Chapter Black Hole Thermodynamics and Perturbative Quantum Gravity Dmitri V . Fursaev

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Chapter “Black Hole Thermodynamics and

Perturbative Quantum Gravity”

Dmitri V. Fursaev

Abstract An introduction to generalized thermodynamics of quantum black holes,

in the one-loop approximation, is given. The material is aimed at graduate students.

The topics include: quantum evaporation of black holes, Euclidean formulation of

quantum theory on black hole backgrounds, the Hartle-Hawking-Israel state, gener-

alized entropy of a quantum black hole and its relation to the entropy of entangle-

ment.

Keywords

Black holes, thermodynamics, statistical mechanics, quantum entanglement.

1 Introduction

Although the perturbative quantum gravity approach has a limited range of appli-

cability, its use in the last decades led to some conceptual issues which are to be

addressed in the full-ﬂedged quantum gravity theory. The most important issues in-

clude understanding evaporation of quantum black holes and resolution of the infor-

mation loss paradox, as well as ﬁnding a microscopic origin of black hole entropy.

Black holes are speciﬁc solutions of the Einstein equations,

Rµν −1

2gµν R=8πG

c4Tµν ,(1)

Dmitri V. Fursaev

Dubna State University, Universitetskaya str. 19, Dubna, Moscow Region, Russia, e-mail:

fursaev@theor.jinr.ru

arXiv:2210.06081v1 [gr-qc] 12 Oct 2022

2 Dmitri V. Fursaev

which describe regions of a space-time where the gravitational ﬁeld is so strong that

nothing, including light signals, can escape them. The interior of a black hole is hid-

den from an external observer. The boundary of the unobservable region is called

the horizon. In (1) we use standard notations Rµν ,R,Tµν for the Ricci tensor, the

scalar curvature, and the stress-energy tensor of matter, respectively. Gis the New-

ton constant. The Schwarzschild and Kerr black holes are solutions to the vacuum

equations (1) with Tµν =0.

In recent years our understanding of physics near the black hole horizon received

important experimental evidences on the base of direct detection of gravitational

waves from binary black hole mergers [1] and observations of shadows of the super-

massive black holes [2].

The perturbative quantum gravity, in this Chapter, is treated in the one-loop ap-

proximation or as a theory of free quantum ﬁelds on black hole geometries. By

explaining quantum effects near black holes in these rather restricted models we

come to important insights which have been a matter of intensive discussions in a

large number of publications.

The concrete aim of this Chapter is to give a self-consistent introduction to gen-

eralized thermodynamics of quantum black holes, accessible to graduate students.

The material is organized as follows. We start in Sec. 2with a brief description of

black hole solutions by focusing mostly on the Killing structure of the black hole

horizon and near-horizon features which are needed do deﬁne the ﬁrst law of black

hole mechanics. Quantization of free ﬁelds on external backgrounds is presented

in Sec. 3. The essence of the Hawking effect is discussed in Sec. 4, by using the so

called s-mode approximation. Thermodynamics of classical black holes is discussed

in Sec. 5. The basic concept, the Hartle-Hawking-Israel state, which we use to study

quantum black holes, is introduced in Sec. 6. We also give here some elements of a

spectral theory of second order elliptic operators and deﬁne the Euclidean effective

action. From a point of view of stationary observers quantum matter near black hole

horizon is in a high-temperature regime. Hence some features of high-temperature

hydrodynamics in gravitational ﬁelds are discussed in Sec. 7. Finally, in Sec. 8we

consider generalized thermodynamics of quantum black holes, and, in particular,

generalized black hole entropy. Quantum corrections to the entropy are discussed in

detail. We introduce the notion of entanglement entropy and show that the general-

ized entropy is partly related to entanglement of states across the black hole horizon.

Section 9contains concluding comments .

We include in this Chapter almost all required deﬁnitions and try to show how

basic relations can be derived . We use the system of units where ¯

h=c=kB=1

(kBis the Boltzmann constant), the Lorentzian signature is deﬁned as (−,+,+,+),

geometrical conventions coincide with [59].

Chapter “Black Hole Thermodynamics and Perturbative Quantum Gravity” 3

2 Necessary deﬁnitions

We start with a brief description of basic properties of black hole geometries in the

near-horizon approximation. For a comprehensive introduction to black hole physics

see [59],[32]. A metric of a neutral rotating black hole, which is most interesting

from the point of view of physical applications, is the Kerr solution to the Einstein

equations (1) in vacuum, Tµν =0,

ds2=gtt dt2+grrdr2+2gtϕdtdϕ+gϕϕ dϕ2+gθ θ dθ2,(2)

gtt =−1−2MGr

Σ,gθθ =Σ,grr =Σ

∆,(3)

gtϕ=−2MGra

Σsin2θ,gϕϕ = ((r2+a2)2−a2sin2θ∆)sin2θ

Σ,(4)

Σ=r2+a2cos2θ,∆=r2−2MGr +a2.(5)

Metric (2) is written in the Boyer–Lindquist coordinates. The Kerr solution is

asymptotically ﬂat at large r. By analyzing its behavior at large rone concludes

that Mis the mass of the source, J=Ma is its angular momentum. It is supposed

that MG >a.

