Semi-classical thermodynamics of quantum extremal surfaces

BRX-TH-6690

Semi-classical thermodynamics of quantum extremal

surfaces in Jackiw-Teitelboim gravity

Juan F. Pedraza,1,2 Andrew Svesko,1 Watse Sybesma3 and Manus R. Visser4

1Department of Physics and Astronomy, University College London,

London, WC1E 6BT, United Kingdom2Martin Fisher School of Physics, Brandeis University

Waltham MA 02453, USA3Science Institute, University of Iceland

Dunhaga 3, 107 Reykjavık, Iceland4Department of Theoretical Physics, University of Geneva

24 quai Ernest-Ansermet, 1211 Geneve 4, Switzerland

E-mail: [email protected], [email protected], [email protected],

[email protected]

Abstract: Quantum extremal surfaces (QES), codimension-2 spacelike regions which ex-

tremize the generalized entropy of a gravity-matter system, play a key role in the study of the

black hole information problem. The thermodynamics of QESs, however, has been largely un-

explored, as a proper interpretation requires a detailed understanding of backreaction due to

quantum fields. We investigate this problem in semi-classical Jackiw-Teitelboim (JT) gravity,

where the spacetime is the eternal two-dimensional Anti-de Sitter (AdS2) black hole, Hawk-

ing radiation is described by a conformal field theory with central charge c, and backreaction

effects may be analyzed exactly. We show the Wald entropy of the semi-classical JT theory

entirely encapsulates the generalized entropy – including time-dependent von Neumann en-

tropy contributions – whose extremization leads to a QES lying just outside of the black hole

horizon. Consequently, the QES defines a Rindler wedge nested inside the enveloping black

hole. We use covariant phase space techniques on a time-reflection symmetric slice to derive

a Smarr relation and first law of nested Rindler wedge thermodynamics, regularized using

local counterterms, and including semi-classical effects. Moreover, in the microcanonical en-

semble the semi-classical first law implies the generalized entropy of the QES is stationary

at fixed energy. Thus, the thermodynamics of the nested Rindler wedge is equivalent to

thermodynamics of the QES in the microcanonical ensemble.

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Contents

1 Introduction 2

2 JT gravity and semi-classical corrections 7

2.1 Classical JT and the eternal AdS2 black hole 7

2.2 Backreaction and vacuum states 9

2.2.1 Normal-ordered stress tensors and vacuum states 12

2.2.2 Backreacted solutions 16

3 Wald entropy is generalized entropy 20

3.1 Wald entropy 20

3.2 von Neumann entropy 23

3.3 Quantum extremal surfaces 27

4 Semi-classical thermodynamics of AdS2-Rindler space 31

4.1 Rindler wedge inside a Rindler wedge 33

4.2 Classical Smarr formula and first law 37

4.2.1 Smarr relation 38

4.2.2 First law of nested AdS-Rindler wedge mechanics 42

4.2.3 An extended first law 45

4.3 Semi-classical Smarr formula and first law 50

4.3.1 Smarr relation 50

4.3.2 First law of nested backreacted AdS-Rindler wedge mechanics 52

4.4 Thermodynamic interpretation: canonical and microcanonical ensembles 54

5 Conclusion 59

A Coordinate systems for AdS2 62

A.1 Conformal gauge identities and the stress-energy tensor 65

B Boost Killing vector from the embedding formalism for AdS2 67

C Covariant phase space formalism for general 2D dilaton gravity 74

C.1 Quasi-local and asymptotic energies with semi-classical corrections 80

D Derivation of the extended first law with variations of couplings 83

E A heuristic argument for the generalized second law 89

– 1 –

1 Introduction

Classical gravity posits black holes obey a set of mechanical laws formally reminiscent of the

laws of thermodynamics, where the horizon area is interpreted as a thermodynamic entropy

obeying a second law. Taking this analogy seriously, Bekenstein linked horizon area A with

entropy SBH via a series of gedanken experiments [1, 2], further established by Hawking who

revealed black holes emit radiation at a temperature TH proportional to their surface gravity κ

[3, 4]. Thus, black holes are genuine thermal systems, with a temperature and entropy, and

obey a first law. Specializing to neutral, static black holes, the first law is

δM = THδSBH , with TH =κ

2πand SBH =

A

4G, (1.1)

and the ADM mass M of the system is identified with the internal energy. When the thermal

system of interest includes a black hole as well as matter in the exterior of the black hole, the

total entropy is quantified by the generalized entropy Sgen, accounting for both the classical

black hole entropy SBH and the entropy of exterior matter Sm [5],

Sgen = SBH + Sm , (1.2)

such that a generalized second law δSgen ≥ 0 holds. Moreover, in the case of an asymptotically

flat black hole surrounded by a fluid, the gravitational first law (1.1) is modified by δHm,

encoding classical stress-energy variations outside of the horizon [6–8]:

δM = THδSBH + δHm . (1.3)

The classical Bekenstein-Hawking entropy formula and the subsequent thermodynamics have

been derived in a variety of ways. Notably, typically one either solves the wave equations of

quantum fields over the fixed classical black hole background [3, 9], or performs a Euclidean

path integral analysis where the logarithm of the partition function is given by the on-shell

Euclidean gravitational action [10]. Neither of these approaches address a fundamental con-

ceptual problem: how does the Hawking radiation know about the (micro)-dynamics of the

event horizon?1 Put another way, Hawking radiation is not standard black body radiation,

i.e., unlike standard electromagnetic radiation of a hot iron caused by energy released from

its atoms, Hawking radiation is understood as excited modes of an external quantum field,

one which is generally not coupled to the background spacetime. Thus, it is unclear why the

Hawking temperature should really be identified with the temperature of the black hole, and,

consequently, why the first law of black hole mechanics should be identified as a first law of

thermodynamics. Indeed, the first law (1.1) is derived from the dynamical gravitational field

equations, without any mention of quantum field theory. It is only after identifying TH ∝ κ

as the black hole temperature that the thermodynamic interpretation of the first law follows.

A crucial ingredient missing from either derivation of black hole thermodynamics is the

effect of backreaction, i.e., the influence quantum matter has on the classical background

1We thank Erik Curiel for bringing this to our attention.

– 2 –

geometry. It is only when these semi-classical effects are included that the aforementioned

conceptual puzzle can be fully addressed since the Hawking radiation backreacts on the geom-

etry, and thus the temperature appearing in the semi-classical first law is naturally interpreted

as the black hole temperature.

Accounting for these backreaction effects requires solving the semi-classical gravitational

field equations, e.g., the semi-classical Einstein equations,

Gµν(g) = 8πG〈Ψ|Tµν(g)|Ψ〉 , (1.4)

where 〈Ψ|Tµν(g)|Ψ〉 is the (renormalized) quantum matter stress-energy tensor, given by

the expectation value of the stress-energy tensor operator in the quantum state of matter

|Ψ〉, and is representative of the backreaction. These gravitational field equations are only

valid in a semi-classical regime with quantum matter fields in a classical dynamical spacetime.

Explicitly solving the semi-classical equations (1.4) is notoriously difficult as it requires one to

simultaneously solve a coupled system of geometry gαβ and correlation functions of quantum

field operators. Typically, in the case of four dimensions and higher, one studies the problem

perturbatively. However, this offers limited insight, especially when backreaction effects are

large.

To clarify, one may formally use the semi-classical Einstein equations (1.4) to derive a

first law which includes semi-classical corrections [8]. The matter Hamiltonian variation δHm

in the first law (1.3) is then replaced by the variation of the expectation value of the quantum

matter Hamiltonian δ〈Hm〉. Invoking the first law of entanglement for the quantum matter

fields, δSm = THδ〈Hm〉, the semi-classical version of the first law (1.3) becomes

δM = THδSgen , (1.5)

where we identified the generalized entropy (1.2). Now TH may be interpreted as the tem-

perature of the backreacted black hole. However, one typically has no computational control

over the backreacted black hole solution.

Important to the thermodynamic interpretation is understanding the correct statistical

ensemble one is considering, involving extremization of the thermodynamic potential of the

ensemble, e.g., for the microcanonical ensemble M is held fixed and Sgen is extremized.

Generally, and particularly in spacetime dimensions D ≥ 4, however, the backreaction in

black hole spacetimes has not yet been be explicitly solved analytically. Consequently, in

such contexts, one cannot generically find the extrema of the generalized entropy or have

complete knowledge of the statistical ensemble.

Progress can be made in special contexts, particularly in two-dimensional dilaton grav-

ity models such as the Callan-Harvey-Giddings-Strominger (CGHS) model [11] or Jackiw-

Teitelboim (JT) gravity [12, 13], since, up to minor ambiguities, the effects of the backreac-

tion are largely fixed by the two-dimensional Polyakov action capturing the contributions of

the conformal anomaly [14]. In fact, the semi-classical extension of both models have had

success in studying the black hole information problem analytically, and address the afore-

mentioned black hole temperature puzzle [15–20] (see [21] for a pedagogical review on both

– 3 –

models). Notably, with backreaction fully accounted for in the extended models, identifying

the temperature of Hawking radiation with the black hole temperature is indeed valid. The

Hawking temperature appears in the semi-classical first law of black hole horizons, where

the classical entropy is replaced by the generalized entropy and the classical ADM energy by

semi-classical energy [18, 20].

Though both (extended) models contain fully analytic black hole solutions incorporat-

ing backreaction and are capable of addressing the problem of black hole formation and

evaporation, in certain respects the JT model is simpler since the background is fixed to be

two-dimensional Anti-de Sitter (AdS2) space and it is only the value of the dilaton field that

tracks gravitational strength. For this reason, in this article we primarily focus on the JT

model, however, many of the results we uncover are expected to generalize to the CGHS

model (and, optimistically, beyond).

The classical JT model can be viewed as the low-energy dynamics of a wide class of

charged, near-extremal black holes and branes in higher dimensions [22–25].2 More recently,

the gravi-dilaton theory was shown to offer a precise realization of holographic AdS2/CFT1

duality, where JT gravity arises as the low-energy gravitational dual to an integrable limit of

the 1D quantum mechanical Sachdev-Ye-Kitaev (SYK) model [29–33]. It is possible to quan-

tize classical JT gravity using canonical/covariant phase space techniques [34–37], or via a

Euclidean path integral, in which the partition function of the theory is dual to some double-

scaled matrix model [38–41]. A particularly exciting aspect of the JT model is that it offers a

potential resolution of the black hole information paradox, in which the fine grained (equiv-

alently, von Neumann or entanglement) entropy of the radiation may be exactly computed

and follows the expected behavior for a unitary Page curve [42–48]).3

The fine grained entropy formula of radiation is inspired by the Anti-de Sitter / Conformal

Field Theory (AdS/CFT) correspondence, where the entanglement entropy Sent of a CFT

reduced to a subregion of the conformal boundary of AdS is dual to the area of an extremal

bulk surface X homologous to the boundary region, in the large N limit of the CFT [55–

57]. In the semi-classical approximation, the classical area term is supplemented by a bulk

entanglement entropy Sbulkent accounting for the entanglement between bulk quantum fields,

including both matter and the metric, [58–60]

Sent(X) = extX

[Area(X)

4G+ Sbulk

ent (ΣX)

]. (1.6)

Thus, the fine grained entropy Sent(X) is given by the generalized entropy (1.2), Sgen(X),

except where now the black hole horizon H has been replaced by the quantum extremal surface

(QES) X [60], and where the matter field entropy Sm is given by the von Neumann entropy of

2The low-energy dynamics of rotating, near-extremal black holes is also described in terms of JT gravity

now coupled to extra matter fields. These extra degrees of freedom describe the deformation or “squashing”

of the near-horizon geometry away from extremality [26–28].3Similar successes were later found to hold in the CGHS model [49, 50], JT de Sitter gravity [51, 52], and

higher-dimensional scenarios [53, 54].

– 4 –

the bulk fields restricted to a bulk codimension-1 slice ΣX in the semi-classical limit.4 In the

limit X coincides with a black hole horizon, we recover the usual generalized entropy (1.2).

Using either “double holography” or a Euclidean path integral analysis invoking the replica

trick, an equivalent expression for the von Neumann entropy of radiation may be derived and

exhibits a dynamical phase transition taking place near the Page time such that the state

of radiation obeys unitary evolution [44, 47]. While understanding the precise details of this

phase transition is not necessary for this article, we are highly influenced by the question of

whether the Page curve follows from a Lorentzian time path integral [61–63].

The aforementioned semi-classical first law [18, 20, 64] only applies to black hole horizons

and not to quantum extremal surfaces. In this article we intend to derive a semi-classical first

law associated to QESs in AdS2. Moreover, a notable observation is the position of the QES

in eternal black hole backgrounds: the QES lies outside of the horizon [45, 49, 50], and has

an associated entanglement wedge – a “nested” Rindler wedge in AdS2 – that differs from

the AdS-Rindler wedge of the black hole. Importantly, it is the entanglement wedge of the

QES which is used to compute the quantum corrected holographic entanglement entropy, and

therefore, as was the case for AdS-Rindler space [65, 66], understanding the thermodynamic

properties of the QES entanglement wedge is worthwhile and necessary.

Motivated by these observations, here we provide a detailed investigation of the (semi-

classical) thermodynamics of surfaces that lie outside of a black hole horizon, particularly

quantum extremal surfaces. More specifically, restricting to scenarios with time-reflection

symmetry, we uncover a first law of thermodynamics for QESs, where the generalized entropy

replaces the usual black hole entropy. When X coincides with the black hole horizon, we

recover a semi-classical first law of black hole thermodynamics [64]. In principle, while this

study should hold for more realistic theories of gravity, we use the remarkable simplicity of the

semi-classical JT model to our advantage where we can solve everything analytically. In this

context the classical “area” term is given by the value of the dilaton evaluated at X, while the

bulk von Neumann entropy is quantified by an auxiliary field used to localize the Polyakov

action, and is understood to represent conformal matter fields living in an eternal AdS2 black

hole background. We demonstrate the semi-classical Wald entropy [67], with respect to the

Hartle-Hawking vacuum state, exactly reproduces the generalized entropy (1.6), including the

von Neumann contribution, and is generally time dependent. We also compute the quantum-

corrected ADM mass, such that together with the entropy we derive a semi-classical Smarr

relation and an associated first law.5

4A point worth clarifying is that the quantum extremal surface is defined as the surface which extremizes

the generalized entropy that includes semi-classical corrections at all orders in 1/G [60], while the authors of

[59] only considered the leading order 1/G semi-classical correction, where X is an ordinary extremal surface

and (1.6) is referred to as the Faulkner-Lewkowycz-Maldacena (FLM) formula.5Recently backreaction was accounted for in a specific effective theory of gravity induced via AdS4/CFT3

brane holography, leading to a quantum BTZ black hole [68], where a first law is uncovered involving the

generalized entropy. This set-up is different from the one we study, however, as their derivation of the first

law relies on AdS/CFT duality while ours does not. See also [69] where conformal bootstrap methods are

used to compute the entanglement entropy of a CFT2 to second order in 1/c corrections, and, upon invoking

– 5 –

The outline of this article is as follows. In Section 2, after reviewing the basics of classical

JT gravity and the eternal AdS2 black hole, we solve the semi-classical JT model in full detail.

In particular, we emphasize the importance of choosing a vacuum state, defined such that its

expectation value of the normal ordered stress tensor vanishes, leading to different solutions for

φ and χ, the dilaton and an auxiliary field localizing the 1-loop Polyakov action, respectively.

With the backreacted model completely solved, in Section 3 we determine the Wald

entropy for the Boulware and Hartle-Hawking vacuum, which is taken to be a thermodynamic

entropy with respect to the Hartle-Hawking vacuum. Furthermore, via the presence of the χ

field, we find the Wald entropy captures the full generalized entropy, that is, including the

von Neumann contribution, and is generally time dependent. Extremization of this entropy

leads to a quantum extremal surface that lies just outside the horizon.

Section 4 is then devoted to the thermodynamics of AdS-Rindler wedges nested inside the

eternal AdS2 black hole, itself an AdS-Rindler wedge. We derive a first law associated with

the nested AdS-Rindler wedges. Using the boost Killing vector associated with the nested

Rindler wedge, and local counterterms to regulate divergences, we compute a semi-classical

Smarr relation and a corresponding first law of thermodynamics of nested Rindler wedges,

where the temperature is proportional to the surface gravity of the nested Rindler horizon.

The global Hartle-Hawking state reduced to the nested Rindler wedge is in a thermal Gibbs

state at this temperature. Allowing for variations of the coupling constants (φ0, G,Λ, c) in the

semi-classical JT action, we also derive a classical and semi-classical extended first law where

we define different counterterm subtracted “Killing volumes” conjugate to these coupling

variations.

We emphasize the classical and semi-classical first laws hold for any nested Rindler wedge.

However, in the microcanonical ensemble the generalized entropy of the nested wedge is ex-

tremized, and hence the thermodynamics of the nested wedge is equivalent to the thermody-

namics of the QES. Consequently, we argue in the microcanonical ensemble it is more natural

to consider the entropy of the QES when semi-classical effects are taken into account. We

summarize our findings in Section 5 and discuss potential future research avenues.

To keep this article self-contained we include a number of appendices providing compu-

tational details and formalism used in the body of the article. Appendix A summarizes the

various coordinate systems of AdS2 and how the stress tensor transforms in these coordinates.

In Appendix B we detail the construction of the boost Killing vector of the Rindler wedge

inside AdS2 using the embedding formalism. Appendix C provides a thorough account of the

covariant phase space formalism for 2D dilaton theories of gravity, and of the construction

of the ADM Hamiltonian with semi-classical corrections. In Appendix D we generalize the

covariant phase formalism with boundaries to allow for coupling variations. We apply this

formalism to JT gravity, and provide a detailed derivation of the extended first law. Lastly,

in Appendix E we provide a heuristic argument for the generalized second law.

AdS3/CFT2, is matched with Sgen to second order corrections in G.

– 6 –

2 JT gravity and semi-classical corrections

To set the stage we first review some basic features of eternal black hole solutions to the

classical JT model. Then we analytically solve the semi-classical JT equations of motion,

accounting for backreaction, paying special attention to the choice of vacuum state.

2.1 Classical JT and the eternal AdS2 black hole

The classical JT action in Lorentzian signature is

IbulkJT =

1

16πG

[∫Md2x√−gφ0R+

∫Md2x√−gφ

(R+

2

L2

)]. (2.1)

where G is the two-dimensional dimensionless Newton’s constant which we retain as a book

keeping device. Further Λ = −1/L2 is a negative cosmological constant, where L is the

AdS2 length scale characterizing the charge of the higher-dimensional black hole system,6 φ

is the dilaton arising from a standard spherical reduction of the parent theory, and φ0 is a

constant proportional to the extremal entropy of the higher-dimensional black hole geometry.

To have a well-posed variational principle with Dirichlet boundary condition, we supplement

the above bulk action with a Gibbons-Hawking-York (GHY) boundary term,7 and to cancel

the divergence in the second GHY term we also subtract a boundary counterterm,

IbdyJT = IGHY

JT + IctJT =

1

8πG

[∫Bdt√−γφ0K +

∫Bdt√−γφ(K − 1/L)

], (2.2)

where B is the spatial boundary of M , and K is the trace of the extrinsic curvature of B with

induced metric γµν . Due to the Gauss-Bonnet theorem the sum of the φ0 terms in the bulk

(2.1) and boundary action (2.2) is proportional to the Euler character of the manifold M , so

these terms are purely topological in two dimensions.

The gravitational and dilaton equations of motion are, respectively,

T φµν = 0 , with T φµν ≡ −2√−g

δIJT

δgµν= − 1

8πG

(gµν−∇µ∇ν −

1

L2gµν

)φ , (2.3)

R+2

L2= 0 . (2.4)

The dilaton equation of motion (2.4) fixes the background geometry to be purely AdS2. In

static Schwarzschild-like coordinates the AdS2 metric takes the form

ds2 = −N2dt2 +N−2dr2 , N2 =r2

L2− µ . (2.5)

6Here we are thinking of a dimensional reduction of an asymptotically flat charged black hole, as in e.g.,

[70]. For a neutral AdS black hole the AdS2 length scale is fixed by the higher-dimensional AdS length scale

(see, e.g., [71]).7Though we won’t need it here, the GHY term can be recast as a Schwarzian theory on the boundary and

is identified with the low-energy sector of the SYK action. For a review, see e.g., [72, 73].

– 7 –

r∗

vu tu

=∞

, UK

=0

v=−∞

, VK

=0

Figure 1: The relevant portion of a Penrose diagram of an eternal non-extremal AdS2 black

hole is presented. The diagonal lines denote the bifurcate Killing horizons and the vertical

lines denote the conformal boundary. The dashed edges indicate that this is only a portion

of the full Penrose diagram. The curved arrows indicate the directions of the Killing flow.

This describes an eternal AdS2 black hole, which is equivalent to AdS2-Rindler space, as

shown in Appendix A. Further, we take the dilaton solution to the gravitational equation of

motion (2.3) to be (see e.g., [74])

φ(r) =r

Lφr . (2.6)

The horizon of the black hole is located at rH = L√µ, where L is the AdS curvature scale

and µ ≥ 0 is a dimensionless mass parameter proportional to the ADM mass (c.f. [64])

Mφr =1

16πG

µφrL

. (2.7)

The conformal boundary is at r →∞, where φr > 0 is the boundary value of φ such that at

a cutoff r = 1ε near the boundary (ε = 0), φ → φr

εL . In the conventions of [64], the boundary

value is given by φr = 1/(JL), where J is an energy scale.8 Throughout this article we will

express the AdS2 black hole in static and Kruskal coordinates. We refer to Appendix A for

the definitions of various AdS2 coordinate systems and the transformations relating them. In

Figure 1 we depict the Penrose diagram of an AdS2 black hole to keep track of the different

coordinates.

Upon Wick rotating the Lorentzian time t coordinate t→ −iτ and identifying τ ∼ τ + β

to remove the conical singularity of the resulting Euclidean cigar, the eternal black hole (2.5)

8Note the dilaton φ and the constant φr are dimensionless.

– 8 –

is in thermal equilibrium at a temperature TH inversely proportional to the period β of the

Euclidean time circle,

TH =N ′(rH)

4π=

√µ

2πL. (2.8)

We observe extremality, TH = 0, occurs in the massless limit µ = 0.

The associated “Bekenstein-Hawking” entropy of the black hole is well known and given

by

SBH =φ0

4G+φH4G

= Sφ0 + Sφr , (2.9)

where Sφ0 = φ0

4G is the entropy of the extremal black hole while Sφr =φr√µ

4G is the non-

extremal contribution to the black hole entropy. Consequently, the thermodynamic variables

Mφr (2.7), TH (2.8), and Sφr (2.9), obey a Smarr relation and corresponding first law [75]

2Mφr = THSφr , δMφr = THδSφr , (2.10)

where only µ is varied in the first law and φ0, φr and G are kept fixed. In Eq. (4.39) we

present a different form of the Smarr relation which does not have a factor of two on the left-

hand side and which contains a new term proportional to the cosmological constant. Further,

in Eq. (4.71) we give the extended first law for the eternal black hole allowing for variations

of the coupling constants.

A chief goal of this article is to uncover the quantum corrected first law and Smarr rela-

tion. To do this we discuss the semi-classical JT model, where we emphasize the importance

of the choice of vacuum state.

2.2 Backreaction and vacuum states

One of the novel and exciting features of JT gravity, similar to the CGHS model [11], is that

1-loop quantum effects may be incorporated and continue to yield fully analytic solutions,

allowing for a complete study of backreaction. These semi-classical effects are entirely cap-

tured by a (bulk) non-local Polyakov action [14] and its associated Gibbons-Hawking-York

boundary term, which take the following form in Lorentzian signature [20]:

Iχ = IPolyχ + IGHY

χ = − c

24π

∫Md2x√−g[(∇χ)2 + χR− λ

L2

]− c

12π

∫Bdt√−γχK . (2.11)

Here χ is a local auxiliary field such that the non-local Polyakov contribution appears local.

The Polyakov term is motivated from the conformal anomaly associated with a classical CFT

having central charge c. The equation of motion for the auxiliary field χ is simply

2χ = R . (2.12)

Upon substituting the solution for χ into the Polyakov action one recovers the standard non-

local expression.9 Lastly, note we have included a cosmological constant term λ/L2 which is

9Specifically, substituting the formal solution χ = 12

∫d2y√−g(y)G(x, y)R(y), where G(x, y) is the Green’s

function for the D’Alembertian operator, into the local form of the Polyakov action (2.11), and ignoring

boundary terms, one recovers the non-local form, IPoly = − c96π

∫d2x√−g(x)

∫d2y√−g(y)R(x)G(x, y)R(y) .

– 9 –

always allowed at the quantum level due to an ambiguity in the conformal anomaly [21]. In

the majority of our results we follow the standard convention and set λ = 0, however, in the

following analysis we demonstrate a special choice of λ removes all semi-classical corrections

to the dilaton in the Hartle-Hawking vacuum state.

The χ field models c massless scalar fields (reflecting a CFT of central charge c), which

are taken to represent the Hawking radiation of an evaporating black hole (e.g., [24]). The

classical limit thus corresponds to c→ 0.10 The semi-classical approximation is only valid in

the regime

φ0/G φ(rH)/G =√µφr/G c 1 . (2.13)

This follows from the following reasoning. Recall the classical JT model arises from a spheri-

cal reduction of a near-extremal black hole. The extremal black hole entropy is Sφ0 ∼ φ0/G,

and deviation from extremality is captured by the entropy Sφr ∼ φ(rH)/G, such that near

extremality implies φ0 φ(rH), with φ0/G 1. The classical solution (2.6) for the dilaton

reveals φ(rH) = (rH/L)φr =√µφr. To keep curvatures small from the higher-dimensional

perspective, one works with large black holes, i.e., rH L or√µ 1. The semi-classical ap-

proximation is understood as quantizing c conformal fields, leaving the background geometry

classical. This is allowed since we are ignoring stringy-like ghost corrections to the dilaton or

metric, provided c 1. However, for c to amount to a correction to the classical result, we

impose φ(rH)/G c. Combining these approximations we find the regime of validity (2.13).

Combining the semi-classical action (2.11) with the classical JT model (2.1) augments

the classical gravitational equation of motion (2.3) with backreaction fully characterized by

〈Tχµν〉 ≡ − 2√−g

δIχδgµν , such that the semi-classical gravitational equation of motion is

T φµν + 〈Tχµν〉 = 0 , (2.14)

with

〈Tχµν〉 =c

12π

[(gµν−∇µ∇ν)χ+ (∇µχ)(∇νχ)− 1

2gµν(∇χ)2 +

λ

2L2gµν

]. (2.15)

Here 〈Tχµν〉 denotes the stress tensor expectation value with respect to some unspecified quan-

tum state |Ψ〉. In any coordinate frame 〈Tχµν〉 yields a conformal anomaly:

gµν〈Tχµν〉 =c

24π

(R+

2λ

L2

), (2.16)

where we used the equation of motion for χ (2.12). Using the on-shell relation R = − 2L2 ,

we note the conformal anomaly is eliminated by selecting λ = 1, i.e., gµν〈Tχµν〉 = 0. It is

worth emphasizing, moreover, since χ does not directly couple to the dilaton φ, the system

continues to admit eternal black hole solutions of the same type as in the classical JT model,

but with the full backreaction taken into account. This is not to say the dilaton will not

receive semi-classical quantum corrections; indeed it will generically, however, the solution

10More precisely, here ~ has been set to unity. To recover factors of ~, one simply replaces c→ c~.

– 10 –

for φ will depend on the vacuum state and value of λ. As we will see momentarily, the choice

λ = 1 eliminates all semi-classical corrections to φ in the Hartle-Hawking state.

Below we will primarily work in conformal gauge, where any two-dimensional spacetime

is conformally flat in some set of light cone coordinates (y+, y−),

ds2 = −e2ρdy+dy− , (2.17)

with a conformal factor e2ρ(y+,y−). In conformal gauge, taking into account backreaction

(2.15), the equations of motion for the dilaton φ (2.4), auxiliary field χ (2.12) and metric

(2.3) become, respectively,

4∂+∂−ρ+e2ρ

L2= 0 , (2.18)

∂+∂−(χ+ ρ) = 0 , (2.19)

− ∂2±φ+ 2(∂±ρ)(∂±φ) = −e2ρ∂±(e−2ρ∂±φ) = 8πG〈Tχ±±〉 , (2.20)

2∂±∂∓φ+1

L2e2ρφ = 16πG〈Tχ±∓〉 , (2.21)

where we employed the conformal gauge identities (A.23), and the components of the stress

tensor for χ (2.15) are

〈Tχ±∓〉 =c

12π∂+∂−χ−

c

48π

λ

L2e2ρ , (2.22)

〈Tχ±±〉 =c

12π[−∂2

±χ+ 2(∂±ρ)(∂±χ) + (∂±χ)2] . (2.23)

Here 〈Tχ±±〉 is shorthand for 〈Ψ|Tχy±y± |Ψ〉. The equation of motion for χ (2.19) is particularly

easy to solve, leading to

χ = −ρ+ ξ , with ξ(y+, y−) = ξ+(y+) + ξ−(y−) , (2.24)

where ξ solves the wave equation ξ = 0 = ∂+∂−ξ. Substituting the χ solution into the

components of the stress tensor (2.22) and (2.23) leads to

〈Tχ±∓〉 = − c

48π

(4∂+∂−ρ+

λ

L2e2ρ

)=

c

48πL2e2ρ(1− λ) , (2.25)

〈Tχ±±〉 =c

12π[∂2±ρ− (∂±ρ)2 + (∂±ξ)

2 − ∂2±ξ] = − c

12π[(∂±ρ)2 − ∂2

±ρ]− c

12πt±(y±) , (2.26)

where we have defined

t+(y+) ≡ ∂2+ξ+ − (∂+ξ+)2 , t−(y−) ≡ ∂2

−ξ− − (∂−ξ−)2 . (2.27)

The functions t±(y±) characterize the quantum matter state and are related to the expectation

value of the normal-ordered stress-tensor, as we will now review, including some additional

new insights.11

11Equivalently, t± can be found from the conformal anomaly and the continuity equation ∇µ〈Tχµν〉 = 0,

where the t± arise as “constants” of integration [20].

