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

c© Sean Patrick Robinson, MMV. All rights reserved.

The author hereby grants to MIT permission to reproduce anddistribute publicly paper and electronic copies of this thesis document

in whole or in part.

Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Department of PhysicsMay 19, 2005

Certified by. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Frank Wilczek

Herman Feshbach Professor of PhysicsThesis Supervisor

Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Professor Thomas GreytakAssociate Department Head for Education

2

Two Quantum Effects in the Theory of Gravitation

by

Sean Patrick Robinson

Submitted to the Department of Physicson May 19, 2005, in partial fulfillment of the

requirements for the degree ofDoctor of Philosophy in Physics

Abstract

We will discuss two methods by which the formalism of quantum field theory can beincluded in calculating the physical effects of gravitation. In the first of these, theconsequences of treating general relativity as an effective quantum field theory willbe examined. The primary result will be the calculation of the first-order quantumgravity corrections to the β functions of arbitrary Yang-Mills theories. These correc-tions will effect the high-energy phenomenology of such theories, including the detailsof coupling constant unification. Following this, we will address the question of howto form effective quantum field theories in classical gravitational backgrounds. Wefollow the prescription that effective theories should provide a description of exper-imentally accessible degrees of freedom with all other degrees of freedom integratedout of the theory. We will show that this prescription appears to fail for a scalar fieldin a black hole background because of an anomaly generated in general covarianceat the black hole horizon. This anomaly is repaired and the effective field theoryis saved, however, by the inevitable presence of Hawking radiation in the quantumtheory.

Thesis Supervisor: Frank WilczekTitle: Herman Feshbach Professor of Physics

3

4

Acknowledgments

The following body of work has benefited from the input of many individuals. Obviousamong these is my thesis advisor, Frank Wilczek. I would like to thank the NobelFoundation for making this an interesting year to write a thesis. I also need torecognize the other members of my thesis committee, Eddie Farhi and Roman Jackiw.Others who made significant contributions to the development of this work, but arenot specifically cited within, include Brett Altschul, Ted Baltz, Serkan Cabi, QudsiaEjaz, Ian Ellwood, Michael Forbes, Brian Fore, Vishesh Khemani, Joydip Kundu,Vivek Mohta, Brain Patt, Dru Renner, Jessie Shelton, and Ari Turner. I especiallythank Michael Forbes for reading an early draft of this thesis. Finally, I would like toacknowledge 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 cautionarylimerick1:

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 occasional error, or three.

This work is supported in part by funds provided by the U.S. Department ofEnergy (D.O.E.) under cooperative research agreement DE-FC02-94ER40818.

1Incidentally,

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

Contents

1 Overture 13

1.1 Quantum General Relativity and Yang-Mills Theory . . . . . . . . . . 141.2 Black Holes and Effective Field Theory . . . . . . . . . . . . . . . . . 17

2 Gravitational Corrections to Yang-Mills β Functions 23

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.1.1 One-loop Divergences . . . . . . . . . . . . . . . . . . . . . . . 262.1.2 Asymptotic Safety . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2 Technical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 282.2.1 Background Field Theory . . . . . . . . . . . . . . . . . . . . 282.2.2 Definition of Newton’s Constant . . . . . . . . . . . . . . . . . 29

2.3 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.4 Expanding the Action . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.4.1 Expanding the Non-Polynomial Terms . . . . . . . . . . . . . 322.4.2 Expanding the Einstein-Hilbert Action . . . . . . . . . . . . . 33

2.4.2.1 Curvature with Background Derivatives . . . . . . . 332.4.2.2 Some Useful Definitions and Identities . . . . . . . . 342.4.2.3 Expansion of Curvature . . . . . . . . . . . . . . . . 34

2.4.3 Expanding the Yang-Mills Action . . . . . . . . . . . . . . . . 362.5 Gauge-Fixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.6 Combining the Pieces . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.7 Compiling the Superfield . . . . . . . . . . . . . . . . . . . . . . . . . 412.8 Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.8.1 Computation of Functional Determinants . . . . . . . . . . . . 462.8.2 Extracting the β Function . . . . . . . . . . . . . . . . . . . . 49

2.9 Enlarging the Matter Sector and the Gauge Group . . . . . . . . . . 502.10 Coupling Constant Unification . . . . . . . . . . . . . . . . . . . . . . 532.11 Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542.12 Commentary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3 Black Hole Effective Field Theory 59

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.1.1 Hawking Radiation . . . . . . . . . . . . . . . . . . . . . . . . 593.1.2 Anomalies and Anomaly Driven Currents . . . . . . . . . . . . 603.1.3 Hawking Radiation and the Conformal Anomaly . . . . . . . . 61

7

3.1.4 Effective Field Theory Framework . . . . . . . . . . . . . . . . 623.2 Spacetime Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.2.1 Spherical Static Metrics . . . . . . . . . . . . . . . . . . . . . 633.2.1.1 Einstein’s Equation . . . . . . . . . . . . . . . . . . . 653.2.1.2 Horizon Structure . . . . . . . . . . . . . . . . . . . 67

3.2.2 Kruskal Extension . . . . . . . . . . . . . . . . . . . . . . . . 693.2.2.1 The Quantum Vacua . . . . . . . . . . . . . . . . . . 743.2.2.2 Euclidean Section . . . . . . . . . . . . . . . . . . . . 75

3.2.3 Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 753.2.3.1 Spherical Harmonics . . . . . . . . . . . . . . . . . . 773.2.3.2 Radial Wave Equation . . . . . . . . . . . . . . . . . 793.2.3.3 Near-Horizon Action . . . . . . . . . . . . . . . . . . 80

3.3 Thermal Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823.3.1 Hypercubic Blackbody Cavity . . . . . . . . . . . . . . . . . . 833.3.2 Flux Versus Energy Density . . . . . . . . . . . . . . . . . . . 863.3.3 Spherical Blackbody Cavity . . . . . . . . . . . . . . . . . . . 88

3.3.3.1 Radial Mode Density . . . . . . . . . . . . . . . . . . 903.3.3.2 Spectral Densities . . . . . . . . . . . . . . . . . . . 92

3.4 Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 953.5 Commentary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 983.6 Blackbody Spectrum from an Enhanced Symmetry? . . . . . . . . . . 99

4 Finale 101

4.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.2 Open Possibilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

8

List of Figures

2-1 Feynman diagrams for two typical processes contributing to the renor-malization of a Yang-Mills coupling at one-loop. . . . . . . . . . . . 25

2-2 The schematic Feynman diagram represented by the functional trace−1

2Tr[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−1

2Tr[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 trace12Tr[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 the-ory evolve as straight lines in a plot of α−1 ≡ 4π/g2 versus log10 (E)when gravitation is ignored. The initial values at MZ0 ≈ 100 GeVare set so that the lines approximately intersect at 1016 GeV. Whengravity is included at one-loop, the three lines curve towards weakercoupling at high energy, but remain unified near 1016 GeV. In Figure2-5(b), g is plotted for the same theory. All three couplings rapidly goto zero near MP, rendering the theory approximately free above thisscale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3-1 Part of the causal diagram of a black hole spacetime, with inset detailof a region near the horizon. . . . . . . . . . . . . . . . . . . . . . . 63

3-2 Sketches of three integrated mass functions and their associated h(r).In 3-2(a) the matter distribution is relatively smooth and vanishes atthe origin, as in a normal star. In 3-2(b) the matter has a densitysingularity at the origin, but is otherwise well behaved. In 3-2(c) apotentially 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 foran asymptotically flat black hole spacetime. The horizon occurs wherethe functions vanish at r = rh. In 3-3(b) the corresponding profileof r∗ is sketched along with the line r∗ = r. Note that r∗ divergeslogarithmically at rh and approaches r at large r . . . . . . . . . . . . 71

9

3-4 A sketch of a typical effective radial scattering potential. The potentialfor any metric qualitatively similar to the one sketched in Figure 3-3will be qualitatively similar to the one sketched here for l > 0 andd > 3. The potential falls off exponentially for negative r∗ and as istypically dominated by the centrifugal term at large r∗ ≈ r, which fallsoff as 1/r2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3-5 The thermal integral Iξ(a) defined in Equation (3.129) is plotted forthe cases ξ = ±1 and ξ = 0. All three lines seem to converge towardse1−a for a≫ 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

10

List of Tables

3.1 A few physically interesting metrics that obey ρ = −P . . . . . . . . . 673.2 The angular state degeneracies for total angular quantum number l, as

determined by Equation (3.64), in a few chosen dimensions. . . . . . 79

11

12

Chapter 1

Overture

In this thesis we shall describe two logically independent lines of research that rep-resent small steps away from ordinary quantum field theory in flat, nondynamicalspacetimes and towards quantum gravity, the as-yet-undiscovered fundamental the-ory of quantum spacetime dynamics. The first strategy we investigate is that of per-turbatively quantizing the small field fluctuations of general relativity, the first theoryof spacetime dynamics historically. This approach is famously limited in power andmuch-maligned, but we will show by a specific example that useful physical predic-tions can nevertheless be obtained in this formalism. The second strategy is to notattempt to give quantum dynamics to spacetime at all, but to instead only use quan-tum theory where it has already proven so successful: in the nongravitational aspectsof matter. In this approach, spacetime is described in a nonquantum way, either asa nondynamical, curved background or using the classical dynamics of general rela-tivity. This formalism is usually called semiclassical gravity. Like quantized generalrelativity, semiclassical gravity is rather restricted in scope and cannot be consideredas more than a limited, but useful, model for quantum gravity. We will use semiclas-sical gravity to describe the behavior of a quantum field theory in the region outsideof a black hole.

The second strategy is often considered more respectable than the first, perhapsbecause it never attempts to be more than a model, and thus its points of failure areboth more understandable and educational. We believe, on the other hand, that bothformalisms are useful as model theories for true quantum gravity as long as they areapplied within their respective regimes of validity. By studying the conditions underwhich a model theory begins to fail, we can learn which aspects of the true theory themodel theory lacks. Since semiclassical gravity and quantized general relativity havedifferent regimes of validity and different failure modes, they are complimentary toolsin the investigation of the properties of quantum gravity. They also have a broadoverlap region of validity, which is the domain of ordinary Minkowski space quantumfield theory. This provides for these models an anchor to known physics, which isoften claimed to be well understood.

In Section 1.1 we will describe the effects of including quantized general relativityin the calculation the β function of a non-Abelian gauge theory. The calculationof 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 fullcalculation and discussion appears in Chapter 2. In Section 1.2 we will describe ourattempts to import the highly successful concepts of Minkowski space effective fieldtheory 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 β functionof the Yang-Mills coupling constant in an arbitrary non-Abelian gauge theory coupledto 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, whichis just quantized general relativity. This effective field theory should be an accuratedescription for the quantum dynamics of spacetime at energy scales below the theory’scutoff scale, the Planck mass, given in four dimensions as MP ≡ G

−1/2N ≈ 1.1× 1019

GeV, where GN is Newton’s constant of universal gravitation.

Before the appreciation for the proper role of effective field theories in physics be-came widespread, common lore held that general relativity and quantum mechanicsare incompatible in terms of describing the physical phenomenon of gravitation. Thiswas primarily because quantum general relativity was finally proven to be perturba-tively nonrenormalizable [5, 6, 7, 8, 9] shortly after the time when renormalizabilityhad become understood as an essential ingredient in quantum field theories of fun-damental interactions. About ten years later, the discovery of quantum theories thatappear to describe gravitation in terms of excited strings [10, 11, 12, 13, 14, forintroductions], rather than local fields, helped to cement the common lore.

Over time, however, appreciation has grown for the fact that even the best quan-tum field theories of reality (that is, the standard model) are, at best, effective theoriescontaining infinite numbers of nonrenormalizable interactions. The ideas of Wilso-nian effective field theory [15, 16, 17] have taken deeper root in the intuition andmade possible the use of nonrenormalizable phenomenological models such as chi-ral perturbation theory [18, 19, 20]. Finally, about 20 years after the proofs of thenonrenormalizability of general relativity, Donoghue [21] made compelling argumentsin favor of taking seriously calculations made with quantized general relativity andtreating the results of these calculations as genuine low-energy predictions of quan-tum gravity. It is in this spirit that we perform our calculation of the Yang-Mills βfunction.

The effective value of a renormalized coupling constant g in a quantum field theorydepends on the energy scale E at which it is probed via a universal function of thetheory known as the Callan-Symanzik β function [22, 23]:

E∂g

∂E≡ β(g, E). (1.1)

14

The remarkable discovery [1, 2] for four-dimensional non-Abelian Yang-Mills theorieswas that these theories obey

β = − g3YM

(4π)2

[

11

3C2(G)− 4

3nfC(r)

]

≡ − b0

(4π)2g3

YM(1.2)

for a gauge group G with nf fermions in representation r. This β is negative as longas nf is not too large. Equation (1.2) integrates to give a running coupling of

1

gYM

(E)2=

1

gYM

(M)2+

b0

(4π)2ln

(

E2

M2

)

, (1.3)

which demonstrates that the negative value of the β function implies asymptotic free-dom: g

YM(E)→ 0 as E →∞, as long as b0 is positive. The only known asymptotically

free theories in four spacetimes dimensions are the non-Abelian gauge theories. Thusa universe with laws of physics governed by non-Abelian gauge theories — as ouruniverse approximately appears to be — becomes simpler and simpler as it is probedat 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, whichwe will sketch briefly in Section 2.2.1. We will let the spacetime background anddimension be arbitrary for as long calculationally feasible. This will require adoptinga definition for Newton’s constant in d dimensions. We choose a definition, describedin Section 2.2.2, which preserves the interpretation of the nonrelativistic gravitationalforce law as describing the areal density of diverging, but conserved, gravitational fluxlines. Then, in Sections 2.3 through 2.7, we perform the detailed expansion of thecoupled Einstein-Yang-Mills action in terms of quadratic fluctuations about nontrivialgauge field and spacetime backgrounds. In particular, in Section 2.5 we gauge-fix thetheory using the Faddeev-Popov [24] procedure and calculate the ghost and gauge-fixing Lagrangians. The gauge chosen to fix general covariance is reminiscent ofthe Rξ gauge [25], except that the original Rξ gauge was for a gauge field in a scalarbackground 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 thebackground effective action and extracting the β function. In Section 2.9 the result isgeneralized to arbitrary gauge groups and matter content. We find that to one-loopaccuracy, the β function is equal to the value calculated in the absence of gravity —such as that given in Equation (1.2) — plus a new term ∆βgrav that is independentof the gauge and matter content. In four spacetime dimensions, this term is given by

∆βgrav(gYM, E) = −g

YM

3

π

E2

M2P

. (1.4)

Note that this term is always negative. It will dominate the running of the couplingwhen the energy is close to the Planck scale and the coupling constant is perturba-tively small. Thus, it appears that the inclusion of quantum gravity effects renderall non-Abelian gauge theories asymptotically free. The integrated running coupling

15

coming from the combination of Equations (1.2) and (1.4) is

1

gYM

(E)2=

1

gYM

(M)2exp

3

π

E2 −M2

M2P

+ 2b0

(4π)2

∫ E

M

dk

kexp

3

π

E2 − k2

M2P

. (1.5)

The logarithmic running of Equation (1.3) becomes modulated by a exponential inE2. This has little effect at low energies, where the exponential is approximatelyequal to one. As E approaches MP, however, the exponential turns on very quicklyin comparison to the logarithm, and the coupling gets driven rapidly to zero. Thisphenomenology comes with the caveat that the interesting physics is occurring veryclose to the cutoff scale of the theory. However, taken at face value, this result seemsto 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 constantunification. That is, we consider a Yang-Mills theory with a simple gauge groupthat is spontaneously broken at some high energy scale such that the theory at lowenergies appears to be a Yang-Mills theory of some product gauge group with severalindependent coupling constants, each with its own β function. Without the contextof the unified theory, the low-energy values of these couplings could be taken to havearbitrary independent values. However, since all the couplings secretly derive froma unified theory at high energy with only a single coupling, the low-energy valuesmust conspire with the β functions in such a way that all the couplings evolve to thesame unified value at the breaking scale. The experimentally measured values of theSU(3)×SU(2)×U(1) couplings of the standard model with minimally supersymmetricmatter content are consistent with such a unification in the real world with a breakingscale of MGUT ≈ 1016 GeV [26]. No matter how many couplings are in the low-energytheory, only two of them may be chosen independently. The rest are then fixed bythe condition of unification.

If the field content of the low-energy theory is changed such that the β functionschange, without a corresponding change in the values of the low-energy couplings,the unification will generically be spoiled. If the addition of gravitation spoiled unifi-cation in this way, it would indicate that the observed unification of standard modelcouplings is a spurious coincidence. Fortunately, as we show in Section 2.10, thisis not the case for four dimensional gauge theories. Although the β functions arechanged in a non-trivial way given by Equation (1.4), we find that theories whichexhibit exact coupling constant unification in the absence of gravity continue to doso with the same values of the low-energy couplings when Equation (1.4) is takeninto account. The values of the unified coupling and the breaking scale are slightlyaltered. For a standard-model-like situation where the measurement scale M and theputative breaking scale M0 obey a hierarchy of the form M ≪ M0 ≪ MP, we findthat the new breaking scale MU is given by

MU ≈M0

[

1 +3

2π

(

M0

MP

)2]

. (1.6)

Finally, in Section 2.11, we make some brief remarks regarding the phenomenology

16

and possible experimental signatures of the calculated gravitational correction to therunning 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 contextof semiclassical gravity, we attempt to formulate an effective field theory for a scalarfield that lives in a black hole background. Our prescription for constructing thistheory ultimately results in a breakdown at the quantum level of the underlying gaugesymmetry of gravitation, general covariance. Demanding that general covarianceholds in the effective theory, as it does in the fundamental theory, forces each partialwave of the scalar field to be in a state with a net energy-momentum flux Φ given by

Φ =κ2

48π, (1.7)

where κ is the surface gravity of the black hole event horizon. If each partial wavemode is occupied with a blackbody frequency spectrum, then Equation (1.7) impliesa temperature of

TH =κ

2π, (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 Hawk-

ing [28, 29] and subsequently rederived by many other methods. Hawking radiation isnow understood as a kinematic effect resulting from the lack of a unambiguous globaldefinition for a particle number basis of Fock space when spacetime is not globallyflat.

Our construction can be thought of as arising from the presumption that thephysics observed by a given experimenter should be describable in terms of the ef-fective degrees of freedom accessible to that experimenter. In the case of ordinaryMinkowski space quantum field theory, one can apply this presumption to an experi-menter with limited energy available to probe highly excited states. In that case, theeffective physics observed by the experimenter is described by the theory in whichstates above the high-energy cutoff have been integrated out, resulting in the stan-dard story of Wilsonian effective field theory [15, 16, 17]. The parameters and degreesof freedom of the low-energy theory may be different from those that appear in thefundamental theory.

We wish to consider an experimenter who lives outside of a static, sphericallysymmetric black hole. Such spacetimes have a global Killing vector (spacetime sym-metry generator) that appears locally like a time translation, but it is only timelikein the region outside the black hole event horizon. Thus, the conserved quantity as-sociated with this symmetry can not be used as an energy outside this region. Sincethe observer cannot see beyond the event horizon of the black hole, however, thisKilling vector should be a perfectly reasonable choice with which to define the energyof quantum states in an effective theory that only describes observable physics. Un-

17

fortunately, the “vacuum state”1 obtained with this definition is exactly the one con-sidered by Boulware [30]. The Boulware vacuum has a divergent energy-momentumtensor due to a pile up at the horizon of would-be outgoing modes (the UP modesin the language of [31]), which take arbitrarily long amounts of coordinate time toescape the near-horizon region.

Our approach differs from most previous work on Hawking radiation in that werecognize the divergent energy of the horizon-skimming modes as an indicator thatthe experimenter who observes these modes will not be able to probe them with finiteenergy. Thus, the proper description of the observed physics is an effective theorywith these modes integrated out. In other words, we choose to take the lessons ofeffective field theory seriously.

The effective theory thus formed no longer has observable divergences, but it nowsuffers 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 covarianceat the quantum level if the number of right-moving and left-moving modes are notidentical — that is, if the theory is chiral. The breakdown of general covariance meansthat the energy-momentum tensor T a

b of the scalar field is not conserved. In the caseof a single chiral scalar field, the anomaly takes the form

∇aTab =

1

96π√−g

ǫcd∂d∂aΓabc, (1.9)

where the Γabc are the Christoffel symbols of the background spacetime.

We show in Section 3.2.3 that in the near-horizon limit, each partial wave behaveslike an independent two dimensional free massless scalar field. In our case, we haveeliminated the horizon-skimming part of each partial wave of the scalar field. So, thiseffective 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 containsall the modes, so it has no anomaly. Some new physics must be introduced into thechiral theory to carry out the job of anomaly cancellation that was formerly performedby the degrees of freedom which were integrated out in the process of forming theeffective theory2. Indeed, we find that demanding general covariance to hold in the

1The word “vacuum” is used here in the Fock space sense, meaning the state in which all mo-mentum modes have zero occupation number. It does not that mean the state has minimal energy;the energy of a state can not be unambiguously defined in a curved spacetime.

2The ability to form a gauge invariant effective theory for a fundamental theory which cancelsanomalies between modes of very different energy seems to run contrary to decoupling theoremswhich state that the only effect the high-energy modes have in the low-energy theory is in therenormalized value of low-energy coupling constants [33]. If the effective theory is at an energyscale where some, but not all, of the modes involved in anomaly cancellation have been integratedout, then decoupling should guarantee that the high-energy modes cannot cancel the remaininganomalies in the effective theory. In theories like the electroweak standard model, which havepotentially anomalous chiral gauge couplings to fermions that gain a wide spectrum of masses viaYukawa couplings to a Higgs field, the problem has been partially solved [34, 35] by the discovery ofa 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 occursbetween 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 fieldin the form of a boundary condition that must be obeyed by each partial wave at theblack hole horizon. The boundary condition can then be used to solve the covariantconservation equation for the energy-momentum tensor over all of spacetime. Theresult is that the energy-momentum tensor must describe a flux of the form given inEquation (1.7) in each partial wave.

Equations (1.7) and (1.8) are derived primarily in Section 3.4. The calculationthere is relatively brief and painless. However, a great deal of formalism needs tobe 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 thatare not directly relevant to the core calculation of Section 3.4 and mostly appear asfootnotes in the main text. These are also summarized below.

In Section 3.2.1, we study the properties of the general static, spherically symmet-ric spacetime in d spacetime dimensions. We compute the components of the Riccitensor and scalar curvatures, as well as the Christoffel symbols, in a natural coordinatesystem. This allows us, in Section 3.2.1.1, to construct and solve the d-dimensionalEinstein’s equations for the most general background matter distributions allowedby the symmetries. The d-dimensional versions of a few simple, well-known four-dimensional spacetimes are listed in Table 3.1. In Section 3.2.1.2 we examine theconditions for the existence of a horizon such that the spacetime would describe ablack hole. We distinguish between event horizons and Killing horizons, but arguethat given some rather general physical conditions, Einstein’s equations imply thatif a event horizon exists at some constant-radius surface of the spacetime, then aKilling horizon must also exist at the same location. We define and compute the sur-face gravity of the horizon. Although much of the core analysis of Chapter 3 will notdepend in any way on the spacetime in question being a solution to Einstein’s equa-tions, we will find it necessary at several points to invoke a coincidence requirementfor Killing horizons and event horizons. Thus, it is reassuring that this requirementis well-motivated by real physics. We also find in this section that the same physicalconditions that lead to the coincidence requirement also imply that at the black holehorizon, the radial pressure of the background matter seen by a static observer mustbe equal to the negative of the observed energy density.