We also need the Schwarzschild solution, which follows from (2)-(5) when a=0,

ds2=gtt dt2+grrdr2+r2(sin2θdϕ2+dθ2),(6)

−gtt =g−1

rr =1−2MG

r.(7)

We denote by Hthe event horizon of a black hole. Has a null hypersurface

located at a constant radial coordinate r. By the deﬁnition, the normal vector lµto

a null hypersurface is null, l2=0. For constant rhypersurfaces lµ=δr

µ. Hence a

surface r=r0is null if grr(r0) = 0, or ∆(r0) = 0. This equation has two roots and H

corresponds to the largest root, r0=rH=MG +p(MG)2−a2. For eternal black

holes (see Fig. (1)) the horizon H=H+∪H−has two components, the future,

H+, and the past H−event horizons. The future light cone of any point on H+

is tangent to H+and is directed inside the black hole. Correspondingly, past light

cones on H−are directed inside the white hole. A detailed discussion of this can be

found in [59],[32].

One can consider observers which rotate with respect to the Boyer–Lindquist

coordinate grid (and therefore with respect to objects at the spatial inﬁnity) with

an angular coordinate velocity dϕ/dt =−gtϕ/gϕϕ . An important property of (2) is

that the only possible value for the angular velocity, when rapproaches rH, is

ΩH=a

a2+r2

.(8)

The parameter ΩHis called the angular velocity of the horizon.

4 Dmitri V. Fursaev

Fig. 1 Carter-Penrose diagrams for Minkowsky space-time (a) and for eternal Schwarzschild black

hole (b). Lines with arrows are integral lines of the Killing-vector ﬁeld. The Killing ﬁeld in the

Minkowsky space-time corresponds to Lorentz boosts. The Killing horizons H±intersect at bifur-

cation 2-surfaces. A constant time section ΣL∪ΣRon the black hole diagram is the Einstein-Rosen

bridge.

Although Kerr solution (2)-(5) looks complicated only few features of the near

horizon geometry are required for studying quantum effects we are interested in.

These features are related to the structure of time-like isometries and properties of

the so called Killing observers. A vector ﬁeld ζµ(x)on a manifold Mis called a

Killing ﬁeld if it generates isometries of M. The Killing ﬁeld obeys the Killing

equation

ζµ;ν+ζν;µ=0.(9)

For the Kerr solution there is a distinguished Killing vector ﬁeld, ζ=∂t−ΩH∂ϕ,

which is null on the horizon

ζ2|H=0.(10)

Since His the null hypersurface Eq. (10) implies that ζis a normal vector to H.

Properties of null hypersurfaces say that integral lines of ζon Hare geodesics:

∇ζζ=kζ.(11)

One can show that ∂ζk=0 on H, and, as a consequence, there is a 2D section, B

of Hwhere the Killing ﬁeld is zero,

ζµ|B∈H=0.(12)

The parameter kin (11) is called the surface gravity of the horizon, the section B

is called the bifurcation surface of the Killing horizons. Examples of Killing ﬁelds

with bifurcating horizons are shown on Fig. (1) for the case of Minkowsky space-

time and eternal Schwarzschild black hole geometry.

The Killing ﬁeld ζis time-like in the wedge to the right from H+and H−. In

this region one can deﬁne a frame of reference of observers whose 4-velocities uµ

are directed along ζ,

Chapter “Black Hole Thermodynamics and Perturbative Quantum Gravity” 5

uµ=ζµ/√B,B=−ζ2=−gtt −2ΩHgtϕ−Ω2

Hgϕϕ .(13)

Such observers are called the Killing observers. For a rotating black hole the given

frame of reference is deﬁned in a domain close to H, where B>0. The congruence

of the trajectories is speciﬁed [46] by the acceleration wµ, the rotation tensor Aµν

and the local angular velocity Ω(x)

wµ=∇uuµ,Aµν =1

2hλ

µhρ

ν(∇ρuλ−∇λuρ),Ω(x) = 1

2Aµν Aµν 1/2

,(14)

where hλ

µ=δλ

µ+uµuλ. Quantities (14) appear under study of quantum systems

in thermal equilibrium with the black hole, see Sec. 7.2. Local angular velocity

determines rotation of the Killing frame with respect to a local inertial frame.

One can use (11) to relate deﬁnition of the surface gravity to the strength of

gravity near the horizon,

k=lim

r→rH√Bw2=rH−MG

a2+r2

.(15)

The right hand side (r.h.s.) of (15) follows from (3)-(5), (13).

It is convenient to change in the Boyer–Lindquist coordinates ϕto ϕ0=ϕ−ΩHt.

In the new coordinates the Killing vector ﬁeld is ζ=∂t, that is, the Killing observers

do not move with respect to the new coordinate grid. Metric (2) can be rewritten as

ds2=−B(dt +aidxi)2+hi j dxidx j,(16)

where xi= (r,ϕ0,θ),aidxi=aϕdϕ0. The non-vanishing components of acceleration

and rotation are wi= (lnB),i/2, Ai j =√B(ai,j−aj,i)/2.

One can check that at small r−rH

B'4k2h rH(r−rH),hrr 'hrH

r−rH

+O((r−rH)2),h(θ)≡Σ(rH,θ)

2rHp(MG)2−a2.

(17)

Here we took into account that hrr =grr. It is convenient to introduce a new coordi-

nate ρ:

r−rH'ρ2

4rH

.(18)

connected with the proper distance Lto the horizon

L(r,θ) = Zr

dr0phrr '√hρ.(19)

In the leading approximation B'hk2ρ2. Since the local angular velocity van-

ishes near the horizon, Ω=O(ρ), terms aidxiin (16) can be neglected near H.

One comes to the following form of near-horizon black hole metric (2) :

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ChapterBlackHoleThermodynamicsandPerturbativeQuantumGravityDmitriV.FursaevAbstractAnintroductiontogeneralizedthermodynamicsofquantumblackholes,intheone-loopapproximation,isgiven.Thematerialisaimedatgraduatestudents.Thetopicsinclude:quantumevaporationofblackholes,Euclideanformulationofquantumtheory...

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