– 11 –

2.2.1 Normal-ordered stress tensors and vacuum states

In what follows it will be useful to know how to transform the components of 〈Tχµν〉 in various

coordinate systems in conformal gauge. This will be crucial in our interpretation of a choice

of vacuum state. First note that since ρ is a component of the metric it transforms as

such. Specifically, under a conformal transformation from (y±) to another set of conformal

coordinates (x±) one has (see, e.g., Eq. (5.68) of [21])

ρ(y+, y−)→ ρ(x+, x−) = ρ(y+, y−) +1

2log

(dy+

dx+

dy−

dx−

). (2.28)

Since χ = −ρ + ξ is a scalar field, but ρ transforms as a tensor, one finds the functions ξ±must transform under a conformal transformation as

ξ±(y±)→ ξ±(x±) = ξ±(x±(y±)) +1

2log

dy±

dx±. (2.29)

This implies the functions t±(y±) (2.27) transform as

t±(x±) =

(dy±

dx±

)2

t±(y±) +1

2y±, x± , (2.30)

where y±, x± is the Schwarzian derivative of y± with respect to x±:

y±, x± ≡ (y±)′′′

(y±)′− 3

2

((y±)′′

(y±)′

)2

, with (y±)′ ≡ dy±

dx±. (2.31)

In fact, t± turns out to be proportional to the expectation value of the normal ordered stress

tensor [21]

〈Ψ| : Tχ±± : |Ψ〉 = − c

12πt±(y±) , (2.32)

where we reiterate that |Ψ〉 is some unspecified quantum state. By normal ordering we mean

: Tχ±±(y±) : ≡ Tχ±±(y±) − 〈0y|Tχ±±(y±)|0y〉 , with |0y〉 the vacuum state with respect to the

(y±) coordinate system, i.e., the vacuum state defined with respect to the positive frequency

modes in coordinates y±, ay|0y〉 = 0.

Crucially, from the definition of normal ordering we have that the vacuum state |0y〉 is the

state such that the expectation value of the normal ordered energy-momentum tensor vanishes,

〈0y| : Tχ±±(y±) : |0y〉 = 0 , ⇔ t±(y±) = 0 . (2.33)

Moreover, from equation (2.30) we see the normal-ordered stress tensor obeys an anomalous

transformation law under a conformal transformation y± → x±(y±)

: Tχ±±(x±) : =

(dy±

dx±

)2

: Tχ±±(y±) : − c

24πy±, x± . (2.34)

Notice that taking the expectation value with respect to |0y〉 on both sides yields

〈0y| : Tχ±±(x±) : |0y〉 = − c

24πy±, x± . (2.35)

– 12 –

The stress-energy tensor itself follows a standard coordinate transformation law

〈Ψ|Tχ±±(x±)|Ψ〉 =

(dy±

dx±

)2

〈Ψ|Tχ±±(y±)|Ψ〉 , (2.36)

because the expectation value of the stress tensor is a rank 2 tensor. This transformation is

consistent with the anomalous transformation law for the normal-ordered stress tensor, since

combining (2.26) and (2.32) with (2.34) yields (2.36) above (following a similar computation

as in Appendix A of [64]). However, under a change of vacuum the stress tensor expectation

value does transform in an anomalous way

〈0y|Tχ±±(x±)|0y〉 = 〈0x|Tχ±±(x±)|0x〉 −c

24πy±, x± , (2.37)

which follows from combining (2.26), (2.32), (2.33) and (2.35).

As noted earlier, the metric and dilaton are unaffected by the introduction of the auxiliary

field χ and the semi-classical cosmological constant term, and thus we still have an eternal

black hole solution. However, as we have just seen, the stress-energy tensor expectation value

takes different forms depending on the vacuum state one is in. The solutions for χ and φ,

found by solving the semi-classical equations of motion with 〈Tχµν〉, are hence state dependent.

Let us thus write down the components of 〈Tχµν〉 for the black hole solution in (static) advanced

and retarded time coordinates (v, u) (A.6), and Kruskal coordinates (UK, UK) (A.17) and find

how the stress tensors transform under a change in coordinates. Below we summarize the

results and we leave the detailed calculations for the interested reader in Appendix A.1.

First note the eternal black hole metric in advanced and retarded time coordinates (v, u)

is given by

ds2 = −e2ρ(v,u)dudv , e2ρ =µ

sinh2[√µ

2L (v − u)], (2.38)

while the line element in Kruskal coordinates is12 (UK, UK),

ds2 = −e2ρ(VK,UK)dUKdVK , e2ρ =4µ

(1 + µL2UKVK)2

. (2.39)

The conformal factors ρ are related in the two coordinate systems via the transformation rule

(2.28), from which

ρ(v, u) = ρ(VK, UK) +

√µ

2L(v − u) = ρ(VK, UK) +

1

2log(− µ

L2UKVK

), (2.40)

where we used (x+, x−) = (v, u) and (y+, y−) = (VK, UK).

Relation (2.29) allows us to relate the function ξ in the different coordinate systems

ξu = ξUK−√µ

2Lu = ξUK

+1

2log

(−√µ

LUK

),

ξv = ξVK+

√µ

2Lv = ξVK

+1

2log

(√µ

LVK

),

(2.41)

12Here we work with the scaled coordinates (A.17).

– 13 –

such that

ξ(v, u) = ξ(VK, UK) +1

2log(− µ

L2UKVK

). (2.42)

Consequently, the transformation rule for the normal-ordered stress tensors (2.34) yields

: Tχvv : = 4π2T 2HV

2K : TχVKVK

: +cπ

12T 2H , (2.43)

: Tχuu : = 4π2T 2U2K : TχUKUK

: +cπ

12T 2H , (2.44)

where we used (√µL )2 = 4π2T 2

H with TH the Hawking temperature (2.8). Upon taking the

expectation value, notice for special states the stress tensor will be purely thermal. We will

return to this point shortly.

Lastly, using the normal ordered relation (2.32) together with the following relations

between tVKand tv, and tUK

and tu:

tv =

(√µ

LVK

)2

tVK− µ

4L2, tu =

(√µ

LUK

)2

tUK− µ

4L2, (2.45)

we find the diagonal stress-energy tensor expectation values in static and Kruskal coordinates

are, respectively,

〈Ψ|Tχuu|Ψ〉 = − cµ

12πL2U2

KtUK, 〈Ψ|Tχvv|Ψ〉 = − cµ

12πL2V 2

KtVK, (2.46)

〈Ψ|TχUKUK|Ψ〉 = − c

48π

1

U2K

− cL2

12πµ

tuU2

K

, 〈Ψ|TχVKVK|Ψ〉 = − c

48π

1

V 2K

− cL2

12πµ

tvV 2

K

. (2.47)

Observe that, depending on the coordinate frame, 〈Ψ|Tχµν |Ψ〉 may diverge in the limit one

approaches the horizon (u→∞, v → −∞ or UK = VK = 0). This feature is in fact dependent

on the choice of vacuum state as we will now describe.

Choice of vacuum: Boulware vs. Hartle-Hawking

Let us now specify the vacuum states considered in this paper. Famously there are two

vacuum states of interest for AdS2 black holes [76]: (i) the Boulware state |B〉 [77] and

(ii) the Hartle-Hawking state |HH〉 [78]. The Boulware vacuum is the state for the eternal

black hole defined by positive frequency modes in static coordinates (v, u). Equivalently, as

in (2.33), the expectation value of the uu and vv components of the normal-ordered stress

tensor vanishes in the Boulware state

〈B| : Tχuu : |B〉 = 〈B| : Tχvv : |B〉 = 0 ⇔ tu = tv = 0 . (2.48)

Consequently, from (2.43) and (2.44), the expectation value of the VKVK and UKUK compo-

nents of the normal-ordered stress tensor are

〈B| : TχVKVK: |B〉 = − c

48π

1

V 2K

, 〈B| : TχUKUK: |B〉 = − c

48π

1

U2K

. (2.49)

– 14 –

Equivalently,

tUK=

1

4U2K

, tVK=

1

4V 2K

. (2.50)

Moreover, the expectation value of the stress tensor becomes

〈B|Tχuu|B〉 = 〈B|Tχvv|B〉 = − cµ

48πL2,

〈B|TχUKUK|B〉 = − c

48πU2K

, 〈B|TχVKVK|B〉 = − c

48πV 2K

.(2.51)

Thus, in Kruskal coordinates, the renormalized stress tensor, normal ordered or not, diverges

at the horizon, as is common for the Boulware state. The expectation value of the stress tensor

in static conformal coordinates, moreover, gives rise to a negative ‘energy’ and is interpreted

as a Casimir energy in the presence of a boundary at the black hole horizon [76].

Alternatively, the Hartle-Hawking vacuum state |HH〉 is defined by positive frequency

modes in Kruskal coordinates (VK, UK), such that the expectation value of the UKUK and

VKVK components of the normal-ordered stress tensor vanishes:

〈HH| : TχUKUK: |HH〉 = 〈HH| : TχVKVK

: |HH〉 = 0 ⇔ tUK= tVK

= 0 . (2.52)

Correspondingly, we have

tu = tv = − µ

4L2. (2.53)

The expectation value of uu and vv components of the normal-ordered stress-tensor then

becomes

〈HH| : Tχuu| : HH〉 = 〈HH : |Tχvv : |HH〉 =cπ

12T 2

H . (2.54)

Lastly, the expectation value of the diagonal components of the stress-tensor in (u, v) and

(U, V ) coordinates vanishes

〈HH|TχUKUK|HH〉 = 〈HH|TχVKVK

|HH〉 = 〈HH|Tχuu|HH〉 = 〈HH|Tχvv|HH〉 = 0 . (2.55)

Thus, with respect to a static observer in (v, u) coordinates, the Hartle-Hawking vacuum state

is interpreted as the state with the energy density of a thermal bath of particles at Hawking

temperature TH. This is completely analogous to an accelerating (Rindler) observer seeing

the Minkowski vacuum as being populated with particles in thermal equilibrium [79–81];

indeed, the AdS2 eternal black hole is simply AdS2-Rindler space. Therefore, the Hartle-

Hawking state allows us to identify the black hole as a thermal system with a temperature,

thermodynamic energy and entropy, unlike the Boulware vacuum. Nonetheless, below we

will solve the semi-classical JT equations of motion with respect to both vacuum states. In

Figure 2 we provide an overview of the physical picture we advocate.

As a final remark, note that had we performed the same analysis in Poincare coordinates

(A.8) (V,U), we would have found the Poincare vacuum state, i.e., the state defined with

respect to positive frequency modes in Poincare coordinates, is equivalent to the Hartle-

Hawking state |HH〉 [76], a feature that persists in higher dimensions [82].

– 15 –

Observer at AdS boundary →T =TH

Hartle-Hawking state

Observer at AdS boundary →T =0

Boulware state

Figure 2: The black circles denote a black hole and the wavy arrows represent Hawking

radiation. In the Hartle-Hawking state (left Figure) a stationary observer at the AdS bound-

ary measures T = TH , whereas a stationary observer at the boundary measures T = 0 in

the Boulware state (right Figure). In the Hartle-Hawking state the black hole is in thermal

equilibrium with all of its surroundings whereas the Boulware state can be interpreted as

the black hole being in thermal equilibrium only with a membrane wrapped tight around the

event horizon.

2.2.2 Backreacted solutions

We now have all the necessary ingredients to solve for the dilaton φ and the auxiliary field

χ in the semi-classical JT model, taking into account the backreaction of Hawking radiation.

We will look to solve the dilaton equation in static coordinates (t, r∗), where we assume φ

is static, i.e., φ = φ(r∗), however we will allow χ to be potentially time dependent. As we

will see below, these assumptions are self-consistent, i.e., they lead to valid and physically

sensible solutions to the equations of motion. Moreover, the time dependence of χ will be

crucial in the next section; in particular, it will allow us to interpret the Wald entropy as the

(time-dependent) generalized entropy.

Using t = 12(v + u) and the tortoise coordinate r∗ = 1

2(v − u), and the relations

∂v =1

2(∂t + ∂r∗) , ∂u =

1

2(∂t − ∂r∗) , ∂u∂v =

1

4(∂2t − ∂2

r∗) , (2.56)

the gravitational equation (2.21) becomes:

2∂u∂vφ+e2ρφ

L2= 16πG〈Tχuv〉 ⇒ ∂2

r∗φ =2µ

L2

1

sinh2(√

µL r∗

) [φ+ (λ− 1)cG

3

], (2.57)

where we assumed ∂tφ = 0. Meanwhile, the two equations in (2.20) become

∂r∗

[sinh2

(√µ

Lr∗

)∂r∗φ

]=

8cG

3

( µ

4L2+ tv(v)

)sinh2

(√µ

Lr∗

), (2.58)

– 16 –

∂r∗

[sinh2

(√µ

Lr∗

)∂r∗φ

]=

8cG

3

( µ

4L2+ tu(u)

)sinh2

(√µ

Lr∗

). (2.59)

In order for the left-hand side to be solely a function of r∗, and since tv(v) (tu(u)) is solely a

function of v (u), we see that tu(u) ≡ τ = tv(v) must be a constant [20].

For tv = tu = τ , the most general solution to the system of differential equations is

φ = −φr√µcoth

(√µ

Lr∗

)+Gc

3

(1 +

4τL2

µ

)(1 +

√µ

Lr∗

)coth

(√µr∗

L

)−Gc

3

(λ+

4τL2

µ

).

(2.60)

In the limit c→ 0 we recover the classical solution for the dilaton.

To proceed, we must specify the choice of vacuum state, thereby fixing the value of τ .

We thus evaluate the general solution (2.60), as well as the possible solutions for the state

dependent χ in the Boulware and Hartle-Hawking vacuum states separately.

Boulware vacuum

The Boulware vacuum |B〉 is defined by (2.48), tu = tv = τ = 0, such that the dilaton (2.60)

simplifies to [21, 83]

φ = −φr√µcoth

(√µ

Lr∗

)+Gc

3

(1 +

√µ

Lr∗

)coth

(√µr∗

L

)− λGc

3. (2.61)

We see φ diverges as φ(r∗) ∼ r∗ as one approaches the horizon, r∗ → −∞.

Meanwhile, from (2.27) we have two possible solutions for ξu and ξv:

ξ(1)u = cu , ξ(1)

v = cv ,

ξ(2)u = c′u − log

(±√µ

Lu+Ku

), ξ(2)

v = c′v − log

(±√µ

Lv +Kv

),

(2.62)

where cu, cv, c′u, c′v,Ku, and Kv are constants, to be fixed by physical boundary conditions. We

will see momentarily these constants can be arranged in such a way that the Wald entropy cap-

tures the full generalized entropy. Note that ξ(v, u) = ξu + ξv only has one time-independent

solution, namely, ξ(1) = ξ(1)u + ξ

(1)v . Using (2.24) χ = −ρ + ξ, and the fact, as a component

of the metric, ρ is always time independent, we find a time-independent solution χ(1) upon

substitution of ξ(1)

χ(1) =1

2log

[1

µsinh2

(√µr∗

L

)]+ C , (2.63)

where C ≡ cu+ cv. A time-dependent solution for χ also exists, via substitution of the choice

ξ(2) = ξ(2)u + ξ

(2)v , namely,

χ(2) =1

2log

[1

µsinh2

(√µr∗

L

)]+ C ′ − log

(±√µ

Lu+Ku

)− log

(±√µ

Lv +Kv

), (2.64)

with C ′ ≡ c′u + c′v. This solution will prove useful when we compare the Wald entropy to the

von Neumann entropy.

– 17 –

From transformations (2.41), we can express the two solutions for ξ in Kruskal coordi-

nates. Considering only the time-indepenent solution ξ(1) yields

ξ(1)UK

= cu −1

2log

(−√µ

LUK

), ξ

(1)VK

= cv −1

2log

(√µ

LVK

). (2.65)

The corresponding χ field is then

χ(1) = −1

2log

4µ(1 + µUKVK

L2

)2

− 1

2log(− µ

L2UKVK

)+ C . (2.66)

To summarize, the eternal AdS2 black hole, characterized by ρ, remains a solution of the

semi-classical equations of motion, for which, with respect to the Boulware vacuum, the full

backreaction is encoded in the dilaton φ (2.61) and the auxiliary field χ (2.63) or (2.64). Notice

φ is the only field explicitly modified from its classical counterpart, such that in the classical

limit c → 0, where physically we can imagine turning off the influence of c background

conformal fields, we recover the classical JT result. The semi-classical effect, moreover, is

non-trivial as in the limit we approach the horizon the solution diverges.

Hartle-Hawking vacuum

The Hartle-Hawking vacuum |HH〉 is defined in (2.52), i.e., tUK= tVK

= 0 or tu = tv = − µ4L2 .

In this case the dilaton (2.60) is only shifted by a constant with respect to the classical

solution [20, 21]

φ(r∗) = −φr√µcoth

(√µ

Lr∗

)+ (1− λ)

Gc

3=rφrL

+ (1− λ)Gc

3, (2.67)

where we recall r is the standard radial Schwarzschild coordinate. Thus, in the Hartle-

Hawking vacuum the dilaton is trivially modified by backreaction. Moreover, selecting the

(Polyakov) cosmological constant to be λ = 1 entirely eliminates the semi-classical modifica-

tion to φ [21]. Almheiri and Polchinski work with the λ = 0 solution [20]. Note that they

consider only the Hartle-Hawking state and discard the Boulware state.

In the Hartle-Hawking vacuum there are three solutions for ξ:

ξ(3)u = cu +

√µ

2Lu , ξ(3)

v = cv −√µ

2Lv ,

ξ(4)u = cu −

√µ

2Lu , ξ(4)

v = cv +

√µ

2Lv , (2.68)

ξ(5)u = c′u − log

[2 cosh

(∓√µ

2Lu+Ku

)], ξ(5)

v = c′v − log

[2 cosh

(±√µ

2Lv +Kv

)],

where there is an inherent sign ambiguity on the linear term differing the ξ(3) solutions from

the ξ(4) solutions.13 Note that both ξ(3) and ξ(4) solutions lead to a time-independent ξ, while

13Here we denote ξ(3) = ξ(3)u + ξ

(3)v . Technically, mixed solutions such as ξ = ξ

(3)u + ξ

(4)v are also allowed, but

we will not consider them in this paper.

– 18 –

the solution ξ(5) is generically time dependent. Furthermore, in the horizon limit the solution

ξ(5) diverges in the same way as ξ(3) and ξ(4),

limr∗→−∞

ξ(5)u = c′u −Ku ∓

√µ

2Lu+ ... , lim

r∗→−∞ξ(5)v = c′v −Kv ±

√µ

2Lv + ... , (2.69)

where we identified cu = c′u −Ku and cv = c′v −Kv.

Using transformation (2.41), the above three solutions read in Kruskal coordinates, re-

spectively,

ξ(3)UK

= cu − log

(−√µ

LUK

), ξ

(3)VK

= cv − log

(√µ

LVK

),

ξ(4)UK

= cu , ξ(4)VK

= cv ,

ξ(5)UK

= kUK− log

[−√µ

LUK +KUK

], ξ

(5)VK

= kVK− log

[√µ

LVK +KVK

],

(2.70)

where we combined c′u ∓ Ku ≡ kUK, c′v ∓ Kv ≡ kVK

, and identified e∓2Ku ≡ KUKand

e∓2Kv ≡ KVK. When KUK

= KVK= 0, the time-dependent solution ξ(5) reduces to the third,

time-independent solution, while for KUK= KVK

= 1 we have ξ(5) → ξ(4) in the horizon limit

UK, VK → 0.

With the ξ solutions in hand, we find three associated χ = −ρ+ ξ fields, namely,

χ(3) =1

2log

[1

µsinh2

(√µr∗

L

)]−√µr∗

L+C = −1

2log

4µ(1 + µUKVK

L2

)2

−log(− µ

L2UKVK

)+C ,

(2.71)

χ(4) =1

2log

[1

µsinh2

(√µr∗

L

)]+

√µr∗

L+ C = −1

2log

4µ(1 + µUKVK

L2

)2

+ C , (2.72)

χ(5) =1

2log

[1

µsinh2

(√µr∗

L

)]−√µr∗

L− log

[(1 +KUK

e√µu/L

)(1 +KVK

e−√µv/L

)]+ k ,

= −1

2log

4µ(1 + µUKVK

L2

)2

− log

[(−√µUK

L+KUK

)(√µVK

L+KVK

)]+ k , (2.73)

where in the fifth solution we defined k ≡ kUK+ kVK

. Observe that for KUK= KVK

= 1 the

solutions χ(4) and χ(5) approach the same constant on the horizon, while for KUK= KVK

= 0

the field χ(5) reduces to the static solution χ(3), which blows up at the black hole horizon. In

[20] Almheiri and Polchinski took only solution χ(4) into account, because they required χ to

be constant on the horizon, but they neglected the fifth solution which is also constant on the

horizon for some KUKand KVK

. Thus, different choices of integration constants KUK,KVK

lead to potentially dramatically different physics. As we show below, there exists yet another

choice of integration constants for the fifth solution, such that the Wald entropy is equal to

the complete generalized entropy, including time dependence.

– 19 –

3 Wald entropy is generalized entropy

Having exactly solved the problem of backreaction in an eternal AdS2 black hole background,

here we begin to study semi-classical corrections to the thermodynamics, starting with the

entropy. Semi-classical corrections to the black hole entropy and associated first law of ther-

modynamics were accounted for in [20, 64], where it was shown that the generalized entropy

appears in the first law [20] and obeys a generalized second law [64]. In contrast, here we

consider the entropy of quantum extremal surfaces (QES) [60], defined as the surface extrem-

izing the generalized entropy, which is generally not the black hole horizon. Moreover, we

demonstrate the Wald entropy of the semi-classical JT model entirely captures the generalized

entropy, and, for certain physical scenarios, includes time dependence. We won’t dwell on the

time dependence, as we are primarily interested in systems in thermal equilibrium (such that

we focus about a point of time reflection symmetry), however, we believe this observation

may be useful in studying the black hole information problem, specifically the dynamics of

entanglement wedge islands. We will have more to say about this connection in Section 5.

There are two points worth emphasizing here. First, the eternal AdS2 black hole is

an AdS-Rindler wedge whose thermodynamics has been extensively studied before in higher

dimensions D > 2. In particular, the Bekenstein-Hawking entropy of higher-dimensional

AdS-Rindler space, a.k.a. the massless “topological” black hole, may be identified with the

entanglement entropy of a holographic CFT in the vacuum reduced to a boundary subregion

of the pure AdS spacetime [65]. As we will show, the quantum extremal surface lies outside

of the eternal black hole horizon, and is itself the bifurcation point of the Killing horizon of a

nested Rindler wedge; thus the QES also defines a topological black hole. Second, in D > 2 the

thermodynamics of a QES corresponds to (i) semi-classical “entanglement” thermodynamics

when the region of interest is a subset of the full AdS boundary, or (ii) semi-classical black

hole thermodynamics, when the region of interest is the full boundary. In D = 2 there is

no distinction between (i) and (ii), since there are no subregions to the (0 + 1)-dimensional

boundary, so this distinction is not relevant for the present article.

3.1 Wald entropy

For diffeomorphism invariant gravity theories other than general relativity, the entropy for

stationary black holes is quantified by the Wald entropy functional SWald, or equivalently the

Noether charge associated with the Killing vector field generating the bifurcate horizon of the

black hole normalized to have unit surface gravity κ = 1 [67, 84],

SWald = −2π

∫HdA

∂L∂Rµνρσ

εµνερσ , (3.1)

where εµν is the binormal satisfying εµνεµν = −2, dA the infinitesimal area element of the

codimension-2 Killing horizon H, and L is the Lagrangian density defining the theory. For

two-dimensional theories the integral is replaced with evaluating the integrand at the horizon.

The Wald entropy can be associated to the AdS2 black hole horizon H, since this is a bifurcate

– 20 –

Killing horizon. In fact, any point outside the black hole horizon in two dimensions may be

viewed as a bifurcation point of the Killing horizon of a nested Rindler wedge, and hence the

Wald entropy may also be evaluated on these points, as we will do later on.14

Black hole entropy generically receives quantum corrections when backreaction is taken

into account. In the present case the semi-classical effects are encoded in the dilaton φ and

auxiliary field χ. Using

∂L∂Rµνρσ

=∂(LJT + Lχ)

∂Rµνρσ=

(φ0 + φ

32πG− c

48πχ

)(gµρgνσ − gµσgνρ) , (3.2)

the Wald entropy for the semi-classical JT model is

SWald =1

4G(φ0 + φ)|H −

c

6χ|H , (3.3)

with χ|H = (−ρ+ ξ)|H . The precise form of φ and χ depends on the choice of vacuum state.

The Hartle-Hawking vacuum is a thermal density matrix when restricted to the exterior region

of the black hole [79–81]

ρHH = trext|HH〉〈HH| = 1

Ze−H/TH , (3.4)

where H is the Hamiltonian generating evolution with respect to the Schwarzschild time t.

Hence the Wald entropy for the Hartle-Hawking vacuum is a thermodynamic entropy. How-

ever, the Boulware vacuum is not thermal when restricted to the exterior of the black hole,

and therefore it is dubious to assign a thermodynamic interpretation to the Wald entropy for

the Boulware state. Nonetheless, for completeness below we will compute the Wald entropy

for both vacuum states, finding that not all solutions lead to sensible values of entropy.

Boulware vacuum

In the Boulware vacuum state, where the solution of the dilaton takes the form (2.61), we

have, before evaluating on the horizon,

SWald = Sφ0 +1

4G

[−φr√µcoth

(√µ

Lr∗

)+Gc

3

(1 +

√µ

Lr∗

)coth

(√µr∗

L

)− λGc

3

]− c

6χ .

(3.5)

Then, considering the time-independent solution χ(1) (2.63) and taking the limit r∗ → −∞,

we find the black hole entropy including semi-classical backreaction effects is given by

S(1)Wald = Sφ0 + Sφr − (λ+ 1)

c

12+c

6log(2

√µ)− c

6C +

c

12

√µ

Lr∗|r∗→−∞ , (3.6)

where Sφ0 + Sφr = φ0

4G +φr√µ

4G is the classical entropy of the black hole. With the additional

semi-classical corrections, we see the entropy is diverging like r∗ to negative infinity when

14For extremal surfaces in higher dimensions, which are not bifurcation surfaces of Killing horizons, the

holographic entanglement entropy functional is not equal to the Wald entropy for generic higher curvature

theories of gravity, but is modified with extrinsic curvature terms [85–87].

– 21 –

evaluated at the black hole horizon. This divergent result might be related to the negative

Casimir energy (2.51) for the Boulware vacuum and to the fact that the Wald entropy cannot

be interpreted as a thermodynamic entropy in this case.

Hartle-Hawking vacuum

In the Hartle-Hawking vacuum state the dilaton solution is given by (2.67), such that

SWald = Sφ0 +1

4G

[−φr√µcoth

(√µ

Lr∗

)+ (1− λ)

Gc

3

]− c

6χ . (3.7)

From the solutions for ξ in (2.68), we have in the horizon limit r∗ → −∞ the entropy

corresponding to χ(3) diverges to negative infinity, while the entropy coorresponding to χ(4)

S(4)Wald = Sφ0 + Sφr + (1− λ)

c

12+c

6log(2

√µ)− c

6C (3.8)

is constant on the horizon, since the divergences in −ρ(r∗) and ξ(4)(r∗) cancel each other.

When λ = 0 and C = 0, this is precisely the entropy found in [20],15 which can be inter-

preted as a generalized entropy, with χ capturing the von Neumann entropy of conformal

matter fields. Further, in [20] the generalized entropy was shown to obey a first law, if µ or

equivalently TH is being varied,

dM = THdS(4)Wald . (3.9)

Here, M is the semi-classically corrected ADM energy, which we rederive in Appendix C.1

from the renormalized boundary stress tensor,

M = Mφr +Mc =1

16πG

µφrL

+c√µ

12πL. (3.10)

Note that the semi-classical first law (3.9) reduces to the classical result (2.10) for c→ 0.

Lastly, substituting in χ(5) leads to a time-dependent Wald entropy (before evaluating

on the black hole horizon UK, VK → 0)

S(5)Wald = Sφ0 + Sφr +

c

12(1− λ) +

c

12log

4µ(1 + µUKVK

L2

)2

UK,VK→0

+c

12log

[(−√µ

LUK +KUK

)2(√µLVK +KVK

)2]∣∣∣∣UK,VK→0

− c

6k .

(3.11)

As we will now show, a specific choice of constants KUKand KVK

allows us to identify the time-

dependent Wald entropy with the full time-dependent generalized entropy, and, moreover, an

extremization procedure leads to a QES other than the black hole horizon.

15The apparent factor of two mismatch in the logarithm – they find a term c6

log(4√µ) – is explained by a

different convention for the mass parameter. By replacing√µ → 2

√µ, φ0 → 1, φr → 1/2 in our expression

for S(4)Wald it matches with equation (5.27) in [20].

– 22 –

3.2 von Neumann entropy

Ordinarily, when considering quantum fields outside of a stationary black hole horizon, the

Wald entropy represents the UV divergent contribution of the entanglement entropy between

field degrees of freedom inside and outside the horizon [88], and is separate from any addi-

tional matter von Neumann entropy. Together, the gravitational Wald entropy and matter

von Neumann entropy form the generalized entropy Sgen attributed to the black hole. An

important lesson garnered from gravitational thermodynamics, however, is that entropy can

be assigned to surfaces other than black hole horizons. Of particular interest are extremal

surfaces [55, 56], wherein the context of AdS/CFT, the gravitational entropy is identified

with the entanglement entropy of a dual holographic CFT. When quantum corrections are

included, the holographic entropy formula is modified by a von Neumann entropy term of

bulk quantum fields [59, 60], such that the holographic entanglement entropy is interpreted

as the generalized entropy. In this context, the generalized entropy is attributed to a quan-

tum extremal surface X, the surface which extremizes Sgen, where the classical gravitational

entropy SWald captures only a portion of the full generalized entropy.