In Section 3.2.2, we construct the analog for this spacetime of the Kruskal exten-sion [38] of the four dimensional Schwarzschild black hole. Unlike the Schwarzschildcase, the resulting metric for the general case is not obviously non-singular at the hori-zon, but we prove by construction that it is. In the process, we show that the choiceof the Kruskal coordinates is quite constrained. As in the original Kruskal extension,the time translation symmetry of the original coordinates now manifests as a boostsymmetry. Further, we observe that translations in the Kruskal U and V coordinatesbecome spacetime symmetries at the past and future event horizons, respectively. Theexistence of the Kruskal extension requires the coincidence of Killing and event hori-zons discussed in Section 3.2.1.2. As a bonus, we construct the Painleve-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, weuse our new Kruskal extension to reproduce the famous arguments originally madeby Unruh [41] for the four dimensional Schwarzschild black hole, thus proving thatHawking radiation can be understood for this spacetime according to standard argu-ments. To further push this point, in Section 3.2.2.2 we follow [42] in constructing thenear-horizon coordinates for the Euclideanized version of the black hole, and showthat the Euclideanized horizon exhibits conical singularity which is resolved by settingthe period of Euclidean time to be 2π/κ. This agrees with Equation (1.8) by standardarguments that associate the period of Euclidean time with inverse temperature. Theexistence of the near-horizon coordinates also requires the coincidence of Killing andevent horizons.

In Section 3.2.3, we consider the partial wave decomposition of an arbitrary scalarfield theory in this d-dimensional black hole spacetime and solve the scalar wave equa-tion by separation of variables. In Section 3.2.3.1, we derive the (d−2)-spherical har-monics, which are an alternating product of Legendre P and Q functions. Demandingregular solutions of the wave equation quantizes the arguments of the Legendre func-tions and therefor sets the spectrum of angular momentum quantum numbers. Wealso find the degeneracy of angular momentum states with total angular momentumquantum number l in d dimensions to be:

Dd(l) =(2l + d− 3)(l + d− 4)!

l!(d− 3)!. (1.10)

In Section 3.2.3.2, we tackle the radial equation, which is not solvable in general. Inthe flat space limit, it is solved by Bessel functions; we define the (d − 2)-sphericalBessel functions. In the near-horizon limit, the radial equation in tortoise coordinatesbecomes the 1 + 1 dimensional massless free wave equation for each partial wave. Fi-nally, in Section 3.2.3.3, we apply the partial wave decomposition to the action of ascalar field with arbitrary self-interactions. We find that whether an interaction isimportant in the far-field limit is controlled by whether it would be power-countingrenormalizable in a quantum theory. We also find that near the horizon, the van-ishing of the metric functions implies the vanishing of all terms in the action exceptfor the tortoise d’Alembertian term. Thus, the near-horizon action is that of an infi-nite collection of free, massless, 1 + 1 dimensional scalar fields in 1 + 1 dimensionalMinkowski 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 multi-component 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 ρ andblackbody flux Φ for p massless fields of statistics ξ in d spacetime dimensions attemperature T is given by

Φ =Vol (Bd−2)

Vol (Sd−2)ρ, (1.11)

where Bn is the unit n-ball and Sn is the unit n-sphere. The d-dimensional Stefan-

20

Boltzmann law is given by

Φ = p

(

1− 2

2d

)(1−ξ)/2ζ(d)(d− 1)!

2d−2(d− 2)Γ(

d−22

)

πd/2T d. (1.12)

In Section 3.3.3, we engage in the unexpectedly challenging calculation of the black-body spectrum of angular momentum modes. No closed form expression is ultimatelyfound, but various limits indicate that each partial wave behaves nearly like a 1 + 1dimensional blackbody, including the correct statistics. We also remark on the differ-ence between canonical momenta of angular variables and actual angular momenta,which is understood using the Cartan subalgebra of SO(d−1). This forms a pleasingconnection between group theory and mechanics.

Finally, in Section 3.6, we speculate on the existence of an enhanced spacetimesymmetry giving rise to a thermal spectrum whose temperature is fixed by theanomaly mechanism of our core calculation. We also make a speculative remarkabout the possible anyon-like behavior of Euclidean black holes.

21

22

Chapter 2

Gravitational Corrections to

Yang-Mills β Functions

In this chapter, we will calculate the one-loop contributions of virtual gravitons tothe running of Yang-Mills coupling constants using background field methods inMinkowski space. We find that this renders all Yang-Mills couplings asymptoticallyfree, independent of any additional matter content. We also find that that the addi-tion of gravity to a theory which previously displayed coupling constant unificationat a high energy MGUT does not upset this unification. Rather, the unification energyis shifted up by approximately 3

2πM3

GUT/M2P, 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 gen-eral relativity can be taken as a low-energy effective theory for the true theory ofquantum gravity, just as the nonrenormalizable chiral Lagrangian of mesons is a low-energy effective theory of the strong interactions. In this sense, quantized generalrelativity cannot be taken as a fundamental theory and its predictions should not betrusted above the built-in scale of the theory, MP ≡ G

−1/2N ≈ 1019 GeV, just as chiral

perturbation theory should not be believed above its scale, fπ ≈ 130 MeV. An inter-esting 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, namelyQCD, 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 evenat 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 providean explanation as to why the appropriate description at low energy does not lookanything 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 thethe chiral Lagrangian may be necessary for understanding its role as a description ofstrong interactions, knowledge of the ultraviolet completion is usually not necessary ifone wants to use chiral perturbation theory simply to calculate the low-energy behav-ior of mesons (where “low-energy” here means below fπ). However, these predictedbehaviors should still be taken as genuine low-energy predictions of QCD. Likewise,as compellingly argued by Donoghue [21], the predictions of quantized general relativ-ity at scales below MP should be taken as unambiguous predictions of whatever thetrue theory of quantum gravity may be. Thus, one could refer to quantized generalrelativity as perturbative quantum gravity.

Armed with this attitude, several authors have used quantized general relativity tocalculate the one-loop corrections to the nonrelativistic Newtonian potential, mostlyin the Born approximation [21, 46], but some by other methods [47, 48]. By readingthe numerator of the results of such calculations, one can interpret an “effectiveNewton’s constant,” altered from its bare value by short-range quantum effects. Ofcourse, the short range in question is very close to the scale at which the theoryshould be cutoff, and even then the effects are weak. So, unlike the running of theQED and QCD couplings, the predicted form of gravitational running is not expectedto be experimentally falsifiable anytime soon.

On the other hand, allowing for the virtual production of any new species ofparticle will change the rate at which couplings run. That is, the form of the Callan-Symanzik β function [22, 23] — the logarithmic derivative of a coupling constant withrespect to the renormalization scale — should be altered with each new field added tothe 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 magnitudefor laboratory experiments, but it may disturb certain high-energy predictions of thetheory. For example, the experimentally measured values of the standard model cou-pling constants seem to conspire together with the theoretically calculated coefficientsof the gauge coupling β functions (augmented with minimal supersymmetry) to givea unification near MGUT ≈ 10−3MP [26]. This unification is highly sensitive to theinput parameters, as it is equivalent to getting three lines to meet at a point, up toexperimental uncertainties. Even a small perturbation of the β function coefficientscould push unification out of the experimentally measured range. If virtual gravitonswere to upset unification in this way, it would be somewhat disturbing, as the uni-fication of standard model gauge couplings is a necessary prediction of all realisticgrand unified theories.

With this in mind, we set about calculating the the scale-dependence of a non-supersymmetric pure Yang-Mills theory coupled to quantized general relativity in3 + 1 dimensions, to one-loop order. We will use effective action background fieldmethods, since they are quite natural for gravity and known to be useful for gaugefields. In principle, an arbitrary background spacetime could be used, in which casethe renormalization of Newton’s constant and the cosmological constant could bestudied, too. Since renormalization depends only on ultraviolet physics, however,and all spacetimes are locally Minkowski, we will restrict our attention here to thecase of Minkowski spacetime and, thus, zero cosmological constant. However, in the

24

gYM

← E g

YMgYM

gYM

v E

Mp

← EvEMp

Figure 2-1: Feynman diagrams for two typical processes contributing to the renor-malization of a Yang-Mills coupling at one-loop. Curly lines represent gluons. Doublelines represent gravitons. The three-gluon vertex is proportional to g

YM, while the

gluon-graviton vertex • is proportional to E/MP.

interest of generality, calculations will be carried out in an arbitrary background withnon-zero cosmological constant for as long as is feasible.

The form of the gravitational contribution to the β function can be guessed with-out calculation, since all the new one-loop Feynman diagrams of interest are essentiallya three-gluon vertex with two legs connected by a graviton (See Figure 2-1). Sincethe gluon vertex has strength g

YMand gravitons couple to energy-momentum, one

expects1 the inclusion of gravitons to add a term to the β function like

∆βgrav(gYM, E) = a0gYM

E2/M2P (2.2)

at energy scale E. Sections 2.3 through 2.8 of this thesis consist of the calculation ofthe unknown coefficient a0.

Once the calculational method is presented for the case of a single gauge field, itcan be extended to the case of multiple gauge fields with interacting matter almostby 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 con-stant unification. Section 2.9 discusses the implications of gravitational correctionsin such theories. One hindrance of applying the results to interesting supersymmetrictheories, such as the minimally supersymmetric standard model, lies in the fact that

1The guess of Equation (2.2) is for the case when gYM

is dimensionless, which demands d = 4.Generically, each term in the β function would be multiplied by an additional Ed−4 to account forthe units of g

YM. So, in the absence of gravity, the β function takes the form

β(gYM

, E) = b0g3YM

Vol (Sd−1)

2(2π)dEd−4, (2.1)

where b0 is a number determined by the theory and the unitless, d-dependent numerical factor hasbeen arbitrarily chosen for convenience.

25

we are using only bosonic gravity in our analysis. Such an application only makessense in a model where supersymmetry is treated as being broken above the Planckscale in the gravitational sector, but 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 oftheoretical physics. The attempts to overcome the many technical difficulties of a rel-ativistic local quantum field theory of gravity are legion. None have been completelysuccessful, but the effort spent on this problem has not been wasted, as several tech-niques originally developed to cope with gravity have ultimately proven essential else-where in field theory (for example, the Faddeev-Popov method [24]). In [5], ’t Hooftand Veltman applied the techniques they had previously used to prove the renormal-izability of spontaneously broken gauge theories [25] to the problem of a scalar fieldcoupled to quantum general relativity. They showed that the one-loop divergences inthis theory can only be cancelled by counterterms that don’t appear in the originalaction. In other words, the theory is nonrenormalizable. To cancel the divergencesto all orders in renormalized perturbation theory would require an infinite number ofcounterterms whose coefficients would require an infinite number of experiments todetermine, thus spoiling the scientific predictability of the theory.

To some degree, part of the nonrenormalizability of the theory can be attributedto the matter to which it is coupled. This is because pure general relativity, withoutmatter, is actually renormalizable at the one-loop level; new counter terms only ariseat the two-loop level. One might hope, therefore, that some special combination ofmatter fields and general relativity might be proven renormalizable, in effect choosinga specific theory or family of theories as unique in this status. This is unfortunatelynot the case. Dirac fields [7], Maxwell fields [6], and Yang-Mills fields [8, 9] all yieldone-loop divergences like the scalar field.

So, the quantum theory we plan to compute with has unavoidable one-loop di-vergences. Since we wish to calculate the Yang-Mills β function to one-loop order,we should start with a one-loop renormalized action for general relativity. In fourdimensions this can be written as the “curvature-squared” action:

SG

=

∫

d4x√−g

−2Λ

16πGN

+R

16πGN

+ α1R2 + α2RµνR

µν

, (2.3)

where Λ is a cosmological constant, g is the determinant of the metric, R is theRicci scalar curvature, and Rab is the Ricci tensor curvature. The allowed “Riemann-squared” term has here been eliminated by rewriting it as a sum of the other twocurvature-squared terms and an unwritten topological density. The case of α1 =α2 = 0 is the standard Einstein-Hilbert action for general relativity. The coefficientsαi have units of action, and can thus be combined with GN and the speed of light c toform a length

√

GNαi/c3 = ℓP

√

αi/~, where ℓP ≈ 1.6×10−35 m is the Planck length.The primary physical effect of the curvature-squared terms is in Yukawa correctionsto the non-relativistic Newton’s law with characteristic lengths given by ℓP

√αi [49],

26

where we have returned to ~ = 1 units. A conservative modern limit on the lengthscale of such forces is approximately 1 mm [50], which is much larger than ℓP. Thelimit αi ≤ 1064 is thus extremely weak.

Since the curvature-squared corrections to the quantum action have such a smalleffect on observable physics, we will ignore them throughout the rest of this thesis.Assuming αi . 1, these terms will only become important at energies very close toand 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 the-ory. This is because all that the nonrenormalizability proofs [5, 6, 7, 8, 9] reallyshow is that perturbation theory in terms of small fluctuations around a free fieldtheory 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 exhibitsan ultraviolet fixed point with only a finite number ultraviolet attractive directions.This condition ensures that there exists a finite-dimensional critical subspace in theinfinite-dimensional space of all allowed coupling constants such that renormaliza-tion group trajectories along this subspace are confined to it. The theory on thecritical surface is then parameterized by a finite number of couplings and has a welldefined continuum limit given by the ultraviolet fixed point. Theories for which theultraviolet 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 noless suited as nonperturbative fundamental theories because of this fact.

An example nontrivial asymptotically safe theory is that of a scalar field φ infive dimensions obeying the symmetry φ → −φ [51]. This theory has no allowedrenormalizable interactions beyond a mass term, but it does exhibit an interactingWilson-Fisher ultraviolet fixed point. This fixed point has only two attractive direc-tions, and is thus asymptotically safe. Another theory with a nontrivial ultravioletfixed point is that of fermions interacting via a four-fermion term in less than fourdimensions [53].

This is now evidence that four dimensional quantum general relativity may beasymptotically safe with two attractive directions given approximately by Newton’sconstant and the cosmological constant [54]. Most of the study of this fixed pointhas been carried out with the exact renormalization group equations for the so-calledeffective average action [48] of Reuter, but the fixed point has also been found withother methods, such as the proper time renormalization group [55]. These flow equa-tions are infinite-dimensional nonlinearly coupled first-order partial differential equa-tions. In practice, the equations must be approximated by truncating to some finite-dimensional subspace. The observed fixed point seems to be stable against changes ofthe truncation and acceptably insensitive to the choice of regulator and gauge. It also

27

appears to persist for realistic matter content [56] and spacetime dimensions rangingfrom 2 + ǫ up to perhaps six [57].

In this chapter, we will be treating quantum general relativity coupled to matteras a perturbatively nonrenormalizable effective field theory with a cutoff. The argu-ments of this section show that perhaps this theory is in fact valid to arbitrarily highscales due to asymptotic safety. If it is also true that only Newton’s constant andthe cosmological constant are essential couplings at the fixed point, then the approx-imation of dropping the curvature-squared terms made in Section 2.1.1 is justifiedeven at high energy scales. Also, if the theory is asymptotically safe, then many ofthe caveats that would need to be stated regarding calculating and interpreting theresults 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 goingto attempt. The application of the method to the calculation of one-loop Yang-Millsβ functions without gravity is textbook fare. Indeed, our use of the method will followvery closely to [58, Section 16.6]. A background expansion is always necessary at somelevel for perturbative gravitational calculations, since these involve metric excitationswhich are small fluctuations about Minkowski spacetime, or some other spacetime,and perturbing about the singular state with vanishing metric would prove a poorapproach. The background method is thus a convenient choice, since it accommodatesthis expansion naturally.

While ultimately equivalent to calculations that could be done with Feynman di-agrams, the background method arranges the calculation differently. Whereas Feyn-man diagrams compute results process-by-process, the background method computesthem species-by-species. This is again convenient for us, since we want to examine theeffect 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 toevaluating a simple Gaussian functional determinant. If we were to attempt calcu-lating the β function to higher-loop accuracy, this method would lose much of itsadvantage over Feynman diagram techniques.

The recipe for the background field method as we will be using it (to extract βfunctions) is as follows:

1. Write down the classical action. Identify the operators whose coefficients arethe renormalizable parameters of interest.

2. Expand each field that contributes to the operators of interest as a quantumfluctuation 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 willintroduce Fedeev-Popov ghosts [24].

28

4. Ignore all terms that are higher than quadratic order in quantum fields. If de-sired, use the classical equations of motion for the background fields to eliminatesome terms. The action is now a background-dependent Gaussian functional ofquantum fields.

5. Using the generating functional, functionally integrate out all of the quantumfields into functional determinates. What remains should be a gauge-invariantfunctional of the classical background fields. This is the exponential of theone-loop effective action.

6. Evaluate the functional determinates as the exponential of a polynomial seriesin the background fields. As usual, interpret the divergent integrals encoun-tered with a convenient scheme, such as minimal subtraction, that introducesdependence on a mass scale E. The only terms in the series that need to beretained are those that correspond to the original operators of interest.

7. Interpret the mass-scale dependent coefficients of the effective action as therunning couplings in the limit that the mass-scale is differentially close to therenormalization scale.

8. Solve for the β functions using dg ≈ βdE/E.

9. Integrate the β 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 dimen-sions which reflects the physical interpretation of the nonrelativistic gravitationalforce law as describing the density of diverging, conserved field lines, commensuratewith its origin in a Gauss law. These units are slightly different from certain otherconventions [14, for example].

Start with the simplest generally covariant actions for a dynamical spacetimemetric, the Einstein-Hilbert action:

S =

∫

ddxLG

=1

K2

∫

ddx√−gR. (2.4)

The overall coefficient is defined as K−2 so that perturbative metric excitations have

a canonically normalized kinetic energy operator when a factor of K is absorbed intothe field definition. When coupled to matter, this action yields the Einstein equation:

Rab − 12gabR =

K2

2Tab, (2.5a)

or equivalently

Rab =K2

2

(

Tab −1

d− 2T a

a

)

. (2.5b)

29

The energy-momentum tensor Tab is defined as

Tab = 2δLmatter

δgab. (2.6)

The factor of 2 in the numerator ensures that tensor so defined — that is, by variationwith respect to the metric, of which there is usually one power in a kinetic matterLagrangian — actually represents the canonically normalized energy-momentum thatis derived as the Noether current of translations—that is, by variations with respectto matter fields, of which there are usually two powers in a kinetic matter Lagrangian.

Consider an approximately Newtonian spacetime as seen by a nonrelativistic ob-server with timelike velocity ua. Contracting Equation (2.5b) with uaub and identi-fying the Newtonian potential ϕ as the field whose gradient gives the acceleration offree-falling particles, we get

∇2ϕ = −K2(d− 3)

2(d− 2)ρ, (2.7)

where ∇2 is the Euclidean Laplacian and ρ = −T 00 is the physical energy density seen

by the observer. We have used R00 = gijR0i0j = −∇2ϕ, where i and j run over spatialindices, uaua = −1, and T a

a ≈ −ρ, since pressures are negligible in the Newtonianlimit. Equation (2.6) gives a force law between two point masses of

|~F (r)| = m1m2

Vol (Sd−2)rd−2

K2(d− 3)

2(d− 2). (2.8)

Of course, we have Vol(Sd−2) = 2π(d−1)/2

Γ((d−1)/2).

Since the physical essence of the 1/rd−2 law is that there is a total flux generatedat some small distance which is conserved as it spreads out over the area of a sphereat a larger distance r, we take the force law to represent the ratio of the areas of the

spheres at these different distances. Thus, the factor K2(d−3)

2(d−2)should be thought of as

the area of a sphere of some special unit radius, K2(d−3)

2(d−2)≡ Vol (Sd−2)ℓd−2

P . We then

define GN = ℓd−2P = M2−d

P as the basic unit for counting square areas. Thus,

K2 = 2

d− 2

d− 3Vol (Sd−2)GN. (2.9)

So, this choice for the d and GN dependence of K gives a nonrelativistic Newton’s

law in d dimensions of ~F (r) = m1m2

Vol (Sd−2ℓP

)

Vol (Sd−2r )

r, where Vol (Snr ) is the volume of an Sn

of radius r.

30

2.3 Setup

With the exception of the overall normalization of the gravitational field, we follow thedefinitions and conventions of [59] for gravitational quantities as closely as possible.Boldface lowercase roman indices are gauge group indices, while normally facedlowercase roman indices are spacetime indices.

The dynamics for a non-Abelian gauge field coupled to a gravity in d spacetimedimensions is given by the sum of the Einstein-Hilbert and Yang-Mills Lagrangians:

L = LG

+ LYM

=1

K2

√−g[R− 2Λ]− 1

4g2YM

√−ggacgbdFaabFa

cd, (2.10)

where K2 is defined in Section 2.2.2, Λ is a cosmological constant, g = det gab, R is

the Ricci scalar, gYM

is the gauge coupling,

Faab ≡ ∆aAa

b −∆bAaa + fabcAb

aAcb (2.11)

is the field strength, Aaa is the gauge field, fabc are the structure constants of the

non-Abelian gauge group G, and ∆a is the derivative operator obeying ∆agbc = 0,i.e. the covariant derivative. Since Fa

ab is antisymmetric under a↔ b, the Christoffelconnections arising from the derivatives in Equation (2.11) will cancel against eachother2. Thus, the covariant derivatives here could be safely replaced with partialderivatives or any other torsion-free derivatives.

The Lagrangian (2.10) is non-polynomial in gab, and the configuration gab = 0is unachievable, so we expand gab about an arbitrary classical background gab withquantum fluctuations hab:

gab = gab + hab. (2.12)

Indices are now raised and lowered with the background metric. We need to re-expressL in terms of hab and gab, up to quadratic order in hab. Higher order terms in hab willonly contribute to higher-loop processes. Once this is done, hab will look like a tensorquantum field that lives in a classical curved spacetime. We also need to expand Aa

a

around classical background configuration aaa:

Aaa = aa

a + Aaa, (2.13)

where aaa obeys the classical equation of motion

DaFaab = 0. (2.14)

F aab is the appropriately named function of classical fields only, and Da = ∇a− iaa

atar .

Here tar is a gauge group generator for the representation r, and ∇ is the torsion-free

derivative operator obeying∇agbc = 0. (2.15)

2We assume that gravity is torsion-free

31

Of course, aaa is in the adjoint representation, so that in Equation (2.14),

[tcG]ab = −ifabc. (2.16)

Performing the expansions (2.12) and (2.13) on the Lagrangian (2.10) is an ex-tended calculation. It is presented in detail for the case of an arbitrary backgroundspacetime in Section 2.4.

