the experimental
signatures thereof in colliders, cosmic rays, and atomic systems? Also, how the does
101
calculated correction effect the fitting of experimental data with uncertainties to the
hypothesis of coupling constant unification?
An obvious program is to extend the calculation to Yukawa couplings and scalar
self-interactions. It would be rather astonishing if these couplings were also rendered
asymptotically free by gravitation, since they can not be asymptotically free in four
dimensions without gravity. Mass renormalization could also be studied, but this is
not as interesting since we expect gravity-induced renormalization to be important
only at scales much higher than any conventional particle physics masses.
The derivation in Section 3.4 of Hawking radiation as a mechanism that cancels
a gravitational anomaly also leaves some open questions. Obvious among these is
whether it can be generalized to rotating or otherwise non-spherical black holes. Also,
it remains to be seen if the Killing frequency spectrum of the 1+ 1 dimensional theory
can be calculated from the anomaly such that the thermal nature of the radiation is
completely elucidated. One might further wonder if this mechanism has any relevance
to the black hole information paradox or whether it can be used to calculate black
hole entropy. The answer to these last questions is probably "no" because Hawking
radiation is a kinematic effect, while information and entropy issues necessarily involve
dynamics. However, these questions do deserve a deeper investigation.
Our study of black hole effective field theory was motivated by rather general
concerns regarding an observer's ability to describe physics in terms of the degrees
of freedom which he can experimentally probe. The essential difficulty in doing this
was encountered most directly in a semiclassical black hole background, but the same
problem should generically arise in any generally covariant quantum field theory. Perhaps the anomaly cancellation mechanism discovered here can be generalized to allow
a generally covariant formulation of quantum field theory in which local quantum field
theories formulated on local coordinate patches of spacetime are stitched together to
form a global theory using anomaly-driven currents that act as a kind of connection.
If such a theoretical framework existed it would, in a certain sense, make no essential
distinction between fundamental and effective degrees of freedom.
4.3
Conclusion
We hope that we have shown in this thesis that much can be learned within the
limited scope of quantum general relativity and semiclassical gravity if the logical
boundaries of these models are properly respected. Furthermore, we believe that the
work presented here points in a promising way towards features that should ultimately
be reproducible in a more fundamental theory of quantum gravity.
102
_
_·_
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