Working directly with the semi-classical JT model, and without invoking AdS/CFT,

here we prove the semi-classical Wald entropy computed above is exactly equivalent to the

generalized entropy, where the dilaton gives the gravitational entropy and the semi-classical

correction associated to the χ field accounts for the matter von Neumann contribution,

SWald = SφWald + SχWald =1

4G(φ0 + φ)− c

6χ = SBH + SvN = Sgen . (3.12)

It remains to be shown that the Wald entropy associated to the time-dependent χ field is

equal to the time-dependent von Neumann entropy, which we will do in this section. Then,

in the next section, upon extremizing the Wald/generalized entropy, we uncover quantum

extremal surfaces which lie just outside the bifurcate horizon of the black hole.

Since the auxiliary χ field is a stand in for a collection of external conformal matter

fields, recall the von Neumann entropy of a 2D CFT with central charge c in vacuum over an

interval [(x1, y1), (x2, y2)] in a curved background in conformal gauge ds2 = −e2ρ(x,y)dxdy is

(see e.g., [18])

SvN =c

6log

[1

δ1δ2(x2 − x1)(y2 − y1)eρ(x1,x1)eρ(x2,y2)

]=c

6(ρ(x1, y1) + ρ(x2, y2)) +

c

6log

[1

δ1δ2(x2 − x1)(y2 − y1)

].

(3.13)

Here δ1,2 are independent UV regulators located at the endpoints of the interval. Since the

von Neumann entropy depends on the vacuum state, and should be evaluated in coordinates

defining the vacuum, it will be different for the Boulware versus the Hartle-Hawking vacuum.

Let us now show the semi-classical contribution encased in χ has the general structure of

the von Neumann entropy (3.13),

SχWald = − c6χ = SvN . (3.14)

– 23 –

It has been noted previously in [20, 64] that the (static) entanglement entropy with one end-

point at the black hole horizon is captured by a static choice of χ, given by our equation (2.72).

Here we instead recognize the integration constants appearing in χ give us the freedom to

match SWald = Sgen, including the full time dependence of the entanglement entropy. For

example, with respect to the Hartle-Hawking vacuum, we may identify constants in χ(5) (2.73)

k = −1

2log

4µ(1 +

µUBK VBK

L2

)2

− 1

2log

[1

(√µδ/L)2(

√µδB/L)2

],

KUK=

√µ

LUBK , KVK

= −√µ

LV B

K ,

(3.15)

such that the von Neumann entropy SHHvN in Kruskal coordinates (y, x) = (VK, UK) is

SHHvN = − c

6χ(5) =

c

12log

4µ(1 + µUKVK

L2

)2

+c

12log

4µ(1 +

µUBK VBK

L2

)2

+

c

12log

[1

δ2δ2B

(UBK − UK)2(V BK − VK)2

].

(3.16)

The two endpoints in Kruskal coordinates are taken to be, respectively, (y1, x1) = (VK, UK)

and (y2, x2) = (V BK , UBK ), where B is some generic endpoint, and δ and δB are the UV

regulators located at the two endpoints. The von Neumann entropy above is generically time

dependent, but it becomes independent of the times t and tB at the two endpoints when

UK

VK=UBKV B

K

or t = tB , (3.17)

since then the argument of the logarithm in (3.16) only depends on the tortoise coordinates:

(UBK − UK)(V BK − VK) = L2

µ

[−e2

√µr∗,B/L + 2e

√µ(r∗,B+r∗)/L − e2

√µr∗/L

]. Although the entan-

glement entropy is static in this case, it nonetheless leads to a QES just outside the horizon,

after extremizing the generalized entropy with respect to r∗. In fact, the relation (3.17) is one

of the conditions for the existence of the QES, which follows from extremizing the generalized

entropy with respect to UK and VK, see equation (3.29) below.

There are two singular cases of (3.17) where a non-trivial QES fails to exist though. The

first is when UBK = V BK = 0, such that the point B is located at the black hole horizon, in

which case the extremum of the generalized entropy lies at the black hole horizon. Second,

for UK = VK = 0, such that the first endpoint is located at the black hole horizon, the surface

extremizing the generalized entropy is again the black hole horizon. The Wald entropy S(4)Wald

(3.8) associated to the field χ(4) is an example of the second case (where in addition we have

UBK = −L/√µ = −V BK ) which is the reason why a non-trivial QES was not found before, in

e.g. [20, 64], from extremizing the Wald entropy.

It is worth emphasizing the semi-classical contribution to the Wald entropy, for the right

choice for the integration constants in χ, yields the entanglement entropy. This demonstrates

– 24 –

the Wald formalism is fully capable of attaining the logarithmic term, in contrast to what

was argued in the context of the flat space RST model [89]; in fact, the same line of reasoning

used above shows the generalized entropy in the RST model is fully characterized by the

semi-classical Wald entropy.

Note we can also match the Wald entropy with the von Neumann entropy in the Boulware

vacuum. To do so we identify the integration constants appearing in the time-dependent

solution χ(2) in (2.64) as

C ′ = −1

2log

[1

µsinh2

(√µ

2L(vB − uB)

)]− 1

2log

[1

(√µδ/L)2(

√µδB/L)2

],

Ku = −√µuB

L, Kv =

√µvB

L,

(3.18)

such that the von Neumann entropy SvN in static conformal coordinates (v, u) takes the form

SBvN =

c

12log

[1

µsinh2

(√µ

2L(v − u)

)]+

c

12log

[1

µsinh2

(√µ

2L(vB − uB)

)]+

c

12log

[1

δ2δ2B

(vB − v)2(uB − u)2

].

(3.19)

In a moment we will show this von Neumann contribution leads to a QES, extremizing the

generalized entropy in Boulware vacuum, which also lives outside the black hole horizon.

Fixing constants of integration

Thus, generically we can arrange the integration constants in the solutions for χ such that the

semi-classical Wald entropy is exactly equal to the generalized entropy, including time depen-

dence. So far, however, we matched these constants in an ad hoc manner. Here we present

a derivation justifying our choice of integration constants. We do this in two steps. First,

we generically solve the equation of motion for χ imposing a Dirichlet boundary condition in

flat space, and then we perform a Weyl transformation back to an arbitrary curved space-

time, leading to a general solution for χ in conformal coordinates. Next we place this generic

derivation of χ into the context of semi-classical JT gravity by providing an interpretation of

the endpoint (V BK , UBK ) where the Dirichlet boundary condition is imposed.

Recall the equation of motion for χ in (2.12), 2χ = R, which defines the solution of χ

in an arbitrary curved spacetime. We expect SvN = − c6χ, where the general solution of χ

in conformal gauge is χ = −ρ + ξ, and for the Hartle-Hawking vacuum the time-dependent

χ should be given by (2.73). Before analyzing the solution of the χ equation of motion in

curved spacetime, let us first perform a Weyl transformation and solve (2.12) in flat space.

The Weyl transformation effectively amounts to setting R = 0, or equivalently ρ = 0. The

solution for χ in flat space thus becomes

χflat = ξ , (3.20)

where it is important to recall that in a general curved spacetime χ transforms as a scalar

whereas ξ does not.

– 25 –

The boundary condition we impose on this solution is a Dirichlet boundary condition

at the endpoint (V BK , UBK ) in Kruskal coordinates. When the inverse Weyl transformation is

performed, we take the endpoint B to be close to the conformal boundary of AdS, but it does

not matter where it is located near the boundary. Explicitly, the boundary condition is

lim(VK,UK) → (V BK ,UBK )

ξ = C , (3.21)

where C is some real constant that does not depend on (VK, UK). Typically in the literature

[20, 64] a boundary condition is imposed on χ in curved space, instead of on ξ in flat space,

where specifically the flux of χ is required to go to zero at the boundary. It will turn out

that we can also put the same requirement on χ, but it is important for what follows that we

require (3.21) for ξ as well.

Since we are solving a harmonic equation ξ = 0 with a Dirichlet boundary condition,

without any loss of generality we know that

ξ(VK, UK, VB

K , UBK ) = ξ(UBK − UK, VB

K − VK) . (3.22)

This forces the solution for χflat to be of the form

χflat = ξ = −1

2log

[1

δ4(UBK − UK)2(V B

K − VK)2

], (3.23)

where δ is some arbitrary length scale which regulates the limit (3.21). We recognize this can

be written as a two-point correlation function of primary operators ∂χ

χflat = log[δ2〈∂χ(VK, UK)∂χ(V B

K , UBK )〉flat

], (3.24)

where δ is another arbitrary length scale. An elementary result from conformal field theory

instructs us how the two-point correlation function transforms under a Weyl transformation.16

Undoing the Weyl transformation, we obtain the full curved spacetime expression for χ,

χ = −1

2log

[e2ρe2ρB

δ4(UBK − UK)2(V B

K − VK)2

], (3.25)

which coincides with the regulated von Neumann entropy (3.16) up to a constant. Notice

that even if e2ρB takes a singular value somewhere at the boundary, we can always regulate

this divergence such that χ goes to zero on the boundary. As a last remark we point out that

from the holographic perspective the cut-off δ takes care of the regularization of the endpoints

and as such is able to generate the conformal factors that appear in the logarithm through

an appropriate change of coordinates of the holographic boundary coordinate, see e.g. [44].

16Let O be primary operators with weight ∆O, then under Weyl transformations we can relate correlation

functions as 〈O(x)O(y)〉Ω2gµν = Ω(x)−∆OΩ(y)−∆O 〈O(x)O(y)〉gµν . In the case considered in this work ∆O = 1.

– 26 –

von Neumann contribution to generalized entropy in general 2D gravity

The argument we presented above to derive the von Neumann entropy from the χ fields

was rather general. Let us now argue how general its conclusions are. Any two-dimensional

theory of gravity with a minimally coupled CFT with large central charge c gives a conformal

anomaly at the semi-classical level, captured by the Polyakov term. The Polyakov term (2.11)

can always be localized by introducing χ fields whose equation of motion is given by (2.12).

Furthermore, the Wald entropy receives a functional contribution from χ, independent of the

value of the dilaton or metric.

In general one has to make a choice for what the point (V BK , UBK ) denotes. We generically

require the product UBKVB

K to define a timelike surface on which the metric is constant, as is

the case for, e.g., the conformal boundary in AdS or near I+ in Minkowski spacetime. Note

the interpretation of this timelike surface depends on the context of the problem of interest.

For example, the case of 2D de Sitter gravity is special as there the asymptotic boundary

I+ is spacelike, instead of timelike or null. Nevertheless one can, from an algebraıc point of

view, perform the same manipulations on a timelike surface close to the poles of de Sitter.

Once the meaning of the endpoint (V BK , UBK ) has been established one can employ the

Dirichlet condition above and, aided by the fact that any two-dimensional spacetime is Weyl

equivalent to flat spacetime, we generically arrive at (3.25) for the Hartle-Hawking state.

Thus, we find SχWald = − c6χ = SvN for any 2D gravity theory coupled to a CFT. Two

remarks worth mentioning are that, first, the above derivation follows through for different

vacua as well, modifying only the contribution of the two-point function in (3.25), however,

the general structure remains the same. Second, as we only made assumptions about the

existence of a timelike surface on which the metric is constant, the argument presented here

holds for static and non-static backgrounds alike.

3.3 Quantum extremal surfaces

Having verified the semi-classical Wald entropy is equivalent to the generalized entropy, let

us now look for the surfaces which extremize the entropy. These surfaces are known as

quantum extremal surfaces (QES) [60], and are generalizations of the Ryu-Takayanagi surfaces

necessary to compute quantum corrected holographic entanglement entropy. We will not need

this interpretation below, however, we are certainly motivated by it. Rather, the point we

aim to emphasize here is that, when including the backreaction of the Hawking radiation of a

stationary black hole in thermal equilibrium, it is natural to associate thermodynamics with

a QES, such that the bifurcate Killing horizon can be viewed as the classical limit of the QES.

Since the form of the Wald entropy is dependent on the vacuum state, so too will the

QES depend on the choice of vacuum. Let us thus extremize the Wald entropy in both

Hartle-Hawking and Boulware vacua and look for the QESs in either case.

– 27 –

Hartle-Hawking QES

Consider the time-dependent generalized entropy SHHgen given by (3.7), where the semi-classical

contribution is the von Neumann entropy SHHvN in (3.16), and where we set λ = 0. Specifically,

SHHgen = Sφ0 +

1

4G

[φr√µ

(1− µ

L2UKVK

1 + µL2UKVK

)+Gc

3

]+ SHH

vN . (3.26)

We extremize with respect to the Kruskal coordinates UK and VK of the first endpoint, while

keeping the second endpoint B fixed,

∂UKSHH

gen = ∂VKSHH

gen = 0 . (3.27)

We find

∂VKSHH

gen =3L3µ3/2UK(V B

K − VK) +Gc Lφr (L2 + µUKVK)(L2 + µUKVB

K )

6G Lφr

(L2 + µUKVK)2(VK − V BK )

= 0 , (3.28)

and similarly for ∂UKSHH

gen = 0.

Subtracting the two extremization conditions we uncover the following relation:

0 = ∂UKSHH

gen − ∂VKSHH

gen ⇔ UK

VK=UBKV B

K

. (3.29)

Meanwhile, adding the two extremization conditions leads to

0 = 2ε+6µ

L2

(1− VK

V BK

)VKU

BK +

2εµ

L2

[1 +

VK

UBK+

µ

L2

(VK)2UBKV B

K

]VKU

BK . (3.30)

where we used (3.29) and introduced the small parameter,

ε ≡ Gc√µφr

1 , (3.31)

which follows from the semi-classical regime of validity (2.13). The effect of the cubic term

can be neglected (it changes the location of the QES at subleading order in ε) and we can

instead solve for VK with the simplified quadratic equation, such that for small ε we have the

following two solutions for VK:

V(1)

K ≈ −L2

3µ

ε

UBK+O(ε2) , V

(2)K ≈ V B

K +(L2 + 2µUBKV

BK )

3µUBK+O(ε2) . (3.32)

The second solution is near the conformal boundary which is deemed unphysical, since it does

not coincide with the black hole horizon in the classical limit c → 0, thus leaving us with a

single QES located at

V QESK ≈ −L

2

3µ

ε

UBK, UQES

K ≈ −L2

3µ

ε

V BK

. (3.33)

– 28 –

In terms of the Schwarzschild coordinate, the QES has the radial position

rQES = L√µ

(1− µ

L2UKVK

1 + µL2UKVK

)≈ rH

(1 +

2ε2

9e−

2√µ

Lr∗,B

), (3.34)

where r∗,B = L2√µ log

(− µL2U

BKV

BK

)is the tortoise coordinate of the second endpoint near the

conformal boundary. Importantly, in the limit we approach the conformal boundary, r∗,B → 0

or − µL2U

BKV

BK = 1, the QES still exists.

Substituting the QES (3.33) back into the generalized entropy (3.26) we find:

Sgen

∣∣QES

= Sφ0 + Sφr +c

12+c

6log(2

√µ) +

c

6log

(−µUBKV

BK

L2

)+

c

12log

[4µ(

1 + µL2U

BKV

BK

)2]

+cL2ε

9µUBKVB

K

+O(ε2) . (3.35)

Thus, to leading order in ε, the generalized entropy of the QES is simply that of the black

hole. Critically, however, since UK < 0 and VK > 0, the ε correction is negative; indeed in the

conformal boundary limit − µL2U

BKV

BK = 1, where the divergence in the generalized entropy is

regulated by a cutoff, the above becomes

Sgen

∣∣QES

r∗,B→0= Sgen

∣∣H− cε

9+

√µφr

18Gε2 +O(ε2) , (3.36)

where we have included the O(ε2) for further clarity. Thence, the generalized entropy evalu-

ated at the QES is less than the entropy evaluated on the eternal black hole horizon,

Sgen

∣∣QES

< Sgen

∣∣H. (3.37)

A few comments are in order. First, notice in the classical limit ε → 0 we observe the QES

is located at the bifurcate Killing horizon. This is consistent with the fact the classical Wald

entropy is extremized by the black hole horizon, while semi-classical effects alter the position

of the extremal surface. In fact, in the case of the static background considered here, we see

the QES lies just outside of the horizon. Specifically, from (3.34), the QES is approximately

located at a distance rH

(Gc√µφr

)2 rH above the horizon. This suggests, in the context of

an eternal black hole background at least, the QES may be viewed as a stretched horizon or

membrane [90], which are well known to obey a first and second law of thermodynamics. We

emphasize, however, the stretched horizon of a black hole is located outside the horizon even

when one does not take into account semi-classical effects (whereas the QES is located at

the event horizon classically). Moreover, in dynamical scenarios such as an evaporating black

hole, the QES is found to be located inside of the horizon [42]. Thus, the QES is generally

different from a stretched horizon.

Next, note the condition (3.29) attained from extremality, yields the Schwarzschild time t

is fixed to the time at B, tB,

t = tB . (3.38)

– 29 –

Equivalently, in static null coordinates we find (u − uB) = −(v − vB). As we mentioned

below (3.17), substituting this time symmetry back into the time-dependent solution for

χ (2.73) we find the logarithmic contribution – where the time dependence resides – solely

becomes a function of (u − uB), such that time dependence drops out. Thus, restricted to

a time slice t = tB, the generalized entropy becomes time independent. Nonetheless, as

can be explicitly verified, had we implemented (3.29) into Sgen initially and performed the

same extremization procedure as above we would have recovered the same position of the

QES, further indicating the QES arises from semi-classical effects and not time dependence

of the von Neumann entropy. This observation is also relevant for the interpretation of the

generalized entropy as a thermodynamic entropy, a fact we use in the next section.

It is worth comparing our set-up and extremization of the generalized entropy to that

in [64]. The set-up in [64] distinguishes between two scenarios: (i) the semi-classical JT

model, with χ representing c scalar fields, with the same form of the generalized entropy as

in (3.3), defined at the horizon, however, where χ is vanishing at the asymptotic boundary of

a collapsing AdS2 black hole (not connected to a thermal bath, such that outgoing radiation

bounces off the boundary, falling back into the black hole), and (ii) working directly with c

scalar fields ψi obeying Dirichlet boundary conditions, which has another generalized entropy

Sψgen, given by the classical contribution of the black hole and the von Neumann entropy of the

fields ψi outside the horizon. Crucially, contrary to our set-up, where we find a QES outside

of the horizon and Sgen|H > Sgen|QES, the authors of [64] find Sψgen is extremized for points

other than the horizon, however, discard such solutions since they claim Sψgen is larger at these

points than at the bifurcate horizon, Sψgen|H < Sψgen|QES. This may be an artifact of working

with the χ system, which is the c → ∞ limit of the ψi system, or perhaps a consequence of

working with different boundary conditions for χ.

Lastly, as a passing comment, note we found a QES outside of the horizon in the eternal

AdS2 background, akin to the entanglement wedge islands exterior to the horizon in static

background systems in [45], used to resolve a particular form of the information paradox

by deriving a unitary Page curve. It is thus tempting to interpret our findings above as

likewise locating islands outside of the horizon. However, in [45] a thermal bath is joined at

the conformal boundary, such that the Hawking radiation disappears into the bath and the

AdS2 black hole evaporates, which is contrary to our set-up. Nonetheless, the similarities are

striking and we will revisit this topic in the discussion.

Boulware QES

Before moving on to the next section where we study the thermodynamics of the QES, let

us point out the generalized entropy in the Boulware state is also extremized by a quantum

extremal surface. In the Boulware vacuum, the generalized Wald entropy (3.5) with λ = 0 is,

SBgen = Sφ0+

1

4G

[−φr√µcoth

(√µ

2`(v − u)

)+Gc

3

(1 +

√µ

2L(v − u)

)coth

(√µ(v − u)

2L

)]+SB

vN ,

(3.39)

– 30 –

with SBvN equal to (3.19). We now extremize (3.39) with respect to both v and u,

∂vSBgen = ∂uS

Bgen = 0 . (3.40)

The rest of the analysis is nearly identical as before. Adding the derivatives we again find

(u−uB) = −(v−vB), while, upon substituting in this time symmetry condition, the difference

yields,

0 = 1− ε

3− ε

6

√µ

L(−2u+ vB + uB) +

ε

2sinh

[√µ

L(−2u+ vB + uB)

]− 4ε

3

L√µ

u− uBsinh2

[√µ

2L(−2u+ vB + uB)

].

(3.41)

Consequently, the position of the QES in advanced and retarded coordinates (v, u) is

vQES ≈1

2(uB + vB) +

L√µ

log

(√ε

2

),

uQES ≈1

2(uB + vB)− L

√µ

log

(√ε

2

).

(3.42)

Equivalently, in Kruskal coordinates

√µ

LV QES

K ≈ 1

2

(−V B

K

UBK

)1/2√ε ,

√µ

LUQES

K ≈ 1

2

(−UBKV B

K

)1/2√ε , (3.43)

or in Schwarzschild coordinates,

rQES ≈ rH(

1 +ε

2

). (3.44)

Thus, as in the Hartle-Hawking vacuum, in the classical limit ε→ 0 we find rQES → rH , but

generally rQES lies outside of the horizon. Moreover, if we evaluate the generalized entropy at

the QES we find it leads to a value smaller than the generalized entropy when evaluated on

the bifurcate horizon, however, both are formally negatively divergent on the horizon. The

overall point here is that while the Boulware vacuum does not describe a system in thermal

equilibrium, the generalized entropy in the Boulware vacuum is nonetheless extremized by a

QES outside of the horizon.

4 Semi-classical thermodynamics of AdS2-Rindler space

Above we showed the Wald entropy SWald associated with the semi-classical JT model is ex-

actly equivalent to the generalized entropy, including the von Neumann entropy contribution

associated with a CFT with central charge c outside of the eternal black hole. Moreover, we

found a quantum extremal surface lying just outside of the horizon which extremizes the Wald

entropy and, crucially, leads to a lower value of the entropy than when SWald is evaluated on

– 31 –

the bifurcate horizon of the black hole. Thus, we may attribute a thermodynamic entropy to

the QES, suggesting the QES may also obey a first law.

The thermodynamics of the QES on display may be seen as a consequence of the fact

the entanglement wedge of the QES is a Rindler wedge.17 First, we recall the AdS2 black

hole background is simply AdS2-Rindler space, such that the black hole thermodynamics

is a consequence of the thermal behavior of AdS-Rindler space, namely, that the density

matrix of the Hartle-Hawking vacuum state reduced to the Rindler wedge is a thermal Gibbs

state. It is this realization which leads to the first law of black hole thermodynamics, where

the temperature is given by the Hawking temperature, the energy by the ADM mass, and

the entropy by the Wald entropy. By extension, since the entanglement wedge associated

with the QES is also a Rindler wedge, the corresponding thermal character leads to a first

law of thermodynamics of quantum extremal surfaces, where the entropy is given by the

generalized entropy, i.e., the semi-classically corrected Wald entropy, and the asymptotic

energy is identified with a semi-classical modification of the ADM mass (we will comment on

the temperature momentarily). Therefore, the first law of QESs is understood as the natural

semi-classical generalization of the first law of black hole thermodynamics. In this section we

derive this semi-classical first law of QES thermodynamics.

An important feature to point out is that, since the QES lies outside of the black hole,

the QES (right) Rindler wedge is nested inside of the (right) Rindler wedge associated with

the black hole, where, in the limit the QES and bifurcate horizon coincide, c→ 0, the Rindler

wedges are identified. Thus, we are in fact interested in studying the thermal behavior of

the nested Rindler wedge. Crucially, the global Hartle-Hawking state restricted to the nested

Rindler wedge is a again a thermal Gibbs state. In fact, we will show below the static observers

in the eternal black hole and in the nested Rindler wedge both see the same Hartle-Hawking

vacuum as a thermal state, but at different temperatures. Indeed, the surface gravity of the

nested Rindler horizon is the parameter which naturally appears in the first law of the nested

wedge, and is identified as the temperature of the entanglement wedge of the QES when in

equilibrium.

Before moving on to the derivation of the semi-classical first law, it is worth mentioning

prior important studies of AdS-Rindler thermodynamics. In the seminal work [65], it was

shown the entanglement entropy of a CFT vacuum state in flat space Rd reduced to a ball

of radius R, via a conformal mapping, is equal to the thermal entropy of a Gibbs state in

the hyperbolic cylinder R × Hd−1, where the temperature is inversely proportional to the

radius of the ball. When the CFT is holographic, the thermal CFT state is then dual to

a massless hyperbolically sliced black hole, where the thermal entropy of the Gibbs state is

17Given a “bulk” asymptotically AdS spacetime with boundary subregion A, the entanglement wedge EA[91–93] is the domain of dependence of any (achronal) codimension-1 bulk spatial surface with boundary

A∪ΓA, where ΓA is the extremal codimension-2 bulk region homologous to A, i.e., the HRT surface [56]. The

entanglement wedge is generally larger than the causal wedge [94], the intersection of bulk causal future and

past boundary domain of dependence of A, however, the two wedges sometimes coincide, e.g., for spherical

boundary subregions A in vacuum AdS, in which case EA is equal to the AdS-Rindler wedge.

– 32 –

identified with the Bekenstein-Hawking entropy of the black hole at the same temperature.

For holographic CFTs dual to higher curvature theories of gravity, the Bekenstein-Hawking

entropy is replaced by the Wald entropy.18 The massless hyperbolic black hole, of course,

is simply (d + 1)-dimensional AdS-Rindler space, such that the Rindler wedge is identified

with the entanglement wedge of the ball on the boundary flat space. Thus, via the Casini-

Huerta-Myers map, the vacuum entanglement entropy of a holographic CFT reduced to a

ball is equal to the Wald entropy of a AdS-Rindler eternal black hole. Furthermore, the first

law of entanglement entropy [96, 97] for ball-shaped regions in the vacuum boundary CFT is

holographically dual to the first law of the AdS-Rindler horizon [66].

The first law we derive below may be interpreted as an exact semi-classical extension of

the first law of the AdS-Rindler wedge, however, there are some conceptual differences with

previous works. First, our work is not based on the AdS/CFT correspondence. Indeed, we

work directly in a (1 + 1)-dimensional AdS-Rindler space, for which the dual one-dimensional

CFT is obscure. Therefore we do not invoke the existence of a holographic CFT. Second, due

to the semi-classical effects of backreaction, we uncover a first law associated with a Rindler

wedge nested inside the AdS-Rindler black hole. We will return to discuss the similarities

between our semi-classical extension and the work of [65, 66, 96] in the discussion.

4.1 Rindler wedge inside a Rindler wedge

Since the quantum extremal surface lies outside of the AdS2 black hole horizon, we are moti-

vated to study the thermodynamics associated with a Rindler wedge nested inside the eternal

AdS2 black hole, whose exterior is itself a Rindler wedge (see Figure 3). Here we summarize

the main aspects of the geometric set-up, more details may be found in Appendix B.

One way to see that the eternal AdS2 black hole is AdS-Rindler space is by the coordinate

transformation (A.20)

κσ =√µt/L , % =

L√µ

√r2

L2− µ , (4.1)

such that the line element (2.5) becomes

ds2 = −κ2%2dσ2 +

(%2

L2+ 1

)−1

d%2 . (4.2)

Here κ is the surface gravity of the AdS-Rindler horizon associated to the boost Killing

vector ∂σ. However, the surface gravity depends on the normalization of the boost Killing

vector, which is not unique since the boost generator cannot be normalized at spatial infinity

in (AdS-)Rindler space, like the stationary Killing vector of asymptotially flat black holes.

The radial coordinate % lies in the range % ∈ [0,∞), with the horizon located at % = 0 and the

asymptotic boundary at % = ∞. Lines of constant σ parametrize the worldlines of Rindler

observers with proper acceleration a =√

1/%2 + 1/L2.

18In fact, the statement holds for holographic CFTs dual to arbitrary higher derivative theories of gravity

in higher and lower dimensions, including JT gravity [95].

– 33 –

V+

V−

ΣB

HH

ζ

α

α

Figure 3: Nested Rindler wedges. The larger AdS-Rindler wedge (shaded in blue) envelopes

a smaller Rindler wedge (shaded in green). The bifurcation points of the larger and smaller

(right) Rindler patches generally do not coincide, but they do in the limit when the boundary

time interval goes to infinity, α → ∞. In this limit the boost Killing vector ζ of the nested

Rindler wedge becomes proportional to the time-translation Killing vector ∂t of the black

hole metric. We have illustrated a nested Rindler wedge whose extremal slice Σ is located at

t = 0, but we also consider nested wedges centered at t = t0 6= 0.

Note we need not work with coordinates (σ, %) explicitly since the Schwarzschild coor-

dinates also cover AdS-Rindler space. Indeed, the proper Rindler time-translation Killing

vector ∂σ is proportional to the Schwarzschild time-translation Killing vector ∂t; lines of con-

stant t likewise represent the worldlines of uniformly accelerating observers, such that the

σ = 0 slice coincides with the t = 0 slice. Consequently, we often refrain from working with

coordinates (σ, %), and instead opt for the advanced and retarded null coordinates (v, u).

Below we let (t, r) denote the Schwarzschild coordinates adapted to the nested Rindler

region with associated advanced and retarded time coordinates (v, u), defined in a completely

equivalent way as for the larger Rindler wedge. The nested AdS-Rindler wedge is the domain

of dependence of the time slice Σ bounded by the bifurcation point B (defined in Eq. (4.9))

and the conformal boundary at r →∞, which has vanishing extrinsic curvature and is hence

extremal. We focus on the “right” AdS-Rindler wedge, neglecting the “left” AdS-Rindler

wedges, and we consider AdS-Rindler patches whose extremal slice Σ is not located on the

t = 0 time slice, but rather on an arbitrary t0 6= 0 slice.

Boost Killing vector of the nested Rindler wedge

The generator of proper Rindler time t translations is the boost Killing vector of the nested

Rindler wedge. We are interested in how this Killing vector is expressed in terms of the

– 34 –

coordinates of the enveloping black hole. This problem is addressed in Appendix B using

the embedding formalism, where AdS2 is embedded as a hyperboloid in a (−,−,+) signature

Minkowski space. The boost Killing vector ζ = ∂t takes the following form in the advanced

and retarded time coordinates (v, u) of the AdS2 black hole

ζ =Lκ/√µ

sinh(√µα/L)

[(cosh(

√µα/L)−cosh(

√µ(v−t0)/L))∂v+(cosh(

√µα/L)−cosh(

√µ(u−t0)/L))∂u

].