2.4 Expanding the Action

In this section, we show the tedious details involved in a particular method for ex-panding both the Einstein-Hilbert action and the Yang-Mills action in terms of per-turbations about arbitrary backgrounds. The results for the gravitational portionfollow closely to results presented in [5], although the calculational method differssignificantly.

2.4.1 Expanding the Non-Polynomial Terms

Given the metric gab, we take the expansion as in Equation (2.12), with gab beingan arbitrary background metric. We will now expand the inverse metric, gab usingthe math fact that given a matrix M = 11 + A, where 11 is the identity, then M−1 =∑∞

n=0 [−A]n. This gives

gab = gab − hab + hach

cb +O(

h3)

. (2.17)

We will also need the expansion of√−g:

√−g =√−g exp

12Tr ln[δa

b + hab ]

.

We now Taylor series the logarithm, evaluate the trace, and Taylor series the expo-nential: √−g =

√−g[1 + 12h + 1

8(h2 − 2habhab)] +O

(

h3)

, (2.18)

where h ≡ haa. To sum up, we define

gab ≡ gab + Iab, (2.19)√−g ≡ √−g[1 + D], (2.20)

where

Iab = −hab + hach

cb +O(

h3)

, (2.21)

D = 12h + 1

8(h2 − 2habhab) +O

(

h3)

. (2.22)

D and Iab are infinite order polynomials (i.e. non-polynomials) in hab. In fact, theyare the source of all the non-polynomial graviton interactions. In practice we willtruncate these polynomials at second order or less in h, but by leaving them as D

32

and Iab for now, we will be able to both keep the calculation more organized and keeptrack of the influence of the non-polynomial terms.

2.4.2 Expanding the Einstein-Hilbert Action

In this section, we expanding the Einstein-Hilbert action in terms of a quantumperturbation about an arbitrary background to second order in the quantum fieldsusing the method of background derivatives.

2.4.2.1 Curvature with Background Derivatives

We make the definitions, as in Wald [59]:

[∆a, ∆b]vc ≡ Rabcdvd, (2.23a)

Rab ≡ Racbc, (2.23b)

R ≡ Rabgab, (2.23c)

where va is an arbitrary vector and ∆a is the derivative operator commensuratewith the metric gab, that is ∆agbc = 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

∆avb = ∂av

b + Γbacv

c, (2.24a)

∆avb = ∂avb − Γcabvc, (2.24b)

∆avbc = ∂avbc − Γdabvdc − Γd

acvbd, (2.24c)

and so forth for arbitrary tensors v, where

Γcab = 1

2gcd(∂agbd + ∂bgad − ∂dgab). (2.25)

Inserting this form of ∆a into Equation (2.23a), we get the standard result

Rabcd = ∂bΓ

dac − ∂aΓ

dbc + Γe

acΓdeb − Γe

bcΓdea. (2.26)

However, we can also use the slightly non-standard expression

∆avb = ∇av

b + Cbacv

c, (2.27)

which relates ∆a to some other derivative ∇a. Manipulations similar to those thatled to Equations (2.25) and (2.26) now lead to

Ccab = 1

2gcd(∇agbd +∇bgad −∇dgab) (2.28)

andRabc

d = Rabcd +∇bC

dac −∇aC

dbc + Ce

acCdeb − Ce

bcCdea, (2.29)

33

where[∇a,∇b]vc ≡ Rabc

dvd. (2.30)

If there is another invertible symmetric tensor gab, perhaps numerically close gab,for which the derivative satisfying ∇agbc = 0 is a known quantity3, then the aboveconstructions can become very helpful. We formally define a tensor hab such that

gab = gab + hab. (2.31)

With this choice of derivative, we have

Ccab = 1

2gcd(∇ahbd +∇bhad −∇dhab). (2.32)

Of course, we want to choose the arbitrary symmetric tensors gab and hab defined inEquation (2.31) to be the background metric and fluctuation, respectively, as definedin Equation (2.12).

2.4.2.2 Some Useful Definitions and Identities

We defineHc

ab ≡ 12(∇ah

cb +∇bh

ca −∇chab). (2.33)

ThengabHc

ab = ∇ahac − 1

2∇ch ≡ Cc (h ≡ gabhab), (2.34)

where Cc is the harmonic gauge factor.

Contracting Equation (2.33) over an upper and lower index gives

Hcac = 1

2∇ah. (2.35)

Another useful relation is

hab∇cHcab+gabHc

dbHdca = −1

4hab(∇2gacgbd−2∇c∇agbd)hcd+

14∇c(2hab∇ahbc−hab∇chab).

(2.36)

2.4.2.3 Expansion of Curvature

Combining Equations (2.19), (2.32), and (2.33) gives

Ccab = Hc

ab + IcdH

dab. (2.37)

3The proof that the derivative operator which annihilates the metric exists and is unique can beextended 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 ∇a suchthat ∇avbc = 0. Thus, we can unambiguously take the ∇a in the above equations to be the operatorthat satisfies ∇agbc = 0.

34

Equation (2.29) then becomes

Rabcd = Rabc

d +∇bHdac −∇aH

dbc

+∇b(Ide He

ac)−∇a(Ide He

bc) + HeacH

deb −He

bcHdea +O

(

h3)

. (2.38)

Applying Equation (2.23b) to Equation (2.38) gives

Rab = Rab +∇cHcab −∇aH

ccb

+∇c(IceH

eab)−∇a(I

ceH

ecb) + Hc

abHece −Hc

ebHeca +O

(

h3)

, (2.39)

where Rab is the appropriately named function of gab. Using Equations (2.23c), (2.19),(2.20), and (2.39) we find

√−gR =√−g(1 + D)(gab + Iab)Rab

=√−g

R +[

DR + IabRab +∇c(gabHc

ab − gcdHeed)]

+[

DIabRab + Dgab∇cHcab + Iab∇cH

cab −D∇bHc

cb − Iab∇aHccb

+gab(HcabH

ece −Hc

ebHeca) +∇c(g

abIceH

eab − gacIb

eHeab)]

+O(

h3)

.

By using Equations (2.21), (2.22), (2.34), (2.35), and (2.36) and pulling the totaldivergences to the outside, this becomes

√−gR =√−g

R −Gabhab − 14hab

[

12(4Gab + gabR)gcd − (4Gac + gacR)gbd

−gacgbd∇2 + gabgcd∇2 + 2gac∇d∇b − 2gcd∇a∇b]

hcd

+ total divergence +O(

h3)

, (2.40)

where we’ve made the standard definition for the Einstein tensor:

Gab ≡ Rab − 12gabR. (2.41)

General spacetimes allow for a cosmological constant. We include this in thegravitational Lagrangian by adding to Equation (2.40) the term

−2√−gΛ =

√−g

−2Λ− gabΛhab

−14hab

[

12(4gabΛ− 2gabΛ)gcd − (4gacΛ− 2gacΛ)gbd

]

hcd

+O(

h3)

.

(2.42)

The final line of Equation (2.42) has been arranged in a form that makes obvious howit should be added to Equation (2.40). The final result is

K2L

G=√−g[R− 2Λ] =

√−g

[R − 2Λ]− [Gab + gabΛ]hab

14hab

[

12

(

4[Gab + gabΛ] + gab[R− 2Λ])

gcd − (4[Gac + gacΛ] + gac[R− 2Λ]) gbd

−gacgbd∇2 + gabgcd∇2 + 2gac∇d∇b − 2gcd∇a∇b]

hcd

+ total divergence +O(

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 thatapplying the variation of Equation (2.13) to the field strength given in Equation (2.11)results in

Faab = F a

ab + DaAab −DbA

aa + fabcAb

aAcb , (2.44)

where F aab is the appropriately named function of classical fields only and Da =

∇a − iaaat

ar . Using only the symmetry of the metric factors and applying Equation

(2.44), we find

gacgbdFaabFa

cd = gacgbd[F aabF

acd + 2F a

ab(DcAad −DdA

ac )

+ (DaAab −DbA

aa)(DcA

ad −DdA

ac ) + 2fabcF a

abAbc Ac

d +O(

A3, A4)

]. (2.45)

However, using the antisymmetry of the field strengths, applying Equations (2.19)and (2.22) of Sections 2.4.1, and inserting Equation (2.45) gives

√−ggacgbdFaabFa

cd

=√−g

[

F aabF

aab + 4F aabD

aAab + 2DaAab (D

aAab −DbAaa) + 2fabcF aabA

baAcb]

+[

−2(F aabF

aadhbd − 1

4F a

abFaabh)− 4(hbd − 1

4hgbd)F a

ab(DaAa

d −DdAaa)]

+[

2(F aabF

aadhbch

cd − 1

4F a

abFaabhcdhcd)− (F a

abFaadhhb

d − 14F a

abFaabh2)

−18F a

abFaab(h2 − 2hcdhcd) + F a

abFacdh

achbd]

+O(

fields3)

. (2.46)

This can be simplified by completing total derivatives and using

Tab ≡ −2√−g

δLYM0

δgab=

1

g2YM

(F aacF

acb − 1

4gabF

acdF acd), (2.47)

where LYM0 is the appropriately named function of classical fields only. Equation

(2.46) becomes

− 4g2YML

YM=√−ggacgbdFa

abFacd

=√−g

[F aabF

aab]− 4[DaF aabA

ab]− 2[AabD

2Aab − AabDaDbAaa − fabcF a

abAbaAcb]

− 2g2YM

[T abhab]− 4[

F acb(DcAaa −DaAa

c )− 12gabF a

cdDcAad

]

hab

+hab

[

2g2YM

T acgbd − g2YM

T abgcd − 18F a

efFaef(gabgcd − 2gacgbd) + F aacF abd

]

hcd

+ total divergence +O(

fields3)

. (2.48)

Before writing the fully expanded action, the contributions to the action due to gauge-fixing need to be considered.

2.5 Gauge-Fixing

We regard gab as fixed with respect to diffeomorphisms and aaa as fixed with respect to

the gauge group G. We attribute the variations of gab andAaa to transformations of hab

36

and Aaa, respectively. Thus, we take the induced infinitesimal gauge transformations

due to diffeomorphisms to be

δhab = ∇aηb +∇bηa +∇aηchcb +∇bη

chca + ηc∇chab, (2.49a)

δAaa = Aa

c∇aηc + ηc∇cA

aa, (2.49b)

δaaa = aa

c∇aηc + ηc∇ca

aa. (2.49c)

Acting with G, we get

δAaa = Daα

a + fabcAbaαc, (2.50a)

δhab = 0. (2.50b)

We need to fix the gauges on hab and Aaa. We take the background-covariant

gauge-fixing conditions

Ca(h, A) ≡ Ca(h)− ξK2

g2YM

F aabAab = 0, (2.51)

Ga(A) ≡ DaAaa = 0, (2.52)

whereCa(h) ≡ ∇bh

ab − 12∇ah (h ≡ ha

a). (2.53)

By using the Faddeev-Popov [24] method and choosing Feynman-’t Hooft gauge fac-tors, these each add a term to the Lagrangian

∆Lgf:h =− 1

2K2

√−gCaCa

=− 1

2K2

√−g(

CaCa − 2ξ K2

g2YM

F aabA

abCa + ξ2 K4

g4YM

F aabA

abF bacA

bc)

, (2.54)

∆Lgf:A =− 1

2g2YM

√−gGaGa. (2.55)

respectively. Equation (2.51) is similar to an Rξ gauge [25], which is here engineeredto cancel unpleasant graviton-gluon cross-terms that will appear later. We will even-tually find that a convenient choice of gauge is ξ = 1, whereas ξ = 0 reproduces thetraditional harmonic gauge.

Equation (2.54) can be expanded using

− 12CaCa = −1

4hab[−1

2gabgcd∇2 + gcd∇a∇b + gab∇c∇d − 2gac∇b∇d]hcd

+ total divergence. (2.56)

Likewise, Equation (2.55) becomes

−12GaGa = −1

2DaA

aaDbAab

= 12AaaDaDbA

ab + total divergence. (2.57)

37

Of course, these gauge-fixing terms appear along with the Faddeev-Popov ghostLagrangians through the application of Equations (2.49) to Equation (2.51) and Equa-tions (2.50) to Equation (2.52). The form of the ghost Lagrangian in a non-Abeliangauge theory is a standard result, and it is only slightly altered in a curved backgroundspacetime. We get

∆LFP:A = −√−gαa δGa(A + δA)

δαbαb

= −√−gαa[

δabD2 − fabc(DaAca + Ac

aDa)]

αb. (2.58)

Ghosts also arise from the gauge-fixing of general covariance. Since both aaa and Aa

a

transform as vectors, so does the combination F aabA

ab. That is,

δ(F aabA

ab) = F abcA

ac∇aηb + ηb∇b(F

aacA

ac). (2.59)

The ghost Lagrangian is then given by

∆LFP:h = −√−gηa δCa(h + δh, A + δA)

δηbηb

= −√−gηa[

gab∇2 − [∇a,∇b] + hab∇2 + Cb∇a +∇bCa − hca[∇c,∇b]

+2gadHdbc∇c − ξ K2

g2YM

(F abcA

ac∇a +∇b(FaacA

ac))]

ηb. (2.60)

To second order in quantum fields, we have:

∆LFP:h =√−gηa

[

−gab∇2 + [∇a,∇b] +O (h) +O (A)]

ηb, (2.61)

∆LFP:A =√−gαa

[

−δabD2 +O (A)]

αb. (2.62)

2.6 Combining the Pieces

The classical fields gab and aaa are governed by classical version of Equation (2.10),

L0 = LG0 + L

YM0 =1

K2

√−g[R − 2Λ]− 1

4g2YM

√−gF aabF

aab. (2.63)

By varying this action with respect to these fields, we get the Euler-Lagrange equa-tions that govern their dynamics:

Gab + gabΛ = − K2

√−g

δLYM0

δgab≡ K2

2Tab, (2.64)

andDaF

aab = 0. (2.65)

We will enforce Equation (2.65) on aaa, but we will not enforce Equation (2.64). In

this way we can study the behavior of an arbitrary gauge field in any spacetime back-

38

ground, but we ignore the warping of spacetime by the background gauge field. Thatis, we’re making a “test field” approximation. In contrast, if we were to enforce bothEquations (2.64) and (2.65) and then restrict attention to, for example, Minkowskispace, we would be forced to consider only gauge field configurations with vanish-ing stress-energy. The most satisfying approach would be to force gab and aa

a to bearbitrary simultaneous solutions of Equations (2.64) and (2.65), but the necessarycalculational techniques required for analysis subsequent to this point have provenelusive.

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

L = L0 + LO(h) + LO(h2) + LO(A2) + LO(Ah) + . . . , (2.66)

where

L0 =1

K2

√−g[R− 2Λ]− 1

4g2YM

√−gF aabF

aab, (2.67a)

LO(h) =− 1

K2

√−g[

Gab + gabΛ− K2

2T ab]

hab, (2.67b)

LO(h2) =− 1

4K2

√−ghab

[

K2

√−gL0(

12gabgcd − gacgbd) + K

2

g2YM

F aacF abd

+ 2(Gab + gabΛ− K2

2T ab)gcd − 4(Gac + gacΛ− K

2

2T ac)gbd

−gacgbd∇2 + gabgcd∇2 + 2gac∇d∇b − 2gcd∇a∇b]

hcd, (2.67c)

LO(A2) =− 1

2g2YM

√−g[−AabD

2Aab + AaaD

bDaAab + Aa

afabcF cabAb

b ], (2.67d)

LO(hA) =1

2g2YM

√−gF acd

[

−gabDcAad + 2gad(DcAab −DbAac)]

hab. (2.67e)

Note that LO(A) was eliminated by the equation of motion (2.65).

Equation (2.67e) can be brought into a more symmetric form. By integrating byparts on the first and third terms of Equation (2.67e), enforcing Equation (2.65), anddropping total divergences, we get

LhA = − 1

g2YM

√−ghab

[

gbcF aadDd + DaF abc]

Aac −

1

g2YM

√−gCaFaabAa

b . (2.68)

Repeating the procedure on the first term of Equation (2.68) gives the similar form,

LhA = − 1

g2YM

√−gAac

[

−gcbF aadDd −DaF acb]

hab −1

g2YM

√−gCaFaabAa

b . (2.69)

39

We now symmetrize Equations (2.68) and (2.69) to get

LhA = − 1

2g2YM

√−g

hab

[

DaF abc + gbcF aadDd

]

Aac

+Aac

[

−DaF acb − gcbF aadDd

]

hab + 2CaFaabAa

b

. (2.70)

Equation (2.54) will give contributions to LO(h2), LO(hA), and LO(A2), whereasEquation (2.55) will only contribute to LO(A2). Adding the gauge fixing contributionsto Equation (2.67c) using Equation (2.56) we get

LO(h2) + ∆Lgf:hO(h2)

= − 1

4K2

√−ghab

[

(gacgbd − 12gabgcd)(−∇2 − K2

√−gL0)− 2gbd[∇a,∇c] + K2

g2YM

F aacF abd

+2(Gab + gabΛ− K2

2T ab)gcd − 4(Gac + gacΛ− K2

2T ac)gbd

]

hcd, (2.71)

where we’ve used the fact that

hab

(

gab∇c∇d − gcd∇a∇b)

hcd = ∇a(

h∇bhab − hab∇bh)

−∇ch∇dhcd +∇ahab∇bh

= ∇a(

h∇bhab − hab∇bh)

(2.72)

is a total divergence, which we’ve dropped. Adding the gauge fixing contributions toEquation (2.67d) using Equation (2.57) we get

LO(A2) + ∆Lgf:A + ∆Lgf:hO(A2)

= − 1

2g2YM

√−gAaa

[

−gabδab[D2]− δab[∇a,∇b] + 2fabcF cab + ξ2 K2

g2YM

F aacF

bbc]

Abb ,

(2.73)

where we’ve used Equation (2.16) and

[Da, Db] = [∇a,∇b]− iF aabt

ar . (2.74)

Finally we can add the gauge fixing contributions to Equation (2.70).

LO(hA) + ∆Lgf:hO(Ah)

= − 1

2g2YM

√−g

hab

[

DaF abc + gbcF aadDd

]

Aac + Aa

c

[

−DaF acb − gcbF aadDd

]

hab

+2(1− ξ)CaFaabAa

b

. (2.75)

Bringing together Equations (2.67a), (2.67b), (2.71), (2.75), (2.73), (2.58), and (2.60)we get

L ≈ L0 + Lh + Lh2 + LhA + LA2 + Lη2 + Lα2 + . . . , (2.76)

40

where

L0 =1

K2

√−g[R− 2Λ]− 1

4g2YM

√−gF aabF

aab, (2.77a)

Lh =− 1

K2

√−g[

Gab + gabΛ− K2

2T ab]

hab, (2.77b)

Lh2 =− 1

4K2

√−ghab

[

(gacgbd − 12gabgcd)(−∇2 − K2√−g

L0)

+ K2

g2YM

F aacF abd − 2gbd[∇a,∇c]

+2(Gab + gabΛ− K2

2T ab)gcd − 4(Gac + gacΛ− K2

2T ac)gbd

]

hcd, (2.77c)

LhA =− 1

2g2YM

√−g

hab

[

DaF abc + gbcF aadDd

]

Aac

+Aac

[

−DaF acb − gcbF aadDd

]

hab + 2(1− ξ)CaFaabAa

b

, (2.77d)

LA2 =− 1

2g2YM

√−gAaa

[

−gabδab[D2]− δab[∇a,∇b]

+2fabcF cab + ξ2 K2

g2YM

F aacF

bbc]

Abb , (2.77e)

Lα2 =√−gαa

[

−δabD2]

αb, (2.77f)

Lη2 =√−gηa

[

−gab∇2 + [∇a,∇b]]

ηb. (2.77g)

These expressions can be evaluated for Minkowski space by taking g = −1, ∇ → ∂,and Rab = R = Λ = 0. The ellipses in Equation (2.76) indicate terms of higherthan quadratic order in quantum fields. Equation (2.77g) will only contribute onlyto the renormalization of K (or simply to an infinite constant in the effective action,in the Minkowski space limit), so it will be ignored from here in. Physically, theterms in Equations (2.77) that are proportional to background field strengths repre-sent magnetic-moment-type interactions of the dynamical fields with the backgroundfields.

2.7 Compiling the Superfield

The term LhA deserves special attention. It is equivalent to

δ

δgab

[

δLδaa

c

Aac

]

hab =δ

δaac

[

δLδgab

hab

]

Aac =

1

2

δ

δgab

[

δLδaa

c

Aac

]

hab +δ

δaac

[

δLδgab

hab

]

Aac

.

(2.78)If Equation (2.77d) were in this more symmetric form, we could rewrite the quadratic

terms in L by using the superoperator VΦ ≡ 12

δ2Lδφδφ

, where φ ≡

g, a

†is a classical

41

superfield. That is, we wish to write

Lh2 + LhA + LA2 = LΦ2 = 12Φ† δ2L

δφδφΦ, (2.79)

where Φ ≡

h, A

†is the superfield of quantum fluctuations.

Equation (2.77d) is the sum of two parts. One of these has differential operatorsacting only on hab while the other part has differential operators acting only on Aa

a.If we now choose ξ = 1, then these two parts are conjugate to each other. This isexactly what we want for the “off-diagonal” terms in VΦ. With this choice of gauge,then, we have

LΦ2 =√−gΦ†VΦΦ =

√−g

hab

Aae

V abcdh V abbf

+

V aecd− V abef

A

hcd

Abf

, (2.80)

where

V abcdh =− 1

4K2

[

Tabcd(−∇2 − K2√−g

L0) + K2

2g2YM

(

F aacF abd + F aadF abc)

− 1

2

(

gbd[∇a,∇c] + gbc[∇a,∇d] + gad[∇b,∇c] + gac[∇b,∇d])

− (Gac + gacΛ− K2

2T ac)gbd − (Gbc + gbcΛ− K2

2T bc)gad

− (Gad + gadΛ− K2

2T ad)gbc − (Gbd + gbdΛ− K2

2T bd)gac

+(Gab + gabΛ− K2

2T ab)gcd + (Gcd + gcdΛ− K

2

2T cd)gab

]

, (2.81a)

V ababA =− 1

2g2YM

[

−gabδab[D2]− δab[∇a,∇b] + 2fabcF cab + K2

g2YM

F aacF

bbc]

, (2.81b)

V abac+ =− 1

4g2YM

[

gbcF aadDd + gacF abdDd + DaF abc + DbF aac]

, (2.81c)

V acab− = +

1

4g2YM

[

gcbF aadDd + gcaF abdDd + DaF acb + DbF aca]

. (2.81d)

We have defined the tensor

Tabcd ≡ 1

2(gacgbd + gadgbc − gabgcd). (2.82)

We will later need the inverse of T in d dimensions, which can be determined from

Tabef

(

Tefcd + 1

2d−4d−2

gefgcd)

= Iabcd, (2.83)

where Iabcd is the identity on the 12d(d+1) dimensional space of symmetric 2-tensors4.