(4.3)

Equivalently, in Schwarzschild coordinates (t, r) we have

ζ =κ

sinh(√µα/L)

[L√µ

cosh

(√µ

Lα

)− r/L√

r2

L2 − µcosh

(√µ

L(t− t0)

) ∂t

+ L

√r2

L2− µ sinh

(√µ

L(t− t0)

)∂r

].

(4.4)

The parameter α may be interpreted as a boundary time interval. This can be seen via the

following coordinate relationship between enveloping black hole coordinates (v, u) and the

nested Rindler wedge coordinates (v, u):

√µ

Lu = log

[e√µt0/L + eκu+

√µ(t0+α)/L

e√µα/L + eκu

],

√µ

Lv = log

[e√µt0/L + eκv+

√µ(t0+α)/L

e√µα/L + eκv

], (4.5)

where (v, u) → (v, u) in the limit α → ∞ (upon identifying κ =√µ/L). In the nested

coordinates the boundary vertices, where the nested AdS-Rindler wedge intersects the global

conformal AdS boundary, are located at V± = u = v = ±∞, with the plus sign for the

future vertex and the minus sign for the past vertex. Inserting these points in the coordinate

transformation above we find that the vertices are at

V± = t = t0 ± α, r =∞ (4.6)

in Schwarzschild coordinates, hence α is the boundary time interval between Σ and V±. In

the limit α→∞ the boundary time interval is identified with the entire AdS boundary such

that the nested Rindler wedge coincides with the exterior of the black hole. Moreover, in this

limit we see the Killing vector ζ (4.4) is proportional to the generator of Schwarzschild time

translations,

ζ∣∣α→∞ →

κL√µ∂t . (4.7)

The past and future AdS-Rindler Killing horizons are located at the null surfaces where ζ

becomes null, ζ2∣∣H = 0, which in Schwarzschild coordinates are given by

H =

cosh

(√µ

L(t− t0 ± α)

)=

r/L√r2/L2 − µ

, (4.8)

– 35 –

where the plus sign corresponds to the past horizon and the minus sign to the future horizon.

From the expression (4.3) for ζ in terms of null coordinates we see the future horizon of the

right Rindler wedge is at u = t0 + α and the past horizon is at v = t0 − α.

Further, the boost Killing vector ζ vanishes at the future and past vertices V± and at

the bifurcation point B, the intersection of the future and past AdS-Rindler horizons, which

is characterized in Schwarzschild coordinates by

B = t = t0, r = L√µ coth(

√µα/L) . (4.9)

The boundary intersection points V± and the bifurcation surface B are thus fixed points of

the flow generated by ζ. Observe that in the α→∞ limit the bifurcation surface B coincides

with the bifurcation point of the black hole event horizon.

A few more comments are in order. First, since ζ is the sum of two future null vectors

in the interior of the (right) Rindler wedge, it is timelike and future directed. Moreover, ζ

also acts outside the right Rindler wedge, and is in fact null on four different Killing horizons

u = t0 ± α, v = t0 ± α. It is also worth remarking, that in the form (4.3) ζ is identical to

the conformal Killing vector preserving a spherical causal diamond in a maximally symmetric

spacetime [8]. Thirdly, note the normalization of ζ is chosen such that the (positive) surface

gravity, defined via ∇µζ2 = −2κζµ on the (future) Killing horizon, is equal to κ, which we

keep arbitrary in the main body of the paper (in Appendix B we set κ = 1).

Some useful derivative identities

There are some useful observations we should remark on before we move to derive the Smarr

relation and first law associated with the nested Rindler wedge. First, in order to derive the

first law we will need to know the Lie derivative of the dilaton φ with respect to the boost

Killing vector ζ. Explicitly, since φ is time-independent in Schwarzschild coordinates,

Lζφ = ζr∂rφ =κφr

sinh(√µα/L)

√r2

L2− µ sinh

(√µ

L(t− t0)

)⇒ Lζφ

∣∣Σ

= 0 , (4.10)

where Σ is defined as the time slice t = t0. Thus, the boost Killing vector is an instantaneous

symmetry of the dilaton at Σ, which will prove crucial momentarily.

In what follows we also need to know the covariant derivative of the Lie derivative of φ,

∇ν (Lζφ)∣∣Σ

= −uνκφr√µ

L sinh(√µα/L)

= −uνκφrL

√r2BL2− µ , (4.11)

where we used the value of α in terms of the bifurcation point (4.9), and identified uν as the

future-pointing unit normal to Σ:

uν =1√

r2/L2 − µ∂νt , uν = −

√r2

L2− µδtν . (4.12)

– 36 –

Given the proper length ` of Σ between the bifurcation point B of the nested Rindler horizon

and the asymptotic boundary

` =

∫ ∞rB

dr√r2/L2 − µ

= λ(∞)− λ(rB), with λ(r) = L arctanh

(r/L√

r2/L2 − µ

), (4.13)

we can define the derivative of the dilaton at B with respect to the proper length ` by

φ′B ≡∂φB∂`

= − ∂φB∂λ(rB)

= −φrL

√r2BL2− µ , (4.14)

where in the last equality we used φB = φrL rB. Comparing with (4.11) we thus find

uν∇ν(Lζφ)∣∣Σ

= −κφ′B . (4.15)

Importantly, note that φ′B is constant on Σ, a crucial property used to prove the existence of

a geometric form of the first law for AdS-Rindler space in (classical) JT gravity.19

Moreover, since they will be useful momentarily, it is straightforward to show

Lζχ∣∣∣Σ

= 0 , (4.16)

∇ν(Lζχ)∣∣∣Σ

= −κ√µ

L

tanh(√µα/2L)

rL −

√r2

L2 − µuν = −κ

L

rBL −

√r2BL2 − µ

rL −

√r2

L2 − µuν . (4.17)

The first equation holds for the time-independent as well as time-dependent solutions for

χ, whereas the second equation is specific to the time-dependent solution χ(5) (2.73), with

constants (3.15) and where t0 = tB. We observe uν∇ν(Lζχ)∣∣∣Σ

is not constant on Σ. This

result will play a role in the derivation of the semi-classical first law.

4.2 Classical Smarr formula and first law

Here we derive the classical Smarr relation and associated first law for the nested AdS2-

Rindler wedge described above, leaving its semi-classical extension for the next section. Our

derivation relies on the covariant phase space formalism [67, 84, 99–101], particularly the

methodology of Wald used to study the first law of black hole thermodynamics. In Appendix C

we briefly review Wald’s Noether charge formalism and apply it to generic two-dimensional

dilaton-gravity theories. We specialize to classical and semi-classical JT gravity in the next

two sections. For previous applications of the covariant phase formalism to classical CGHS

and JT models, see, e.g., [35, 84, 102–104].

19A similar property is necessary to prove the geometric form of the first law of spherical causal diamonds

in higher-dimensional maximally symmetric spacetimes [8, 98].

– 37 –

4.2.1 Smarr relation

The Smarr identity [105] is a fundamental relation between thermodynamic variables of a

stationary black hole. For example, in the case of a D > 3 dimensional AdS black hole of

mass M and horizon area A, the Smarr relation is [106]

D − 3

D − 2M =

κ

8πGA− 2Λ

(D − 2)8πGΘ , (4.18)

where Θ ≡ 8πG(∂M∂Λ

)A

is the quantity conjugate to the cosmological constant Λ. In the

literature (minus) Θ is better known as the “thermodynamic volume” since −Λ/8πG is in-

terpreted as a bulk pressure. Naively, from expression (4.18) it appears a Smarr relation is ill

defined in D = 2 spacetime dimensions. Nonetheless, a Smarr identity for (1+1)-dimensional

AdS black holes does exist, see for instance (2.10), but it has to be derived separately for 2D

dilaton gravity (see also [107, 108]).

Here we derive the Smarr identity for the (1 + 1)-dimensional nested Rindler wedge by

equating the Noether current jζ associated with diffeomorphism symmetry generated by the

boost Killing vector ζ to the exterior derivative of the associated Noether charge Qζ , and

integrating over the Cauchy slice Σ at t = t0. A similar generic method was used in the

context of Lovelock black holes [109], spherical causal diamonds in (A)dS [8], and for higher

curvature gravity theories coupled to scalar and vector fields [110], all of which are based on

Wald’s Noether charge formalism [67]. We mostly follow the notation of [8].

We recall the Noether current 1-form jζ is defined as

jζ ≡ θ(ψ,Lζψ)− ζ · L , (4.19)

where θ is the symplectic potential 1-form, ψ denotes a collection of dynamical fields (namely,

the metric gµν and dilaton φ), L is the Lagrangian 2-form, whose general field variation is

δL = Eψδψ + dθ(ψ, δψ), and δζψ = Lζψ is the field variation induced by the flow of vector

field ζ. For diffeomorphism covariant theories, for which δζL = LζL, the Noether current is

closed for all ζ when the equations of motion hold, Eψ = 0, and hence jζ can be cast as an

exact form,

jζ = dQζ , (4.20)

where Qζ is the Noether charge 0-form.

The Smarr relation follows from the integral version of (4.20),∫Rjζ =

∫RdQζ =

∮∂R

Qζ , (4.21)

where, generically, R is a codimension-one submanifold with boundary ∂R. The second

equality follows from Stokes’ theorem. For a black hole with a bifurcate Killing horizon, ζ is

taken to be the horizon generating Killing field, and R the hypersurface extending from the

bifurcation surface to spatial infinity. Using Lζgµν = 0 one arrives at the Smarr relation (4.18)

for stationary black holes [8]. In our case, where ζ is the boost Killing vector associated

– 38 –

with the nested Rindler wedge, we have in addition Lζφ = 0 (4.10) only along the t = t01-dimensional hypersurface Σ extending from the bifurcatation point B to the asymptotic

boundary. Thus, ∫Σjζ =

∮∞Qζ −

∮BQζ , (4.22)

where the orientation of the Noether charge integral on B is chosen to be outward, towards

spatial infinity. The Noether charge integral over either location in two dimensions can be

replaced by evaluating Qζ at either point∮∞Qζ = limr→∞Qζ and

∮BQζ = limr→rB Qζ .

Let’s first focus on the right-hand side of (4.22). As derived in Appendix C, the Noether

charge (C.13) for JT gravity is

QJTζ = − 1

16πGεµν [(φ+ φ0)∇µζν + 2ζµ∇ν(φ+ φ0)] , (4.23)

where εµν is the binormal volume form for ∂Σ, which we may write in terms of the timelike

and spacelike unit normals, uµ and nµ, respectively, such that εµν = 2u[µnν] (where we used

ε∂Σ = 1 in two dimensions). At the bifurcation point B the boost Killing vector vanishes,

ζ|B = 0, and ∇µζν |B = κεµν , where κ is the surface gravity. Since εµνεµν = −2, we find∮

BQJTζ =

κ

8πG(φ0 + φB) , (4.24)

where φB is the value of the dilaton at the bifurcation point. In the limit the nested Rindler

horizon coincides with the black hole horizon, H → H, we recover the “area” term. That is,

had we been working with general relativity, this contribution gives κAH/8πG, such that one

interprets φ0 + φH as the horizon “area” of the higher-dimensional black hole from which JT

gravity is reduced. Meanwhile, the Noether charge evaluated at the intersection of the spatial

boundary B and the extremal slice Σ, given by the point t = t0, r = rB in Schwarzschild

coordinates, is

QJTζ

∣∣∣t0,rB

=κφ0 coth(

√µα/L)

8πGL√µ

rB −κφ0

8πG√µ

√r2B/L

2 − µsinh(

√µα/L)

+κ√µ

8πGcoth(

√µα/L)φr , (4.25)

which diverges as rB →∞. Thus, the first term on the right-hand side (4.22) is∮∞QJTζ = lim

rB→∞

κφ0 tanh(√µα/2L)

8πGL√µ

rB +κ√µφr coth(

√µα/L)

8πG. (4.26)

We will return to the divergent contribution momentarily.

We now evaluate the left-hand side of (4.22), where we need the form of the Noether

current jζ . We can either work directly with the current (C.12) specific to JT gravity, or, use

general the form (4.19) together with the symplectic potential (C.5) and the Lagrangian

LJT =ε

16πG

[(φ0 + φ)R+

2

L2φ

], (4.27)

– 39 –

where ε is the volume form for the full spacetime manifold. In the current context it is simpler

to work with the latter since ζ is a Killing field, such that Lζgµν = ∇νLζgµν = 0, and the

symplectic potential θ(ψ, δψ) (C.5) for JT gravity is linear in δgαβ and ∇νδgαβ, hence

θJT(ψ,Lζψ) = 0 . (4.28)

The Noether current (4.19) is therefore

jJTζ = −ζ · LJT =

φ0

8πGL2(ζ · ε) , (4.29)

where the Lagrangian is taken on-shell in the last equality by inserting R = − 2L2 . Conse-

quently, the left-hand side of the relation (4.22) is∫ΣjJTζ =

φ0

8πGL2

∫Σ|ζ|d` , (4.30)

where we used∫

Σ ζ ·ε =∫

Σ |ζ|d`, with d` being the infinitesimal proper length d` = dr√r2/L2−µ

,

and |ζ| ≡√−ζ · ζ is the norm of the Killing vector ζ. It is straightforward to show this integral

is divergent as r →∞.

To remove the undesired divergences in the integrals of jJTζ and QJT

ζ |∞, one can either

introduce local (boundary) counterterms [111, 112], or regulate with respect to the extremal

(µ = 0) background. The latter method is a standard regularization technique in the Smarr

formula for AdS black holes [106], however, here we use the local counterterm method, which

was employed in [113] to derive a finite Smarr formula. Specifically, as in (2.2) we add a

boundary term to the JT action, such that the total action becomes

IJT =

∫MLJT −

∮BbJT , where bJT = − εB

8πG

[(φ0 + φ)K − φ

L

](4.31)

is the boundary Lagrangian 1-form on the timelike boundary B of M , which contains both the

Gibbons-Hawking boundary term (the first term proportional to K) and a local counterterm

(the second term). Here, εB is the volume form on B, and K is the trace of the extrinsic

curvature of B, which restricted to t = t0 in Schwarzschild coordinates takes the form

K = γµν∇µnν∣∣∣Σ

=

rBL2 cosh(

√µα/L)− 1

L

√r2B/L

2 − µ√(r2B/L

2 − µ)

cosh2(√µα/L)− 2 rBL

√r2B/L

2 − µ cosh(√µα/L) + r2

B/L2

,

(4.32)

with nν the unit normal vector (C.34) to boundary B. Importantly, vector nµ is not necessar-

ily proportional to ∂µr everywhere, since by definition it is orthogonal to the vector field ζµ,

leading to the complex expression for K above. In the α→∞ limit the trace of the extrinsic

curvature simplifies to K = rB/L2√

r2B/L

2−µand the normal to nµ =

√r2B/L

2 − µ∂µr .

Next, we regulate the divergences in the on-shell integral identity (4.22) by subtracting

on both sides of the equation (the integral of) the 0-form ζ · bJT at spatial infinity. This term

– 40 –

arises as follows. Subtracting an exact form from the Lagrangian, L → L − db, is known to

modify the symplectic potential, θ → θ − δb, the Noether current, jζ → jζ − d(ζ · b), and

the Noether charge, Qζ → Qζ − ζ · b [84]. In the present case, we add a boundary term to

the action only at the spatial boundary B, and we take the limit B →∞ to the asymptotic

timelike boundary. This regulates the Noether current and charge at infinity, and not at the

inner boundary B of Σ,20 such that the regularized integral identity becomes:∫Σjζ −

∮∞ζ · b =

∮∞

(Qζ − ζ · b)−∮BQζ . (4.33)

Specifically,

ζ · bJT

∣∣∞ = − lim

rB→∞

κφ0 tanh(√µα/2L)

8πGL√µ

rB −κ√µφr

16πGcoth(

√µα/2L) , (4.34)

where we used ζ · εB∣∣Σ

= −ζµuµ. Adding this to QJTζ yields a finite total Noether charge at

infinity and hence the regulated version of (4.26) becomes∮∞

(QJTζ − ζ · bJT) =

κ√µφr

16πGtanh(

√µα/2L) = Eφrζ , (4.35)

where Eφrζ is the asymptotic energy (C.39), which in the limit α→∞ reduces to the classical

ADM energy (2.7) of the black hole, limα→∞Eφrζ = Mφr .

Meanwhile, the regulated Noether current is now∫ΣjJTζ −

∮∞ζ · bJT ≡

φ0

8πGL2ΘGζ , (4.36)

where we have introduced the counterterm subtracted “G-Killing volume” ΘGζ , formally de-

fined by

ΘGζ ≡

∫Σζ · ε− 8πGL2

φ0

∮∞ζ · bJT . (4.37)

We refer to ΘGζ as a Killing volume since the first integral is the proper volume of Σ locally

weighted by the norm of the boost Killing vector. It is analogous to the background subtracted

“Killing volume” Θ, a.k.a. “thermodynamic volume”, in ordinary AdS black hole mechanics

[8, 106]. However, we will see in Section 4.2.3 that ΘGζ is not the conjugate quantity to

the cosmological constant in the extended first law, as is usual with the thermodynamic

volume, but rather to Newton’s constant. That is why we have added a superscript G on ΘGζ .

Explicitly, we find

ΘGζ = −

κL2√µφr2φ0

coth(√µα/2L)− κL2 . (4.38)

20At the bifurcation point B we find that ζ · b is nonzero, since ζ → 0 while K → ∞ at B, and is given by

ζ · bJT|B = − κ8πG

(φ0 + φB) = −∮BQ

JTζ . We could have added it to the regularized integral identity, since it

yields the same Smarr formula, but we think it is more natural to only add boundary terms at infinity.

– 41 –

Therefore, combining (4.35) and (4.36) we arrive to the classical Smarr relation for nested

Rindler wedges:

Eφrζ =κ

8πG(φ0 + φB)− φ0Λ

8πGΘGζ . (4.39)

Inserting expression (4.38) for the Killing volume we obtain a different form of the Smarr law

Eφrζ +κ√µφr

16πGcoth(

√µα/2L) =

κ

8πGφB . (4.40)

From this form of the Smarr law, in the limit α → ∞ where rB → rH , and identifying

κ =√µ/L, we recover the original Smarr relation (2.10) for the eternal AdS2 black hole. In

particular the left-hand side of (4.40) becomes equal to 2Mφr in the α → ∞ limit. Notably,

the Smarr relation is not of the standard form in higher dimensions (4.18), and is not identical

to the (1 + 1)-dimensional Smarr relation in Eq. (4.7) of [108], which differs by a factor of

two from our equation (4.39). The difference between the 2D Smarr formula in [108] and our

Smarr relation is that in the former one rescales the D-dimensional Newton’s constant G by

(D − 2) so as to have a well defined D → 2 limit. Here we made no such rescaling.

4.2.2 First law of nested AdS-Rindler wedge mechanics

The Smarr relation and the first law of black hole mechanics can be derived from each other

for stationary black holes in general relativity. In fact, using Euler’s scaling theorem for

homogenous functions, Smarr [105] originally used the first law of black holes to derive his

relation, identifying the ADM mass as the homogenous function. Here we derive the first

law for the nested AdS-Rindler wedge, which is an extension of the first law (2.10) for the

eternal AdS2 black hole. As it turns out to be difficult to derive this first law from the Smarr

relation (4.39), we follow Wald’s method of deriving the first law of black holes [67] by varying

the integral identity (4.21). The variations under consideration are arbitrary variations of the

dynamical fields ψ (the metric and dilaton) to nearby solutions, whilst keeping the Killing

field ζ, and the extremal slice Σ of the unperturbed wedge fixed.

On-shell, the variation of the Noether current 1-form (4.19) is formally given by the

fundamental variational identity [67, 84]

δjζ = ω(ψ, δψ,Lζψ) + d(ζ · θ(ψ, δψ)) , (4.41)

with ω(ψ, δ1ψ, δ2ψ) ≡ δ1θ(ψ, δ2ψ) − δ2θ(ψ, δ1ψ) the symplectic current 1-form. This follows

from the variation of the Lagrangian 2-form, δL = Eψδψ + dθ(ψ, δψ), and from Cartan’s

magic formula applied to the symplectic potential, Lζθ = d(ζ · θ) + ζ · dθ. Substituting δjζinto the variation of the integral identity (4.21) we obtain∫

Σω(ψ, δψ,Lζψ) =

∮∂Σ

[δQζ − ζ · θ(ψ, δψ)] . (4.42)

where we have taken the hypersurface R to be the extremal slice Σ. By Hamilton’s equations

the left-hand side is equal to the variation of the Hamiltonian Hζ generating evolution along

– 42 –

the flow of ζ,

δHζ =

∫Σω(ψ, δψ,Lζψ) , (4.43)

hence we find an integral variational identity stating the Hamiltonian variation is a boundary

charge on-shell

δHζ =

∮∂Σ

[δQζ − ζ · θ(ψ, δψ)] . (4.44)

We first evaluate the right-hand side. As before, ∂Σ consists of two points, the bifurcation

point B and a point at the asymptotic boundary B →∞. At spatial infinity on Σ the boost

Killing vector (4.4) is proportional to the Schwarzschild time-translation Killing vector, i.e.,

limr→∞,t→t0 ζ = A∂t. As reviewed in Appendix C, the variation of the form δQζ − ζ · θevaluated at spatial infinity is equal to the asymptotic energy δEφrζ .21 Explicitly, referring to

(C.39) we find that the asymptotic energy is equal to the product of the normalization A and

the ADM mass associated to the time-translation generator ∂t, and its variation is hence

δEζ =

∮∞

[δQJTζ − ζ · θJT(ψ, δψ)] = δ

∮∞

(QJTζ − ζ · bJT) = δ(AMφr) , (4.45)

where A ≡ (κL/√µ) tanh(

√µα/2L) and Mφr is the classical mass (2.7) of the static black

hole. In the last equality we inserted equation (4.35). Notice when A = 1, i.e., when

α → ∞ and κ =√µ/L, we recover the expected variational result for black holes [84, 114].

Furthermore, if the form δQζ − ζ · θ on the right-hand side of the variational identity is

evaluated at the bifurcation point B (where ζ = 0, and κ is constant) we have from the

Noether charge (4.24),∮B

[δQJTζ − ζ · θJT(ψ, δψ)] = δ

∮BQJTζ =

κ

8πGδφB , (4.46)

where the orientation of the integral is outward, towards the AdS boundary. If we allow for

variations of the couplings φ0 and G, then the Noether charge variation at B would be equal

to κ2π δ(

φ0+φBG ). Thus, ∮

∂Σ[δQJT

ζ − ζ · θJT(ψ, δψ)] = δEφrζ −κ

8πGδφB . (4.47)

Let us now turn to the left-hand side of (4.42), where we must evaluate the symplectic

current ω on the Lie derivative of the fields along the Killing vector ζ. In general relativity

21Following [104], in Appendix C we subtract the combination δC(Lζψ)−LζC(δψ), evaluated at the spatial

boundary B, from both sides of equation (4.42), where the 0-form C is defined via the restriction of the

symplectic potential to B, θ|B = δb + dC, and Dirichlet boundary conditions are imposed. This results in a

different definition of the asymptotic energy variation, since [δQζ−ζ ·θ(ψ, δψ)−δC(ψ,Lζψ)+LζC(ψ, δψ)]∣∣B

=

δ[Qζ−ζ ·b−C(ψ,Lζψ)] where we used Cartan’s magic formula and the restriction of θ on the asymptotic spatial

boundary. With weak Dirichlet conditions, δφ|B = 0 = γαµγβ

νδγµν |B , the C term contributes non-trivially to

the energy, but with stronger boundary conditions, δφ|B = 0 = δγµν |B , the C term vanishes identically. In the

main body of the paper we take the latter perspective, and in the Appendix we allow for weaker conditions.

– 43 –

the symplectic current ω(g, δg,Lζg) is linear in Lζg and hence vanishes when ζ is a Killing

vector, whereas in JT gravity, due to the nonminimal coupling of the dilaton, the current

ω(g, δg,Lζg) is generically nonzero for a Killing vector ζ since the dilaton is not everywhere

invariant under the flow of ζ. The full symplectic current (C.10), specialized to JT gravity, is

ωJT(ψ, δ1ψ, δ2ψ) =εµ

16πG

[(φ0 + φ)Sµαβνρσδ1gρσ∇νδ2gαβ +

1

2gµβgανgρσδ1gρσδ2gαβ∇ν(φ0 + φ)

+ (gµβgαν − gµνgαβ)[δ1(φ0 + φ)∇νδ2gαβ − δ1(∇ν(φ0 + φ))δ2gαβ]− [1↔ 2]], (4.48)

where the tensor Sµαβνρσ is given by (C.11). Identifying δ1 ≡ δ and δ2 ≡ Lζ and using

Lζgµν = ∇νLζgαβ = 0 and Lζφ|Σ = 0 (4.10), we find the expression above dramatically

reduces when evaluated on the Lie derivative along ζ and restricted to Σ:

ωJT(ψ, δψ,Lζψ)∣∣∣Σ

=εµu

µ

16πG(uνhαβ − uβhαν)(∇νLζφ)δgαβ = −

κφ′B16πG

hαβδgαβ(u · ε) . (4.49)

Here we decomposed the metric on Σ, gµν = −uµuν + hµν , where uν is the future-pointing

unit normal to Σ and hµν is the induced metric. In the second equality we used identity (4.15)

together with εµ|Σ = −uµ(u · ε). Finally, we note (u · ε)hαβδgαβ = 2δ(u · ε) = 2δ(d`), where

d` is the infinitesimal proper length. Since both φ′B in (4.14) and κ are constant over Σ, we

may pull them out of the integral such that,

δHφrζ ≡

∫ΣωJT(ψ, δψ,Lζψ) = −

κφ′B8πG

δ` . (4.50)

Notice this term arises since uν∇νLζφ, given by (4.15), does not vanish at Σ. Thus, combining

(4.47) and (4.50), we arrive to the classical mechanical first law of the nested Rindler wedge,

δEφrζ =κ

8πG

(δφB − φ′Bδ`

). (4.51)

As a reminder to the reader we recall the notation used here: Eφrζ is the asymptotic energy

associated to the boost Killing vector ζ of the nested AdS-Rindler patch, κ is the surface

gravity, φB is the value of the dilaton at the bifurcation point B of the nested AdS-Rindler

Killing horizon, φ′B is the derivative of the dilaton at B along `, and ` is the proper “volume”

of the extremal slice Σ with endpoints at the bifurcation point and spatial infinity.

Several comments are in order. First, notice in the limit α→∞ we have φ′B = 0 and hence

we recover the usual first law (2.10) for black holes in JT gravity. More generally, the proper

length variation contribution is non-zero and is analogous to the proper volume variation of

a spherical region Σ in the first law of (A)dS causal diamonds [8], where, however, ζ is the

conformal Killing vector preserving the diamond. It would be interesting to understand this

analogy in more detail. We further emphasize the δ` contribution is absent in the higher-

dimensional first law of AdS-Rindler space (c.f. [66]).

Importantly, as a consistency check, we verify the first law holds if all variations are

induced by changing the proper length ` of Σ. The asymptotic energy stays the same if the

– 44 –

proper length varies, δ`Eζ = 0, hence the left-hand side of the first law vanishes, while the

dilaton variation becomes δ`φB = ∂φB∂` δ` = φ′Bδ`, thus the right-hand side is also zero. This

explains why the coefficient of δ` is given by −φ′B.Further, a first law for JT gravity was previously derived in Eq. (5.7) of [115], where

the AdS-Rindler wedge was extended to a causal diamond, and was shown to encode the

dynamics of the kinematic space of a two-dimensional CFT. Notably, the variation of the

asymptotic energy is absent, since there is no asymptotic infinity, whereas a variation of the

matter Hamiltonian is included, identical to the term δHmζ in (4.52) below. Moreover, the

variation of the metric is taken to be zero, such that there is no variation of the proper length

as in our first law.

There are also a number of ways in which the first law (4.51) may be generalized or

extended. Firstly, one may consider adding in classical matter contributions, where now the

variation of the Hamiltonian (4.43) includes a contribution coming from the matter Hamilto-

nian δHmζ , characterized by an additional matter energy-momentum tensor Tm

µν which vanishes

in the AdS background, and can be cast as δHmζ = −

∫Σ δ(Tµ

ν)mζµεν [7, 8]. The classical

first law with matter Hamiltonian variation reads

δEφrζ =κ

8πG

(δφB − φ′Bδ`

)+ δHm

ζ . (4.52)

Secondly, we can consider variations of coupling constants and other parameters, e.g., φ0, G,

and Λ, thereby leading to an extended first law. We discuss this generalization below.

4.2.3 An extended first law

Above we kept all couplings and parameters fixed. It is often natural to include variations

with respect to the couplings λi of the theory, which can enrich the interpretation of the first

law. For example, when the cosmological constant Λ and Newton’s constant G are allowed

to vary, the first law for static, neutral AdS black holes is extended to [106, 116]

δM =κ

8πGδA− M

GδG+

Θ

8πGδΛ , (4.53)

where the quantity Θ conjugate to Λ is the same as the one in the Smarr equation (4.18).

Standardly, Λ is interpreted as a pressure, such that (minus) Θ plays the role of “ther-

modynamic volume”, which leads to an extended first law of black hole thermodynamics,

defining the study of extended black hole thermodynamics or black hole chemistry. Including

a pressure-volume work contribution enriches the phase space structure of black holes (for a

review see [117]).

In the context of the AdS/CFT correspondence, gravitational coupling variations in the

AdS bulk are dual to central charge variations of the boundary CFT [106, 113, 118–121].

For example, according to the AdS/CFT dictionary, the (generalized) central charge a∗d in

a CFT, defined as the universal coefficient of the vacuum entanglement entropy for ball-

shaped regions, is dual to LD−2/G for Einstein gravity in asymptotically AdS spacetimes

[122]. Recall the CFT vacuum state reduced to a ball on flat space is dual to AdS-Rindler

– 45 –

space, such that the first law of entanglement is dual to the first law of AdS-Rindler space

[65, 66]. AdS-Rindler space is identical to the massless static hyperbolic AdS black hole, so

the Smarr formula (4.18) reduces to (D− 2)κA = 2ΘΛ and the extended first law (4.53) can

be written as [123, 124]

δM =κ

8πδ

(A

G

)− κA

8πG

[(D − 2)

δL

L− δG

G

], (4.54)

where we used δΛ/Λ = −2δL/L. Crucially, the combination of variations between brackets

on the right-hand side is dual to the variation of the (generalized) central charge a∗d of the

CFT. Therefore, the extended bulk first law of the massless hyperbolic AdS black hole can

be identified with the extended first law of entanglement [120]

2π

κδ〈Hball〉 = δSent −

Sent

a∗dδa∗d , (4.55)

where Sent is the vacuum entanglement entropy of the CFT across the ball, dual to A/4G

due to the Ryu-Takayanagi formula [55], and Hball is the modular Hamiltonian defining the

CFT vacuum state reduced to the ball, and the variation of its expectation value δ〈Hball〉 is

identified with δM in the bulk. If the generalized central charge is kept fixed, δa∗d = 0, then Eq.

(4.55) reduces to the standard first law of entanglement δ〈Hball〉 = δSent, where the surface

gravity is conveniently normalized as κ = 2π [96, 97]. The extended first law of entanglement

was verified to hold for arbitrary coupling variations and specific higher derivative theories of

gravity in [123] and further generalized to arbitrary bulk theories in high and low dimensions

in [95], including an extended first law for JT gravity (though, notably, the δ` variation is

absent; see also Appendix G of [125] for a derivation using covariant phase techniques).

Before moving on to the extended first law of the nested Rindler wedge, it is worth briefly

reviewing the derivation of the extended first law in the covariant phase space formalism

[123, 126]. Moreover, we refine this derivation here by adding appropriate boundary terms

to the action. In Appendix D we present more details of the derivation of the extended first

law, where we also allow for a nonzero C(ψ, δψ) which appears in the covariant phase space

formalism with boundaries [104].

Consider the bulk off-shell Lagrangian d-form L = Lbulk(ψ, λi)ε, which depends on fields

ψ and couplings λi, and whose variation is given by

δL = Eψδψ + EλiL δλi + dθ(ψ, δψ) , (4.56)

where EλiL ≡∂Lbulk∂λi

ε and summation over i is understood. Allowing for such coupling varia-

tions modifies the on-shell integral identity (4.42) as follows [123, 126]∫Σω(ψ, δψ,Lζψ) =

∮∂Σ

[δQζ − ζ · θ(ψ, δψ)] +

∫Σζ · EλiL δλi , (4.57)

where the ‘δ’ is a general place holder for the variation with respect to both the couplings

and dynamical fields ψ. When δ is a variation solely with respect to the dynamical fields, we

attain the standard first law of black hole mechanics (without coupling variations).

– 46 –

Next, we specialize to the geometric setup where Σ is a Cauchy slice with inner boundary

at the bifurcation surface B of a bifurcate Killing horizon and an outer boundary at spatial

infinity. At the outer boundary B → ∞ the symplectic potential is a total variation (if we

set C(ψ, δψ) = 0, see footnote 21) [67, 84]

θ∣∣B→∞ = δψb = δb− Eλib δλi , (4.58)

where δψb is the variation of b only with respect to the dynamical fields, and Eλib ≡∂Lbdy

∂λiεB,

with Lbdy being the off-shell boundary Lagrangian density. We also include local counterterms

in the 1-form b to regulate any divergences, so b = bGHY + bct. Combined, the variational

identity (4.57) becomes∫Σω(ψ, δψ,Lζψ) = δ

∮∞

(Qζ − ζ · b)− δ∮BQζ +

∫Σζ · EλiL δλi +

∮∞ζ · Eλib δλi . (4.59)

This identity can be seen as a generalization of the extended Iyer-Wald formalism described

in [123, 126] to the case of spacetimes with boundaries.

In the context of classical JT gravity, the relevant coupling constants defining the theory

are λi = φ0, G,Λ. Leaving the computations to Appendix D, we obtain the following result

for the extended first law of the nested AdS-Rindler wedge

δEφrζ =κ

8πδ

(φ0 + φB

G

)−κφ′B8πG

δ`+Θφ0

ζ

8πGL2δφ0 −

φ0ΘGζ

8πGL2

δG

G−

ΘΛζ

8πGL2

δΛ

Λ, (4.60)

where we have introduced a counterterm subtracted “φ0-Killing volume”

Θφ0

ζ ≡∫

Σζ · ε− L2

∮∞Kζ · εB , (4.61)

and a counterterm subtracted “Λ-Killing volume”

ΘΛζ ≡

∫Σφζ · ε− L

2

∮∞φζ · εB . (4.62)

Notice the dilaton arises inside the integrals above because of its non-minimal coupling to the

metric in the JT Lagrangian. Although the definition includes a “counterterm” the result for

ΘΛζ is generally divergent and is given by (D.23), where we have introduced a cutoff.

We can perform some consistency checks of the extended first law (4.60) by considering

specific variations. Firstly, the variations with respect to φ0 cancel between the first and the

third term on the right-hand side,

κ

8πGδφ0 +

Θφ0

ζ

8πGL2δφ0 = 0 −→ Θφ0

ζ = −κL2 . (4.63)

We have verified the last identity explicitly using the definition of the φ0-Killing volume.

Secondly, the variations of Newton’s constant cancel between the left and right-hand side,

δGEφrζ = − κ

8πG(φ0 + φB)

δG

G−

φ0ΘGζ

8πGL2

δG

G−→ Eφrζ =

κ

8πG(φ0 + φB) +

φ0ΘGζ

8πGL2, (4.64)

– 47 –

where we used Eφrζ ∼ 1/G. This identity is precisely the classical Smarr law (4.39). Thirdly,

verifying the extended first law if the variations are induced by changing the cosmological

constant (or AdS length) is a bit more subtle, since the δL variation coming from the Λ-

Killing volume is divergent as rB →∞. However, this divergence is precisely cancelled by the

rB →∞ divergence appearing in the δL` term, see (D.13). In fact, equating the variation of

Eφrζ with respect to L to the sum of these two separately divergent δL variations yields yet

another nontrivial relation:

δLEφrζ = −

κφ′BL

8πGδL`+

ΘΛζ

4πGL2

δL

L−→ Eφrζ =

κφ′BL

8πG

∂`

∂L−

ΘΛζ

4πGL2. (4.65)

This follows from observing δL` = ∂`∂LδL and δLE

φrζ = −Eφrζ

δLL , since from (C.39) we see

Eφrζ ∼ κ and by dimensional analysis we have κ ∼ 1/L. We check this identity more explicitly

in (D.26). Note that it also involves the asymptotic energy, just like the Smarr relation, but

not the entropy of the bifurcation point, in contrast to the standard Smarr relation. Thus,

the extended first law has uncovered three nontrivial relations for the three Killing volumes,

one of which (4.64) is the Smarr relation.

We emphasize that here, unlike in the case of pure gravity theories, we find three generally

different “Killing volumes”, ΘGζ , Θφ0

ζ , and ΘΛζ , defined in (4.37), (4.61), and (4.62), which

appear in the extended first law as conjugate variables to Newton’s constant G, φ0, and the

cosmological constant Λ, respectively. None of them separately have the interpretation of a

thermodynamic volume, since they are not conjugate to the bulk pressure P = −Λ/8πG (note

we are also considering variations of G here). The reason we have three Killing volumes is a

consequence of the fact we are studying a non-minimally coupled dilaton-gravity theory with

three different coupling constants, since if we set φ = 1 the volume integrals in the definitions

of Θφ0

ζ , ΘGζ and ΘΛ

ζ match (neglecting the boundary integrals at infinity).

Furthermore, let us discuss the extended first law for the eternal AdS2 black hole. In the

α → ∞ limit, where φ′B = 0 such that the δ` term in the first law vanishes, we recover the

classical extended first law of the black hole

δMφr =κ

8πδ

(φ0 + φH

G

)+

Θφ0

ζ,α→∞8πGL2

δφ0 −φ0ΘG

ζ,α→∞8πGL2

δG

G−

ΘΛζ,α→∞

8πGL2

δΛ

Λ, (4.66)

where the Killing volumes associated with the eternal black hole are (setting κ =√µ/L)

Θφ0

ζ |α→∞ = −√µL , ΘGζ |α→∞ = −µLφr

2φ0−√µL , and ΘΛ

ζ |α→∞ = −µLφr4

. (4.67)

In this limit we observe the volume ΘGζ is a combination of the volumes ΘΛ

ζ and Θφ0

ζ , namely,

ΘGζ = 2ΘΛ

ζ /φ0 + Θφ0

ζ . Moreover, explicitly, in terms of the variables of the black hole, we

have the extended first law

δMφr =

√µ

8πLδ

(φ0 + φH

G

)−√µ

8πGLδφ0 +

(µφr

16πGL+

√µφ0

8πGL

)δG

G− µφr

16πGL

δL

L. (4.68)

– 48 –

Simplifying the expression, and in addition allowing for variations of φr, yields

δ(MφrL) =

√µ

8πδ

(φHG

)−MφrL

δ(φr/G)

φr/G. (4.69)

Interestingly, in terms of the dimensionless mass Mφr = MφrL, the dimensionless inverse

Hawking temperature βH = 2π/√µ, the entropy Sφr =

√µφr/4G above extremality, and the

gravitational coupling a = φr/G, the black hole extended first law can be written as22

βHδMφr = δSφr −1

2

Sφraδa . (4.71)

This is a generalization of the standard first law (2.10) to nonzero coupling variations. It

is very similar to the extended first law of entanglement (4.55), dual to the extended first

law for higher-dimensional AdS-Rindler space, except for the factor one half in the δa term.

This factor does not appear in the extended first law of entanglement since the modular

Hamiltonian vanishes in the CFT vacuum, so δa∗d〈Hball〉 = 0, and hence the left-hand side

of (4.55) vanishes identically. The right-hand side of (4.55) also vanishes identically because

the vacuum CFT entanglement entropy of a ball in flat space is proportional to the central

charge, Sent ∼ a∗d, hence δa∗dSent = Senta∗dδa∗d. On the other hand, in the AdS2-Rindler extended

first law (4.71) the energy variation induced by changing the coupling a is nonzero, since

δaMφr =Mφra δa, and the entropy variation takes the same form as the entanglement entropy

variation, δaSφr =Sφra δa. Inserting these variations into (4.71) yields a true identity due to

the classical Smarr law (2.10) Mφr = 12THSφr .

To summarize, the coupling variation of the entropy for JT gravity is still of the form

δaSφr =Sφra δa, with a = φr/G, just like in the first law of entanglement (in contrast to

the findings of [95]). However, the coefficient of the δa term in the extended first law for

AdS2 black holes differs from the coefficient of the δa∗d term in the extended first law of

entanglement. Further, the coupling a∗d in the extended first law of entanglement has the

interpretation of a generalized central charge in d ≥ 2 dimensional CFTs, so in analogy it

would be interesting to find a microscopic interpretation of the JT gravitational coupling a

in the dual double-scaled random matrix theory [38] or in the SYK model [31].

Lastly, it is worth recalling JT gravity arises from a spherical reduction of a charged

higher-dimensional black hole, where φ represents the transversal area of the sphere and Λ

is a function of the charge q and higher-dimensional AdS length. Therefore, in a certain

sense, the extended first law can be seen as the dimensionally reduced extended first law

22Alternatively, we could include the variation of the entropy Sφ0 = φ0/4G of the extremal black hole into

the extended first law, but then we have to subtract off a term proportional to the variation of φ0/G

βHδMφr = δ(Sφ0 + Sφr )− Sφ0

aφ0

δaφ0 −1

2

Sφraφr

δaφr , (4.70)

where now there are two couplings aφ0 = φ0/G and aφr = φr/G. Note the variation of the black hole entropy

with respect to the couplings is δai(Sφ0 + Sφr ) =Sφ0aφ0

δaφ0 +Sφraφr

δaφr .

– 49 –

of a charged static black hole. For example, in the case of a spherical reduction of a four-

dimensional AdS-RN black hole of charge q and AdS length L4, the AdS length of the 2D

black hole is given by L = q3/2L4 [70]. Consequently, the variation with respect to L encodes

the variation of the black hole charge and higher-dimensional AdS length, δLL = 32δqq + δL4

L4. It

would be interesting to study the relation between our extended first law for JT gravity and

the extended first law of a higher-dimensional charged black hole in more detail.

4.3 Semi-classical Smarr formula and first law

Having derived a classical Smarr relation and first law for the nested Rindler wedge, here we

aim to determine their semi-classical extension. Mathematically this amounts to accounting

for the quantum corrections to the dilaton φ and including the auxiliary field χ.

4.3.1 Smarr relation

Let us first derive the Smarr relation including semi-classical backreaction effects. The general

procedure used to derive the classical Smarr law (4.40) is the same, where now we need to

compute the Noether current and charge associated to the Polyakov action and to the 1-loop

quantum corrections to dilaton φ. As we will see shortly, the Smarr relation is generalized

accordingly, where the classical entropy is replaced by the generalized entropy, and a semi-

classically corrected asymptotic energy.

We again make use of the regulated integral identity (4.33). First, we consider the right-

hand side of the identity. Working in the Hartle-Hawking state, the semi-classical correction

to the dilaton is only a constant (2.67), φc = Gc3 (where we have set λ = 0). Thus here we

primarily focus on the effect of the auxiliary field χ. Using the generic expression for the

Noether charge (C.13), the contribution from the χ field is

Qχζ =c

24πεµν [χ∇µζν + 2ζµ∇νχ] . (4.72)

Evaluating this at the bifurcation surface B leads to∮BQχζ = − κ

2π

cχB6

, (4.73)

which we recognize as the Wald entropy associated to the χ field (times the temperature).

Note we did not need to specify the precise form of χ such that this expression even holds

in the Boulware vacuum. We ignore the Boulware vacuum solution, however, as it does not

describe a system in thermal equilibrium.

Next we evaluate∮∞Q

cζ , where Qcζ is the total Noether charge due to the semi-classical

corrections in φ and χ, Qcζ = Qφcζ +Qχζ . In addition we subtract the term ζ ·bc = ζ ·bφc +ζ ·bχat infinity, where bφc is the local counterterm associated with φc (see (4.31) with φ replaced

by φc = Gc3 and φ0 = 0), and bχ is the Polyakov boundary counterterm 1-form [20]

bχ =c

12πχKεB +

c

24πLεB . (4.74)

– 50 –

Working in the Hartle-Hawking state and using the static solution χ(4) (2.72) we find∮∞

(Qcζ − ζ · bc

)=

cκ

12πtanh(

√µα/2L) = Ecζ , (4.75)

where Ecζ is the semi-classical correction to the asymptotic energy in (C.52). In the limit

α → ∞, with κ =√µ/L we recover the semi-classical correction to the eternal black hole

mass Mc =c√µ

12πL [20, 64].

Consider now the left-hand side of the identity (4.33). For the semi-classical correction

to the dilaton, we need to compute∫Σjφcζ −

∮∞ζ · bφc =

∫Σ

[θJT(φc,Lζφc)− ζ · Lφc ]−∮∞ζ · bφc , (4.76)

where bφc is the boundary counterterm Lagrangian 1-form (4.31) with φ0 = 0 and φ → φc,

and similarly for Lφc . Since Lφc = 0 on-shell, and θJT(φc,Lζφc) = 0, and ζ · bφc |∞ = 0 we

find ∫Σjφcζ −

∮∞ζ · bφc = 0 . (4.77)

Meanwhile, the contribution from χ to the left-hand side is∫Σjχζ −

∮∞ζ · bχ =

∫Σ

[θχ(ψ,Lζψ)− ζ · Lχ]−∮∞ζ · bχ . (4.78)

From (C.5) and using Lζχ|Σ = 0 (4.16), we see the symplectic potential for χ will not

contribute: θχ(ψ,Lζψ)|Σ = 0. Further, with the Polyakov Lagrangian 2-form (setting λ = 0

in (2.11))

Lχ = − c

24πε[χR+ (∇χ)2] , (4.79)

and the boundary 1-form (4.74), it is straightforward to show23∫Σjχζ −

∮∞ζ · bχ = − c

12πL2Θcζ , (4.80)

where we have introduced the counterterm subtracted “semi-classical c-Killing volume” Θcζ

analogous to (4.37), on shell given by

Θcζ ≡

∫Σ

(χ− L2

2(∇χ)2

)ζ · ε+

∮∞

(L2χK +

L

2

)ζ · εB . (4.81)

Or, more explicitly, for the time-independent solution χ(4) (2.72) it is equal to

Θcζ =

κL2

sinh(√µα/L)

+κL2

2

[1 + 2(

√µα/L− coth(

√µα/L))− log

(1

µsinh2(

√µα/L)

)].

(4.82)

23Here for convenience we use the static χ(4) solution, however, similar expressions hold for the generally

time-dependent χ(5) solution.

– 51 –

Combining (4.73), (4.75), and (4.80) together with the classical Smarr relation (4.39), we

arrive at the semi-classical Smarr law for nested AdS-Rindler wedges

Eφrζ + Ecζ =κ

2π

(1

4G(φ0 + φB)− c

6χB

)− φ0Λ

8πGΘGζ +

cΛ

12πΘcζ . (4.83)

As we will argue momentarily, the first term on the right-hand side is recognized to be TSgen.

In the limit α → ∞, we uncover the semi-classical Smarr relation for the AdS2 black hole

(setting κ =√µ/L)

Mφr +Mc =

√µ

2πL

(1

4G(φ0 + φH)− c

6χH

)− φ0Λ

8πGΘGζ,α→∞ +

cΛ

12πΘcζ,α→∞ . (4.84)

For the time-independent χ(4) the α→∞ limit of the semi-classical Killing volume is

Θcζ,α→∞ = −

√µL

2(1− 2 log(2

√µ)) . (4.85)

The semi-classical Smarr law for the black hole can thus be simplified to

2Mφr +Mc =

√µ

2πL(Sφr + Sc) +

cΛ

12πΘcζ,α→∞ , (4.86)

with Sφr =φr√µ

4G and Sc = c12 + c

6 log(2√µ). This is the semi-classical extension of the standard

Smarr formula (2.10) for the eternal black hole.

4.3.2 First law of nested backreacted AdS-Rindler wedge mechanics

Now we derive the first law of the nested Rindler wedge, taking into account the full back-

reaction. The steps are morally the same as deriving the classical first law (4.51), where we

make use of the integral variational identity (4.42). Since in the Hartle-Hawking state the

semi-classical correction to the dilaton is a constant, it trivially modifies the classical first

law. We therefore mainly discuss the χ contribution.

We first focus on evaluating the right-hand side of (4.42). As in the classical case, using

the Noether charge at spatial infinity (4.75) and at the bifurcation point (4.73) we find∮∂Σ

[δQcζ − ζ · θ(ψ, δψ)] = δEcζ −

κ

2πδ

(φc4G− c

6χB

), (4.87)

where the second term is the variation of the semi-classical contribution to the Wald entropy.

Moving on, the quantum correction φc to the dilaton does not modify the left-hand side

of (4.42) since it is a constant. Thus, the only addition comes from the auxiliary field χ. The

symplectic current 1-form (C.10) with respect to the Polyakov action is

ωχ(ψ, δψ,Lζψ)∣∣∣Σ

= − c

24πεµ

[(gµβgαν − gµνgαβ)∇ν(Lζχ)δgαβ − 2∇µ(Lζχ)δχ

], (4.88)

– 52 –

where we used Lζgµν = 0 and Lζχ|Σ = 0. The same algebraic steps leading to (4.49) yield

ωχ(ψ, δψ,Lζψ)∣∣∣Σ

=c

24π(hαβδgαβ − 2δχ)(uν∇νLζχ)d`

=cκ√µ

24πLtanh(

√µα/2L)

(hαβδgαβ − 2δχ)

rL −

√r2

L2 − µd` ,

(4.89)

where we used the derivative identity (4.17). Since uν∇νLζχ is not constant on Σ we cannot

explicitly integrate the symplectic current over Σ. Therefore, we write the integral over the

symplectic current (4.89) formally, via Hamilton’s equations, as the Hamiltonian variation

associated to the χ field

δHχζ =

∫Σωχ(ψ, δψ,Lζψ) . (4.90)

Thus, altogether, combining (4.87) and (4.90) with the classical first law (4.51), the semi-

classical first law is

δ(Eφrζ + Ecζ) =κ

2πδ

(φB4G− cχB

6

)−κφ′B8πG

δ`+ δHχζ . (4.91)

Importantly, note the first term on the right-hand side is simply TδSgen, evaluated at the

bifurcation point of the nested Rindler wedge B opposed to the black hole horion. In the next

section we will see in the microcanonical ensemble the semi-classical first law is equivalent to

extremizing the generalized entropy, such that the semi-classical first law of nested Rindler

wedges becomes the semi-classical first law of quantum extremal surfaces.

Extended semi-classical first law

We conclude this section by briefly commenting on extending the semi-classical first law to

include coupling variations, where we now also consider variations of the central charge c. The

central charge appears both in the Polyakov action as well as in the semi-classical correction

to the dilaton. As detailed in Appendix D, the semi-classical extended first law is

δ(Eφrζ + Ecζ) =κ

8πδ

(φ0 + φB

G− cχB

6

)−κφ′B8πG

δ`+ δHχζ

+Θφ0

ζ

8πGL2δφ0 −

φ0ΘGζ

8πGL2

δG

G−

ΘΛζ + ΘΛ,c

ζ

8πGL2

δΛ

Λ−

cΘcζ

12πL2

δc

c+

cΘBζ

48πL

δΛ

Λ,

(4.92)

where we have introduced the counterterm subtracted “semi-classical Λ-Killing volume”

ΘΛ,cζ = φc

(∫Σζ · ε− L

2

∮∞ζ · εB

), (4.93)

where φc comes out of the integrals because it is constant and ΘBζ ≡

∮∞ ζ · εB is the Killing

volume of B. Both ΘΛ,cζ (D.30) and ΘB

ζ are formally divergent, however, it can be checked

– 53 –

these divergences precisely cancel, see (D.34). Using the α→∞ limit of the Killing volumes

(4.67) and (4.85), we attain

δ(Mφr +Mc) =

√µ

2πLδ

(φ0 + φH

4G− cχH

6

)−√µ

8πGLδφ0 +

(µφr

16πGL+

√µφ0

8πGL

)δG

G

− (Mφr +Mc)δL

L+

(Mc −

c√µ

24πL−

c√µ

12πLlog(2

√µ)

)δc

c,

(4.94)

where we note the δHχζ contribution has dropped out in this limit. This is the semi-classical

generalization of the extended first law (4.68). We have inserted the time-independent solution

χ(4) here, for which the semi-classical mass Mc is given by (C.53).

4.4 Thermodynamic interpretation: canonical and microcanonical ensembles

Now we are going to provide a thermodynamic interpretation to the classical and semi-classical

first laws of the nested Rindler wedges. We keep all the coupling constants φ0, G,Λ, c fixed

in this section. Specifically, in the classical limit, we make the following identifications with

the temperature T and entropy SBH

T =κ

2π, SBH =

φ0 + φB4G

. (4.95)

The classical first law (4.51) of nested Rindler horizons can thus be written as

δEφrζ = TδSBH + δHφrζ , (4.96)

where δEφrζ is the variation of the classical asymptotic energy and δHφrζ the variation of the

JT Hamiltonian generating time evolution along the flow of ζ. The asymptotic energy can be

understood as the internal energy of the system, whereas the thermodynamic interpretation

of δHφrζ is less obvious. It is a gravitational energy variation which appears on the same

side of the first law as TδSBH, and hence might be interpreted as a gravitational work term.

We recall the Hamiltonian variation (4.50) is negative, δHφrζ = − κφ′B

8πGδ`, thereby it thus

contributes negatively to the asymptotic energy variation, just like the work term δW in the

first law of thermodynamics, dE = δQ− δW , contributes negatively to the change in internal

energy, suggesting the identification δW = −δHφrζ .

In the semi-classical case the temperature identification remains the same, while the

entropy is replaced by the generalized entropy of the bifurcation point B

Sgen =1

4G(φ0 + φB)− c

6χB . (4.97)

The semi-classical first law (4.91) thus becomes

δ(Eφrζ + Ecζ

)= TδSgen + δHφr

ζ + δHχζ , (4.98)

where we note δHφrζ = δHφ

ζ , since δHφcζ = 0. Here δEcζ is the semi-classical contribution to

the asymptotic energy variation, and δHχζ is the Hamiltonian variation associated to the χ

– 54 –

field in the Polyakov action. The role of the asymptotic energy variation and the Hamiltonian

variation in the thermodynamic first law is the same as in the classical case.

The above identification (4.95) for the temperature follows from the fact an asymptotic

observer in the nested AdS-Rindler wedge sees the Hartle-Hawking state as a thermal state

with temperature T , as we will now show.

A comment on vacuum states in adapted coordinates

Solving the fully backreacted semi-classical JT model required us to a pick a specific vacuum

state of the quantum matter. Since we are interested in studying thermodynamics, we solved

the system with respect to the Hartle-Hawking state, a thermal state according to a static

observer in (v, u) coordinates such that the black hole radiates at the Hawking temperature

TH =√µ

2πL , with√µ/L the surface gravity evaluated at the future Killing horizon. The

QES naturally led us to consider a nested AdS-Rindler wedge characterized by the boost

Killing vector ζ, where the Killing horizon has surface gravity κ. To justifiably attribute

thermodynamics to this system, we are interested to know what a static observer in adapted

coordinates (v, u) would detect outside of the QES in the global Hartle-Hawking state.

The global Hartle-Hawking state of the eternal black hole is thermal with respect to the

static coordinates (v, u), cf. Eq. (2.54),

〈HH| : Tχuu : |HH〉 = 〈HH| : Tχvv : |HH〉 =cπ

12T 2

H . (4.99)

To write down a thermodynamic first law for the nested Rindler wedge, accounting for effects

of backreaction, as done above, requires us to know the form of the solutions φ and χ, which

depend on the choice of vacuum state. We choose the global Hartle-Hawking state, since

this is known to be a thermal state in the exterior of the black hole, but we now show it is

also thermal in the nested Rindler wedge. To this end, we compute the expectation value of

the normal-ordered stress tensor in the nested coordinates (v, u) in the Hartle-Hawking state.

Recall the transformation rule (2.34), such that

: Tχuu :=

(du

du

)2

: Tχuu : − c

24πu, u , (4.100)

and similarly for vv and vv components. Taking the expectation value of both sides with

respect to the Hartle-Hawking vacuum, and using (4.99) together with the coordinate relation

(4.5), we obtain

〈HH| : Tχuu : |HH〉 =

(du

du

)2 cπ

12T 2

H −c

24πu, u =

cπ

12

( κ2π

)2, (4.101)

for all κ and t0 6= 0. In other words, asymptotic (v, u) observers see the Hartle-Hawking state

as a thermal state at the temperature T = κ/2π, where the surface gravity is associated to

the nested AdS-Rindler horizon. This surface gravity, and hence this temperature, appears

in both the classical and semi-classical thermodynamic first laws.

– 55 –

To be clear, the Hartle-Hawking state |HH〉 above is the global Hartle-Hawking state of

the enveloping black hole, i.e., the vacuum state with respect to global Kruskal coordinates

(VK, UK). Naturally, one may wonder whether the Hartle-Hawking state defined with respect

to the Kruskal coordinates adapted to the nested wedge (VK, UK) also appears thermal and

coincides with the global Hartle-Hawking state. To check the latter, one may compute the

expectation value of the normal ordered TχUKUKwith respect to the “adapted” Hartle-Hawking

state |HH〉 using the relation (2.35). A straightforward calculation yields

〈HH| : TχUKUK : |HH〉 = − c

24πUK, UK = 0 , (4.102)

where UK = − 1κe−κu. Therefore, the “adapted” and global Hartle-Hawking states are one and

the same: |HH〉 = |HH〉! The Hartle-Hawking vacuum can thus be written as a thermofield

double state between left and right AdS-Rindler wedges of the eternal black hole or the left

and right nested Rindler wedges, where, however, the associated temperature depends on the

location where the spacetime is cut in half. Correspondingly, the reduced density matrix is

thermal in the exterior region, where the temperature is dependent on the Rindler wedge

one has access to. Note that we are not saying the vacuum state |0u,v〉 with respect to (v, u)

coordinates is equivalent to the vacuum state |0u,v〉 in (v, u) coordinates, i.e., the Boulware

vacuum states of the different wedges are generally different, 〈0u| : Tχuu : |0u〉 6= 0, as can be

shown from a calculation similar to the above. Only in the α → ∞ limit, when the nested

Rindler wedge coincides with the eternal black hole, do the Boulware vacua match.

It is worth comparing the above comment to the recent findings of [127], which considers

an infinite family of nested Rindler wedges in Minkowski spacetime. The Minkowski (iner-

tial) vacuum appears thermal with respect to each nested Rindler wedge, at a temperature

proportional to the Unruh temperature of the corresponding wedge. The above calculation

(4.101) is a realization of the observation made in [127] for a single nested AdS-Rindler wedge,

and a similar result is expected for subsequent nested AdS-Rindler wedges.

Generalized second law

There is additional evidence suggesting we should interpret SBH and Sgen as thermodynamic

entropies: both quantities, under particular assumptions, obey a second law. That is, upon

throwing some matter into the system, the total entropy of the system monotonically increases

along the future Killing horizon defining the nested wedge, such that, semi-classically, one has

the generalized second law,dSgen

dλ ≥ 0, where λ is an affine parameter along a null generator of

the future Killing horizon associated with the boost Killing vector ζ. We provide a heuristic

derivation of the generalized second law in Appendix E. Thus, akin to black holes, we note

the nested Rindler wedge obeys a generalized second law of thermodynamics.