4The projector onto the space of traceless tensors in 4 dimensions is PT = 12 (I + T), while the

projector onto the space of “pure trace” tensors is PS = 12 (I − T). In a spacetime of dimension d,

42

Finally, we can rewrite the source term (2.77b) as

Lh = LΦ =√−g

12T ab

0

·

hab

Aaa

≡ √−gJ · Φ. (2.84)

Thus we haveL ≈ L0 + LΦ + LΦ2 + Lα2 . (2.85)

2.8 Renormalization

Beginning from this section, we specialize all calculations to Minkowski space.

We define the partition function, Z, as

Z ≡∫

DgDAeiR

ddxL[g,A] =

∫

DgDAeiS[g,A]

=

∫

DhDAeiS[g,h,a,A] ≡ eiW[g,a], (2.86)

where, in the last line, we have defined the effective action, W, as a functional of theclassical fields. To evaluate W, we first gauge fix and then manipulate terms as inSections 2.3–2.7:

eiW[g,a] =

∫

DhDADηDαeiS[g,h,a,A]+∆Sgf :h+∆Sgf :A+∆SF P :h+∆SF P :A

≈∫

DhDADαeiS0[g,a]+Sh+Sh2+ShA+SA2+Sα2

=

∫

DΦDαeiS0[g,a]+SΦ+SΦ2+Sα2.

We can eliminate the SΦ term by writing

LΦ + LΦ2 = J · Φ + Φ†VΦΦ =[

Φ + 12V −1

Φ J]†

VΦ

[

Φ + 12V −1

Φ J]

− 14J †V −1

Φ J . (2.87)

Then we shift Φ→ Φ′ = Φ + 12V −1

Φ J , which has no effect on DΦ, that is DΦ = DΦ′.Renaming Φ′ to Φ, this leaves

eiW[g,a] ≈∫

DΦDαein

S0[g,a]−14

R

ddxJ †V −1Φ J+SΦ2+Sα2

o

.

In this expression, both eiS0 and the J -term are constants and the Gaussian

the T appearing in these projector equations is given by Tabcd = Iabcd − 2dgabgcd. The T appearing

in Equation (2.81a), however, is given properly by Equation (2.82) in any spacetime dimension. Theidentification of these objects for d = 4 is a coincidence.

43

integrals over Φ and α can be evaluated as functional determinants:

eiW[g,a] ≈ exp

iS0 −i

4

∫

ddxJ †V −1Φ J

det[VΦ]−12 det[Vα]+1

= exp

iS0 −i

4

∫

ddxJ †V −1Φ J − 1

2Tr ln[VΦ] + Tr ln[Vα]

, (2.88)

where Equation (2.77f) tells us that Vα = −D2. Thus,

iW[g, a] ≈ iS0[g, a]− i

4

∫

ddxJ †V −1Φ J − 1

2Tr ln[VΦ] + Tr ln[Vα]. (2.89)

Note that each term in Equation (2.76) is invariant under both diffeomorphisms anda formal gauge transformation of the background gauge field where Aa

a is treated asadjoint matter. Since each term of Equation (2.76) is a gauge invariant functional ofthe classical field, so must be each term of W. Thus each term must be proportionalto∫

ddx√−gF a

abFaab, at least to first order.

We begin evaluating Equation (2.89) by turning our attention to the VΦ term.First we extract out an overall normalization constant from the superoperator in thefollowing way:

VΦ =

V abcdh V abbf

+

V aecd− V abef

A

=

g2YM

2K2 Tabgh 0

0 1

∂2

2g2YM

Ighcd + Mghcdh Oghbf

+

Oaecd− δabgef + Mabef

A + Nabef

, (2.90)

where

Mabcdh =−K

2Iabcd(−∂2)−1L0 + K2

g2YM

(

Tab

ef + 12

d−4d−2

gabgef

)

(−∂2)−1[F aecF afd]

+ 2K2(

Tab

ef + 12

d−4d−2

gabgef

)

(−∂2)−1[T ecgfd − 14T efgcd − 1

4T cdgef ], (2.91a)

MababA =− gabδab

(

1 + (−∂2)−1D2)

+ 2fabc(−∂2)−1F cab, (2.91b)

Nabab = K2

g2YM

(−∂2)−1[F aacF

bbc], (2.91c)

Oabac+ = 2K2

g2YM

(−∂2)−1[gfcF aedDd + DeF afc](

Tab

ef + 12

d−4d−2

gabgef

)

, (2.91d)

Oacab− =(−∂2)−1[−gfcF aedDd −DeF acf ]Iab

ef . (2.91e)

44

Observe that each term in Mh and N is quadratic in F aab. So,

−12Tr ln[VΦ] =− 1

2Tr ln[N]− 1

2Tr ln

[

I +

Mh O+

O− MA + N

]

≈− 12Tr ln[N]− 1

2Tr

[

Mh O+

O− MA + N

−12

M2h + O+O− MhO+ + O+(MA + N)

O−Mh + (MA + N)O− (MA + N)2 + O−O+

+ . . .

]

.

(2.92)

We can drop the Tr ln[N] term, as it simply represents an infinite constant. Workingto the order of the fields that appear in the the classical action, we can also drop theM2

h , MAN , N2, MAO, and NO terms. The ellipses in Equation (2.92) indicate thatall the excluded terms are also of higher order in the classical fields. Thus we have,

−12Tr ln[VΦ] ≈ −1

2Tr

[

Mh O+

O− MA + N

− 1

2

O+O− 00 M2

A + O−O+

]

= −12Tr[Mh − 1

2O+O−]− 1

2Tr[MA − 1

2M2

A + N − 12O−O+]

= −12Tr[Mh − O+O−]− 1

2Tr[MA − 1

2M2

A + N ], (2.93)

where the last line follows from trace cyclicity.

The MA and M2A terms from the second trace, along with the S0 and Vα terms in

Equation (2.89), are exactly what would be found for a Yang-Mills theory evaluatedto the one-loop level in the absence of gravitation. Evaluating these terms for d = 4and inserting the form of S0 therefore gives

iW[g, a] ≈ −12Tr[Mh − O+O−]− 1

2Tr[N ]− i

4

∫

ddxJ †V −1Φ J

− i

4

∫

ddx

[

1

g2YM

+b0

(4π)2ln

E2

M2

]

F aabF

aab, (2.94)

where b0 = 113C2(G), E is the background energy scale, and M is a renormalization

scale at which gYM

(E)|E=M = gYM

is imposed. Ignoring gravity, this would lead toβ(g

YM) ≈ − b0

(4π)2g3

YM.

We now return to evaluating the remaining terms in the effective action. We’llstart with the J -term. Expanding VΦ as in Equation (2.90), we get

− i

4

∫

ddxJ †V −1Φ J ≈ −

i

4

∫

ddxJ †[

I−

Mh O+

O− MA + N

+

M2h + O+O− MhO+ + O+(MA + N)

O−Mh + (MA + N)O− (MA + N)2 + O−O+

+ . . .

]

×

−4K2T−1(−∂2)−1 00 −2g2

YM(−∂2)−1

J . (2.95)

45

&%'$

vp→

k →

k →K

2

g2

YM

Figure 2-2: The schematic Feynman diagram represented by the functional trace−1

2Tr[Mh]. A momentum p circulates in a virtual graviton loop coupled to external

gluons of momentum k.

Since J =

12T ab, 0

†, we can see that all terms in Equation (2.95) have four or

more powers of F aab. Since we are only working to the order of fields appearing in L0,

none of these terms make any contribution to the effective action.

The final terms, −12Tr[Mh −O+O−]− 1

2Tr[N ], are each evaluated in turn in Sub-

section 2.8.1, using an ultraviolet cutoff, ΛUV , in d dimensions.

2.8.1 Computation of Functional Determinants

The term−12Tr[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 evaluatedas follows:

−12Tr[Mh]

=

∫

ddxIabcdK2

2

(−∂2)−1IabcdL0 − 1

g2YM

(

Tab

ef + 12

d−4d−2

gabgef

)

(−∂2)−1[F aecF afd]

− 2(

Tab

ef + 12

d−4d−2

gabgef

)

(−∂2)−1[T ecgfd − 14T efgcd − 1

4T cdgef ]

=− 1

4g2YM

K2

2

(

12d(d + 1)− 4 + 2d−4

d−2− (d− 4)(d + 1)

)

∫

ddx

(−∂2)−1[F aabF

aab]

(2.96)

where we’ve used the fact that

gabTab = − 1

4g2YM

(d− 4)F aabF

aab = (d− 4)L0. (2.97)

46

p→

k →

k → ⋆

K2

g2

YM

Figure 2-3: The schematic Feynman diagram represented by the functional trace−1

2Tr[N ]. A momentum p circulates in a virtual gluon loop coupled to external

gluons of momentum k.

We now evaluate the integral as

∫

ddx

(−∂2)−1[F aabF

aab]

=

∫

ddk

(2π)dF a

abFaab

∫

ddp

(2π)d

1

p2

=

∫

ddk

(2π)dF a

abFaab

∫

dpEddΩ

(2π)dpd−1

E

−i

p2E

=

∫

ddk

(2π)dF a

abFaab −2iπd/2

(2π)dΓ(d/2)

∫ ΛUV

E

dpEpd−3

E

=−2i

(4π)d/2Γ(d/2)

(

Λd−2UV− Ed−2

d− 2

)∫

ddxF aabF

aab. (2.98)

So, finally

− 12Tr[Mh] = −i

1

4

[

−K2(

12d(d + 1)− 4− (d− 4)(d + 1− 2

d−2))

g2YM

(4π)d/2Γ(d/2)

Λd−2UV−Ed−2

d− 2

]

×∫

ddxF aabF

aab. (2.99)

The term −12Tr[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→

k →

k →N

K

gYM

HK

gYM

Figure 2-4: The schematic Feynman diagram represented by the functional trace12Tr[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

−12Tr[N ] = − K2

2g2YM

∫

ddxδabgab

(−∂2)−1[

F aacF

bbc]

= −i1

4

[ −4K2

g2YM

(4π)d/2Γ(d/2)

Λd−2UV− Ed−2

d− 2

]∫

ddxF aabF

aab. (2.100)

The trace 12Tr[O+O−] represents a process where the external gauge field emits

and reabsorbs a virtual graviton, as in Figure 2-4. This becomes

12Tr[O+O−] =

1

2

∫

ddx

Oabac+ Oa

−cab

=K2

g2YM

∫

ddx

(−∂2)−1[gfcF aedDd + DeF afc](

Tab

ef + 12

d−4d−2

gabgef

)

×(−∂2)−1[−gicFahgD

g −DhFaci]Iab

hi

≈ K2

2g2YM

∫

ddk

(2π)d

ddp

(2π)d

−i

p2[gdeF acf(k)(−p− k)f + kcF ade(k)]

×(

Tab

cd + 12

d−4d−2

gabgcd

) −i

(p + k)2[gbeF

aag(−k)pg + kaF

aeb(−k)]

.

In passing to momentum space, we dropped the gauge field part of the covariantderivatives, as these will only produce terms which are of high powers in the classicalfield. To the order of fields in which we are working, the classical equation of motion

48

for aaa, Equation (2.14), becomes kaF

aab(k) = 0. Applying this, we get

12Tr[O+O−] ≈ K

2

2g2YM

∫

ddk

(2π)d

ddp

(2π)d

F aab(k)F a

cd(−k)1

p2(p + k)2

×[(

d + 1− 2

d− 2

)

gacpbpd + k2gacgbd

]

.

The k2 term produces a higher derivative term in the effective action whose propertieswe are not interested in. Ignoring it, we get

12Tr[O+O−] ≈ −i

1

4

−K2(

−4− (d− 4) 4d(d−2)

)

g2YM

(4π)d/2Γ(d/2)

Λd−2UV−Ed−2

d− 2

∫

ddxF aabF

aab.

(2.101)

2.8.2 Extracting the β Function

Combining Equations (2.99), (2.100), and (2.101) of Subsection 2.8.1 together gives

−12Tr[Mh −O+O−]− 1

2Tr[N ] ≈ −i

1

4

[ −K2C(d)

g2YM

(4π)d/2Γ(d/2)

Λd−2UV−Ed−2

d− 2

]∫

ddxF aabF

aab,

(2.102)where

C(d) = 12d(d + 1)− 4− 1

d(d− 4)(d− 1)(d + 2) (2.103)

is a function that is positive-valued for all integers 1 < d < 8 and negative-valuedfor d = 1 and integers d ≥ 8. Note that C(4) = 6, which is the maximum for realpositive d.

The ultraviolet divergence in Equation (2.102) is regulated by counter terms —which we’ve not been writing — whose values are determined at energy scale E bya renormalization condition at scale M . Subtracting divergences by including theseterms gives:

−12Tr[Mh−O+O−]− 1

2Tr[N ] ≈ −i

1

4

[ −K2C(d)

g2YM

(4π)d/2Γ(d/2)

Md−2 −Ed−2

d− 2

]∫

ddxF aabF

aab.

(2.104)Evaluating this for d = 4 gives

W[g, a] ≈ −1

4

∫

d4x

[

1

g2YM

+K2

g2YM

3

(4π)2(E2 −M2) +

b0

(4π)2ln

E2

M2

]

F aabF

aab. (2.105)

Thus, when E is differentially close to M — and only then — we define the one-looprunning coupling constant at scale E by

1

gYM

(E)2=

1

gYM

(M)2+

K2

gYM

(M)2

3

(4π)2(E2 −M2) +

b0

(4π)2ln

E2

M2. (2.106)

49

Under these conditions we find

d

(

1

g2YM

)

=K

2

g2YM

3

(4π)2(2EdE) +

b0

(4π)2

2dE

E, (2.107)

and thus,

β(gYM

, E) ≡ dgYM

d lnE= −1

2g3

YME

d(1/g2YM

)

dE= −3

K2

(4π)2g

YME2 − b0

(4π)2g3

YM. (2.108)

Using the definition K2 = 16π/M2P, we find that the unknown coefficient of Equation

(2.2) is now determined to be a0 = − 3π≈ 0.95.

The form of the running coupling can now be found by integrating the β function.

Equation (2.107) can be integrated using the integrating factor exp

−3 K2

(4π)2E2

.

This yields

1

gYM

(E)2=

1

gYM

(M)2exp

3 K2

(4π)2(E2 −M2)

+2b0

(4π)2

∫ E

M

dk

kexp

3 K2

(4π)2(E2 − k2)

.

(2.109)Notice that for E > M , g

YM(E) will always be less than the value it would have had

in the absence of gravitation. Indeed, as E → ∞, gYM

(E) → 0 independent of thevalues of b0 and g

YM(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 addinggravity. Remember, however, that any discussion of the theory at or above MP isdubious at best.

It is interesting that quantum gravity perturbations can cause gauge couplings torun even in theories that exhibited exact conformal invariance, and thus vanishingβ function, before gravity was added. Two notable examples in four dimensions arepure U(1) electromagnetism and N = 4 super-Yang-Mills [60, 61, 62]. For thesetheories, the exponential integral in Equation (2.109) has zero coefficient, so we areleft with

α(E) ≡ gYM

(E)2

4π= α(0) exp

−3 K2

(4π)2E2

. (2.110)

The couplings in such theories run down from their infrared values as Gaussians ofwidth

√

π/6MP, corresponding to a 10% reduction in α 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 notexpand our new scalar and spinor fields as quantum perturbations about a classical

50

background. Therefore, every term in the Lagrangian containing these fields is ofquadratic order or more in quantum fields. Thus, adding these fields doesn’t changeany of the terms already calculated in Equation (2.76). Each new field simply suppliesone new term which is quadratic in the matter field and zeroth order in all the otherfields. These terms have no effect on the functional integration over hab and Aa

a asdescribed in Section 2.8. That is, adding matter to our previous discussion will notchange 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 absenceof gravity. Namely, they add to the logarithmic running. So, adding matter to ourprevious discussion only changes the coefficient b0 in the β function and has no effecton the gravitational term. Moreover, the matter contribution is exactly what it wasin the absence of gravity.

The addition of new gauge fields is slightly more complicated. Each new fieldcontributes a term to L0 and corresponding terms to Lh (via T ab) and Lh2. Likewise,the ith gauge field brings in a LA2

iterm and an LhAi

cross term. Of course, gauge-fixing each new symmetry brings in a new gauge-fixing term and a correspondingghost field. We also need to augment the gauge condition of Equation (2.51) withan additional term for each new gauge field. This adds additional couplings of thegravitational ghosts to the new gauge fields, but these terms will drop out to secondorder in the quantum fields. Each LA2

iand LhAi

receives a similar contribution fromthe cross terms generated by gauge-fixing. Gauge-fixing in this way also generatesnew cross terms between gauge fields. With so many graviton-gluon and gluon-gluoncross terms, we need to enlarge the definition of the superfield to include the gravitonand each of the new gauge fields.

The net effect of all this is that to second order in the fields, we get

L ≈ L0 + LΦ + LΦ2 +N∑

i=1

Lα2i+ . . . , (2.111)

where N is the number of gauge symmetries,

L0 =

N∑

i=1

− 1

4g2i

F a(i)abF

aab(i) , (2.112a)

LΦ =J · Φ =

12T ab

0...0

·

hab

Aa(1)a...

Aa(N)a

, (2.112b)

LΦ2 =

hab

Aa(1)e...

Aa(N)e

†

V abcdh V abbf

+1 · · · V abbf+N

V aecd−1 V abef

A1· · · V abef

1N...

.... . .

...

V aecd−N V abef

N1 · · · V abefAN

hcd

Ab(1)f...

Ab(N)f

, (2.112c)

Lα2i

=αia[

−δabD2]

αbi , (2.112d)

51

and

V abcdh = − 1

4K2

[

Tabcd(−∂2 −K

2L0) +N∑

i=1

K2

2g2i(F aac

(i) F abd(i) + F aad

(i) F abc(i) )

+ K2

2(T acgbd + T adgbc − T abgcd + T bcgad + T bdgac − T cdgab)

]

, (2.113a)

V ababAi

=− 1

2g2i

[

−gabδab[D2] + 2fabc(i) F cab

(i) + K2

g2iF aa

(i) cF bbc

(i)

]

, (2.113b)

V abac+i =− 1

4g2i

[

gbcF aad(i) Dd + gacF abd

(i) Dd + DaF abc(i) + DbF aac

(i)

]

, (2.113c)

V acab−i =

1

4g2i

[

gcbF aad(i) Dd + gcaF abd

(i) Dd + DaF acb(i) + DbF aca

(i)

]

, (2.113d)

V ababij =

K2

2g2i g

2j

F aa(i)cF

bcb(j) . (2.113e)

Equations (2.112) and (2.113) are written for a Minkowski space background. Theenergy-momentum tensor, T ab, now refers to the total energy-momentum tensor ofthe background fields,

Tab ≡N∑

i=1

1

g2i

(F a(i)acF

a(i)b

c − 14gabF

acd(i) F a

(i)cd). (2.114)

The covariant derivatives appearing in the above equations include a term for eachgauge symmetry. However, the ghosts and gauge fields that they act on are all inadjoint representations of their respective symmetries and singlet representations ofevery other symmetry. Thus, these derivatives don’t produce any unexpected crossterms.

We now need to evaluate the effective action. The LΦ term can again be shiftedaway, as in Equation (2.87), producing a constant shift in the effective action. Weagain find, as in Equation (2.95), that this shift makes no contribution to lowest orderin the field strengths. After extracting an overall normalization from the remainingquadratic terms and performing functional integration, we need to expand in powersof the fields. We again find that all of the terms of V 3

Φ and most of the terms inV 2

Φ 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 survivingterm of the operator whose trace we are evaluating is of exactly the same form asthose 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 thegraviton contribution, which is instead equal to a sum over multiple copies of the formpresented in Section 2.8. Thus we only need to evaluate N copies of the integralsalready shown in Section 2.8.1. Once this is done and terms are recombined to givethe N running gauge couplings, we find that each one get renormalized separatelyand in exactly same way as in our analysis of a single gauge field.

52

Combining these observations about adding matter and extending the gauge sym-metry, we can now see how the analysis applies to a theory with arbitrary gauge andmatter content. The β function for each gauge coupling is equal to what it would bein 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 supersymmet-ric. In this case one might include gravitinos, which we have not included in thiscalculation, since including only gravitons breaks supersymmetry.

We noted at the end of Section 2.8 that the addition of gravity to a pure Yang-Mills theory rendered the coupling asymptotically free. This, in itself, is not soimpressive, since all pure Yang-Mills theories are already asymptotically free. We cannow, however, make a much stronger statement about the gauge theory coupled tomatter. Since the running is of the exact same form in the presence of matter aswithout it — only the value of b0 is changed — we can now see that the additionof gravity renders all gauge couplings asymptotically free even in the presence ofarbitrary numbers and types of matter fields. Of course, the meaning and utilityof asymptotic freedom is somewhat ambiguous here, since none of these calculationsshould 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 mattercontent such that at some high energy all couplings take on the same value. Definingthe symbol yi = 1/g2

i for the ith gauge coupling, each one runs as

yi(E) = yi0 +

bi0

(4π)2ln

E2

M2. (2.115)

The condition for unification is that there exists an energy E0 such that for any pairi, j we have

−(4π)2

2

(

yi0 − yj

0

bi0 − bj

0

)

= lnE0

M. (2.116)

When gravity is added to the theory, the form of the running follows Equation (2.109):

yi(E) = yi0 exp

3 K2

(4π)2(E2 −M2)

+ 2b0

(4π)2

∫ E

M

dk

kexp

3 K2

(4π)2(E2 − k2)

. (2.117)

The condition for unification at some energy EU now becomes that for any pair i, j,we have

−(4π)2

2

(

yi0 − yj

0

bi0 − bj

0

)

exp

3 K2

(4π)2(E2

U −M2)

=

∫ EU

M

dk

kexp

3 K2

(4π)2(E2

U − k2)

.

(2.118)But, since we assumed the theory exhibited unification before gravity was added, theratio (∆y0/∆b0)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

lnE0

M=

∫ EU

M

dk

kexp

3 K2

(4π)2(M2 − k2)

(2.119)

is an intersection point for all N gauge couplings. Note that since the exponentialfunction is always positive, the integral in Equation (2.119) is a monotonically in-creasing function of EU. Thus, Equation (2.119) has exactly one solution and thegauge couplings intersect at exactly one point.

If E0 ≪ MP, then it should be true that EU ≈ E0. Assuming this is the case,then Equation (2.119) can be approximated by

lnE0

M≈∫ E0

M

dk

kexp

3 K2

(4π)2(M2 − k2)

+EU − E0

E0

exp

3 K2

(4π)2(M2 −E2

0)

. (2.120)

Thus,

EU ≈ E0 + E0

∫ E0

M

dk

k

[

1− exp

3 K2

(4π)2(M2 − k2)

]

exp

−3 K2

(4π)2(M2 −E2

0)

.