Equilibrium conditions and thermodynamic ensembles

In standard thermodynamics, the stationarity of free energy at fixed temperature follows from

the first law of thermodynamics, dE = δQ− δW , and Clausius’ relation, δQ = TdS. Indeed,

– 56 –

assuming δW = 0 for the moment, the first law and Clausius’ relation yield dE = TdS,24

with E = E(S). This implies the Helmholtz free energy F ≡ E − TS, with F = F (T ),

is stationary at fixed temperature dF |T = −SdT = 0. Thus, the stationarity of the free

energy arises from the first law, from which the free energy or thermodynamic potential

actually may be identified. The same line of reasoning has been applied to the first law of

black hole mechanics, where the Helmholtz free energy of a static, neutral black hole system

corresponds to F = M − THSBH, such that the first law implies F is stationary at a fixed

Hawking temperature TH.

From the classical thermodynamic first law (4.96), the (Helmholtz) free energy F clζ is

identified as

F clζ = Eφrζ − TSBH −Hφr

ζ . (4.103)

Note that the absolute value of Hφrζ appears here instead of its variation.25 Upon invoking

the first law it is straightforward to verify the free energy is stationary at fixed temperature

δF clζ

∣∣T

= −SBHδT = 0 . (4.104)

In the limit α→∞, where the nested Rindler wedge becomes the full exterior of the eternal

AdS black hole and φ′B = 0, we recover the standard result for black holes. The free energy

of the eternal black hole in our notation is F = Mφr − THSBH.

In the semi-classical regime, the quantum corrected (Helmholtz) free energy and its sta-

tionarity condition are given by

F semi-clζ = Eφrζ + Ecζ − TSgen −Hφr

ζ −Hχζ , δF semi-cl

ζ

∣∣T

= −SgenδT = 0 , (4.105)

where the stationarity condition follows from the semi-classical first law (4.98). This es-

tablishes the nested AdS-Rindler wedge is an equilibrium state, even when semi-classical

corrections are taken into account.

The variable dependence and stationarity condition for the free energy also define the

statistical ensemble the system is characterized by. The canonical ensemble is defined to be the

ensemble at fixed temperature, i.e., a system in thermal equilibrium, such that the Helmholtz

free energy is stationary in equilibrium in this ensemble. Moreover, thermodynamic stability

requires the Helmholtz free energy be minimized at equilibrium in the canonical ensemble with

respect to any unconstrained internal variables of the system. By analogy, the free energy

F clζ and its semi-classical counterpart define a canonical ensemble at fixed temperature T .

Different ensembles may be transformed into one another via an appropriate Legendre

transform of a particular thermodynamic potential. Indeed, in standard thermodynamic par-

lance, the Helmholtz free energy F (T ) is simply a Legendre transform of the internal energy

24Below we colloquially call this the thermodynamic “first law”, which is standard terminology in black hole

thermodynamics literature.25In the covariant phase space formalism, the variation of the Hamiltonian δHζ is a well-defined quantity,

however, Hζ is not precisely determined. Typically one imposes “integrability conditions” (see, e.g., Eq. (80)

of [84]) such that Hζ is ambiguous up to an additive constant. Here, following [104], the integrability follows

from the boundary condition θ|B = δb+ dC, a consequence of requiring the variational principle is well posed.

– 57 –

E of the system with respect to the entropy S, F = E − TS. Likewise, the microcanoical

entropy S(E) defining the microcanonical ensemble is equivalent to the (negative) Legendre

transform of βF with respect to β = T−1, S(E) = −(βF − Eβ).26 Furthermore, in stan-

dard thermodynamics, the stationarity of free energy at fixed temperature in the canonical

ensemble is equivalent to the stationarity of entropy at fixed energy in the microcanonical

ensemble. This is because dF |T = dE−TdS and dS|E = dS−βdE, so dF |T = −TdS|E , and

hence dF |T = 0 is equivalent to dS|E = 0.

Transforming between different ensembles also holds for self-gravitating systems, includ-

ing black holes [128]. In this context, the microcanonical description is specified by fixing the

energy surface density as boundary data, while the canonical ensemble fixes the surface tem-

perature. Changing the boundary data thus corresponds to a Legendre transform. Therefore,

as in the case of black holes in general relativity [128], or more general theories of gravity

[129], a suitable Legendre transformation of the free energy F clζ casts the classical entropy

SBH as the microcanonical entropy, describing the nested Rindler wedge in the microcanonical

ensemble. Specifically,

SBH = −βF clζ + β(Eφrζ −H

φrζ ) . (4.106)

Likewise, transforming F semi-clζ allows us to describe the microcanonical ensemble for semi-

classical JT gravity, where the generalized entropy Sgen is equal to the microcanonical entropy

Sgen = −βF semi-clζ + β(Eφrζ + Ecζ −H

φrζ −H

χζ ) . (4.107)

It now follows from the semi-classical first law (4.98) that the generalized entropy is stationary

at fixed asymptotic energy Eζ ≡ Eφrζ + Ecζ and Hamiltonian Hζ ≡ Hφrζ +Hχ

ζ :

δSgen

∣∣Eζ ,Hζ

= 0 . (4.108)

This is the equilibrium condition in the microcanonical ensemble for semi-classical JT gravity

in the nested AdS-Rindler wedges.

To summarize, in the semi-classical regime the two thermodynamic ensembles and cor-

responding stationarity conditions are defined by

canonical ensemble: fixedT, δF semi-clζ = 0 , (4.109)

microcanonical ensemble: fixedEζ , Hζ , δSgen = 0 . (4.110)

Crucially, since quantum extremal surfaces extremize the generalized entropy, the thermody-

namics of nested AdS-Rindler wedges reduces to the thermodynamics of quantum extremal

surfaces in the microcanonical ensemble, when backreaction effects are taken into account.

It is worth emphasizing that the classical and semi-classical first laws, (4.96) and (4.98),

respectively, hold for any nested AdS-Rindler wedge. The above discussion on ensembles,

however, leads to a fundamental insight into semi-classical thermodynamics of gravitating

26This may also be derived at the level of the partition functions, where the canonical partition function

Z(β) is equal to the Laplace transform of the microcanonical partition function Ω(E), with S(E) = log Ω(E).

– 58 –

systems: in the microcanonical ensemble it is more natural to consider the entropy of quantum

extremal surfaces than black hole horizons when semi-classical effects are included. This is

because it is the QES, and not the black hole horizon, which extremizes the generalized

entropy. In this way, in the microcanonical ensemble the semi-classical first law may be

genuinely regarded as a first law of thermodynamics of quantum extremal surfaces.

Finally, the stationarity of generalized entropy at fixed energy Eζ and Hζ for nested

AdS-Rindler wedges is strikingly similar to Jacobson’s “entanglement equilibrium” proposal

[98] for small causal diamonds: the generalized (entanglement) entropy (a.k.a. Sgen) of causal

diamonds is stationary in a maximally symmetric vacuum at fixed proper volume. This

equilibrium condition was derived in [8] from the semi-classical first law of causal diamonds

in maximally symmetric spacetimes, applied to small diamonds. The first law of causal

diamonds contains the volume variation of the maximal slice, which is proportional to the

variation of the gravitational “York time” Hamiltonian. In order to obtain the stationarity

of generalized entropy in this setting, the volume or, equivalently, the York time Hamiltonian

needs to be held fixed. This suggests the entanglement equilibrium condition should be

interpreted in the microcanonical ensemble, as recently acknowledged in [130].

5 Conclusion

In this article we considered the full backreaction due to Hawking radiation in an eternal AdS2

black hole using the semi-classical JT model. Importantly, we showed the time-dependent

auxiliary field χ, which localizes the Polyakov action modelling the Hawking radiation, leads

to a time-dependent Wald entropy which is equal to the generalized entropy under appro-

priate identifications. This time-dependent solution emphasized here appears to have been

overlooked, however, we believe it will prove useful in studying the Hawking information

puzzle in certain contexts, as the entropy associated with χ is precisely identified with a

(time-dependent) von Neumann entropy.

Extremizing the (time-dependent) generalized entropy along the time slice t = tB led

to a crucial insight: the appearance of a quantum extremal surface that lies just outside of

the black hole horizon, consistent with previous investigations on the black hole information

paradox in eternal backgrounds, e.g., [45, 49, 50]. The QES arises purely due to semi-classical

effects, such that in the classical limit the QES coincides with the bifurcation surface of the

eternal black hole. Notably, we find the QES is always present and, moreover, the generalized

entropy evaluated at the QES is smaller than Sgen evaluated at the black hole horizon, contrary

to what was observed previously [64]. Since the QES lies outside of the horizon, such that

the QES describes an AdS-Rindler wedge nested inside the eternal black hole, we derived

the classical and semi-classical Smarr relations and associated first laws of the nested Rindler

wedge. Actually, we established first laws for generic nested AdS2-Rindler wedges, for which

the bifurcation point is at an arbitrary location outside the black hole. Notably absent in

higher dimensions, we showed the first laws naturally incorporate the variation of the proper

length, which is absent in the limit the nested wedge concurs with eternal AdS black hole.

– 59 –

From the first law we uncovered the semi-classical free energy in the canonical ensemble, and,

via a Legendre transform, identified the microcanonical entropy as the generalized entropy.

This leads to the important observation that it is more natural in the microcanonical ensemble

to assign thermodynamics to quantum extremal surfaces over black hole horizons when semi-

classical effects are included, as it is the QES which extremizes the generalized entropy.

There are several exciting prospects and applications of this work. Firstly, while here we

focused on JT gravity in AdS, our methods may be appropriately applied to JT gravity in de

Sitter space (see, e.g., [51]), and the semi-classical RST [16] model of 2D dilaton-gravity in

flat space, for which [49, 50] will be of use. Indeed, a chief technical result of this article is the

covariant phase formalism for a wide class of 2D dilaton-gravity theories, which will prove

useful in deriving the Smarr relation and first law in the RST model. Second, we extended

the first laws derived here by allowing for coupling variations. As with higher-dimensional

black holes, these new terms enrich the phase space of the black hole thermodynamics and it

would be interesting to study this in more detail.

A principal question in the context of gravitational thermodynamics is which surface

does the entropy increase with respect to (c.f. [131, 132] for a thorough examination of

this question). For example, in the case of JT gravity, the classical Wald entropy of the

black hole monotonically increases along the black hole event horizon as well as the apparent

horizon, however, when the semi-classical correction is included the generalized entropy of the

black hole only monotonically increases along the event horizon and not along the apparent

horizon27 [64]. In Appendix E we provided a heuristic argument suggesting Sgen obeys a

second law along the future Killing horizon of the nested Rindler wedge. For the sake of

consistency with our thermodynamic interpretation, it would be worthwhile to have a more

detailed proof of the generalized second law initiated here.

As noted before the first law of entanglement entropy is holographically equivalent to the

first law of (AdS-Rindler) black hole thermodynamics [96]. The relation between the first laws

of entanglement and AdS-Rindler thermodynamics was used in [66] to further show the first

law of holographic entanglement encodes the linearized Einstein’s equations. Similarly, the

semi-classical first law of AdS-Rindler space, where bulk quantum corrections are included

dual to 1/N corrections in the CFT, is equivalent to the (linearized) semi-classical Einstein’s

equations [135]. It would be interesting to see whether we can use the semi-classical first law

uncovered here as an input and derive gravitational equations of motion for semi-classical JT

gravity, following an analysis similar to [115], where one invokes the proposal of entanglement

equilibrium [98].

A compelling feature of JT gravity (and its supersymmetric extensions), defined as a

27More precisely, one considers the future holographic screen, i.e., the codimension-1 spacelike hypersurface

foliated by marginally trapped surfaces whose area, given by the dilaton in the 2D context, is the apparent

horizon and obeys the classical focusing conjecture. When semi-classical corrections are included, the general-

ized entropy Sgen does not increase monotonically along the apparent horizon. Alternatively, the generalized

entropy does increase along a future Q-screen [133, 134], the analog of the future holographic screen where the

classical area is replaced by the generalized entropy and obeys a quantum focusing conjecture.

– 60 –

Euclidean path integral over manifolds of constant negative curvature, is that it may be

completely captured by a double-scaled random matrix model [38, 39], including all non-

perturbative effects [40, 41]. Thus, JT gravity provides an example of an exactly solvable

model of 2D quantum gravity, whose thermodynamics is only now being investigated (c.f.

[136–139]) using matrix model techniques. Most recently, it was shown the matrix models

provide an explicit and detailed description of the underlying microscopic degrees of freedom

[140], allowing for a deeper understanding of the quantum properties of black holes. It would

be interesting to see how the semi-classical thermodynamics studied here incorporates this

microscopic physics.

A particularly tantalizing application of this work is its ability to potentially address

the information paradox in eternal backgrounds using entanglement wedge islands. As briefly

described in the introduction, the fine grained gravitational entropy is given by the generalized

entropy, where the QES represents an extremum of this entropy. Invoking either double

holography, or the replica trick, the fine grained radiation entropy in an evaporating 2D black

hole background may be computed, which can be used to show the radiation entropy follows

a unitary Page curve, where the entropy undergoes a dynamical phase transition near the

Page time. At times before the Page time, the entropy follows the Hawking curve, eventually

turning over at the Page time where von Neumann entropy includes the degrees of freedom

of an entanglement wedge island, the (typically) disconnected portion to the codimension-1

entanglement wedge of the radiation, bounded by the QES. In the case of eternal black holes,

an information paradox still exists, and the island region remains outside of the horizon

[45, 49, 50].

The derivation involving the replica trick [46, 47, 61] is specifically compelling as it does

not require AdS/CFT holography, and argues the phase transition may understood as an

exchange in dominant saddle points in a Euclidean path integral, where at times after the

Page time, the Euclidean path integral used in the replica trick is dominated by a “replica

wormhole” saddle. Understanding this phase transition precisely in Lorentzian time remains

an open question, though for recent work focused on addressing this problem, see [62, 63].

Given that we uncovered a time-dependent generalized entropy, and that the entropy may

be identified as the thermodynamic potential in a microcanonical ensemble, it would be very

interesting to apply this observation to the question at hand. Moreover, at least in eternal

backgrounds, we would then be able to attribute the thermodynamics of quantum extremal

surfaces to the associated islands. Studying this exciting avenue is currently underway [141].

For dynamical settings, such as an evaporating black hole, perhaps the Iyer-Wald definition

of dynamical entropy [84] can be used.

Lastly, let us briefly comment on the prospect of whether our work is applicable to higher-

dimensional models. Indeed, JT gravity is known to describe the physics of nearly extremal,

charged four dimensional black holes, however, the question remains whether the techniques

developed here and elsewhere are achievable for higher-dimensional systems. In the context

of this article, properly addressing this question would involve solving the full backreacted

Einstein’s equations in higher dimensions, a notoriously difficult and open problem. Some

– 61 –

progress can be made, however, as it is straightforward to compute the Wald entropy of the

leading 1-loop quantum effective action induced by the trace anomaly in four dimensions

[142]. It may be interesting to see how much of the generalized entropy the Wald functional

encodes in this context, and this may shed new light on the black hole information puzzle in

higher dimensions.

Acknowledgments

JP and AS are supported by the Simons Foundation through It from Qubit: Simons collabora-

tion on Quantum fields, gravity, and information. WS is supported by the Icelandic Research

Fund (IRF) via a Personal Postdoctoral Fellowship Grant (185371-051). MV is supported

by the Republic and canton of Geneva and the Swiss National Science Foundation, through

Project Grants No. 200020- 182513 and No. 51NF40-141869 The Mathematics of Physics

(SwissMAP).

A Coordinate systems for AdS2

Here we list various coordinate systems for the eternal AdS2 black hole and the transforma-

tions between them.

Static coordinates: A general solution to the vacuum equations of motion (2.3) and (2.4)

in static Schwarzschild coordinates is (see, e.g., [74])

ds2 = −N2dt2 +N−2dr2 , N2 =r2

L2− µ , φ(r) =

r

Lφr , (A.1)

where we emphasize the dilaton φ is purely classical. Introducing the tortoise coordinate r∗,

r∗ = −∫ ∞r

dr′

r′2/L2 − µ=

L

2√µ

log

(r − L√µr + L

√µ

),

r = L√µcoth(−√µr∗/L) ,

(A.2)

the line element in static coordinates becomes [64]

ds2 = e2ρ(−dt2 + dr2∗) , e2ρ(r∗) =

µ

sinh2(√µr∗/L)

. (A.3)

Here r∗ ranges from r∗ ∈ [−∞, 0], where r∗ = −∞ is the location of the horizon and r∗ = 0

is the asymptotic boundary. In these coordinates the dilaton φ becomes

φ(r∗) = φr√µcoth(−√µr∗/L) . (A.4)

In terms of advanced and retarded time coordinates (v, u)

v = t+ r∗ , u = t− r∗ , (A.5)

– 62 –

the static metric (A.3) is cast in conformal gauge

ds2 = −e2ρ(v,u)dudv , e2ρ =µ

sinh2[√µ

2L (v − u)], (A.6)

where the dilaton is now

φ(v, u) = φr√µcoth

(−√µ

2L(v − u)

). (A.7)

The future horizon is at u = ∞, while the past horizon is at v = −∞. As discussed in

[76], this system is characterized by the Boulware vacuum state, i.e., the state annihilated by

modes which are positive frequency with respect to the static coordinates (v, u). As a final

comment, note that the surface gravity κH =√µL defining the Hawking temperature TH = κH

2π

appears in the above coordinate transformations, as well as those below.

Poincare coordinates: Introducing a set of lightcone coordinates (V,U) which are related

to coordinates (u, v) via [76]

V =L√µe√µ

Lv , U =

L√µe√µ

Lu , (A.8)

the metric (A.6) now takes the Poincare form

ds2 = −4L2dUdV

(V − U)2. (A.9)

The asymptotic boundary is located at V = U , while the future horizon is located at U =∞and the past horizon at V = 0. The classical dilaton becomes

φ(V,U) = φr√µcoth

[1

2log

(U

V

)]= φr

√µU + V

U − V. (A.10)

Kruskal coordinates: The line element in Schwarzschild form (A.1) only covers one of the

exterior regions of the black hole at once, however, one can perform a Kruskal extension

to analytically continue to the other exterior, as well as future and past regions. Kruskal

coordinates VK, UK in terms of (t, r) coordinates are

VK =

√r − L√µr + L

√µe√µ

Lt , UK = −

√r − L√µr + L

√µe−√µ

Lt , (A.11)

then

ds2 = − 4L2dUKdVK

(1 + UKVK)2. (A.12)

The inverse coordinate transformation is

t =L

2√µ

log

(− VK

UK

), r = L

√µ

1− UKVK

1 + UKVK

. (A.13)

– 63 –

The past horizon is located at VK = 0 and the future horizon at UK = 0. In Kruskal

coordinates the classical dilaton is

φ(VK, UK) = φr√µ

1− UKVK

1 + UKVK

. (A.14)

Since Kruskal coordinates cover all of AdS2, the metric (A.12) characterizes the global struc-

ture of AdS2 in conformal coordinates. As pointed out in [76], the vacuum state with respect

to this set of coordinates is equivalent to the Hartle-Hawking state.

We can relate (dimensionless) Kruskal coordinates to static coordinates (v, u) by

VK = e√µ

Lv , UK = −e−

√µ

Lu , (A.15)

and to conformal Poincare coordinates (A.8) as

VK =

√µ

LV , UK = − L

√µ

1

U. (A.16)

Note we will often work with scaled (dimensionful) Kruskal coordinates as in [76],

VK =L√µe√µ

Lv , UK = − L

√µe−√µ

Lu , (A.17)

in which case the Kruskal line element becomes

ds2 = − 4µdUKdVK

(1 + µL2UKVK)2

. (A.18)

AdS-Rindler coordinates: A line element for AdS2-Rindler space is

ds2 = −κ2%2dσ2 +

(%2

L2+ 1

)−1

d%2. (A.19)

Surfaces of constant σ parametrize the worldlines of accelerating observers with a > 1/L.

Thus, σ is the proper Rindler time. The AdS-Rindler horizon is located at % = 0 and the

asymptotic boundary at % =∞. Further, κ is the surface gravity of the future Rindler horizon,

which is kept arbitrary here. In the flat space limit L→∞ one recovers flat 2D Rindler space.

AdS-Rindler coordinates (σ, %) may be directly related to Schwarzschild coordinates (t, r)

via

κσ =√µt/L , % =

L√µ

√r2

L2− µ , (A.20)

so that Schwarzschild time t is a rescaled proper Rindler time, and hence the eternal AdS2

black hole is equivalent to AdS2-Rindler space. If we set the surface gravity equal to κ =√µ/L

then the two times σ and t coincide. Further, since conversely we have r = L√µ√

%2

L2 + 1,

the static dilaton is cast as

φ(%) = φr√µ

√%2

L2+ 1 . (A.21)

– 64 –

Defining the tortoise coordinate as %∗ = 1κarcsinh[L/%], AdS-Rindler space (A.19) can be

written in conformal gauge,

ds2 = e2ρ(%∗)(−dσ2 + d%2∗) , e2ρ(%∗) =

κ2L2

sinh2(κ%∗)= κ2%2 . (A.22)

The horizon % = 0 is now located at %∗ = −∞, while the conformal boundary is at %∗ = 0.

Note that this tortoise coordinate is just a rescaling of the Schwarzschild tortoise coordi-

nate (A.2), namely κ%∗ =√µr∗/L.

A.1 Conformal gauge identities and the stress-energy tensor

In conformal gauge ds2 = −e2ρdy+dy− the following identities are useful for working out the

equations of motion

g+− = −1

2e2ρ , g+− = −2e−2ρ , Γ+

++ = 2∂+ρ , Γ−−− = 2∂−ρ , (A.23)

R = −2ρ = 8e−2ρ∂+∂−ρ , Φ = −4e−2ρ∂+∂−Φ , (∇Φ)2 = −4e−2ρ(∂+Φ)(∂−Φ) .

for a scalar field Φ (for us Φ denotes either φ or χ).

In conformal static coordinates (A.5) (v, u), the expectation value of the stress-energy

tensor is

〈Tχuv〉 = 〈Tχvu〉 =c

48πL2e2ρ(1− λ) ,

〈Tχuu〉 = − c

48πL2µ− c

12πtu(u) ,

〈Tχvv〉 = − c

48πL2µ− c

12πtv(v) ,

(A.24)

where 〈Tχµν〉 ≡ 〈Ψ|Tχµν |Ψ〉 for some unspecified quantum state |Ψ〉. Meanwhile, in null Kruskal

coordinates (A.17) (VK, UK), it is straightforward to verify

〈TχVKUK〉 = 〈TχUKVK

〉 =c

48πL2e2ρ(1− λ) ,

〈TχUKUK〉 = − c

12πtUK

(UK) ,

〈TχVKVK〉 = − c

12πtVK

(VK) .

(A.25)

We observe in either static or Kruskal coordinates for λ = 1 the off-diagonal components

〈Tχ±∓〉 vanish.

Using the transformation

ρ(v, u) = ρ(VK, UK) +

√µ

2L(v − u) = ρ(VK, UK) +

1

2log(− µ

L2UKVK

), (A.26)

we find

ξu = ξUK−√µ

2Lu = ξUK

+1

2log

(−√µ

LUK

), (A.27)

– 65 –

ξv = ξVK+

√µ

2Lv = ξVK

+1

2log

(√µ

LVK

), (A.28)

such that

ξ(v, u) = ξ(VK, UK) +1

2log(− µ

L2UKVK

). (A.29)

Then with the following Schwarzian derivatives:

c

24πVK, v =

c

24πUK, u = − cµ

48πL2,

c

24πv, VK =

c

48πV 2K

,c

24πu, UK =

c

48πU2K

,(A.30)

the transformation rule for the normal-ordered stress tensors (2.34) yields

: Tχvv : =

(dVK

dv

)2

: TχVKVK: − c

24πVK, v =

(√µ

LVK

)2

: TχVKVK: +

Nµ

48πL2

= 4π2T 2HV

2K : TχVKVK

: +cπ

12T 2H ,

(A.31)

: Tχuu : =

(dUK

du

)2

: TχUKUK: − c

24πUK, u =

(√µ

LUK

)2

: TχUKUK: +

cµ

48πL2

= 4π2T 2HU

2K : TχUKUK

: +cπ

12T 2H .

(A.32)

Lastly, the normal ordered relation (2.32) gives

〈: Tχvv :〉 = − c

12πtv(v) , 〈: TχVKVK

:〉 = − c

12πtVK

(VK) , (A.33)

and similarly for uu and UKUK components, such that tVKand tv, and tUK

and tu are related

by

tv =

(√µ

LVK

)2

tVK− µ

4L2, tu =

(√µ

LUK

)2

tUK− µ

4L2. (A.34)

Consequently, the diagonal stress-energy tensors in (A.24) and (A.25) are, respectively,

〈Tχuu〉 = − cµ

12πL2U2

KtUK, 〈Tχvv〉 = − cµ

12πL2V 2

KtVK, (A.35)

〈TχUKUK〉 = − c

48π

1

U2K

− cL2

12πµ

tuU2

K

, 〈TχVKVK〉 = − c

48π

1

V 2K

− cL2

12πµ

tvV 2

K

. (A.36)

It is worth comparing to the relation between stress tensors in static and conformal Poincare

coordinates (A.8)

: Tχvv : =

(dV

dv

)2

: TχV V : − c

24πV, v =

(√µ

LV

)2

: TχV V : +cµ

48πL2

= 4π2T 2HV

2 : TχV V : +cπ

12T 2

H ,

(A.37)

– 66 –

: Tχuu : =

(dU

du

)2

: TχUU : − c

24πU, u =

(√µ

LU

)2

: TχUU : +cµ

48πL2

= 4π2T 2HU

2 : TχV V : +cπ

12T 2

H .

(A.38)

Using the relations

〈: Tχvv :〉 = − c

12πtv , 〈: TχV V :〉 = − c

12πtV , (A.39)

equation (A.37) leads to a relation between tv and tV ,

tv =

(√µ

LV

)2

tV −µ

4L2, tu =

(√µ

LU

)2

tU −µ

4L2, (A.40)

we find

〈Tχuu〉 = − cµ

12πL2U2tU , 〈Tχvv〉 = − cµ

12πL2V 2tV ,

〈TχUU 〉 = − c

48π

1

U2− cL2

12πµ

tuU2

, 〈TχV V 〉 = − c

48π

1

V 2− cL2

12πµ

tvV 2

.(A.41)

Thus, the relations between stress-energy tensors in static and Poincare coordinates and static

and Kruskal coordinates are identical. This is indicative of the fact the Kruskal and Poincare

vacuum states coincide [76].

B Boost Killing vector from the embedding formalism for AdS2

A quick method to compute the Killing vectors of AdS, particularly the boost Killing vector

of AdS-Rindler space, follows from the embedding formalism. Locally two-dimensional AdS

space can be embedded in three-dimensional flat space R2,1 with signature (−−+) and metric

ds2 = −(dT 1)2 − (dT 2)2 + dX2, (B.1)

where AdS2 is realized as a hyperboloid in the embedding space

− (T 1)2 − (T 2)2 +X2 = −L2. (B.2)

The embedding space induces a metric on this hyperboloid, corresponding to the metric of

AdS2. Furthermore, the symmetry group SO(2, 1) that preserves this hyperboloid is the

isometry group of AdS2. In embedding coordinates, the rotation generator J and the two

boost generators B1,2 of SO(2, 1) are given by

J = T 1∂T 2 − T 2∂T 1 , B1 = X∂T 1 + T 1∂X , B2 = X∂T 2 + T 2∂X , (B.3)

and satisfy the commutation relations

[J,B1] = −B2, [J,B2] = B1, [B1, B2] = J. (B.4)

We can now compute the Killing vectors of AdS2 from (B.3) by inserting the embeddding

coordinates for specific AdS2 coordinate systems. In particular, here we restrict ourselves to

– 67 –

the AdS2 metric in Schwarzschild coordinates (A.1), for which the embedding coordinates are

given by

T 1 =r√µ, T 2 =

L√µ

√r2

L2− µ sinh

(√µt

L

), X =

L√µ

√r2

L2− µ cosh

(√µt

L

), (B.5)

and also to the metric in retarded and advanced time coordinates (v, u) (A.6), for which the

embedding coordinates are

T 1 = Lcosh

[√µ

2L (u− v)]

sinh[√

µ2L (u− v)

] , T 2 = Lsinh

[√µ

2L (u+ v)]

sinh[√

µ2L (u− v)

] , X = Lcosh

[√µ

2L (u+ v)]

sinh[√

µ2L (u− v)

] . (B.6)

In Schwarzschild coordinates the isometry generators are

J =r/√µ√

r2/L2 − µcosh(

√µt/L)∂t − L

√r2/L2 − µ sinh(

√µt/L)∂r , B2 =

L√µ∂t ,

B1 = −r/√µ√

r2/L2 − µsinh(

√µt/L)∂t + L

√r2/L2 − µ cosh(

√µt/L)∂r . (B.7)

Thus, B2 is the time-translation Killing vector of the eternal AdS black hole. Meanwhile, in

retarded and advanced null coordinates (v, u), the isometry generators take the form28

J =L√µ

[cosh(√µv/L)∂v + cosh(

√µu/L)∂u] , B2 =

L√µ

(∂v + ∂u) ,

B1 = − L√µ

[sinh(√µv/L)∂v + sinh(

√µu/L)∂u] . (B.8)

Let us now compute the boost Killing vector field of an AdS2-Rindler wedge nested inside

an eternal AdS2 black hole, i.e., a Rindler wedge inside a Rindler wedge. Denote (t, r) as

the Schwarzschild coordinates adapted to the nested AdS-Rindler wedge. Correspondingly,

the adapted advanced and retarded time coordinates are denoted as (v, u), and are related

to Schwarzschild coordinates in the same way as the eternal black hole, (A.5). Thus, barred

(adapted) coordinates, e.g., (t, r), describe the generally smaller Rindler wedge nested inside

the larger Rindler wedge, described by, say, (t, r) coordinates. Succinctly,

Enveloping eternal black hole : (t, r) and (v, u) coordinates , (B.9)

Nested AdS-Rindler wedge : (t, r) and (v, u) coordinates . (B.10)

Note the future and past Killing horizons of the smaller Rindler wedge are located at u→∞and v → −∞, respectively, which generally do not coincide with the future and past horizon

of the AdS2 black hole. Moreover, the timeslice t = 0 is equal to the t = 0 slice.

In adapted coordinates, the boost Killing vector is proportional to ∂t, i.e., the generator

of proper Rindler time translations. Ultimately, however, we want to express this boost

28It is useful to know the inverse relationship (u− v) = 2L√µ

arccoth(T 1/L) and (u+ v) = 2L√µ

arctanh(T 2/X).

– 68 –

Killing vector in terms of the coordinates of the enveloping black hole, (t, r) or (v, u). To do

so, let us relate (v, u) coordinates to (v, u) coordinates, which may be conveniently addressed

in the embedding space formalism. As proven in [65] and explained below, the small and

large AdS-Rindler wedges can be transformed into each other by an isometry, corresponding

to a boost in the (T 1, X) plane in embedding space. Below we mainly follow the derivation

of the boost Killing vector in Appendix C.2 of [124], but we use different coordinates and we

extend the analysis to nested Rindler wedges centered at arbitrary time slices.

Consider the following boost in embedding space:

(T 1)′ = cosh(β)T 1 − sinh(β)X, (X)′ = cosh(β)X − sinh(β)T 1, (B.11)

where β ∈ (−∞,∞) is the rapidity. Notice when β = 0 we recover (T 1)′ = T 1 and X ′ = X.

The boost Killing vector of the transformed AdS-Rindler wedge is given by the boost generator

in the (T 2, X ′) plane, which we denote as ζ = X ′∂T 2 +T 2∂X′ . We can now express ζ in terms

of the umprimed embedding coordinates. Indeed, using (B.11) we see ζ becomes the following

linear combination of isometry generators

ζ = cosh(β)B2 − sinh(β)J. (B.12)

Observe if β = 0 we recover ζ = B2.

Since we are interested in the size of the nested Rindler wedge, we want to know how β

is related to the size of the Rindler wedge at the boundary, characterized by a dimensionful

parameter α we define momentarily. We begin by considering the following coordinate trans-

formation between the enveloping AdS-Rindler wedge and the nested AdS-Rindler wedge:

T 1 = L cosh(β)cosh

[κ2 (u− v)

]sinh

[κ2 (u− v)

] + L sinh(β)cosh

[κ2 (u+ v)

]sinh

[κ2 (u− v)

] ,X = L sinh(β)

cosh[κ2 (u− v)

]sinh

[κ2 (u− v)

] + L cosh(β)cosh

[κ2 (u+ v)

]sinh

[κ2 (u− v)

] ,T 2 = L

sinh[κ2 (u+ v)

]sinh

[κ2 (u− v)

] ,(B.13)

where κ is the (arbitrary) surface gravity of the Killing horizon of the nested wedge, associated

to the boost Killing vector ∂t, which is analogous to the (fixed) surface gravity√µ/L of

the Killing horizon of the enveloping black hole, associated to the time-translation Killing

vector ∂t. We arrived to the above transformation by first inverting the boost (B.11), such

that T 1 = cosh(β)(T 1)′+ sinh(β)X ′ and X = sinh(β)(T 1)′+ cosh(β)X ′. We then substituted

the primed embedding coordinates for (v, u) of the smaller wedge, given by (B.6) with the

replacements (T 1, T 2, X) → ((T 1)′, (T 2)′, X ′), (v, u) → (v, u) and√µ/L → κ. Substituting

the unprimed embedding coordinates (B.6) for the enveloping eternal black hole into the

left-hand side of the above transformation, with the help of Mathematica, we uncover the

following simple transformation between the retarded and advanced time in the two wedges

coth

[√µ

2Lu

]= eβ coth

[κ2u], coth

[√µ

2Lv

]= eβ coth

[κ2v]. (B.14)

– 69 –

Next, consider the limit to the bifurcation point B of the nested AdS2-Rindler horizon, u→∞,

v → −∞, and define a parameter α to be the values of the retarded and advanced times at B,

i.e., u = α and v = −α,

u =∞, v = −∞ → sinhβ =1

sinh(√µα/L)

coshβ =cosh(

√µα/L)

sinh(√µα/L)

. (B.15)

Notice the limit α→∞ corresponds to the rapidity β = 0, while α = 0 corresponds to β →∞.

The null boundaries of the nested Rindler wedge intersect the AdS boundary at t = ±α in

static coordinates. The parameter α therefore represents a (boundary) time interval, such

that in the α→∞ limit the time interval is the entire time boundary and the nested Rindler

wedge coincides with the Rindler wedge of the enveloping black hole.

Replacing rapidity β with α in (B.14) we now have

coth

[√µ

2Lu

]= coth

[√µ

2Lα

]coth

[κ2u], coth

[√µ

2Lv

]= coth

[√µ

2Lα

]coth

[κ2v], (B.16)

or equivalently,

√µ

Lu = log

[1 + eκu+

√µα/L

e√µα/L + eκu

],

√µ

Lv = log

[1 + eκv+

√µα/L

e√µα/L + eκv

]. (B.17)

Consequently, the boost Killing vector ζ (B.12) in terms of α is

ζ =1

sinh(√µα/L)

(cosh(√µα/L)B2 − J) . (B.18)

In null coordinates (v, u),

ζ =1

sinh(√µα/L)

L√µ

[(cosh(

√µα/L)−cosh(

√µv/L))∂v+(cosh(

√µα/L)−cosh(

√µu/L))∂u

],

(B.19)

from which we observe ζ becomes null on the future and past Killing horizons u = α and

v = −α, respectively. In Schwarzschild coordinates (t, r), ζ takes the form

ζ =1

sinh(√µα/L)

L√µ

cosh(√µα/L)− r/L√

r2

L2 − µcosh(

√µt/L)

∂t + L

√r2

L2− µ sinh(

√µt/L)∂r

.

(B.20)

We see in the limit α → ∞ the boost Killing vector reduces to the isometry generator

B2 = (L/√µ)∂t, as expected. Moreover, the boundary vertices t = ±α, r = ∞ and

the bifurcation point t = 0, r = L√µ coth(

√µα/L) are fixed points of the flow of ζ. We

emphasize that here ζ is normalized such that the surface gravity κ = 1 at the future horizon.

We can restore a generic surface gravity κ 6= 1 by replacing ζ → κζ.

– 70 –

Nested wedge on arbitrary time slice

Thus far we have considered a nested AdS-Rindler wedge centered at the t = 0 slice. More

generally one could consider an AdS-Rindler wedge where the bifurcation point is located at

some other slice t0 6= 0. To see this, first note that ζ becomes null on the Killing horizon

when the square of its norm,

ζ2 = − L2/µ

sinh2(√µα/L)

(r2

L2− µ

)cosh

(√µ

L(t+ α)

)− r/L√

r2

L2 − µ

cosh

(√µ

L(t− α)

)− r/L√

r2

L2 − µ

,

(B.21)

vanishes. This occurs at

cosh

(√µ

L(t± α)

)=

r/L√r2/L2 − µ

, (B.22)

where the plus sign corresponds to the past Killing horizon, and the minus sign to the future

Killing horizon. This observation suggests the following generalization to an AdS-Rindler

wedge which is centered at some arbitrary time t0 slice, instead of just t = 0,

cosh

(√µ

L(t− t0 ± α)

)=

r/L√r2/L2 − µ

. (B.23)

The associated AdS2 Killing vector is then

ζ =1

sinh(√µα/L)

[L√µ

cosh

(√µ

Lα

)− r/L√

r2

L2 − µcosh

(√µ

L(t− t0)

) ∂t

+ L

√r2

L2− µ sinh

(√µ

L(t− t0)

)∂r

],

(B.24)

or in terms of (v, u) coordinates

ζ =1

sinh(√µα/L)

L√µ

[(cosh

(√µ

Lα

)− cosh

(√µ

L(v − t0)

))∂v

+

(cosh

(√µ

Lα

)− cosh

(√µ

L(u− t0)

))∂u

].

(B.25)

In fact, this expression for ζ corresponds in the embedding space to the following linear

combination of isometry generators

ζ =1

sinh(√µα/L)

[cosh(√µα/L)B2 − cosh(

√µt0/L)J − sinh(

√µt0/L)B1] . (B.26)

This new ζ is associated to a shifted AdS-Rindler wedge centered at t0 6= 0, and can be

obtained through another isometry transformation of the AdS-Rindler wedge centered at

– 71 –

t = 0. Specifically, if we perform a subsequent boost in the (T 2, X) plane then the extremal

slice of the nested Rindler wedge shifts from t = 0 to some arbitrary t = t0 6= 0.

To see this explicitly, consider a boost in the (T 2, X) plane for a different rapidity γ

which we will identify momentarily,

(T 2)′ = cosh(γ)T 2 − sinh(γ)X, (X)′ = cosh(γ)X − sinh(γ)T 2. (B.27)

The boost Killing vector of the transformed AdS-Rindler wedge is now, analogous to (B.12),

ζ = cosh(β)B′2 − sinh(β)J ′ , (B.28)

where B′2 = X ′∂(T 2)′ + (T 2)′∂X′ = X∂T 2 + T 2∂X = B2 and J ′ = T 1∂(T 2)′ − (T 2)′∂T 1 =

cosh(γ)J + sinh(γ)B1. Thus, the transformed boost Killing vector can be written as

ζ = cosh(β)B2 − sinh(β) cosh(γ)J − sinh(β) sinh(γ)B1 , (B.29)

so, upon comparing to (B.26), we identity γ =√µt0/L. Thus, the AdS2 Killing vector ζ for

t0 6= 0 is a twice boosted Killing vector in the embedding space, where rapidity β is related

to a boundary time interval α, and γ to t0.

Let us now derive the transformation between the (v, u) coordinates of the nested wedge

centered at an arbitrary time slice and the (v, u) coordinates of eternal black hole, thereby

extending the transformation (B.17) to t0 6= 0. The two successive boosts in the (T 2, X) and

(T 1, X) planes lead to the following transformation in embedding space

(T 1)′ = cosh(β)T 1 + sinh(β) sinh(γ)T 2 − sinh(β) cosh(γ)X ,

(T 2)′ = cosh(γ)T 2 − sinh(γ)X ,

X ′ = − sinh(β)T 1 − cosh(β) sinh(γ)T 2 + cosh(β) cosh(γ)X ,

(B.30)

and the inverse transformation is

T 1 = cosh(β)(T 1)′ + sinh(β)X ′ ,

T 2 = sinh(β) sinh(γ)(T 1)′ + cosh(γ)(T 2)′ + cosh(β) sinh(γ)X ′ ,

X = sinh(β) cosh(γ)(T 1)′ + sinh(γ)(T 2)′ + cosh(β) cosh(γ)X ′ .

(B.31)

Then, following the analogous steps which led us to (B.14), we find the relation:

coth

[√µ

2Lu

]=eκu(eβ+γ + eβ + eγ − 1

)+ eβ+γ + eβ − eγ + 1

eκu (eβ+γ − eβ + eγ + 1) + eβ+γ − eβ − eγ − 1, (B.32)

and similarly for v and v. Next we take the limit (u → ∞, v → −∞) to the bifurcation

point B of the AdS-Rindler Killing horizon to express the rapidities in terms of the static

coordinates of B , (tB = t0, r∗,B = α),

γ =√µt0/L, eβ = coth(

√µα/2L) . (B.33)

– 72 –

The relation between α and β is the same as in (B.15), so by replacing the rapidities in (B.32)

with variables t0 and α we find

√µ

Lu = log

[e√µt0/L + eκu+

√µ(t0+α)/L

e√µα/L + eκu

],

√µ

Lv = log

[e√µt0/L + eκv+

√µ(t0+α)/L

e√µα/L + eκv

],

(B.34)

which is desired extension of (B.17) to t0 6= 0.

As a final consistency check, let us verify ζ in the embedding formalism reduces to

the usual boost Killing vector in Poincare coordinates. The embedding coordinates for the

Poincare patch are

T 1 =1

2z(L2 − τ2 + z2) , T 2 =

Lτ

z, X =

1

2z(L2 + τ2 − z2) , (B.35)

for which the following inverse relations are useful z = L2

T 1+Xand τ = LT 2

T 1+X. In these

coordinates it is straightforward to show the generators (B.3) become

J =1

2L

[(L2 + τ2 + z2)∂τ + 2τz∂z

],

B1 = −τ∂τ − z∂z ,

B2 =1

2L

[(L2 − τ2 − z2)∂τ − 2τz∂z

].

(B.36)

Further, we need the relations between the rapidities β and γ and the Poincare coordinates

(zB = R, τB = τ0) of the bifurcation point of the nested Rindler wedge,

R =L

cosh(β) + cosh(γ) sinh(β), τ0 = L

sinh(γ)

cosh(γ) + coth(β), (B.37)

or

cosh(β) =L2 +R2 − τ2

0

2LR, sinh(β) cosh(γ) =

L2 −R2 + τ20

2LR, sinh(β) sinh(γ) =

τ0

R. (B.38)

Note for the special case γ = 0 the relations reduce to R = e−βL and τ0 = 0, derived

in [65] (just below their Eq. (3.18)). The generic case γ 6= 0 follows from inserting the

primed embedding coordinates for (v, u) of the nested wedge and the unprimed embedding

coordinates for (z, τ) of the Poincare patch into the dubble boost transformation (B.31), and

inserting v = −u (restricting to the τ = 0 slice) and u → ∞ (restricting to the bifurcation

point). Next we substitute the generators (B.36) into our boosted expression (B.29) for ζ and

use the relations (B.38). The final result is

ζ =1

2R

[(R2 − (τ − τ0)2 − z2)∂τ − 2(τ − τ0)z∂z

]. (B.39)

This is the boost Killing vector in AdS2 in Poincare coordinates, which is well known in the

literature, e.g., in Eq. (3.1) of [66] (their normalization is different since they set κ = 2π).

– 73 –

C Covariant phase space formalism for general 2D dilaton gravity

In Section 4, where we derived the Smarr relation and associated first law of Rindler-AdS2,

we made extensive use of the covariant phase space formalism [67, 84, 100, 101, 129]. Thus,

to keep this article self-contained, here we work out the covariant phase space formalism for

generic two-dimensional dilaton-gravity theories, including the Polyakov action. For previous

applications of the covariant phase formalism for specific two-dimensional dilaton-gravity

models, namely, the classical CGHS and JT models, see, e.g., [35, 84, 102–104].

Lagrangian formalism

Let ψ = (gµν ,Φ) denote a collection of dynamical fields, where gµν is an arbitrary background

metric of a (1 + 1)-dimensional Lorenztian spacetime M and Φ represents any scalar field

on M , specifically, the dilaton φ or the local auxiliary field χ appearing in the Polyakov

action. Consider the following covariant29 Lagrangian 2-form depending on ψ:

L = L0ε[RZ(Φ) + U(Φ)(∇Φ)2 − V (Φ)

], (C.1)

where ε is the spacetime volume form, L0 is some coupling constant, R is the Ricci scalar

in 1 + 1 dimensions, and Φ is a scalar field. This is the most general Lagrangian for two-

dimensional Einstein gravity coupled non-minimally to a dynamical scalar field (c.f. [143]).

Here we keep functions Z(Φ), U(Φ) and V (Φ) generic, however, it is worth highlighting three

specific models characterized by the Lagrangian30 (C.1):

1. JT model : L0 = 116πG , Z(φ) = φ+ φ0, U(φ) = 0 and V (φ) = 2φΛ ,

2. CGHS model : L0 = 116πG , Z(φ) = φ2, U(φ) = 4 and V (φ) = −4λ2φ2 ,

3. Polyakov term: L0 = − c24π , Z(χ) = χ, U(χ) = 1 and V (χ) = λ2

L2 ,

where G is the gravitational Newton’s constant, Λ, λ and λ are cosmological constants, and

c is the central charge of massless scalar fields (quantifying the semi-classical correction).

The variation of the Lagrangian (C.1) yields

δL = EΦδΦ + Eµνδgµν + dθ(ψ, δψ), (C.2)

29By covariant we mean Lagrangian L transforms as a 2-form under diffeomorphisms acting only on dy-

namical fields, as opposed to local Lagrangians where the L transforms under diffeomorphisms acting on both

dynamical and (non-dynamical) background fields.30If the function Z(Φ) is invertible, then under a suitable transformation, Z(Φ) may be re-

placed by Φ. Subsequently, under the field redefinition Φ = e−2ϕ, the Lagrangian becomes L =

L0εe−2ϕ

[R+ U(ϕ)(∇ϕ)2 − V (ϕ)

], where now U(ϕ) = 4e−2ϕU(e−2ϕ) and V (ϕ) = e2ϕV (e−2ϕ), a more con-

ventional form of the CGHS model.

– 74 –

with d as the spacetime exterior derivative and δψ representative of field variations, and where

the equations of motion Eµν and EΦ are, respectively,

Eµν = L0

[1

2V (Φ)gµν + Z(Φ)gµν −∇µ∇νZ(Φ) + U(Φ)

(∇µΦ∇νΦ− 1

2(∇Φ)2gµν

)],

EΦ = L0

[RZ ′(Φ)− V ′(Φ)− 2∇µ [U(Φ)∇µΦ] + U ′(Φ)∇µΦ∇µΦ

], (C.3)

where we used δε =(

12gµνδgµν

)ε. The symplectic potential 1-form θ for a theory of the type

(C.1) is generally given by [84]

θ = εµ

[2Pµαβν∇νδgαβ − 2(∇νPµαβν)δgαβ + 2L0U(Φ)(∇µΦ)δΦ

], (C.4)

where Pµαβν ≡ ∂L∂Rµαβν

such that for L = εZ(Φ)R, one has Pµαβν = Z2 (gµβgαν − gαβgµν)ε.

Consequently,

θ = L0εµ

[Z(Φ)(gµβgαν − gµνgαβ)∇νδgαβ + (gαβ∇µZ(Φ)− gβµ∇αZ(Φ))δgαβ + 2U(Φ)∇µΦδΦ

].

(C.5)

Here we used that in two dimensions, Rµν = 12Rgµν , and Z ′(Φ) = dZ

dΦ , etc.

For spacetimes M with boundary ∂M the pullback of the potential θ to the boundary

can be written as [104, 129]

θ∣∣∂M

=[p(Φ)δΦ +

1

2τµνδgµν

]ε∂M + δb+ dC , (C.6)

with

p(Φ) ≡ 2L0[Z ′(Φ)K + U(Φ)nµ∇µΦ], τµν ≡ 2L0γµνnα∇αZ(Φ) , (C.7)

and

b = −2L0ε∂MZ(Φ)K , C = c · ε∂M , cµ = −L0Z(Φ)γµλnνδgλν . (C.8)

Here ε∂M is the volume form on ∂M , K is the trace of the extrinsic curvature Kµν = 12Lnγµν

of the timelike boundary, and γµν = −nµnν + gµν is the induced metric on ∂M , with nµ the

unit normal to ∂M . Moreover, we used εµ|∂M = nµε∂M , δε∂M =(

12γ

µνδgµν)ε∂M , and that

in two dimensions the extrinsic curvature obeys Kµν = Kγµν , such that31

δb = L0ε∂M

[− 2Z ′(Φ)KδΦ− Z(Φ)(gµνnλ∇λδgµν − nα∇βδgαβ)

− nα(∇βZ(Φ))δgαβ +Dµ[Z(Φ)γµνnαδgαν ]

].

(C.9)

The quantity b is a local 1-form and C is a local 0-form on the boundary ∂M , which are both

covariant under diffeomorphisms which preserve the location of the (spatial) boundary [104].

31It is also useful to know δnµ = 12nµn

αnβδgαβ , and

δK = −1

2Kµνδgµν +

1

2gµνnλ∇λδgµν −

1

2nα∇βδgαβ −

1

2Dµ(γµνnαδgνα) ,

where Dµ is the hypersurface covariant derivative, obeying γµνDµ = Dν .

– 75 –

Further, τµν ≡ 2√−γ

δIδγµν

with action given by (C.18) is the (generalized) Brown-York stress

tensor [144]. We will use the (renormalized) Brown-York stress tensor below to derive the

quasi-local and asymptotic energy for the semi-classical JT model.

The symplectic current 1-form, defined as ω(ψ, δ1ψ, δ2ψ) ≡ δ1θ(ψ, δ2ψ) − δ2θ(ψ, δ1ψ), is

explicitly given by

ω = L0εµ

[Z(Φ)Sµαβνρσδ1gρσ∇νδ2gαβ +

1

2gµβgανgρσδ1gρσδ2gαβ∇νZ(Φ)

+ (gµβgαν − gµνgαβ)(δ1(Z(Φ))∇νδ2gαβ − δ1(∇νZ(Φ))δ2gαβ)

+ U(Φ)∇µΦgαβδ1gαβδ2Φ + 2δ1(U(Φ)∇µΦ)δ2Φ− [1↔ 2]],

(C.10)

where Sµαβνρσ is given by

Sµαβνρσ = gµρgασgβν − 1

2gµνgαρgβσ − 1

2gµαgβνgρσ − 1

2gαβgµρgσν +

1

2gαβgµνgρσ . (C.11)

To derive (C.10), one can directly work with the potential θ in (C.5) and the definition of ω,

or, alternatively, one can utilize that a non-minimally coupled theory of gravity of the Brans-

Dicke type is equivalent f(R) gravity. Following the latter approach, the non-minimally

coupled contribution in ω follows from, e.g., Eq. (E.17) of [145] with f ′(R) replaced by Z(Φ).

Moving on, let ζ be an arbitrary smooth vector field on M , i.e., the infinitesimal generator

of a diffeomorphism. The Noether current 1-form jζ associated with ζ and arbitrary field

configuration ψ is defined as jζ ≡ θ(ψ,Lζψ)− ζ ·L, with Lζ being the Lie derivative along ζ.

For the theory (C.1), explicitly we have

jζ = εµ

[2L0∇ν(Z(Φ)∇[νζµ] + 2ζ [ν∇µ]Z(Φ)) + 2Eµνζ

ν], (C.12)

where we can again use that non-minimally coupled scalar-Einstein theory is equivalent to

f(R) theory. From the current, we easily read off the Noether charge 0-form, defined as

jζ = dQζ on-shell,

Qζ = −L0εµν [Z(Φ)∇µζν + 2ζµ∇νZ(Φ)] , (C.13)

where εµν is the volume form for the codimension-0 surface ∂Σ, which is a cross section of

the spatial part of ∂M .

Note that the definitions of the aforementioned quantities L, θ and Qζ are ambiguous up

to the addition of exact forms. Specifically, adding a total derivative dµ to L does not change

the equations of motion, and the symplectic potential and Noether charge are defined up to a

closed and hence locally exact form, denoted by dY and dZ respectively. These ambiguities

potentially alter other quantities as well, and can be summarized as follows [84]

L→ L+ dµ ,

θ → θ + δµ+ dY (ψ, δψ) ,

Qζ → Qζ + ζ · µ+ Y (ψ,Lζψ) + dZ .

(C.14)

– 76 –

Moreover, the symplectic current and the form δQζ − ζ · θ(ψ, δψ) are also ambiguous [145]

ω → ω + d[δ1Y (ψ, δ2ψ)− δ2Y (ψ, δ1ψ)] (C.15)

δQζ − ζ · θ(ψ, δψ)→ δQζ − ζ · θ(ψ, δψ) + δY (ψ,Lζψ)− LζY (ψ, δψ) + d[δZ + ζ · Y (ψ, δψ)]

Nonetheless, these ambiguities all cancel between the left- and right-hand side in the following

on-shell fundamental variational identity

ω(ψ, δψ,Lζψ) = d [δQζ − ζ · θ(ψ, δψ)] . (C.16)

Next, we briefly discuss a recent refinement of the covariant phase formalism for spacetimes

with boundaries [104]. Let us decompose the boundary as ∂M = B ∪ Σ− ∪ Σ+, where Σ−and Σ+ are past and future boundaries, respectively, and B (denoted by Γ in [104]) is the

spatial boundary.32 We require that the action is stationary under arbitrary variations of the

dynamical fields up to terms at the future and past boundary of the manifold. In order for

the variational principle to be well posed we require Dirichlet boundary conditions only on B,

since boundary conditions at Σ± would also fix the initial or final state, which is too strong.

From (C.6) it follows that fixing Φ and the pullback of gµν to the spatial boundary B, i.e.,

δΦ|B = 0 = γαµγβ

νδgµν |B, is sufficient to restrict the symplectic potential to the form33

θ∣∣B

= δb+ dC . (C.17)

Therefore, we supplement the action by the integral of (minus) b at ∂M , which thus serves

as a Gibbons-Hawking-York (GHY) boundary term,34

Itot =

∫ML−

∫∂M

b . (C.18)

Then, using (C.2) and (C.17), the variation of the total action becomes [104]

δItot =

∫MEψδψ +

∫Σ+−Σ−

(θ − δb− dC) . (C.19)

This shows the total action (C.18) is stationary up to boundary terms at Σ±. Since the

combination Ψ ≡ θ − δb − dC appears a boundary term in the variation of the action, it is

more natural to define the symplectic current instead as

ω(ψ, δ1ψ, δ2ψ) ≡ δ1Ψ(ψ, δ2ψ)− δ2Ψ(ψ, δ1ψ) . (C.20)

32In the main text we neglect the past and future boundaries, and only treat the spatial boundary B at

infinity.33We could have imposed a stronger boundary condition, δγµν |B = 0, by which C(ψ, δψ) = 0 (C.8) on B,

since γµλnνδgλν = −γµνδnν = γµλnνδγλν = 0. This amounts to a partial gauge fixing of the theory with the

weaker boundary condition γαµγβ

νδγµν |B = 0 (see footnote 25 in [104]). We do this in the main text, but in

the appendices we allow for a nonzero C (see footnote 21).34Our notation can be matched with that of [104] by replacing b→ −`.

– 77 –

The b term drops out due to the antisymmetrization, hence the new symplectic current is

ω(ψ, δ1ψ, δ2ψ) = δ1θ(ψ, δ2ψ)− δ2θ(ψ, δ1ψ)− d[δ1C(ψ, δ2ψ)− δ2C(ψ, δ1ψ)] , (C.21)

where the first two terms combine into the old symplectic current ω(ψ, δ1ψ, δ2ψ). By con-

struction the symplectic current vanishes on the spatial boundary, ω|B = 0. We note the

new definition for the symplectic current can be obtained by (partly) fixing the ambiguity in

(C.15) as Y = −C (there is a remaining ambiguity that involves a simultaneous shift in θ

and C). Now, the on-shell variational identity (C.16) for the new symplectic current (C.21)

becomes∫Σω(ψ, δψ,Lζψ) =

∮∂Σ

[δQζ − ζ · θ(ψ, δψ)− δC(ψ,Lζψ) + LζC(ψ, δψ)] . (C.22)

One should be careful, however, not to add the C term at an inner boundary of Σ, for example

at the bifurcation surface of a Killing horizon, since C does not extend covariantly into M .

For the most part we will ignore this subtlety below.

Hamiltonian formalism

Thus far we have only discussed the Lagrangian formalism. However, in our study of thermo-

dynamics we need a notion of energy and hence we must make contact with the Hamiltonian

formalism. We very briefly review this here and in the following subsection describe the

quasi-local and asymptotic energy for the semi-classical JT model in both Boulware and

Hartle-Hawking vacuum states.

The power of the covariant phase space formalism is that it provides a way to under-

stand the Hamiltonian dynamics of covariant Lagrangian field theories. The phase space P,

defined as the set of solutions to the equations of motion,35 is a symplectic manifold supplied

with a closed and non-degenerate 2-form Ω, called the symplectic form.36 In this context,

the infinitesimal variation δ is viewed as the exterior derivative of differential forms on the

configuration space – the set of off-shell field configurations on spacetime obeying boundary

conditions on B – such that symplectic potential θ and C are interpreted as one-forms on the

configuration space. Moreover, the symplectic current ω is related to Ω via

Ω =

∫Σω , (C.23)

for Σ a Cauchy slice of M . We denote the induced metric on Σ by hµν = uµuν + gµν , with

uµ the unit normal to Σ.

35This definition differs from the standard, non-covariant, definition of phase space of a dynamical system,

which labels the set of distinct initial conditions on a given timeslice to solve the systems equations of motion.36More accurately, in the context of theories with local symmetries, where the initial value problem is

generally ill defined, e.g., general relativity and Maxwell theory, one instead works with the pre-phase space.

Physical phase space P is given by the quotient of pre-phase space under the action of the group of continuous

transformations whose generators are the zero modes of Ω [100]. Thus, more rigorously, Ω is really the pre-

symplectic 2-form, and ω the pre-symplectic current.

– 78 –

Of interest is the Hamiltonian Hζ which generates time evolution along the flow of the

diffeomorphism generating vector field ζ. Hamilton’s equations read δHζ = Ω(ψ, δψ,Lζψ).

Therefore,

δHζ =

∫Σω(ψ, δψ,Lζψ) =

∮∂Σ

[δQζ − ζ · θ(ψ, δψ)− δC(ψ,Lζψ) + LζC(ψ, δψ)] , (C.24)

where the identity (C.22) was used. If we impose θ|B = δb + dC and make use of Cartan’s

“magic” formula,37 the Hamiltonian variation simplifies to

δHζ =

∮∂Σ

[δQζ − ζ · δb− δC(ψ,Lζψ)] . (C.25)

In Appendix D we include variations of coupling constants in this equation. Thus, since

δζ = 0, the Hamiltonian Hζ itself is

Hζ =

∫∂Σ

[Qζ − ζ · b− C(ψ,Lζψ)] + cst . (C.26)

The constant represents a standard ambiguity to the energy of any Hamiltonian system. Below

we will set this arbitrary constant to zero. Notice, the “integrability” of the Hamiltonian,

allowing us to move from δHζ to Hζ , arises as a consequence of the boundary condition (C.17),

which appeared from demanding the action Itot is stationary up to terms at the future and

past boundaries.