(2.121)Now explicitly using MP ≫ E0 ≫M , we get

EU ≈ E0

[

1 +3

2π

(

E0

MP

)2]

. (2.122)

Examining Equation (2.122) for the numerical values E0 = MGUT ≈ 1016 GeV andMP ≈ 1019 GeV reveals that the new intersection point is shifted from E0 by less thana part per million. The one-loop coupling constant flows for a theory with runningquantitatively similar to the minimally supersymmetric standard model with gravityare shown in Figure 2-5(a). The dramatic switching-off of the Yang-Mills interactionnear MP is made more apparent when the couplings themselves are plotted, ratherthan 4π/g2, as in Figure 2-5(b).

2.11 Phenomenology

In this section, we will briefly comment on a few possible experimental implicationsof the preceding analysis. First, however, we will discuss the physical content of theresult.

As apparent from Figure 2-5(b), the Yang-Mills couplings turn off so quickly nearMP that free field theory (zeroth-order perturbation theory) should become an excel-lent 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 onlyYang-Mills couplings (that is, no Yukawa couplings, masses, of scalar self-couplings),this implies that all physics near MP is well described by free fields coupled to grav-

54

(a)

0

10

20

30

40

50

60

70

2 4 6 8 10 12 14 16 18 20

α−

1(E

)

log10(E/GeV)

(b)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

2 4 6 8 10 12 14 16 18 20

g(E

)

log10(E/GeV)

Figure 2-5: In Figure 2-5(a), the three Yang-Mills couplings of an MSSM-like theoryevolve as straight lines in a plot of α−1 ≡ 4π/g2 versus log10 (E) when gravitationis ignored (dashed lines). The initial values at MZ0 ≈ 100 GeV are set so thatthe 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 remainunified near 1016 GeV. In Figure 2-5(b), g is plotted for the same theory. All threecouplings rapidly go to zero near MP, rendering the theory approximately free abovethis 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 poten-tially divergent high-energy dynamics is sequestered in the gravitational sector. It isdifficult to say how seriously this discussion should be taken, however, since all of themost interesting statements are being made near or above the theory’s cutoff. Thesestatements can only be strictly meaningful if the renormalization group for the quan-tum effective action flows to a nonperturbative ultraviolet fixed point, as discussedin 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 effectpredictions of GUT phenomenology, such as the proton lifetime or other “undesirable”consequences of GUT theories. We believe, however, that all effects are likely tobe in the parts per million range or smaller, which is an accuracy that GUT scaleexperiments are not likely to reach for some time.

We should consider the case where the measured values of low-energy couplingshave experimental uncertainties. In this case, the hypothesis of coupling constantunification can be tested against some accepted goodness-of-fit parameter, such asχ2. Accordingly, the best-fit values of the unification scale and coupling will have as-sociated fitting uncertainties and potential cross-correlations. When gravity is takeninto account in the β function, we expect the best-fit unification values to change ac-cording to the discussion of Section 2.10. An important open question whose answeris not obvious at this point is how the fitted uncertainties and χ2 will be affectedby the inclusion of gravity. Can gravitation make non-unifying models (such as thenon-supersymmetric standard model) unify to within experimental error, or can itpush generic almost-unified models (such as the minimally supersymmetric standardmodel) out of a statistically acceptable range? The first of these possibilities wouldsuggest that coupling constant unification is not evidence for low-energy supersym-metry. The second possibility would either rule out many supersymmetric GUTs (ifthe coupling constants are no longer unified) or make the conditions for unification sostrict that if unification persisted with the addition of gravity, the case for low-energysupersymmetry would be significantly strengthened.

There has been some speculation in the literature, mostly inspired by the possi-bility of building large extra dimensions in string theory, that the fundamental scaleof quantum gravity may in the multi-TeV range rather than at 1019 GeV. It has beennoted that among the predictions of such scenarios is the creation of microscopic blackholes in high-energy colliders and cosmic rays experiments. The present calculationsuggests that such experiments should also observe a reduced strength of standardmodel interactions, presumably disturbing the properties of extensive air showers andhadronic 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 Equa-tion (2.110). There, we saw that the fine structure constant of a theory with vanishingβ function in the absence of gravity runs down from its zero-energy value as a Gaus-sian of width

√

π/6MP. 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. Logarithmicrunning should only set in well above threshold, where the electron mass can be ig-nored. With gravity included, the fine structure constant should run as a Gaussianat low energies, with possible effects on atomic spectra and electromagnetic nucleardecays. However, even with a speculatively low Planck scale of 1 TeV, the shift in thefine structure constant at threshold would be a part in 1012, which does not appearto be within experimental reach.

2.12 Commentary

We have found that for a very generic class of four dimensional gauge theories, theaddition of gravity adds a term to the one-loop β function of the form

∆βgrav(gYM, E) = −g

YM

3

π

E2

M2P

. (2.123)

The new term renders all Yang-Mills couplings asymptotically free. In fact, the cou-plings 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 certainunorthodox theories of particle physics. For a theory whose gauge couplings exhibitedunification before gravity was added, the unique unification point is maintained, butshifted to a slightly higher energy (EU/E0 ≈ 1 + 3

2πE2

0/M2P) 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 interpretit somewhat close to its cutoff. This is the strongest critique of the calculation. Ifquantized general relativity has an ultraviolet fixed point in its renormalizationgroup flow as discussed in Section 2.1.2, however, the results given here may, infact, be reliable to higher energies.

• Quantized general relativity should properly include terms of order R2 at one-loop, as discussed in Section 2.1.1. These terms usually have little effect onobservable physics, but they may begin to become important near MP. Also,the coefficients of these operators should be allowed to run, as should K andΛ. The running of these coefficients could become important in about thesame range as where the gravitational contribution to the Yang-Mills β functionbecome important. Again, however, an ultraviolet fixed point may mitigatethese problems.

• Restricting the background spacetime configuration raises the problem of backreaction. 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, itshould be allowed to warp the background spacetime.

The caveats just given are issues of principle. There are also several warnings to bemade of a less fundamental nature with regard to model building:

57

• We live in an approximately Robertson-Walker spacetime, not Minkowski, whichis well modeled as having a small positive cosmological constant. We do notexpect large scale structure to have dramatic effects on the ultraviolet physicsdiscussed here, however, so this is a minor point.

• We have not considered extra dimensions as they are often implemented inrealistic models of particle physics, whether they be compactified, orbifolded,deconstructed, warped, or otherwise of popular interest. We have only con-sidered the case where all matter fields, gauge fields, and gravity propagate inthe same bulk space and none of the fields are confined to lower-dimensionaldefects.

• We have made an inadequate treatment of supersymmetry in the gravitationalsector by not including gravitinos. If we imagine that below MP supersymmetryis broken only in the gravitational sector, while remaining valid in the gauge andmatter sector, then the analysis given here is intact. However, this restrictionmay be too unnatural for serious application to some models.

58

Chapter 3

Black Hole Effective Field Theory

In this chapter, we will show that in order to avoid a breakdown of general covarianceat the quantum level, the total flux in each outgoing partial wave of a quantum fieldin a black hole background must be equal to that of a 1+1 dimensional blackbody atthe Hawking temperature. Specifically, we attempt to formulate an effective quantumfield theory only in the region outside of a black hole and discover that this theory isanomalous unless the stated radiation flux is included.

In Section 3.2, we will turn our attention to the physics of d-dimensional black holespacetimes in classical general relativity and the behavior of scalar wave equations insuch spacetimes. Then, in Section 3.3, we will derive some useful but nonstandardresults regarding the properties of blackbody radiation in spherical coordinates andin d spacetime dimensions. Finally, in Section 3.4, we will attempt to formulate aneffective field theory of scalar field modes that live only outside a black hole eventhorizon.

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 andgeneral relativity. On the other hand, potential sources of conflict between the centralprinciple of general relativity, general covariance, and quantum theory may exist incertain situations in the form of gravitational anomalies. Both the anomaly andHawking radiation result from, in a certain sense, ambiguities of the quantizationprocess 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 spacetimethat contains an event horizon — for example, a black hole. One finds that the oc-cupation number spectrum of quantum field modes in the vacuum state is that of ablackbody at a fixed temperature given by the surface gravity of the horizon. Theliterature contains several derivations of Hawking radiation, each with strengths and

59

weaknesses. Hawking’s original derivation [29, 28] is very direct and physical, butit relies on hypothetical properties of modes that undergo extreme blue shifts, andspecifically assumes that their interactions with matter can be ignored. Derivationsbased on Euclidean quantum gravity are quick and elegant, but the formalism lacksa secure microscopic foundation [42]. Derivations based on string theory have a log-ically consistent foundation, but they only apply to special solutions in unrealisticworld-models, and they do not explain the simplicity and generality of the resultsinferred from the other methods [63, 64]. In all these approaches, the Hawking radi-ation appears as a rather special and isolated phenomenon. Here we discuss anotherapproach, 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 classicalaction and the procedure of quantization (see [65] for a review). Anomalies in globalsymmetries can signal new and interesting physics, as in the original application toneutral pion decay π0 → γγ [66, 67] and in ’t Hooft’s resolution of the U(1) problemof QCD [68, 69]. Anomalies in gauge symmetries, however, represent a theoreticalinconsistency, leading to difficulties with the probability interpretation of quantummechanics due to a loss of positivity1. Cancellation of gauge anomalies gives powerfulconstraints on the charge spectrum of the standard model, which were importanthistorically [70, 71].

A gravitational anomaly [32] is a gauge anomaly in general covariance, taking theform of non-conservation of the energy-momentum tensor. A gravitational anomalycan only occur in theories with chiral matter coupled to gravity in spacetimes ofdimension 4k + 2, for integer k. The chiral matter could be either a chiral fermion ora 2k-form with an (anti-)self-dual field strength. An important case is the self-dualscalar field in 1 + 1 dimensions. This is a scalar field constrained to obey

∂aφ = ǫab∂bφ. (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]

∇aTab =

1

96π√−g

ǫcd∂d∂aΓabc. (3.2)

The energy-momentum tensor of the chiral scalar is therefore not conserved in curvedspacetimes [32].

There are several cases in physics where anomalies have been connected to theexistence of current flows. Pair creation in an electric field has been related to achiral anomaly [73]. The existence of exotic charges on solitons, with or without the

1A third kind of anomaly is a “conformal anomaly”, which is a quantum contribution to the traceof the energy-momentum tensor T a

a . This is the source of scale-dependent renormalization effects inotherwise scale-free theories.

60

existence of zero modes, has been related to anomalous charge flows that arise inbuilding up the soliton adiabatically [74, 75].

Especially closely related to our problem is the connection between anomaly can-cellation and the existence of chiral edge states in the quantum Hall effect [76, forexample]. The effective action modeling the electromagnetic field in this state involvesthe Chern-Simons Lagrangian density confined to the sample area:

S ∝∫

d3xΘA(x)ǫabcFab(x)Ac(x), (3.3)

where the ΘA function is 1 within the sample region A and 0 outside. The gaugetransformation Aa → Aa + ∂aα does not leave this action invariant, because afterintegration by parts a term appears from the derivative acting on ΘA. The extraterm is confined to the boundary and is proportional to α there. This variationtakes the same form as the anomaly of a 1 + 1 dimensional massless, charged, chiralfermion field on the boundary. Since the theory we are modeling is gauge invariant, anadequate effective theory must cancel the boundary gauge variations. This motivatesone to expect the existence of massless chiral edge states, whose anomaly cancelsthe boundary gauge variation. Such states do in fact arise, as can be proved frommore microscopic considerations. We will demonstrate a similar phenomenon forgravitational anomaly cancellation, with Hawking radiation playing a role analogousto 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 touse an anomaly in conformal symmetry to derive important constraints on the energy-momentum tensors of quantum fields in a black hole background2. This anomalyappears as a contribution to the trace T a

a of the energy momentum tensor in a theorywhere it vanishes classically. By requiring finiteness of the energy-momentum tensorof massless fields as seen by a freely falling observer at the horizon in 1 + 1 dimen-sional Schwarzschild background metric and imposing the anomalous trace equationeverywhere, one finds an outgoing flux given by GNm

2

∫∞2GNm

drr2 T

aa (r), where m is the

black hole mass, which is in quantitative agreement with Hawking’s result. This is abeautiful observation, but it is quite special, and might be regarded an isolated cu-riosity. 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-scatteringfor 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 overthe exterior. Also, as a conceptual matter, the central role ascribed to conformalsymmetry seems rather artificial in this context.

2For 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 regionoutside the horizon. The relevant dynamics of the interior (that is, the part thateffects the exterior) is assumed to be captured by an account of the horizon, regardedas a dynamical system. At the classical level, there is a very useful effective membranetheory of the horizon, which can be derived in a fairly straightforward way from theclassical action [31, 79].

A delicate issue arises, however, when one moves to the quantum theory. Toidentify the ground state of a quantum field (say, for definiteness, a free field), onenormally 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 naturaldefinition of time, for which translation t → t + t0 leaves the metric invariant. Thistime coordinate becomes mathematically ill-defined at the horizon, and the “groundstate” associated with its use (the Boulware state [30]) is physically problematic,since in it a freely-falling observer would, upon passing through the horizon, feela singular flux of energy-momentum. The singular contribution arises from modesthat propagate nearly along the horizon at high frequency. In the Boulware state,these modes have non-trivial occupation. The Unruh vacuum [41], which is non-singular, is defined instead by associating positive energy to these modes, so they areunoccupied. Mathematically, it is implemented by associating positive energy withoccupation of modes that are positive frequency with respect to translation of theKruskal 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, towhich we should join, does not contain the offending modes: in effect, that they canbe integrated out.

There is an apparent difficulty with this, however. Having excluded propagationalong one light-like direction, the effective near-horizon quantum field theory becomeschiral. But chiral theories contain gravitational anomalies, as discussed above. In ourcontext the original underlying theory is generally covariant, so failure of the effectivetheory to reflect this symmetry is a glaring deficiency. Analogy to the quantum Halleffect, as in Section 3.1.2, suggests that one might relieve the problem by introducinga compensating real energy-momentum flux whose divergence cancels the anomalyat the horizon. We will show that the energy-momentum associated with Hawkingradiation originating at the horizon does the job. One can extend the discussion toconstruct an effective theory for the interior as well as the exterior bulk, separated bya chiral bilayer membrane near the horizon. The primary features of our frameworkare 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 constantelectric field.

62

Figure 3-1: Part of the causal diagram of a black hole spacetime, with inset detailof a region near the horizon. Dashed arrows indicate unoccupied modes, while solidarrows indicate occupied modes. The white region is the infinitesimal slab near thehorizon where outgoing modes are eliminated.

3.2 Spacetime Preliminaries

We will be studying the behavior of scalar fields in static, spherically symmetricbackground space times. It will be useful to have certain facts on hand about thesespacetimes, 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, sphericallysymmetric spacetime in d dimensions in a suitably intuitive coordinate system. Wechoose coordinates (t, r, Ω) such that Ω are the d − 2 angular coordinates θi on theunit sphere Sd−2 with metric dΩ2 ≡ sijdθidθj . Spherical symmetry allows settinggri and gti to zero and to declaring the remaining t-r components of the metric tobe angle-independent. The static condition allows setting gtr to zero and makes themetric components t-independent. We can further choose r to be the coordinate thatmeasures the areas of spheres in the sense that a spatial surface of constant coordinater and t has an area3 Vol(Sd−2)rd−2. The general metric is now given by

ds2 = −f(r)dt2 + h(r)−1dr2 + r2dΩ2, (3.4)

3The word “area” is used here only because Sd−2 is a boundary of a spatial region. Whenappropriate, we refer to the volume of such a spatial boundary as an “area” or “surface area” inorder 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 ofthe metric is diagonal and given by

dΩ2 =

d−2∑

i=1

(

(dθi)2

d−2∏

j=i+1

sin2 θj

)

. (3.5)

Thus, θ1 is the azimuthal angle ranging over [0, 2π] and the rest are polar anglesranging over [0, π]4.

The determinant of the metric (3.4) is given by

g = −f

hr2(d−2) det Ω. (3.6)

The Christoffel symbols are

Γttr = Γt

rt =f ′

2f, (3.7a)

Γrtt =

f ′h

2, (3.7b)

Γrrr = − h′

2h, (3.7c)

Γrij = −rhsij , (3.7d)

Γijr = Γi

rj =1

rδij , (3.7e)

Γijk = same as unit S

d−2, (3.7f)

with all other components vanishing. The primes in Equation (3.7) indicate differen-tiation with respect to r. The Ricci tensor is given by

Rab = ∂cΓcab − ∂aΓ

ccb + Γc

cdΓdab − Γc

adΓdcb. (3.8)

All the off-diagonal components of Rab vanish. The rest are given by

Rtt =1

2hf

[

f ′′

f+

f ′h′

2fh− 1

2

(

f ′

f

)2

+f ′

f

d− 2

r

]

, (3.9a)

Rrr = −1

2

[

∂2f

f+

f ′h′

2fh− 1

2

(

f ′

f

)2

+h′

h

d− 2

r

]

, (3.9b)

Rij = sij

[

(d− 3)(1− h)− rh∂r ln√

fh]

. (3.9c)

4Note that this ordering of angles is reversed from some conventions.

64

For completeness, we list the Ricci scalar

R = −h

[

f ′′

f− f ′

f∂r ln

√

f

h+

2(d− 2)

r∂r ln

√

fh

]

+(d− 2)(d− 3)

r2(1− h) (3.10)

and Einstein tensor

Gab ≡ Rab −1

2gabR, (3.11)

which has components

Gtt =d− 2

2rd−2f∂r

[

rd−3(1− h)]

, (3.12a)

Grr =d− 2

r∂r ln

√

f

h− Gtt

fh, (3.12b)

Gij = sijr2h

[

f ′′

2f− ∂r ln

√

f

rd−4∂r ln

√

f

h+

1

r∂r ln

√

fh− d− 4

d− 2

Gtt

fh

]

.(3.12c)

Since Rab and gab 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 interestingto consider spacetimes that are solutions of the d-dimensional Einstein equations forsome given background matter source energy-momentum tensor Tab:

Gab =K

2

2Tab, (3.13a)

or equivalently,

Rab =K2

2

(

Tab −1

d− 2gabT

cc

)

. (3.13b)

Recall that K2 was defined in Section 2.2.2 as K2 = 2d−2d−3

Vol(Sd−2)GN. Since Gab

is diagonal, Equation (3.13a) implies that Tab 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 canwrite Tab as

Ttt = ρ(r)f(r), (3.14a)

Trr = P (r)/h(r), (3.14b)

Tij = S(r)r2sij. (3.14c)

The physical energy density of background matter measured by a static observer inthese coordinates is given by ρ. Likewise, P gives the radial pressure measured bythis observer. For many matter models, the tangential pressure S is not independentof ρ and P . For example, a perfect, static, isotropic fluid obeys S = P , but this factwill be neither relevant here nor true in general. The trace of the energy-momentum

65

tensor isT c

c = −ρ + P + (d− 2)S. (3.15)

The tt-component of Equation (3.13a) can be integrated immediately, using Equa-tions (3.12a) and (3.14a), to give

h = 1− C

rd−3− K2

(d− 2)rd−3

∫ r

xd−2ρ(x)dx, (3.16)

where C is constant of integration parameterizing physically different solutions withthe same given matter distribution. A non-zero C is equivalent to placing an addi-tional point mass at the origin. The explicit appearance of C could be absorbed intothe lower limit of the integration. This expression can be simplified by introducingthe “integrated mass” function

m(r) ≡ Vol(Sd−2)

∫ r

0

xd−2ρ(x)dx, (3.17)

where we allow for a possible point-mass singularity in ρ at the origin. This functionhas the intuitive — but, in general, incorrect5 — interpretation as the total massinside a sphere of radius r. Equation (3.16) then simplifies to

h(r) = 1− 2GN

(d− 3)

m(r)

rd−3. (3.18)

Given the solution for h(r), the rr-component of Equation (3.13a) can also beintegrated, using Equations (3.12b) and (3.14b), to give

f(r) = Ah(r) exp

K2

d− 2

∫ r x[ρ(x) + P (x)]

h(x)dx

. (3.19)

The integration constant A allows an arbitrary rescaling of t. It can be set to any valuewithout 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 = ∞.Equation (3.19) illustrates that if the matter source obeys P = −ρ, as it does forvacuum, pure cosmological constant, or electrostatic field, we get the simple resultf(r) = h(r), everywhere. A few simple metrics of physical interest that obey ρ = −Pare listed in Table 3.1.

The exact forms of ρ(r) and P (r) will depend on the dynamics or equation ofstate of the matter source in question, which we won’t be concerned with here. It isoften useful for the intuition to consider the case of f(r) = h(r), although the generalcase 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 thespacetime, the ADM mass [80], does exist. Also in such spacetimes, a Newtonian limit can be takenand the expression (3.17) does indeed agree with the Newtonian mass.

66

Name f(r) = h(r) ρ = −P

Schwarzschild 1− 2GN

(d−3)m

rd−3 0

Reissner-Nordstrom 1− 2GN

(d−3)m

rd−3 + GN

(d−3)2 Vol (Sd−2)Q2

r2(d−3)Q2

2Vol (Sd−2)2r2(d−2)

de Sitter 1− 2Λ(d−2)(d−1)

r2 2 ΛK2

Schwarzschild-de Sitter 1− 2GN

(d−3)m

rd−3 − 2Λ(d−2)(d−1)

r2 2 ΛK2

Table 3.1: A few physically interesting metrics that obey ρ = −P .

that if a surface of constant r exists such that one of the functions f or h vanishesthere, then the other must vanish there, too. We will always use this condition whenneeded in further sections. Interest in the special case f(r) = h(r) would be wellmotivated, however, by the study of matter distributions that could possibly give riseto 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 physicallyinteresting cases (again, see Table 3.1).

3.2.1.2 Horizon Structure

Two types of horizons will be encountered when studying black holes: event horizonsand 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 spatialregions where timelike trajectories may escape to timelike infinity from regions wherethey may not) to exist for some constant r hypersurface is that the one-form normalto the hypersurface has vanishing norm: gab∂ar∂br = h(r) = 0. A Killing horizon forthe global Killing vector η ≡ ∂t will exist when the norm of η vanishes: gabη

aηb =−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 non-negative. Thus, the sign of f(r) and h(r) must always match. In particular, the zerosof f(r) and h(r) coincide. The only possible exception to this might occur when theintegral in Equation (3.19) has a positive pole. Assuming the background matter isnonsingular near a given event horizon, Equation (3.19) reads

f(r) ∼ (r − rh)1+

K2rh[ρ(rh)+P (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 exponentis non-negative, f(rh) will vanish and the zeros of f(r) and h(r) will match. Theexponent can only be negative if h(r) approaches 0 with a slope of sign opposite tothat of ρ + P , and even then the exponent may still be positive.

As will be discussed below, most spacetimes that describe stars or black holes willbe asymptotically flat and obey the null energy condition. This means that near theoutermost event horizon both h′ and ρ + P are non-negative. Thus, it is the casethat for sufficiently well behaved matter sources, at least the outermost event horizonwill indeed obey f(rh) = 0. Since this is the only event horizon that can be observed

67

from infinity, it is sufficient and unambiguous for physics outside of rh to refer tothis hypersurface as the event horizon6. The event horizons of static, sphericallysymmetric spacetimes are therefore also Killing horizons. We can then simply talkabout “horizons” without confusion.

The fact that the event horizon is also a Killing horizon means, among otherthings, that in the static coordinates (3.4), a patch of spacetime near the horizonlooks locally like Minkowski spacetime in Rindler coordinates. If this were not thecase, analysis of the behavior of wave modes near the horizon would be much moredifficult (see Section 3.2.3). Arguably, the universality of black hole thermodynamicsdepends, at least at the kinematic level (Hawking radiation), on the ubiquitousnessof Killing horizons.

From the form of the metric (3.4), it can be seen that f(r) must remain positivefor t to remain a timelike coordinate. If f(r) does change sign, as it does at aKilling horizon, then the global isometry t → t + t0 can no longer be referred to asa “time translation,” and this spacetime cannot be considered truly static. Rather,such a spacetime is merely stationary. This loss of staticness allows time-irreversibleprocesses to occur. As such, spacetimes of this sort may contain spatial regions thatcan 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, F, is defined byηa∇aηb = −Fηb, evaluated at the horizon with η normalized to 1 at a conventional

point, typically infinity. For the metric(3.4) this gives F = 12f ′∣

∣

∣

rh

. The surface gravity

is so named because in a static, asymptotically flat spacetime it typically measuresthe magnitude of the acceleration of a static particle at the horizon, as seen by anobserver at infinity. However, in our spacetime, this acceleration is actually given by7

κ = 12

√

f ′h′∣

∣

∣

rh

. (3.21)

If we define the corresponding quantity for the event horizon, namely H ≡ 12h′∣

∣

∣

rh

,

then we findκ =

√

FH. (3.22)

We will find that the relevant quantity appearing in the equations of black holethermodynamics is in fact κ, not F.

One of the classical energy conditions often imposed on the choice of energy-momentum tensor is the so-called null energy condition (NEC) (Tabn

anb ≥ 0 for anynull vector na), which demands ρ+P ≥ 0. This condition appears to be necessary for

6Nevertheless, there do exist spacetimes for which some of the zeros of f(r) and h(r) do notcoincide. See, for example, the GHS black hole [81] and similar dilaton black holes. These mayoccur 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 horizonsobeying f(rh) = h(rh) = 0.

7It may be worth noting that the acceleration as observed from infinity of a static point at aconstant coordinate r that is not a horizon is given by 1

2f ′√

h/f .

68

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 normalstar, the integrand is always positive. Thus, the exponential is greater than 1 andf(r) ≥ h(r). If regions of both positive and negative h(r) exist, the situation is morecomplicated and |f(r)| may be less than |h(r)| (but always the same sign). There isno physical significance to an intersection f = h other than f = h = 0, since suchintersections can be moved around arbitrarily by rescaling t.

The null radial component of Equation (3.13), Gabnanb = K2

2Tabn

anb, reads

d− 2

rh

(

f ′

f− h′

h

)

= K2(ρ + P ). (3.23)

Interestingly, examining Equation (3.23) near the horizon using f(rh) = h(rh) = 0we find simply

(ρ + P )∣

∣

∣

rh

= 0. (3.24)

This shows that Einstein’s equations force any form of matter to saturate the NECat a horizon in a static, spherically symmetric spacetime.

Another physically reasonable condition to impose on the background matter isthe weak energy condition (WEC) (Tabu

aub ≥ 0 for any timelike vector ua), whichreads ρ ≥ 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) mayhave an arbitrary number of zeros, as shown in Figure 3-2. Thus, the WEC says littleabout the horizon structure of these spacetimes.

3.2.2 Kruskal Extension

In order to properly understand the global structure of the spacetime under consid-eration and the nature of its horizon, we want to define a Kruskal-like extension ofthis 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 manifestlynonsingular at the horizon. But we would also like the new coordinates, in analogyto the d = 4 Schwarzschild case, to show that the global time translation isometryof 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 toexhibit an explicit on-horizon translation symmetry, which should intuitively existgiven the close similarity of this spacetime to d = 4 Schwarzschild space8.

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 coordi-

8We can also construct a Painleve-Gullstrand [39, 40] coordinate system for this spacetime. By

defining τ = t +∫

dr√

1−hhf , the metric (3.4) becomes

ds2 = −f(r)dτ2 + 2

√

f(1− h)

hdτdr + dr2 + r2dΩ2. (3.25)

69

6

-r

m,h

(a)

6

-r

m,h

rh

(b)

6

-r

m,h

rh

(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 themetric function h(r). Vertical dashed lines indicate the position of the outermost eventhorizon, if it exists. In 3-2(a) the matter distribution is relatively smooth and vanishesat the origin, as in a normal star. In 3-2(b) the matter has a density singularity atthe origin, but is otherwise well behaved. In 3-2(c) a potentially difficult-to-analyzesituation is sketched.

70

6

-r

f ,h

rh

(a)

6

-r

r∗

rh

(b)

Figure 3-3: In 3-3(a) typical profiles for the functions h(r) (solid line) and f(r) (dottedline) are sketched for an asymptotically flat black hole spacetime. The horizon occurswhere the functions vanish at r = rh. In 3-3(b) the corresponding profile of r∗ issketched along with the dotted line r∗ = r. Note that r∗ diverges logarithmically atrh and approaches r at large r.

nates will be poorly behaved at the horizon, but it will be possible to construct arescaled 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 aLorentz boost in the Minkowski-like coordinates associated with the null coordinates(U, V ). This will greatly restrict the possible form of the new coordinates, leavingonly three arbitrary real constants to be fixed in the coordinate transformation. Wewill then check whether the metric is regular at the horizon in these coordinates. Theregularity condition will fix one of the free constants, while the other two are simplyglobal rescalings of U and V that be chosen by convenience. We will then show thattranslations in U or V are on-horizon isometries.

The first step is to introduce the analog of the tortoise coordinate r∗ defined by

∂r∗∂r

=1√fh

. (3.26)

The generic form of r∗ is sketched in Figure 3-3. With this coordinate the metricbecomes

ds2 = f(r)(−dt2 + dr2∗) + r2dΩ2, (3.27)

where r is now thought of as an implicit function of r∗. These are the locallyMinkowski coordinates of a free-falling observer. Radial null geodesics (dΩ2 = ds2 =0) in this spacetime obey

(

∂t

∂r

)2

=1

fh, (3.28)

In these coordinates the spacetime appears stationary, but not static. It also has the nice feature ofhaving spatially flat equal time slices. Finally, they demonstrate that the event horizon at f = h = 0is a non-singular surface. The case of h(r) = f(r) = 1 − 2GNm

r is the original Painleve-Gullstrandcase, which has recently been applied widely [79, 84, 85, 86].

71

which can be combined with Equation (3.26) to show that t± r∗ is a constant alongsuch paths, with the plus sign for ingoing geodesics and the minus sign for outgoinggeodesics. 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 horizonwhere either f(r) or h(r) vanishes. This is the coordinate singularity we hope toeliminate with a coordinate transformation. The singularity is made much worse iff(r) and h(r) do not have the same sign everywhere. We thus impose the physicalchoice of spacetimes discussed in Section 3.2.1. That is, we demand that the zeros off(r) and h(r) coincide.

As outlined above, we wish to construct new null coordinates U(u) and V (v) suchthat the Minkowski-like coordinates (T, R) defined by

U = T + R, (3.30a)

V = T − R (3.30b)

obey

T → γT + γβ(t0)R, (3.31a)

R → γR + γβ(t0)T (3.31b)

when t→ t + t0, for some function β(t0). Of course, 1/γ ≡√

1− β2. This demandsthat

U → γ(1− β)U, (3.32a)

V → γ(1 + β)V , (3.32b)

or equivalently

UV → UV , (3.33a)

V

U→

(

1 + β

1− β

)

V

U. (3.33b)

Equation (3.33a) means that the product UV ≡ A(r) is a time independent functionof r only. Equation (3.33b) means that V/U depends exponentially9 on t as

V

U= B(r)e2kt, (3.34)

9For a sufficiently well-behaved real function a(x) to obey a(x+y) = b(y)a(x) for a given functionb(y), also sufficiently well-behaved, it must be true that a(x) = Aekx and b(y) = eky. This can beproven by writing a(x + y) = ey∂xa(x) and Taylor expanding about y = 0.

72

with B(r) and k arbitrary. That k must be an r-independent constant can be seenfrom Equation (3.33b), which can be solved to give

β(t0) = tanh kt0. (3.35)

Since β is r-independent by construction, so is k.

We now have U2 = (A/B)e−2kt and V 2 = ABe2kt. For U to be a function of uonly and V to be a function of v only, while maintaining simultaneously that A andB are functions of r only, it must be the case that A = cUcV e2kr∗ and B = cV /cU , forarbitrary constants cU , cV . The (U, V ) coordinates now take the form

U = cUe−ku, (3.36a)

V = cV ekv, (3.36b)

and metric in the now reads

ds2 =1

k2cUcV

f(r)e−2kr∗dUdV + r2dΩ2 =f(r)

k2UVdUdV + r2dΩ2. (3.37)

To examine the metric near the horizon, we Taylor expand the metric functionsh(r) and f(r) around r = rh using the notation of Section 3.2.1.2:

f(r) = 2F(r − rh) +∞∑

n=2

f (n)(rh)

n!(r − rh)

n, (3.38a)

h(r) = 2H(r − rh) +

∞∑

n=2

h(n)(rh)

n!(r − rh)

n. (3.38b)

This yields

r∗ =1

2κln

(

r − rh

r0 − rh

)

+Hf ′′(rh) + Fh′′(rh)

4κ(r − r0) + . . . , (3.39)

near the horizon, where r0 is an integration constant. (A sensible choice is r0 =rh + 1/(2F).) Thus the metric is approximately

ds2 ≈ 2F(r0 − rh)

kκ

k2cUcV

[

(r − rh)1− k

κ +O(

(r − rh)2− k

κ

)]

dUdV + r2dΩ2. (3.40)

This metric will only have a non-vanishing determinate at r = rh if k = κ. Aspromised, regularity has fixed the value of the free constant k. We fix the scalefactors cU and cV by aesthetics, choosing cV = −cU = 1/κ. This choice gives as closeas meaningfully possible to a unit determinate at the horizon, such that the (T, R)coordinates are a close as possible to Minkowski coordinates.

73

The final expressions for the Kruskal extension of the metric (3.4) are given by

U = −1

κe−κ(t−r∗), (3.41a)

V =1

κeκ(t+r∗), (3.41b)

and

ds2 = −f(r)e−2κr∗dUdV + r2dΩ2 =f(r)

κ2UVdUdV + r2dΩ2, (3.42)

with r∗ given by Equation (3.26). Neither form of the metric given in Equation (3.42)are obviously nonsingular at the horizon, but we have shown by the constructionsof this section that they are. The horizon at r = rh is now seen to occur at bothU = 0 and V = 0. The horizon is a bifurcate Killing horizon with bifurcation surfacegiven by the Sd−2 at U = V = 0 (t = 0, r = rh). The U = 0 surface corresponds tot = +∞ and is therefore referred to as the future horizon, while the V = 0 surfacecorresponds to t = −∞ and is referred to as the past horizon. Since all of the metriccomponents are t-independent, they depend on U and V only through the productUV . The metric is thus trivially invariant under translation of one coordinate alongthe hypersurface where the other coordinate is zero. This shows that translationalong 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 sym-metric black hole in d dimensions that has all the familiar properties of the Kruskalextension of d = 4 Schwarzschild spacetime. In particular, the fact that the Lorentzboosts of the local Minkowski-like coordinates near the horizon generate t-translationsin the original coordinates of Equation (3.4) shows that near the horizon, the originalcoordinates are more akin to Rindler coordinates than Minkowski ones. Equation(3.39) shows that the logarithmic divergence or r∗, which will ultimately translateinto a logarithmic phase singularity in wave modes defined with respect to the globalKilling vector η, is a universal feature of horizons in this family of spacetimes. Fur-thermore, just as for Schwarzschild spacetime, the final form of the coordinates (3.41)shows that U → U +U0 is an isometry along the past horizon V = 0, and thus ξ ≡ ∂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 ξ-vacuum, for this spacetime10 by definingpositive energy states as those that have positive frequency with respect to ξ on thepast horizon, and letting this vacuum propagate outward and forward in time. Thevacuum state defined in this way differs from the one obtained by defining positiveenergy with respect to the global Killing vector η, which is called the Boulware state

10The quantum vacuum for a star dynamically collapsing to a form black hole, such that themetric of this collapsing spacetime settles down at late times to a stationary black hole metric ofthe form (3.4), is thought to be well represented at times long after the collapse by the ξ-vacuum ofthe eternal black holes we have been studying.

74

or η-vacuum [30]. In fact, Unruh found that the ξ-vacuum is a thermal ensembleof η-frequency states at temperature TH ≡ κ

2π, in exact agreement with the earlier

results of Hawking [29, 28]. The η-vacuum has divergences in its energy-momentumtensor arising from horizon-skimming modes, despite appearing empty11 to static ob-servers. This divergence renders the Boulware state an unphysical candidate vacuum.The Unruh state, on the other hand, is well behaved despite its nonintuitive modeoccupation 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 ξ-vacuum ofthis spacetime given by TH. Furthermore, the η-vacuum of this spacetime also hasdivergences that render it unphysical.

3.2.2.2 Euclidean Section

A relatively easy route to determining the thermal nature of a spacetime is to considerthe Euclidean section of the complexified coordinates, following the method of [42].This can be effectively accomplished by letting t→ iτ . Using the Taylor expansionsof equation (3.38) and defining x2 = (2/H)(r− rh), the near-horizon metric becomes

ds2 ≈ dx2 + κ2x2dτ 2 + r2hdΩ2. (3.43)

This is a metric on the manifold Cone2⊗ S

d−2, with the conical singularity occurringat the horizon. The singularity is removed if the coordinate κτ is periodic in 2π.The period of τ must then be given by 2π

κ. Since the inverse period of Euclidean

time is interpretable as the temperature of a system, we see that the temperatureof 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 theinterpretation of a Euclidean black hole as a non-singular thermal system.

3.2.3 Wave Equation

Consider an interacting classical scalar field φ(x) living in a spacetime with the metric(3.4). This field has an action given by:

S =

∫

ddx√−g

−φ∇2φ−∞∑

n=2

λnφn

. (3.44)

The λn are a set of arbitrary coupling constants. In particular, λ2 ≡ m2 gives themass of a weakly coupled excitation of this field. We will expand φ in the eigenmodes

11Here, “empty” means all modes have zero occupation number.

75

of the free, classical wave equation such that

φ(x) =

∫

dµpapφp(x), (3.45)

∇2φp = m2φp. (3.46)

A sufficiently large set of quantum numbers p label the eigenbasis. The abstract for-mal expression dµp simply represents an appropriate measure over the modes underwhich

∫

dµpφp(x)φp(y) = δd(x− y) and∫

ddx√−gφp(x)φq(x) = δpq. Of course, the

full theory will not necessarily obey Equation (3.46), but the expansion (3.45) is nev-ertheless 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, thecoefficients ap would be promoted to field operators.

We will exploit the high degree of symmetry in the background spacetime anduse separation of variables to examine the normal modes of this field. As remarkedin Section 3.2.2, the wave equation arising from the action (3.44) will take a muchsimpler form in the tortoise coordinates described by Equations (3.26) and (3.27). Forthe sake of generality and to illustrate the nature of the useful cancellations inherentin certain coordinate systems, we now examine the wave modes in a coordinate systemwith metric

ds2 = −f(r)dt2 + j(r)−1dr2 + r2dΩ2, (3.47)

where r = r(r) is an arbitrary radial coordinate. The metric (3.4) will be recoveredwith the choice r = r, j = h, while the tortoise coordinates will be recovered fromr = r∗, j = 1/f .

We factor a given basis function as

φp =1

rd−22 (fj)1/4

R(t, r)Y (Ω). (3.48)

With this factorization, the eigenvalue equation becomes

∇2φp

φp

= m2 =1

R

1

f

[

−∂2t + fj∂2

r −fj

rd−22 (fj)1/4

∂2r

(

rd−22 (fj)1/4

)

[

∂r

∂r

]2

−fj

2∂r

(

[

∂r

∂r

]2)

∂r ln(

rd−22 (fj)1/4

)

]

R

+1

Y

1

r2

d−2∑

i=1

∂i

[

(sin θi)i−1

∂iY]

[

∏d−2j=i+1 sin2 θj

]

(sin θi)i−1. (3.49)

Clearly, all the variables are separable. We first turn our attention to the angularpart. 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 (Ω) = Θ1(θ1)Θ2(θ

2) . . .Θd−2(θd−2). (3.50)

We then get

1

Y

d−2∑

i=1

∂i

[

(sin θi)i−1

∂iY]

[

∏d−2j=i+1 sin2 θj

]

(sin θi)i−1=

d−2∑

i=1

1

Θi

∂i

[

(sin θi)i−1

∂iΘi]

[

∏d−2j=i+1 sin2 θj

]

(sin θi)i−1. (3.51)

By standard separation of variables methods, the above must be equal to a constant,which we name −l(l + d − 3). Further, the angular separation clearly requires thefollowing equations to hold:

∂21Θ1 =− n2

1Θ1, (3.52a)

∂2 [(sin θ2)∂2Θ2]

sin θ2− n2

1

sin2 θ2Θ2 =− n2

2Θ2, (3.52b)

∂3 [(sin θ3)2∂3Θ3]

sin2 θ3− n2

2

sin2 θ3Θ3 =− n2

3Θ3, (3.52c)

. . .

∂i [(sin θi)i−1∂iΘi]

sini−1 θi− n2

i−1

sin2 θiΘi =− n2

i Θi, (3.52d)

. . .

∂d−3

[

(sin θd−3)d−4∂d−3Θd−3

]

sind−4 θd−3− n2

d−4

sin2 θd−3Θd−3 =− n2

d−3Θd−3, (3.52e)

∂d−2

[

(sin θd−2)d−3∂d−2Θd−2

]

sind−3 θd−2− n2

d−3

sin2 θd−2Θd−2 =− l(l + d− 3)Θd−2, (3.52f)

where the ni are constants. Defining x = cos θi and

F (x) = (sin θi)i−22 Θi = (1− x2)

i−24 Θi, (3.53)

we find each function obeys

(1− x2)∂2xF − 2x∂xF +

[

i

2

(

i− 2

2

)

+ n2i −

(i− 2)2/4 + n2i−1

1− x2

]

F = 0, (3.54)

where i = 1 . . . d− 2, n0 ≡ 0, and n2d−2 ≡ l(l + d− 3). This is solved by

F (x) = L

√(i−2)2/4+n2

i−1√(i−1)2/4+n2

i −1/2(x), (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 ofsingle-valuedness and finiteness of the scalar wavefunction restrict both the choiceof Legendre function and the allowed values for the ni. In general, the Legendrefunctions have singularities unless they are P functions with integer parameters or Qfunctions with half-integer parameters. Looking at i = 1, we see that

Θ1 = eim1θ1

, (3.56)

where m1 = n1 is restricted to be any integer by the condition of single-valuedness.We now have, for i = 2,

Θ2 = Lm1√n2

2+1/4−1/2(cos θ2). (3.57)

In order for this function to be finite over all values of θ2 for integer m1, L must bea P function and n2 must obey n2

2 = m2(m2 + 1) for m2 a positive integer obeyingm2 ≥ |m1|. This yields

Θ2 = P m1m2

(cos θ2). (3.58)

For i = 3 we getΘ3 = (sin θ3)−1/2L

m2+1/2√n2

3+1−1/2(cos θ3), (3.59)

which requires that L must be a Q function and n3 must obey n23 = m3(m3 + 2) for

m3 a positive integer obeying m3 ≥ m2. This yields

Θ3 = (sin θ3)−1/2Qm2+1/2m3+1/2(cos θ3). (3.60)

This pattern continues, giving for angle i

Θi = (sin θi)−i−22 L

mi−1+(i−2)/2mi+(i−2)/2 (cos θi), (3.61)

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

Θi = (sin θd−2)−d−42 L

md−3+(d−4)/2

l+(d−4)/2 (cos θd−2), (3.62)

where L = P for d even, L = Q for d odd, and l a positive integer obeying l ≥ 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 thequantum numbers l,m, where m = m1, . . . , md−3. Thus, we write Y = Y m

l (Ω).Only the “highest” angular quantum number, l, will appear in the radial equation,so it will be useful to calculate the degeneracy, Dd(l), of angular states with a givenvalue of l. The construction outlined above make clear that

Dd(l) =

l∑

n=0

Dd−1(n). (3.63)

Starting from D4(l) = 2l + 1, Equation (3.63) can be iterated to find the proper

78

d Dd(l)3 1 for l = 0, 2 otherwise4 2l + 15 (l + 1)2

6 16(2l + 3)(l + 2)(l + 1)

7 124

(2l + 4)(l + 3)(l + 2)(l + 1)8 1

120(2l + 5)(l + 4)(l + 3)(l + 2)(l + 1)

9 1720

(2l + 6)(l + 5)(l + 4)(l + 3)(l + 2)(l + 1)

Table 3.2: The angular state degeneracies for total angular quantum number l, asdetermined by Equation (3.64), in a few chosen dimensions.

expression for higher dimensions12. The exact expression13 for arbitrary d and l is

Dd(l) =(2l + d− 3)(l + d− 4)!

l!(d− 3)!. (3.64)

Some values of Dd(l) are tabulated in Table 3.2.

We chose the normalization of the Y ml such that

∫

dd−2ΩY ml (Ω)Y n

k (Ω) = δlkδmn (3.65)

3.2.3.2 Radial Wave Equation

Separating off the time dependence by writing R(t, r) = eiωtR(r), the eigenvalueequation (3.49) now simplifies to

m2 =1

R

1

f

[

ω2t + fj∂2

r −fj

rd−22 (fj)1/4

∂2r

(

rd−22 (fj)1/4

)

[

∂r

∂r

]2

−fj

2∂r

(

[

∂r

∂r

]2)

∂r ln(

rd−22 (fj)1/4

)

− fl(l + d− 3)

r2

]

R. (3.66)

There is no general solution to this equation, but one can be found for certain specialcases, and approximate solutions can be found in others.

12Of course, one could also start with D3(l) =

1 for l = 02 for |l| > 0

.

13We are aware of three different original proofs of this expression, each developed independentlywhen the problem was presented by the author as a puzzle to a group of fellow students. The methodof 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 “balls-and-buckets” problem [88]. The method of the author uses inspired guess work checked againstEquation (3.63).

79

In Minkowski space (f = j = 1, r = r), Equation (3.66) becomes

[

∂2r −

(l + (d− 4)/2)(l + (d− 2)/2)

r2+ k2

]

R(r) = 0, (3.67)

where k2 = ω2 −m2. This is solved by R(r)/r(d−2)/2 = k(d−2)/2Jl(kr), where Jl(z) =(

π2z

)d−32 jl+(d−3)/2(z) is the lth (d − 2)-spherical Bessel function, and jν(z) is the νth

ordinary, cylindrical Bessel function. This Bessel function will also be the solutionto 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 = r∗, ∂r/∂r∗ =

√fh). The radial wave equation becomes

[

∂2r∗ + ω2 − fV (r∗)

]

R = 0, (3.68)

with

V (r∗) =1

r2

[

(

l + d−42

) (

l + d−22

)

− (d− 2)(d− 4)

4(1− h) +

(d− 2)

4

r∂r(fh)

f

]

+ m2.

(3.69)Using the fact that h and f both vanish when f does, we see that all the terms inV are nonsingular at a horizon, f = h = 0. Thus, the overall factor of f causesthis effective radial scattering potential to vanish near the horizon. In fact, usingf ≈ 2F(r − rH) and h ≈ 2H(r − rH), we find

fV (r∗) ≈ e2κr∗

(

l(l + d− 3) + (d− 2)rHH

r2H

+ m2

)

, (3.70)

which vanishes exponentially fast as the horizon at r∗ → −∞ is approached. A sketchof 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 thatof 1+1-dimensional flat spacetime with coordinates t and r∗. The solutions are simplyplane waves of the given frequency. Using the coordinates of Section 3.2.2, these canbe written as

R(t, r∗) ∼ eiωu + eiωv ∼ |U |iω/κ + |V |−iω/κ. (3.71)

The identification of the near-horizon dynamics of this spacetime with that of stan-dard Schwarzschild-type black hole physics is now essentially complete. Essentiallyall qualitative results for the case of f(r) = h(r) = 1− 2GNm

rhave 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, wecan execute a partial wave decomposition of the action (3.44). The field is expanded

80

6

-r∗

fV (r∗)

←(

r∗=−∞r=rh

)

r ∼ a few rh

Figure 3-4: A sketch of a typical effective radial scattering potential. The potentialfor any metric qualitatively similar to the one sketched in Figure 3-3 will be quali-tatively similar to the one sketched here for l > 0 and d > 3. The potential falls offexponentially for negative r∗ and as is typically dominated by the centrifugal term atlarge r∗ ≈ r, which falls off as 1/r2.

as

φ =1

r(d−2)/2(fj)1/4

∑

l,m

Rlm(t, r)Y ml (Ω). (3.72)

The action becomes

S =

∫

dtdrdd−2Ω

−∑

l,k,m,n

1

fjRlmY m

l

[

−∂2t + fj∂2

r −fj

rd−22 (fj)1/4

∂2r

(

rd−22 (fj)1/4

)

[

∂r

∂r

]2

−fj

2∂r

(

[

∂r

∂r

]2)

∂r ln(

rd−22 (fj)1/4

)

− fk(k + d− 3)

r2

]

RknY nk

−∞∑

n=2

λn

√

f/j

r(n−2)(d−2)/2(fj)n/4

∑

l1...ln

∑

m1...mn

[

Rl1m1Ym1

l1. . . RlnmnY mn

ln

]

=

∫

dtdr

−∑

l,m

1

fjRlm

[

−∂2t + fj∂2

r −fj

rd−22 (fj)1/4

∂2r

(

rd−22 (fj)1/4

)

[

∂r

∂r

]2

−fj

2∂r

(

[

∂r

∂r

]2)

∂r ln(

rd−22 (fj)1/4

)

− fl(l + d− 3)

r2

]

Rlm

−∞∑

n=2

λn

√

f/j

r(n−2)(d−2)/2(fj)n/4

∑

l1...ln

∑

m1...mn

nCml Rl1m1 . . . Rlnmn

, (3.73)

81

where nCml is the set of group theoretic constants obtained by integrating the local

product of n properly normalized d-dimensional spherical harmonics over Sd−2.The radial wavefunction, Rlm, in unitless, just like a canonically normalized 1+ 1

dimensional scalar field. All the units of the original d-dimensional field are carriedby the prefactor 1/r(d−2)/2 in Equation (3.72). The mass units of λn are [λn] =2− (n− 2)(d− 2)/2. Thus, we can write the unit-carrying factors in the interactionseries of Equation (3.73) as

λn

r(n−2)(d−2)/2=

λn

r2−[λn]. (3.74)

This rewriting emphasizes that interactions which are not perturbatively renormal-izable — and are therefore governed by coefficients with negative mass dimension —fall off at large distances faster than 1/r2 in a partial wave decomposition. Super-renormalizable interactions fall off slower than 1/r2 and marginal interactions fall offas 1/r2. The scaling relative to 1/r2 is important because the large-distance scalingof the centrifugal term in the kinetic part of the action is given by 1/r2.

If we now evaluate the action (3.73) for the tortoise coordinates as in Section3.2.3.2, we get

S =

∫

dtdr∗

−∑

l,m

Rlm

[

−∂2t + ∂2

r∗ − fV (r∗)]

Rlm

−f

∞∑

n=3

λn

r2−[λn]

∑

l1...ln

∑

m1...mn

nCml Rl1m1 . . . Rlnmn

, (3.75)

where V is defined as in Equation (3.69) with m2 = λ2. Again, f(r) vanishes ex-ponentially fast near the horizon in tortoise coordinates. So, near the horizon, theaction becomes simply

S ≈∫

d2x∑

l,m

−RlmRlm, (3.76)

where d2x = dtdr∗ and = −∂2t + ∂2

r∗ . This is the action for an infinite collection offree scalar fields in 1+1 dimensions. This description of the dynamics becomes exactat the event horizon, however the notation and coordinates used in this section failthere.

3.3 Thermal Radiation

The theoretical derivation of the blackbody spectrum is standard fare for 3 + 1 di-mensions with wave modes labeled by Cartesian coordinates, but the derivation ford-dimensional spherical coordinates is somewhat less well known. We will illustratethis derivation in this section. For purposes of comparison and introduction of somenecessary mathematics, we will first study the spectrum of blackbody radiation in dCartesian coordinates. Some subtleties of the spherical case will be illuminated by

82

comparison with the Cartesian case. Minkowski space will be used throughout thissection.

3.3.1 Hypercubic Blackbody Cavity

Consider a real scalar field φI(x, t), where I is some kind of p-dimensional polarizationindex representing p internal degrees of freedom14. Further assume that the field issufficiently weakly coupled that each polarization component can be treated as anindependent field obeying an action similar to Equation (3.44) with all interactioncoefficients higher than λ2 ≡M2 set equal to zero. Then each field component obeysthe classical equation of motion

∇2φI(x) = M2φI(x). (3.77)

The solutions to Equation (3.77) may be expressed as a sum over modes labeled byby a wave vector ka obeying k2 = −M2:

φIk = AI

k sin (kaxa) + BI

k cos (kaxa), (3.78)

for arbitrary real coefficients AIk and BI

k . We take the state to be labeled by the d−1spatial components of ka and fix the frequency of each mode by ω2

k ≡ k20 = M2 +kik

i.We now confine the field to live in a cubic box of side length L by demanding

Dirichlet boundary conditions at xi = 0, L for i = 1 . . . d − 1. This demands BIk = 0

andki =

π

Lmi, (3.79)

where mi is a spatial vector whose components are non-negative integers. The fullspace of modes available to the system is given by the tensor product of the spaceof all vectors of integers with the state space of the polarization index. The theoryis then effectively quantized by stating that the energy in each such mode is a non-negative integer multiple of ωk. We denote the integer as nmI . The total energy inthe hypercube is then given by

U =

p∑

I=1

∞∑

m1=0

· · ·∞∑

md−1=0

ωknmI , (3.80)

where we understand that ωk is given by

ωk =

√

√

√

√M2 +π2

L2

d−1∑

i=1

m2i . (3.81)

The set of integers nmI defining the quantum state must obey the appropriate statis-

14Sufficiently 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 behavesessentially like an internal index on a scalar field with p = d− 2.

83

tics15 for the field φI . That is, nmI is unrestricted if the field obeys Bose-Einsteinstatistics, 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 temper-ature T and zero chemical potential, the system will be in a mixed state governed bythe partition function

Q =∑

nmIe−βωk , (3.82)

where β ≡ 1/T . This can be evaluated to give

ln Q = −pξ

∞∑

m1=0

· · ·∞∑

md−1=0

ln(

1− ξe−βωk)

, (3.83)

where ξ = 1 for bosons and ξ = −1 for fermions. The overall factor of p occurs becausethe energy is independent of p, so each polarization mode contributes equally. Theaverage occupation number of a given momentum mode in the thermal state is thengiven by

〈nm〉 =

p∑

I=1

〈nmI〉 = − 1

β

∂

∂ωkln Q =

p

eβωk − ξ. (3.84)

The total energy in the hypercube can now be found by combining the expressions(3.80) and (3.84), or by

〈U〉 = − ∂

∂βln Q =

∞∑

m1=0

· · ·∞∑

md−1=0

ωk〈nm〉 =∞∑

m1=0

· · ·∞∑

md−1=0

pωk

eβωk − ξ. (3.85)

In principle, Equation (3.85) can be evaluated and the summand can be interpreted asthe spectral energy density over the quantum numbers mi. Making this interpretationis problematic, however, in the L → ∞ limit. The problem is illustrated by lookingat the M = 0 case, such that ωk = π

L|m|, and examining the scaling with L:

〈U〉 =pπ

L

∞∑

m1=0

· · ·∞∑

md−1=0

|m|eπβ|m|/L − ξ

L→∞≈ −pπξ

L

∞∑

m1=0

· · ·∞∑

md−1=0

|m|(

1 + ξ + ξ|m|πβ

L+ . . .

)

, (3.86)

which is nonsensical for several reasons. The total energy does not scale with thevolume of the hypercube, Ld−1. In fact, the total energy appears to vanish at largeL as either L−1 or L−2, depending on the statistics. The expansion itself is not evenwell defined, since it is an expansion in the ratio of a divergent quantity to a infinitely

15Of course, the spin-statistics theorem says that if φI is a scalar field, then it obeys Bose-Einsteinstatistics. However, we want to allow for the case where φI is a single real component of largermultiplet, which may obey Fermi-Dirac statistics. In this case, the polarization index I would alsorun over spin.

84

large quantity.

The major mistake in Equation (3.86) is that in the L → ∞ limit, the modedensity of mi states diverges, as can be seen from Equation (3.79). That is, themodes cease to be countable and discrete. As the limit is taken, we should pass froma state labeling in terms of quantum numbers mi to a labeling in terms of physicalmomenta ki, with a mode density determined by the differential limit of Equation(3.79). The sums over mi then become integrals over ki as

〈U〉 L→∞→∫

dd−1kLd−1

(2π)d−1

pωk

eβωk − ξ, (3.87)

where now ωk is understood as√

M2 + kiki. The factors of 2 in the denominator ofthe measure arise because the integrals over the ki run over both positive and negativevalues, 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 candefine the spectral energy density over ki modes.

For the current case of Cartesian coordinates, the failure of Equation (3.86) andits resolution by passing from quantum numbers to physical momenta are obviousand the discussion has been overly pedantic. Similar failures will be encounteredwhen using spherical coordinates to study the spectral density of angular momenta inblackbody radiation, however in that case the nature of the problem and its solutionwill not be as obvious.

Using the spherical symmetry of the infinite volume limit and defining k =√kiki = |ki|, Equation (3.87) becomes

〈U〉V

= ρ = pVol (Sd−2)

(2π)d−1

∫ ∞

0

dkkd−2√

M2 + k2

eβ√

M2+k2 − ξ. (3.88)

This defines the spectral energy density over the magnitude of the spatial momentum,via ρ ≡

∫

dkuk(k), as

uk(k) = pVol (Sd−2)

(2π)d−1

kd−2√

M2 + k2

eβ√

M2+k2 − ξ. (3.89)

Similarly, we can define the spectral energy density over the frequency16 as

uω(ω) = pVol (Sd−2)

(2π)d−1

ω2(ω2 −M2)(d−3)/2

eβω − ξ, (3.90)

where ω runs over [M,∞].

The total energy density can now be evaluated using either Equation (3.89) orEquation (3.90). Simple analytic results can be found for the case M = 0, which willalso apply when T ≫M . In this case, ω = k and Equations (3.89) and (3.90) match.

16Note, dk = ωdω√ω2−M2

.

85

They give

ρ = pT d Vol (Sd−2)

(2π)d−1

∫ ∞

0

dxxd−1

ex − ξ. (3.91)

The integral can be evaluated by pulling the exponential into the numerator, per-forming 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

a

xd−1dx

ex − ξ=

(

1− 2

2d

)(1−ξ)/2

ζ(d)(−1)d−1 dd−1

dαd−1

[

1

α

(

e−αa − e−αb)

]

∣

∣

∣

α=1

=

(

1− 2

2d

)(1−ξ)/2

ζ(d)(d− 1)!d−1∑

n=0

ane−a − bne−b

n!, (3.92)

where ζ(s) is the famous Riemann zeta function, which can be defined for real s > 1as

ζ(s) =∞∑

n=1

1

ns. (3.93)

This series arises in the evaluation of the integral with the ξ = 1, for bosons. Forfermions, ξ = −1, the corresponding series is

R(s) ≡∞∑

n=1

(−1)n−1 1

ns. (3.94)

This series can be evaluated using

ζ(s)−R(s) =2

2s+

2

4s+

2

6s+ . . . =

2

2sζ(s), (3.95)

so that

R(s) =

(

1− 2

2s

)

ζ(s), (3.96)

which is the origin of this factor in Equation (3.92). So, Equation (3.91) becomes

ρ = p

(

1− 2

2d

)(1−ξ)/2ζ(d)(d− 1)!

2d−2Γ(

d−12

)

π(d−1)/2T d. (3.97)

3.3.2 Flux Versus Energy Density

In the previous section, we considered the energy density, ρ, of a massless scalar fieldin a infinitely large hypercubic cavity at temperature T . We now want to calculatethe energy flux (power per unit area), Φ, emitted from a blackbody with this sametemperature. Any number of textbook arguments can be followed to derive an ex-pression for Φ. The key insight in all of them is that thermal equilibrium requiresthe condition of detailed balance on every mode of the system, which is a far morerestrictive condition than simply total energy conservation.

86

At the end of the day, the calculation can be cast as finding the flux through achosen spatial boundary. We chose a set of (d− 1)-dimensional spherical coordinatesfor the spatial momenta at some given point of the boundary using the same set ofangular conventions introduced in Section 3.2.1. We orient the coordinates such thatthe vector normal to the boundary at the chosen point lies at the “north pole” of thecoordinates. Then the calculation of the flux proceeds exactly as the calculation ofthe energy density, with two key differences. First, the polar angle θd−2 only rangesover [0, π/2] instead of [0, π] because we are interested in the flux passing in only onedirection through the surface. Second, there will be an additional factor of cos (θd−2)multiplying the phase space measure to account for the scalar product between theflux vector and the area element of the boundary.

The only change to the mathematics of Section 3.3.1 is in the passing from Equa-tion (3.87) to Equation (3.88). We used there dd−1k = dd−2Ωkd−2dk and

∫

dd−2Ω =Vol (Sd−2). For the flux calculation, instead of Vol (Sd−2) we will encounter

∫

dd−2Ω cos (θd−2)Θ(π/2− θd−2)

=

∫ 2π

0

dθ1

∫ π

0

dθ2 · · ·∫ π

0

dθd−3

∫ π/2

0

dθd−2

cos (θd−2)

d−3∏

i=2

(

sin (θi))i−1

=2π(d−2)/2

(d− 2)Γ(

d−22

) =1

d− 2Vol (Sd−3) = Vol (Bd−2)

=1√

π(d− 2)

Γ(

d−12

)

Γ(

d−22

) Vol (Sd−2), (3.98)

where Θ(x) is the step function and Bn is the n-dimensional unit ball17: the compactsubspace of Rn bounded by Sn−1. The fact that the expression for the energy densitybecomes that for the flux when Vol (Sd−2) is replaced by Vol (Bd−2) makes physicalsense, since B

n is the projection of Sn onto R

n.

We are left with the relationship of flux to energy density as

Φ =Γ(

d−12

)

√π(d− 2)Γ

(

d−22

)ρ =Vol (Bd−2)

Vol (Sd−2)ρ. (3.99)

Thus, the d-dimensional Stefan-Boltzmann law is given by

Φ = p

(

1− 2

2d

)(1−ξ)/2ζ(d)(d− 1)!

2d−2(d− 2)Γ(

d−22

)

πd/2T d. (3.100)

Thermodynamic relations like Equations (3.97) and (3.100) are independent of themode 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

17For example, B2 is the unit disc of volume π bounded by the circle S1 of area 2π. Also, B3 isthe unit 3-ball of volume 4π/3 bounded by the sphere S2 of area 4π.

87

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. SinceEquations (3.99) and (3.100) are somewhat ambiguous for d = 2, we explicitly listthe results for that case here:

Φ =1

2ρ =

(

1

2

)(1−ξ)/2pπ

12T 2. (3.101)

3.3.3 Spherical Blackbody Cavity

We will now repeat the calculation of Section 3.3.1, but now in a spherical cavity ofradius R instead of a hypercubic cavity. We define spherical coordinates as in Section3.2.1 with f = h = 1 and decompose φI in partial waves as in Section 3.2.3. The fieldstill obeys Equation (3.77), but instead of expanding classical solutions in the planewave basis (3.78), we separate variables as in Section 3.2.3:

φI(x) =1

rd−22

eiωktRkl(r)Yml (Ω), (3.102)

where the (d− 2)-spherical harmonics Y ml were defined in Section 3.2.3.1, the radial

part of the field obeys Equation (3.67):

[

∂2r −

(l + (d− 4)/2)(l + (d− 2)/2)

r2+ k2

]

Rkl(r) = 0, (3.103)

and, again, k2 = ω2k −M2.

As discussed in Section 3.2.3.2, Equation (3.103) is solved by

1

r(d−2)/2Rkl(r) = k(d−2)/2Jl(kr) ≡

√k( π

2r

)(d−3)/2

jl+(d−3)/2(kr), (3.104)

where jν(z) is the νth cylindrical Bessel function. The boundary condition at r = Rdemands that Jl(kR) = 0. This quantizes the allowed values of k as

k =λnl

R , (3.105)

where the pure numbers λnl are defined by Jl(λnl) = 0.

The modes are now labeled by the quantum numbers n, l,m, I. The quan-tization of the angular quantum numbers was discussed in Section 3.2.3.1. Thenew radial quantum number, n, takes all positive integer values. Upon second-quantization of the φI , the energy of a singly occupied mode is given by the frequency,ωnl =

√

M2 + λ2nl/R2. Analogously to Equation (3.80), the total energy in the cavity

for a pure quantum state is given by

U =

p∑

I=1

∞∑

n=1

∑

l,m

ωnlnnlmI , (3.106)

88

where the integers nnlmI are the occupation numbers specifying the state18.

Except for the detailed labeling of modes, the partition function for a thermal stateis computed exactly as it was in Equations (3.82) and (3.83). In fact, the expressionsare somewhat simpler because the energy only depends on the two quantum numbersn and l:

lnQ = −pξ∞∑

n=1

∞∑

l=0

Dd(l) ln(

1− ξe−βωnl)

. (3.107)

The angular momentum degeneracy Dd(l) is given by Equation (3.64). The averageoccupation number of a given n, l mode in the thermal state is then given by

〈nnl〉 = − 1

β

∂

∂ωnlln Q =

pDd(l)

eβωnl − ξ. (3.108)

Likewise, the total energy in the cavity is given by

〈U〉 = − ∂

∂βln Q =

∞∑

n=1

∞∑

l=0

pDd(l)ωnl

eβωnl − ξ. (3.109)

This had better be proportional to the volume of the cavity VR = Vol (Bd−1)Rd−1.

As in the case of the hypercubic cavity, we cannot simply take the R → ∞ limitof Equation (3.109), interpret the result as a spectral density unl(n, l), and then sumover n, l to get the total energy. For the reasons discussed in Section 3.3.1, such aprocedure gives meaningless results. One’s intuition from commonly interpreting thel quantum number as the actual, physical angular momentum of a wave mode wouldlead one to believe that in the large R limit, the n integral can be performed in someproperly regulated way, leaving a quantity that can be appropriately interpreted asthe angular momentum spectral density ul(l) which should obey

〈U〉 =∞∑

l=0

ul(l). (3.110)

Unfortunately, this is not the case. Instead, the l quantum number must be regulateddue to a diverging density of states by passing to a new variable ℓ = l/R. This new ℓhas units of linear momentum19 and actually would be the physical linear momentum

18Care should be taken not to confuse the symbols for occupation number and radial quantumnumber. Likewise, the symbol m is being used differently in this section than it was used in Section3.3.1.

19The quantum numbers l,m were never really the angular momentum operator eigenvaluesto begin with. As defined in Section 3.2.3.1, they are more properly referred to as the canonicalmomenta 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 1

2 (d − 1)(d − 2)dimensional. Of course, a wonderful point of connection between group theory and mechanicsoccurs here in that the d− 2 conjugate momenta form a set consisting of the Casimir operator forthe representation and a Cartan subalgebra of SO(d− 1), which is why these quantum numbers areall that is needed to label physical states.

89

if space were an Sd−2 of radius R. Also, with this definition we get

Dd(l)R→∞=

2ℓd−3

(d− 3)!Rd−3. (3.111)

We will approach the radial part of Equation (3.109) by eliminating the sum overn in favor of a sum over physical radial momentum k, which is simply related to λby Equation (3.105). This will require knowing the phase space density ∂n/∂λ in thelimit of large R. We will evaluate now ∂n/∂λ by two independent methods.

3.3.3.1 Radial Mode Density

One approach to finding ∂n/∂λ is to find an approximate explicit expression for λnl

which is valid in the limit R →∞. We begin on this route with an asymptotic seriesexpression for jn(z) of the Hankel type (large argument, fixed order) [89, SectionIII.3.14.1, for example]:

√

πz

2jn(z) = cos

(

z − π(n + 12)

2

)

[ ∞∑

m=0

(−1)mΓ(n + 12

+ 2m)

(2m)!Γ(n + 12− 2m)

(2z)−2m

]

− sin

(

z − π(n + 12)

2

)

[ ∞∑

m=0

(−1)mΓ(n + 12

+ 2m + 1)

(2m + 1)!Γ(n + 12− 2m− 1)

(2z)−2m−1

]

. (3.112)

Thus, we have

(

2

π

)(d−4)/2

Rkl(z/k) =π

2

(

2z

π

)(d−2)/2

Jl(z) =

√

πz

2jl+(d−3)/2(z)

= cos

(

z − π(2l + d− 2)

4

)

[ ∞∑

m=0

(−1)mΓ(l + (d− 2)/2 + 2m)

(2m)!Γ(l + (d− 2)/2− 2m)(2z)−2m

]

− sin

(

z − π(2l + d− 2)

4

)

[ ∞∑

m=0

(−1)mΓ(l + (d− 2)/2 + 2m + 1)

(2m + 1)!Γ(l + (d− 2)/2− 2m− 1)(2z)−2m−1

]

.

(3.113)

Since we will be writing l = Rℓ for large R, we can apply the following math fact: ifb is an integer, then

Γ(a + 1− b)

Γ(a + 1− 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)− 1 · 2][a(a + 1)]

≈[a(a + 1)]b, (3.114)

90

where the last line holds when a≫ b. Then we have

(

2

π

)(d−4)/2

Rkl(z/k)

R→∞≈ cos

(

z − π

2

[

l +d− 2

2

]) ∞∑

m=0

(−1)m

(2m)!

(

[l + (d− 4)/2][l + (d− 2)/2]

2z

)2m

− sin

(

z − π

2

[

l +d− 2

2

]) ∞∑

m=0

(−1)m

(2m + 1)!

(

[l + (d− 4)/2][l + (d− 2)/2]

2z

)2m+1

= cos

(

z − π

2

[

l +d− 2

2

])

cos

(

[l + (d− 4)/2][l + (d− 2)/2]

2z

)

− sin

(

z − π

2

[

l +d− 2

2

])

sin

(

[l + (d− 4)/2][l + (d− 2)/2]

2z

)

= cos

(

z − π

2

[

l +d− 2

2

]

+[l + (d− 4)/2][l + (d− 2)/2]

2z

)

. (3.115)

So,

Jl(z)z→∞≈ 2

π

( π

2z

)(d−2)/2

cos

(

z − π

2

[

l +d− 2

2

]

+[l + (d− 4)/2][l + (d− 2)/2]

2z

)

.

(3.116)This vanishes when z = λnl obeys

λnl −π

2

[

l +d− 2

2

]

+[l + (d− 4)/2][l + (d− 2)/2]

2λnl

= π

(

n− 1

2

)

, (3.117)

such that

∂n

∂λ=

1

π

(

1− [l + (d− 4)/2][l + (d− 2)/2]

2λ2

)

≈ 1

π

√

1− [l + (d− 4)/2][l + (d− 2)/2]

λ2. (3.118)

This last expression holds for large values of λ, such that λ2nl ≫ l2, Thus, the mode

hierarchy implicit here is λ≫ l ≫ 1, or equivalently R = λ/k ≫ Rℓ/k ≫ 1/k. Thismakes sense, since k and ℓ are physical momenta which should have finite values asR→∞.

Equation (3.118) can also be derived by the method of Ari Turner [90], which isto look at the WKB solution to (3.103). This is

Rnl(r) ≈ V (r)−1/4 exp

i

∫ r

r0

√

V (x)dx

, (3.119)

with

V (r) = k2 − [l + (d− 4)/2][l + (d− 2)/2]

r2. (3.120)

91

Taking an appropriate real solution, this will vanish when the phase is πn. So,

πn =

∫ λnl

kr0

√

1− [l + (d− 4)/2][l + (d− 2)/2]

x2dx, (3.121)

which again yields

∂n

∂λ=

1

π

√

1− [l + (d− 4)/2][l + (d− 2)/2]

λ2. (3.122)

3.3.3.2 Spectral Densities

We can now return to Equation (3.109). Using R → ∞, either of Equations (3.118)and (3.122) become

∂n

∂λ=

1

R∂n

∂k=

1

π

√

1− ℓ2

k2. (3.123)

Note that the square root sets either a lower limit on k or an upper limit on ℓ,depending on the order of integration. We get

〈U〉 =

∫ ∞

0

dk

∫ k

0

dℓR∂n

∂k

2pℓd−3Rd−3ωk

(d− 3)!(eβωk − ξ)

=

∫ ∞

0

dk

∫ k

0

dℓ2pRd−1

π(d− 3)!

√

1− ℓ2

k2

ℓd−3ωk

eβωk − ξ

=VR

∫ ∞

0

dk

∫ k

0

dℓ2p

π(d− 3)! Vol (Bd−1)

√

1− ℓ2

k2

ℓd−3ωk

eβωk − ξ

=VR

∫ ∞

0

dkp Vol (Sd−2)

(2π)d−1

kd−2ωk

eβωk − ξ, (3.124)

which is in perfect agreement20 with Equation (3.88). As in Section 3.3.1, this yieldsa spectral energy density over the radial momentum given by Equation (3.89) whichcan 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 ofthe thermal state. The methods of Section 3.3.2 could then be applied to give theblackbody flux.

We still need to calculate the new physical quantity made available by analysis inspherical coordinates: the angular momentum spectral energy density. Reversing the

20The ℓ integral in Equation (3.124) was performed using∫ 1

0 dxxn√

1− x2 =√

π4

Γ[(n+1)/2]Γ[(n+4)/2] . We

also used the identity Γ(

n+22

)

Γ(

n+12

)

= n!√

π2n .

92

order of integration in Equation (3.124), we find

〈U〉VR

= ρ =2p

π(d− 3)! Vol (Bd−1)

∫ ∞

0

dℓ

∫ ∞

ℓ

dk

√

1− ℓ2

k2

ℓd−3ωk

eβωk − ξ

=2p

π(d− 3)! Vol (Bd−1)

∫ ∞

0

dℓℓd−3

∫ ∞

√ℓ2+M2

dω

√ω2 −M2 − ℓ2

ω2 −M2

ω2

eβω − ξ. (3.125)

Some remarks are in order regarding the form of Equation (3.125). Note that thed-dependence has dropped out of the k or ω integrals. This is because these integralsdescribe the thermodynamics of only the t−r section of spacetime, which is indepen-dent of dimension. In fact, for low partial wave modes (ℓ≪M) or large masses, thefrequency form of Equation (3.125) yields

uℓ(ℓ) ≈pDd(ℓ)

Vol (Bd−1)

∫ ∞

M

dω1

π

1√ω2 −M2

ω2

eβω − ξ, (3.126)

suggesting, via Equation (3.90), an infinite collection of massive 1 + 1 dimensional

modes carrying an internal quantum number ℓ with degeneracy pDd(ℓ)Vol (Bd−1)

, each ther-mally occupied according to standard 1 + 1 dimensional physics. This degeneracyfactor is just the normal counting of angular momentum states, taken to the contin-uum limit. Thus Equation (3.126) is telling us that all the original l,m modes withℓ≪M are all uniformly occupied like 1 + 1 dimensional blackbodies.

Another important limit of Equation (3.125) is high temperature (T ≫M, ℓ). Inthis limit, the mass and angular momentum contribute to the ω integral only throughthe lower integration limit, so we can use Equation(3.92) to get

uℓ(ℓ) ≈πpT 2ℓd−3

3(d− 3)! Vol (Bd−1)e−

√ℓ2+M2/T

(

1

2

)(1−ξ)/2

. (3.127)

Equation (3.127) should be valid for both massive and massless fields near ℓ = 0, aslong as T ≫M .

For M = 0, Equation (3.125) yields

uℓ(ℓ) =2p

π(d− 3)! Vol (Bd−1)T 2ℓd−3Iξ(βℓ), (3.128)

where

Iξ(a) =

∫ ∞

a

dx

√x2 − a2

ex − ξ. (3.129)

Note that I+(0) = π2/6 and I−(0) = π2/12. The function Iξ(a) is plotted in Figure3-5. For ℓ≪ T we again find Equation (3.127), evaluated at M = 0. Again factoringout the angular degeneracy factor, Equation (3.127) implies that for ℓ ≪ T each of

93

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

1 2 3 4 5 6

I ξ(a

)

a

Figure 3-5: The thermal integral Iξ(a) defined in Equation (3.129). The solid topcurve indicates the bosonic function I+(a). The dashed bottom curve indicates thefermionic function I−(a). The dotted central curve indicates ξ = 0, corresponding toBoltzmann statistics. All three lines seem to converge towards e1−a for a≫ 1.

the original l,m modes contributes an amount

(

1

2

)(1−ξ)/2pπ

6T 2e−ℓ/T (3.130)

to the energy density ρ. At ℓ = 0, this agrees with Equation (3.101), showing onceagain 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 ofmasses, interactions, and angular momentum in the radial wave equation of a fieldare exponentially suppressed, with the suppression becoming exact at the horizon. Inparticular, this means that all modes of blackbody radiation near an event horizonbehave just like the ℓ = 0 mode, which makes no contribution to the effective radialscattering potential. Thus, we expect that the energy density in every partial waveof d-dimensional blackbody radiation at the event horizon of a d-dimensional blackhole is given exactly by the 1 + 1 dimensional result of Equation (3.101).

Figure 3-5 shows that, at least for large ℓ (ℓ≫ T ), we have

uℓ(ℓ) ∝2p

π(d− 3)! Vol (Bd−1)T 2ℓd−3e−ℓ/T . (3.131)

Equations (3.127) and (3.131) together should accurately reproduce the exact M = 0spectrum given in Equation (3.128) except for ℓ ≈ T . It is interesting to see thatapparently all the effects of statistics are in the small ℓ modes. Also, the Fermi-Dirac

94

factor appearing at small ℓ is 12, 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 blackbodyin the partial wave spectrum. Of course, when summed up over all values of ℓ, theFermi-Dirac factors must combine to give 1 − 2/2d, in order to properly reproduced-dimensional physics. It is not obvious from Equations (3.128) and (3.129) exactlyhow 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. Theeffective theory is formed by eliminating the dangerous horizon-skimming modes byhand — to avoid singularities in the energy-momentum tensor near the horizon —while simultaneously adding a compensating energy-momentum tensor — to avoidanomalies.

Consider the partial wave decomposition of a scalar field in a static, sphericallysymmetric background spacetime In suitable coordinates, the metric of the spacetimecan be written as in Section 3.2:

ds2 = −f(r)dt2 +1

h(r)dr2 + r2d2Ω. (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) arecoincident. In this scenario, we found that physics near the horizon can be describedusing an infinite collection of 1+1 dimensional fields, each propagating in a spacetimewith a metric given by the “r-t” section of the full spacetime metric (3.132). We willalso 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 takethis condition to be localized on a slab of width 2ε straddling the horizon at r = rh

with ε→ 0 ultimately (see Figure 3-1). The energy-momentum tensor in this regionthen exhibits an anomaly of the form (3.2).

For a metric of the form (3.132), the anomaly is purely time-like and can bewritten as

∇aTχab ≡ Ab ≡

1√−g∂aN

ab , (3.133)

where the components of Nab are

N tt = N r

r = 0, (3.134a)

N rt =

1

192π(f ′h′ + f ′′h) , (3.134b)

N tr =

1

192π∂2

r ln(

fαh1−α)

. (3.134c)

95

The parameter α is an arbitrary number with no effect on physics21 because Ab isindependent of N t

b .

The contribution to effective action for the metric gab due to matter fields thatinteract with this metric is given by

W[gab] ≡ −i ln

(∫

D[matter]eiS[matter,gab]

)

, (3.135)

where S[matter, gab] is the classical action functional. Under general coordinate trans-formations the classical action S changes by δλS = −

∫

ddx√−gλb∇aT

ab where T a

b isthe energy-momentum tensor and λ is the variational parameter.

General covariance of the full quantum theory requires δλW = 0. We write thisas

−δλW =

∫

d2x√−gλb∇a

TχabH + To

abΘ+ + Ti

abΘ−

=

∫

d2xλt

∂r(Nrt H) +

(√−gTort −√−gTχ

rt + N r

t

)

∂Θ+

+(√−gTi

rt −√−gTχ

rt + N r

t

)

∂Θ−

+

∫

d2xλr√−g(

Torr − Tχ

rr

)

∂Θ+ +(

Tirr − Tχ

rr

)

∂Θ−

(3.136)

where Θ± = Θ (±r ∓ rh − ε) are scalar step functions and H = 1 − Θ+ − Θ− is ascalar “top hat” function which is 1 in the region between rh ± ε and zero elsewhere.The anomalous chiral physics is described by Tχ

ab via Equation (3.133). The energy-

momentum tensors Toab and Ti

ab are the covariantly conserved energy-momentum ten-

sors outside and inside the horizon, respectively. Constancy in time and Equation(3.133) together restrict the form of the T a

b up to an arbitrary function of r, which isthe trace T a

a , and two constants of integration, K and Q:

T tt = −(K + Q)/f −B(r)/f − I(r)/f + T a

a (r), (3.137a)

T rr = (K + Q)/f + B(r)/f + I(r)/f, (3.137b)√−gT rt = −K + C(r) = −fhT t

r , (3.137c)

where we have defined C(r) ≡∫ r

rh

√−gAt(x)dx, B(r) ≡∫ r

rhf(x)Ar(x)dx, and I(r) ≡

12

∫ r

rhT a

a (x)f ′(x)dx.

A few remarks regarding the evaluation of Equation (3.137) are in order. A tracecould arise from a number of physical sources, among them a conformal anomaly. We

assume, however, that I/f∣

∣

∣

rh

= 12T a

a

∣

∣

∣

rh

is finite. Since we will be concerned with the

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

21Interestingly, for α = 1 the antisymmetric part of Nab is equal to −ǫabR/(192π), where R is thed = 2 Ricci scalar as given in Equation (3.10).

96

that for the diagonal terms in Equation (3.137), the limit r → rh depends on whetherrh is approached from above or below, since f flips signs as the horizon is crossed.The limit on 1/f is exactly antisymmetric, so

limr−rh→0−

1

f= −

(

limr−rh→0+

1

f

)

. (3.138)

Finally,√−g is a finite number at the horizon, given by

√

F/H in the notation ofSection 3.2.1.2.

We can now take the ε→ 0 limit of Equation (3.136). The term ∂r(Nrt H) vanishes

in this limit. Using the relation (3.138) to take all limits from above, and the smallε expansions,

∂aΘ± = δra

[

±1− ε∂r ± 12ε2∂2

r − . . .]

δ(r − rh), (3.139)

the variation (3.136) becomes

δλW =

∫

d2xλt [Ko −Ki] δ(r − rh)

−ε [Ko + Ki − 2Kχ − 2N rt ] ∂δ(r − rh) + . . .

−∫

d2xλr√

F

H

[

Ko+Qo+Ki+Qi−2Kχ−2Qχ

f

]

δ(r − rh)

−ε[

Ko+Qo−Ki−Qi

f

]

∂δ(r − rh) + . . .

. (3.140)

The ellipses represent higher order terms in ε with higher derivatives of δ-functions;the coefficients of these terms are simply repetitions of the ones given above. Thedelta functions in Equation (3.140) indicate that only the on-horizon values of theenergy-momentum tensors will contribute to the possible loss of general covariance.The finite trace terms make no contribution in comparison to the divergent K + Qterms.

Since λt and λr are independent arbitrary variational parameters, each of the fourterms in square brackets in Equation (3.140) must vanish simultaneously, but onlyneed do so at r = rh. These four conditions can be solved to give

Ko = Ki =Kχ + Φ, (3.141a)

Qo = Qi =Qχ − Φ, (3.141b)

where

Φ = N rt

∣

∣

∣

rh

=κ2

48π. (3.142)

These conditions fix the 4 of the 6 constants Q and K. The total energy momentumtensor

T ab = To

abΘ+ + Ti

abΘ− + Tχ

abH (3.143)

becomes, in the limit ε→ 0,T a

b = Tcab + TΦ

ab , (3.144)

97

where Tcab is the conserved energy-momentum tensor which the matter in this theory

would have without any quantum effects, and TΦab is a conserved tensor with K =

−Q = Φ, a pure flux.

As discussed in Section 3.3.2, a beam of massless blackbody radiation moving inthe positive r direction at a temperature T has a flux of the form Φ = π

12T 2. Thus

we see that the flux required to cancel the gravitational anomaly at the horizon hasa form equivalent to blackbody radiation with a temperature given by T = κ/(2π).This is exactly the Hawking temperature for this spacetime, as discussed in Sections3.2.2.1 and 3.2.2.2. Thus, the thermal flux required by black hole thermodynamics iscapable of canceling the anomaly. If we fill each partial wave of the full d-dimensionaltheory so that each one behaves like a 1 + 1 dimensional blackbody source at theHawking temperature, then we reproduce the core of the standard calculation ofblack hole emission. This is exactly what one requires such that when the partialwaves are propagated outwards from the black hole and undergo mode-dependentscattering from effective potential due to spatial curvature outside the horizon, theresultant occupation density at infinity is that of of a d-dimensional blackbody atthe Hawking temperature, modulo greybody factors. As shown in Section 3.3, theresulting radiation observed at infinity is that of a d-dimensional greybody at theHawking temperature.

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 arbitrarydimensions using partial wave analysis. In that framework the connection betweenthe anomaly and the Hawking flux is made through an integral over all of space. Inour framework the connection between the anomaly and the Hawking flux is madethrough a boundary condition at the horizon, which is accurately described using1 + 1 dimensional physics, irrespective of the true dimension.

Comparing the fluxes for thermal radiation of massless bosons and fermions in1 + 1 dimensions, we find, as in Equation (3.101), that the boson flux is twice thatof the fermion flux. This same factor of two appears in the relative values of theconformal anomalies (central charge) and of the gravitational anomalies. There doesnot appear to be any comparably simple correspondence in higher dimensions. How-ever, when the 1 + 1 dimensional field in question is really a single partial wave of ahigher dimensional system, the blackbody angular momentum spectrum discussed inSection 3.3.3.2 guarantees us that this factor of two arising from the 1+1 dimensionalanomaly adds up to an overall (1− 2/2d)−1, which is the proper statistical factor ind dimensions.

In the context of an eternal black hole one can find a role for thermal radiationincoming to the black hole by imposing additional boundary conditions near the pasthorizon (V = 0, in the language of Section 3.2.2) that are symmetric with the oneswe imposed above near the future horizon. This corresponds to the Hartle-Hawkingstate [91].

98

3.6 Blackbody Spectrum from an Enhanced Sym-

metry?

While the arguments advanced here show a pleasing consistency between the existenceof Hawking radiation flux and gravitational anomaly cancellation, they do not inthemselves suffice to show that the spectrum of radiation is thermal. One mighthope to single out the thermal state by imposing an appropriate symmetry. Indeed,thermal states support a form of time-translation symmetry that makes sense evennear the horizon, namely translation by discrete units β of imaginary time.

Specifically, in Section 3.2.2.2 we generalized the Euclidean method of [42] toshow that the Euclidean section of the black holes we are studying contains a conicalsingularity at the horizon unless the period of Euclidean time is taken to be β = 1/TH.If multivalued coordinates are allowed, however, the singularity can be removed bytaking the period to be any integer multiple22 of this factor, β = n/TH, giving adiscrete set of allowed temperatures for this spacetime, Tn = 1

nTH.

Perhaps less objectionable is consideration of the Kruskal coordinate U . We ar-gued in Section 3.2.2.1 that respecting translation in U as a good spacetime symmetryat the past horizon, V = 0, can be seen to give rise to Hawking radiation. We alsonoted, by construction, that t-translation is a global Killing vector. From the defini-tion (3.41a), we see that under a time translation t→ t + a, U → e−κaU . If we wantHawking radiation modes to be properly analytic in the complex frequency plane, weshould demand that that U is invariant under this transformation when a is a pureimaginary number. This demands that a = 2πin/κ, for any integer n. This saysthat the black hole must be invariant under discrete translations of imaginary timeof magnitude β = 2πn/κ, which again gives a discrete set of allowed temperatures,Tn = 1

nTH.

While neither of these arguments is quite satisfying, they do make plausible theidea that an enhanced discrete translation symmetry can fix the thermal nature of theblack hole system, and that temperatures of different magnitude can be accommo-dated as different units for the periodicity by T = 1/β. If we assume that a symmetryof this form exists, then anomaly cancellation fixes the unit. One could certainly wishfor a less formal, more physically enlightening perspective, however.

22With this integer factor n, one must travel in a circle around the origin a distance 2πn inEuclidean 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 τ -x plane of Equation(3.43), but their physical significance is uncertain.

99

100

Chapter 4

Finale

4.1 Summary

In Chapter 2, we found that the addition of gravity to four-dimensional Yang-Millstheories adds a term to the one-loop β functions of the form

∆βgrav(gYM, E) = −g

YM

3

π

E2

M2P

, (4.1)

which renders all Yang-Mills couplings asymptotically free. To some extent, thisresult 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 ofquantum gravity is unexpectedly low. We also found that the gravitational correctiondoes not spoil coupling constant unification in a theory whose Yang-Mills couplingsexhibited unification before gravity was added. The unification point remains uniqueand 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 classof spherically symmetric black holes in arbitrary dimensions can be understood asarising from a gravitational anomaly in the 1 + 1 dimensional effective quantum fieldtheory that governs physics close to the horizon. This helps to raise the connection ofanomalies to Hawking radiation above the level of an isolated curiosity. We did notprove definitively that anomaly cancellation requires the radiation to be thermal, butargued that the radiation has several features in common with blackbody radiationat the Hawking temperature.

4.2 Open Possibilities

As mentioned in Section 2.11, there are several questions raised by the calculationof the gravitational correction to the Yang-Mills β function. For example, how isthe calculation implemented in TeV-scale gravities and what are the experimentalsignatures thereof in colliders, cosmic rays, and atomic systems? Also, how the does

101

calculated correction effect the fitting of experimental data with uncertainties to thehypothesis of coupling constant unification?

An obvious program is to extend the calculation to Yukawa couplings and scalarself-interactions. It would be rather astonishing if these couplings were also renderedasymptotically free by gravitation, since they can not be asymptotically free in fourdimensions without gravity. Mass renormalization could also be studied, but this isnot as interesting since we expect gravity-induced renormalization to be importantonly at scales much higher than any conventional particle physics masses.

The derivation in Section 3.4 of Hawking radiation as a mechanism that cancelsa gravitational anomaly also leaves some open questions. Obvious among these iswhether 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 theorycan be calculated from the anomaly such that the thermal nature of the radiation iscompletely elucidated. One might further wonder if this mechanism has any relevanceto the black hole information paradox or whether it can be used to calculate blackhole entropy. The answer to these last questions is probably “no” because Hawkingradiation is a kinematic effect, while information and entropy issues necessarily involvedynamics. However, these questions do deserve a deeper investigation.

Our study of black hole effective field theory was motivated by rather generalconcerns regarding an observer’s ability to describe physics in terms of degrees offreedom which he can experimentally probe. The essential difficulty in doing thiswas encountered most directly in a semiclassical black hole background, but the sameproblem should generically arise in any generally covariant quantum field theory.Perhaps the anomaly cancellation mechanism discovered here can be generalized toallow a generally covariant formulation of quantum field theory in which local quan-tum field theories formulated on local coordinate patches of spacetime are stitchedinto a global theory by anomaly-driven currents that act as a kind of connection. Ifsuch a framework existed it would, in a certain sense, make no essential distinctionbetween fundamental and effective degrees of freedom.

4.3 Conclusion

We hope that we have shown in this thesis that much can be learned within thelimited scope of quantum general relativity and semiclassical gravity if the logicalboundaries of these models are properly respected. Furthermore, we believe that thework presented here points in a promising way towards features that should ultimatelybe reproducible in a more fundamental theory of quantum gravity.

102

Bibliography

[1] D. J. Gross and F. Wilczek, “Ultraviolet Behavior Of Non-Abelian Gauge The-ories,” Phys. Rev. Lett. 30, 1343 (1973).

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