For the 2D dilaton-gravity model in question, the term on the right-hand side of Hζ

pulled back to ∂Σ are given by

Qζ∣∣∂Σ

= −L0ε∂Σ(uµnν − uνnµ) [Z(Φ)∇µζν + 2ζµ∇νZ(Φ)] , (C.27)

ζ · b|∂Σ = +2L0ε∂ΣζµuµZ(Φ)K , (C.28)

C(ψ,Lζψ)|∂Σ = L0ε∂Σ(uµnν + uνnµ)Z(Φ)∇µζν , (C.29)

where we used εµν |∂Σ = uµnν −uνnµ in (C.13), δgλν → ∇λζν together with γµν = gµν −nµnν

in C (C.8), and (ε∂M )µ = −uµε∂Σ.38 Thus, the Hamiltonian for 2D dilaton gravity is

Hζ = −2L0

∫∂Σε∂Σ [γµνu

µζνnα∇αZ(Φ) + Z(Φ)uµnν∇µζν + γµνuµζνZ(Φ)K] ,

= −∮∂Σε∂Σζ

µuντµν ,

(C.30)

with τµν the Brown-York stress-energy tensor (C.7). To get to the second line we used

nν∇µζν∣∣∂Σ

= −ζν∇µnν , which follows from n · ζ = 0 on ∂M , and we inserted the extrinsic

curvature Kµν = 2∇(µnν) and the identity Kµν = Kγµν in 2D.

37For a form σ, and vector field X Cartan’s magic formula says: LXσ = X · dσ + d(X · σ). Note, when σ is

a zero form, then X · σ = 0.38Notice if ζ is a Killing field, then C(ψ,Lζψ)|∂Σ = 0 by Killing’s equation. We do not use this though in

deriving (C.30).

– 79 –

From here we note the quasi-local momentum and spatial stress vanish, since there is no

extra dimension, while the quasi-local energy density ε is

ε ≡ uµuντµν = −2L0nα∇αZ(Φ) . (C.31)

Thus, another way to express the Hamiltonian (C.30) is

Hζ = −∮∂Σε∂ΣNε , (C.32)

where N = −ζµuµ is the lapse function.

C.1 Quasi-local and asymptotic energies with semi-classical corrections

We now have all of the ingredients to compute the quasi-local and asymptotic energies for the

nested Rindler wedge and the eternal black hole in the semi-classical JT model. Following

[144], the quasi-local energy density is defined as in (C.31), while the asymptotic energy is

equal to the Hamiltonian (C.32) evaluated at spatial infinity. Since C(ψ,Lζψ)|∂Σ = 0 for the

boost Killing vector ζ, in what follows and in the main text we denote Hζ as Hζ .

For this computation we need the unit normals u and n, respectively, to Σ = t = t0and to B in the black hole background. Away from Σ the future-pointing timelike unit vector

uµ can be defined as the velocity vector of the flow of ζ,

uµ =ζµ√−ζ · ζ

, and at Σ uµ =1√

r2/L2 − µ∂µt . (C.33)

By definition uµ is by definition tangent to the spatial boundary B of M . The outward-

pointing spacelike unit vector nµ is defined as the unit normal to B

nµ =nµ√n · n

, and at Σ nµ =√r2B/L

2 − µ∂µr . (C.34)

where the non-normalized normal n satisfies n · ζ = 0 at B, so for convenience we take

n = ζr∂t − ζt∂r. Thus both u and n are defined in terms of the vector field ζ.

Classical energy

Specializing to JT gravity requires us to eliminate unwanted divergences arising in the energy

upon evaluating the Hamiltonian at the conformal boundary. We accomplish this by sup-

plementing the classical and semi-classical JT actions with appropriate local counterterms.

Specifically, the boundary counterterm Lagrangian 1-form for classical JT gravity is

bctJT =

φ

8πGLεB , (C.35)

such that the total boundary Lagrangian 1-form b for JT gravity is

bJT = − εB8πG

(φ+ φ0)K +εB

8πG

φ

L. (C.36)

– 80 –

Feeding this through the derivation of Hζ we are led to the same expression (C.30), however

with the BY stress tensor (C.7) now given by (matching Eq. (3.87) of [104])

τµνφr =1

8πG

(nα∇αφ−

φ

L

)γµν

∂Σ=

φr8πGL

(√r2B

L2− µ− rB

L

)γµν . (C.37)

where we inserted the unit normal nµ (C.34) at ∂Σ in the second equality. The corresponding

classical quasi-local energy ε is

εφr∣∣∂Σ

=φr

8πGL

(rBL−√r2B

L2− µ

). (C.38)

Let us compute the asymptotic energy on the extremal slice Σ associated with the boost

Killing vector (4.4) of a nested Rindler wedge. In the limit we approach spatial infinity

rB → ∞ at t = t0, where the boost Killing vector is ζµ → κL√µ tanh(

√µα/2L)∂µt ≡ A∂

µt , we

can compute the asymptotic energy from (C.32)

Eφrζ ≡ limrB→∞

Hφrζ =

κ√µφr

16πGtanh

(√µα

2L

)= AMφr , (C.39)

where

A ≡ κL√µ

tanh

(√µα

2L

), (C.40)

and Mφr is the classical ADM mass of the black hole, given by (2.7). Further, in the α→∞limit, upon identifying κ =

√µ/L, we recover limα→∞E

φrζ →Mφr since A → 1 in this limit.

Semi-classical energy

Let’s now see how the energy is modified under semi-classical corrections. This requires us to

find the BY tensor solely associated with χ, denoted τµνc,χ, which will likewise be modified due

to a local counterterm (Eq. (3.25) of [64]), such that the 1-form b for the Polyakov action is

bχ =c

12πχKεB +

c

24πLεB . (C.41)

The appropriate BY tensor (C.7) with L0 = − c24π , and Z(Φ) = χ, is found to be

τµνc,χ = − c

12π

(nα∇αχ+

1

2L

)γµν ,

∂Σ= − c

12π

[1√µ

sinh

(−√µr∗B

L

)(√µ

Lcoth

(√µr∗B

L

)+ ∂r∗ξ

)+

1

2L

]γµν ,

(C.42)

where r∗,B is the tortoise coordinate at the surface B, ξ is the function defining t± in the

quantum stress-energy tensor, and the Polyakov cosmological constant is set to λ = 0.

We must also include in the BY tensor a contribution coming from semi-classical correc-

tions to the dilaton, denoted by φc, whose BY tensor τµνc,φ is

τµνc,φ =1

8πG

(nα∇αφc −

φcL

)γµν . (C.43)

– 81 –

Since the backreacted solutions φc, χ and ξ depend on the choice of vacuum state, let us

determine the energy in both the Boulware and Hartle-Hawking vacuum states.

Boulware vacuum: Specifically consider the static solution ξ(1) = c, for which the quasi-

local energy εc,χ associated to χ is easily worked out to be

εc,χ∣∣∂Σ

=c

24πL

[1− 2 cosh

(√µr∗B

L

)], (C.44)

while the quasi-local energy εc,φ associated to φc = Gc3 (1 +

√µL r∗) coth(

√µr∗/L) is

εc,φ∣∣∂Σ

=c

24πL

[(1 +

√µr∗B

L

)coth

(√µr∗B

L

)

+L√µ

sinh

(√µr∗B

L

)∂r∗

((1 +

√µr∗B

L

)coth

(√µr∗B

L

))],

=c

24πL

[(1 +

√µr∗B

L

)tanh

(√µr∗

2L

)+ cosh

(√µr∗B

L

)].

(C.45)

Thus, the total semi-classical quasi-local energy is

εc ≡ (εc,χ + εc,φ)∣∣∂Σ

=c

24πL

[1 +

(1 +

√µr∗B

L

)tanh

(√µr∗B

2L

)− cosh

(√µr∗B

L

)].

(C.46)

Consequently,

Ecζ ≡ limr∗B→0

Hcζ = − cκ

48πtanh

(√µα

2L

)= AMc , (C.47)

where

Mc = −c√µ

48πL. (C.48)

In the α → ∞ limit we identify the above as the semi-classical correction to the asymp-

totic energy, i.e., limα→∞Ecζ → Mc (upon setting κ =

√µ/L), which is proportional to the

expectation value of the stress-tensor (2.51).

Hartle-Hawking vacuum: First consider the static solution ξ(4) =√µr∗/L+ c , for which

the quasi-local energy associated to χ is

εc,χ∣∣∂Σ

=c

12πL[1/2 + sinh(−√µr∗B/L)− cosh(

√µr∗B/L)] , (C.49)

while the quasi-local energy associated to φc = Gc/3 is

εc,φ∣∣∂Σ

=c

24πL. (C.50)

Hence, the total semi-classical quasi-local energy is

εc = (εc,χ + εc,φ)∣∣∂Σ

=c

12πL[1 + sinh(−√µr∗B/L)− cosh(

√µr∗B/L)] , (C.51)

– 82 –

such that

Ecζ ≡ limr∗B→0

Hcζ =

cκ

12πtanh

(√µα

2L

). (C.52)

In the α → ∞ limit we attain the semi-classical correction to the ADM energy of the black

hole, upon setting κ =√µ/L:

Mc =c√µ

12πL, (C.53)

matching what was found using holographic renormalization methods [20, 64].

Next consider the time-dependent solution for χ:

ξ(5)u,v = k − log

[(e−√µu/2L +KUK

e√µu/2L

)(e√µv/2L +KVK

e−√µv/2L

)]. (C.54)

The derivative with respect to the tortoise coordinate r∗ is

∂r∗ξ(5)u,v =

√µ

L

[1

1− VK/V BK

− 1

1− UBK /UK

]. (C.55)

Plugging this into the BY stress tensor (C.42), the total quasi-local energy is

εc∣∣∂Σ

=c

12πL

1 + sinh

(−√µr∗B

L

) 1

1 + 1KVK

e√µv/L

− 1

1 +KUKe√µu/L

− cosh

(√µr∗B

L

) .

(C.56)

Meanwhile, the semi-classical asymptotic energy vanishes as r∗,B → 0 for any α

Ecζ ≡ limr∗,B→0

Hcζ = 0 . (C.57)

D Derivation of the extended first law with variations of couplings

Here we provide a detailed derivation of the extended first law in JT gravity for the nested

AdS-Rindler wedge with boost Killing vector ζ. This amounts to evaluating the fundamental

variational identity in the covariant phase space formalism extended to include coupling

variations [123, 126]∫Σω(ψ, δψ,Lζψ) =

∮∂Σ

[δQζ − ζ · θ(ψ, δψ)] +

∫Σζ · EλiL δλi . (D.1)

Here the ‘δ’ denotes a variation with respect to the coupling constants λi and dynamical fields

ψ = (φ, gµν). The two-form EλiL ≡∂Lbulk∂λi

ε is the derivative of the bulk off-shell Lagrangian

density, defined via L = Lbulk(ψ, λi)ε, with respect to the coupling constant λi, and sum-

mation over i is implied in (D.1). In the context of semi-classical JT gravity the couplings

are λi = φ0, G, L, c. The relation (D.1) was previously invoked in [125] to derive an ex-

tended first law for classical JT gravity, where background subtraction was used to regulate

divergences.

– 83 –

In this appendix we consider a manifold with boundary ∂M , with the same decomposition

∂M = B∪Σ−∪Σ+ as in Appendix C. In the covariant phase space formalism with boundaries

the relevant fundamental variational identity is slightly different compared to (D.1). Below

we derive this new variational identity, thereby simultaneously generalizing the work of [104]

to allow for coupling variations and [123, 126] to the case of spacetimes with boundaries.

Recall the total action (C.18), where L is the covariant bulk Lagrangian defining a theory

of dynamical fields ψ and couplings λi. The variation of the bulk Lagrangian is

δL = Eψδψ + EλiL δλi + dθ(ψ, δψ) . (D.2)

Thus, the variation of the total action is

δItot =

∫M

(Eψδψ + EλiL δλi) +

∫∂M

(θ − δb) . (D.3)

Here δb refers to the total variation of boundary Lagrangian 1-form b, i.e., δb = δψb + δλib.

Following [104], we invoke the boundary condition (C.17) on the symplectic potential, given

by θ|B = δψb+ dC. Implementing this boundary condition and expanding the total variation

δb on B, we find

δItot =

∫M

(Eψδψ + EλiL δλi) +

∫Σ+−Σ−

(θ − δb− dC)−∫Bδλib . (D.4)

Observe that we recover (C.19) when we drop the coupling variations.

Now consider the variation of the Noether current jζ = θ(ψ,Lζψ)− ζ · L. Using δζ = 0,

we have the following on-shell identity

δjζ = δθ(ψ,Lζψ)− Lζθ(ψ,Lζψ) + d[ζ · θ(ψ, δψ)]− ζ · EλiL δλi= ω(ψ, δψ,Lζψ) + d[ζ · θ(ψ, δψ)]− ζ · EλiL δλi ,

(D.5)

where to get to the first line we used the Lagrangian variation (D.2) and Cartan’s magic

formula, and in the second line we identified the symplectic current ω evaluated on the Lie

derivative of ψ with respect to ζ. Since, moreover, the Noether current is closed on-shell and

hence locally exact, jζ = dQζ , such that δjζ = dδQζ , we arrive at the on-shell variational

identity including coupling variations

ω(ψ, δψ,Lζψ) = d[δQζ − ζ · θ(ψ, δψ)] + ζ · EλiL δλi , (D.6)

which agrees with the integral identity (D.1). This is a true identity, but it is not the relevant

fundamental identity in the covariant phase space formalism with boundaries.

Indeed, the appropriate symplectic current ω for spacetimes with boundaries differs by

an exact form from ω and was defined in (C.21) [104]. Inserting the identity (D.6) into this

definition and integrating over a Cauchy slice Σ yields∫Σω(ψ, δψ,Lζψ) =

∮∂Σ

[δQζ−ζ ·θ(ψ, δψ)−δC(ψ,Lζψ)+LζC(ψ, δψ)]+

∫Σζ ·EλiL δλi . (D.7)

– 84 –

This replaces the variational identity (D.1), and it is the generalization of (C.24) to nonzero

coupling variations.

Next, we apply this identity to the case where the Cauchy slice Σ has an inner boundary

at the bifurcation surface B of a Killing horizon and an outer boundary at spatial infinity. We

should be careful not to include the C terms at B, since the boundary condition (C.17) for the

symplectic potential only applies to the spatial boundary B →∞ and not to the bifurcation

surface B, and moreover C does not extend into the bulk in a covariant way. Consequently,

by imposing the bounday condition θ|∞ = δψb + dC and using Cartan’s magic formula and

δψb = δb− δλib, we can rewrite the above as∫Σω(ψ, δψ,Lζψ) = δ

∮∞

[Qζ−ζ ·b−C(ψ,Lζψ)]−δ∮BQζ+

∫Σζ ·EλiL δλi+

∮∞ζ ·Eλib δλi . (D.8)

Here we introduced the notation

δλib = Eλib δλi with Eλib ≡∂Lbdy

∂λiεB , (D.9)

where Lbdy is the boundary of-shell Lagrangian density. We will include local counterterms

in b to regulate any divergences, so b = bGHY + bct. Note in the context of 2D dilaton

gravity and in the case ζ is a Killing vector, we see from (C.29) that C(ψ,Lζψ) = 0 and

δC(ψ,Lζψ) = 0, if the perturbed metric is also stationary, i.e., δ[∇(µζν)] = 0. Thus the C

term in (D.8) can be ignored in this context, which is consistent with the stronger boundary

condition C(ψ, δψ) = 0 that we could have imposed (see footnotes 21 and 33)∫Σω(ψ, δψ,Lζψ) = δ

∮∞

(Qζ − ζ · b)− δ∮BQζ +

∫Σζ · EλiL δλi +

∮∞ζ · Eλib δλi . (D.10)

Below we will apply this variational integral identity to classical and semi-classical JT gravity

in order to derive the extended first laws of nested AdS-Rindler wedges including variations

of the couplings λi = φ0, G, L, c.

Classical extended first law

Let us first evaluate the left-hand side of the integral identity (D.10). From (4.48), where

δ1ψ = δλiψ and δ2ψ = Lζψ, we find39

ω(ψ, δλiψ,Lζψ)∣∣∣Σ

=εµ

16πG(gµβgαν − gµνgαβ)(∇νLζφ)δλigαβ = −

κφ′B16πG

hαβδLgαβd` , (D.11)

where we used the metric is a function of L and not of the other JT couplings. Thus the only

nonzero coupling variation contribution to the left-hand side of (D.10) is∫Σω(ψ, δLψ,Lζψ) = −

κφ′B8πG

δL` . (D.12)

39More accurate would be to use the definition ω(ψ, δλiψ,Lζψ) = δλiθ(ψ,Lζψ)−Lζθ(ψ, δλiψ) which consid-

ers all possible coupling variations. Using θJT(ψ,Lζψ)|Σ = 0 and θJT(ψ, δλiψ) = 0 for λi = (φ0, G) variations,

we find the only non-zero coupling variation in the symplectic current of JT gravity is with respect to the AdS

length L, and so the result is the same in (D.11).

– 85 –

Explicitly we have∫Σω(ψ, δLψ,Lζψ) =

κφr√µδL

8πGL sinh(√µαL )

[cosh

(√µα

L

)− 1 + log

(tanh

(√µα

2L

))+

1

2log

(4r2B

L2µ

)],

(D.13)

which is logarithmically divergent as the cutoff rB goes to infinity.

Next, we evaluate the coupling variations of the Noether charge on the right-hand side

of (D.10). Implicitly summing over each of the coupling variations, where we note the δL

contribution entirely drops out, we have∮∞δλi(Q

JTζ − ζ · b) = −

κ√µφr

16πGtanh(

√µα/2L)

δG

G, (D.14)

∮BδλiQ

JTζ =

κ

8πGδφ0 −

κ

8πG(φ0 + φB)

δG

G, (D.15)

such that combined, with the field variations,

δ

∮∞

(QJTζ − ζ · b)− δ

∮BQJTζ = δEζ −

κ

8πδ

(φ0 + φB

G

), (D.16)

where we used δGEζ = −Eζ δGG .

Lastly, consider the coupling variations of the off-shell Lagrangian and boundary coun-

terterm on the right-hand side of the integral relation (D.10). We have [125]∫Σζ · EλiL δλi =

∫Σ

(R

16πGδφ0 −

4φ

16πGL2

δL

L− LJT

δG

G

)ζ · ε , (D.17)

where EλiL = ∂LJT∂λi

ε, with LJT = 116πG [(φ + φ0)R + 2

L2φ]. Similarly, the coupling variation of

the local boundary Lagrangian (C.36) yields∮∞ζ · Eλib δλi =

∮∞

[K

8πGδφ0 +

φ

8πGL2δL− 1

8πG

((φ0 + φ)K − φ

L

)δG

G

]ζ · εB . (D.18)

Having taken the variations, on-shell the sum of (D.17) and (D.18) yields∫Σζ · EλiL δλi +

∮∞ζ · Eλib δλi =

(φ0

8πGL2

∫Σζ · ε−

∮∞ζ · b

)δG

G(D.19)

+

(− 1

8πGL2

∫Σζ · ε+

1

8πG

∮∞Kζ · εB

)δφ0 +

(− 1

4πGL2

∫Σφζ · ε+

1

8πGL

∮∞φζ · εB

)δL

L.

We recognize the first term on the right-hand side as being proportional to the “G-Killing

volume” ΘGζ (4.37), and we introduce a counterterm subtracted “φ0-Killing volume”

Θφ0

ζ ≡∫

Σζ · ε− L2

∮∞Kζ · εB , (D.20)

– 86 –

and a counterterm subtracted “Λ-Killing volume”

ΘΛζ ≡

∫Σφζ · ε− L

2

∮∞φζ · εB . (D.21)

Notice we have normalized each Killing volume such that the coefficient in front of the volume

integral is equal to unity. Then, the sum simplifies to∫Σζ · EλiL δλi +

∮∞ζ · Eλib δλi =

φ0ΘGζ

8πGL2

δG

G−

Θφ0

ζ

8πGL2δφ0 +

ΘΛζ

8πGL2

δΛ

Λ. (D.22)

Here we replaced δL/L for −δΛ/(2Λ). Explicitly, the Λ-Killing volume ΘΛζ is

ΘΛζ = −

κL2√µφr4 sinh(

√µα/L)

(−1 + cosh

(√µα

L

)+ log

(4r2B

L2µ

)− 2 log

[cosh

(√µα

L

)+ csch

(√µα

L

)]),

(D.23)

where rB is a cutoff which should be taken to infinity. So ΘΛζ also diverges logarithmically as

rB →∞. Furthermore, the φ0-Killing volume can be computed to be

Θφ0

ζ = −κL2 . (D.24)

Putting everything together, we find the classical extended first law of the nested Rindler

wedge is

δEφrζ =κ

8πδ

(φ0 + φB

G

)−κφ′B8πG

δ`+Θφ0

ζ

8πGL2δφ0 −

φ0ΘGζ

8πGL2

δG

G−

ΘΛζ

8πGL2

δΛ

Λ, (D.25)

where here δ includes both field and coupling variations. Notice the variations with respect

to φ0 cancel on the right-hand side, and upon invoking the classical Smarr relation (4.39),

the δG variations cancel between the left- and right-hand sides. The δL variation on the

right-hand side is divergent as rB → ∞, however, this divergence is precisely cancelled by

the rB → ∞ divergence appearing in the δL` term (D.13). In fact, combining these two δL

variations leads to the remarkable simplification:

−κφ′B8πG

δL`+ΘΛζ

4πGL2

δL

L= −

κ√µφr

16πGLtanh(

√µα/2L)δL = −Eφrζ

δL

L. (D.26)

This is consistent with the L variation of the left-hand side of the first law, δLEφrζ = −Eφrζ

δLL ,

which follows from κ ∼ 1/L.

Semi-classical extension

Including semi-classical effects introduces a new coupling, the central charge c, and the aux-

iliary field χ. The auxiliary field χ generally depends on L and the dilaton receives a c

correction, φc = Gc3 (working in the Hartle-Hawking state).

– 87 –

It is straightforward to show the combined coupling and field variation of the Noether

charge yields

δ

∮∞

(Qcζ − ζ · bc)− δ∮BQcζ = δEcζ −

κ

2πδ

(φc4G− cχB

6

), (D.27)

where Qcζ = Qχζ +Qφcζ and bc = bχ + bφc .

We also have contributions from the coupling variations of the Lagrangian 2-forms and

boundary counterterm 1-forms associated with the correction φc and χ. The φc correction

only provides an additional δL variation,∫Σζ · EλiLφc δλi +

∮∞ζ · Eλibφc δλi = −

ΘΛ,cζ

4πGL2

δL

L, (D.28)

where we have defined the counterterm subtracted “semi-classical Λ-Killing volume”

ΘΛ,cζ ≡ φc

∫Σζ · ε− L

2φc

∮∞ζ · εB . (D.29)

This term diverges as rB, more specifically it is given by

ΘΛ,cζ =

cGκL

6

(−2L+

rB√µ

tanh(√µα/2L)

). (D.30)

Moreover, the non-vanishing coupling variations of the Lagrangian 2-form and boundary

counterterm associated with χ include∫Σζ · EcLχδc+

∮∞ζ · Ecbχδc =

(∫Σζ · Lχ +

∮∞ζ · bχ

)δc

c=

c

12πL2Θcζ

δc

c, (D.31)

and ∮∞ζ · ELbχδL =

c

24πL2ΘBζ δL = lim

rB→∞

cκ tanh(√µα/2L)

24π√µL2

rBδL , (D.32)

where Θcζ is the “semi-classical c-Killing volume” (4.81), and

ΘBζ ≡

∮∞ζ · εB (D.33)

is the Killing volume of the asymptotic boundary B → ∞. Combining (D.32) with (D.28)

cancels a common divergence resulting in,

−ΘΛ,cζ

4πGL2

δL

L+

cΘBζ

24πL

δL

L=

cκ

12πLδL , (D.34)

however, we will not make use of this simplification in the extended first law below.

As in the classical case, only variations with respect to L appear in the symplectic current

associated with χ (4.88). Explicitly,

ωχ(ψ, δLψ,Lζψ)∣∣∣Σ

= − c

24πf(r, α)

[hαβδLgαβ − 2δLχ

](u · ε) , (D.35)

– 88 –

where we used ∇µ(Lζχ)|Σ = uµf(r, α) (4.17). Using hαβδLgαβ = 2LδL, and considering the

time-independent solution χ(4) (2.72) for illustrative purposes,40 we find this additional term

(D.35) vanishes in the α → ∞ limit, however, below and in the main text we simply absorb

this contribution into δHχζ .

Putting together (D.27), (D.28), (D.31), and (D.32), we add to the classical extended

first law the following combination of terms

δEcζ =κ

2πδ

(φc4G− cχB

6

)− ΘΛ,c

8πGL2

δΛ

Λ−

cΘcζ

12πL2

δc

c+

cΘBζ

48πL

δΛ

Λ+ δHχ

ζ , (D.36)

where δHχζ =

∫Σ ωχ(ψ, δψ,Lζψ) and here again δ include both field and coupling variations.

If the variation is induced by only changing the central charge c then it reduces to

δcEcζ = Ecζ

δc

c=

κ

2π

(φc4G− cχB

6

)δc

c−

cΘcζ

12πL2

δc

c, (D.37)

which holds because of the semi-classical part of the Smarr relation (4.83). Further, for

variations that only change the cosmological constant the extended first law becomes

δLEcζ = −Ecζ

δL

L= − ΘΛ,c

8πGL2

δΛ

Λ+

cΘBζ

48πL

δΛ

Λ+ δLH

χζ . (D.38)

In the limit α→∞ both the left- and right-hand side reduce to −McδLL = − c

√µ

12πLδLL , due to

relation (D.34) between the Killing volumes and the vanishing of the Hamiltonian variation

limα→∞ δLHχζ = 0.

E A heuristic argument for the generalized second law

Under some particular assumptions about the final state of the background, here we provide

a heuristic derivation of the generalized second law. That is, upon throwing some matter

into the system, the semi-classical generalized entropy monotonically increases along the

future Killing horizon of the nested Rindler wedge. Our approach is similar in spirit to the

presentation given in [64] (see also [89] for a flat space analog), and so we only present the

broad details.

Classical JT and the second law

Consider including infalling matter, characterized by a stress-energy tensor Tmµν . The dila-

ton equation (2.4) remains unchanged such that the background is still AdS2, however, the

gravitational field equation (2.3) becomes

(gµνφ−∇µ∇νφ−1

L2gµνφ) = 8πGTm

µν . (E.1)

40For the time-independent solution for χ (2.72) we have ∇ν(Lζχ)∣∣Σ

=κ√µ

L1

sinh(√µα/L)

uν

( rL+√µ)

. Also, note

that when performing the variation of χ and hαβδLgαβ , one should use the unscaled Kruskal coordinates (A.12)

as it is in these coordinates the L variations of χ and gµν are covariant.

– 89 –

Following [64], assume conformally invariant matter gµνTmµν = 0, such that in conformal gauge

Tm+− = 0, which is infalling, i.e., Tm

++ = 0. The infalling matter, encoded in Tm−− is assumed to

obey the classical null energy condition, Tmµνk

µkν ≥ 0 for any future directed vector kµ = dxµ

dλ

tangent to a null geodesic xµ parametrized by an affine parameter λ; equivalently, Tm−− ≥ 0.

Solving the equation of motion (E.1) under these conditions leads to a positive increase

to the mass of the black hole ∆M = L2φr

∫dy−h(y−)Tm

−−, where h(y−) is a “constant” of

integration arising from solving the field equations, h(y−) = h0(y−)− 16πG∫y−=0 T

m−−, with

h0 being its value when there is no infalling matter.

Accompanied by the increase in mass is an increase in the black hole entropy. That is, the

classical “area” of the black hole, given by the dilaton φ, increases along the (future) event

horizon y+H of the black hole. This is seen as follows. Allow the matter to fall in for some finite

duration in y−, such that after the matter has fallen in and the system reaches equilibrium

it remains an eternal black hole. For an affine parameter λ along the event horizon, the null

generator satisfies dy−

dλ = (y+H−y−)2, such that the first derivative of the Bekenstein-Hawking

entropy SBH = 14G(φ0 + φ) with respect to λ is

dSBH

dλ

∣∣∣H

= (y+H − y

−)2dSBH

dy−

∣∣∣H. (E.2)

Using the −− component of the gravitational field equations (E.1) and the null energy con-

dition Tm−− ≥ 0, the second derivative of SBH with respect to λ satisfies [64]

d2SBH

dλ2

∣∣∣H

=1

4G(y+H − y

−)4∇−∇−φ∣∣∣H

= −2π(y+H − y

−)4Tm−− ≤ 0 . (E.3)

At late times, after the system has reached equilibrium, where dφdy−

∣∣∣H

= 0,41 it follows from

(E.3) that dSBHdλ ≥ 0 at all times. Hence, the classical entropy obeys the second law, i.e., it

monotonically increases along the future event horizon. Mechanically, where φ is interpreted

as the area of the horizon, the second law is completely analogous to the area increase theorem

of black hole horizons. While we won’t review this here, the classical entropy also monotoni-

cally increases along the apparent horizon [64]. Note that the second law also holds from the

viewpoint of the nested Rindler wedge. This follows trivially since the above derivation can

be worked out with respect to the nested coordinates identically, differing only by constant

factors of α which do not influence the sign of the first or second derivative of the entropy.

41This is nothing more than the extremization of the classical entropy. More explicitly, in a generic conformal

gauge ds2 = −e2ρ(y+,y−)dy+dy−, the generic solution for the dilaton φ is [64]

φ(y+, y−) =a+ b(y+ + y−) + cy+y−

(y+ − y−)

φrL,

with black hole mass parameter µ/L2 = b2−ac for real constants a, b, c. The future horizon y+H = − 1

c(b+√µL)

is found by computing ∂y−φ = 0, and similarly for the past horizon.

– 90 –

Semi-classical JT and the generalized second law

At equilibrium, we havedSgen

dλ = 0 along the QES, by definition of the QES. Let us thus

consider the semi-classical JT model where, as above, we include classical infalling conformal

matter obeying the null energy condition, Tm−− ≥ 0. Including infalling matter will alter the

solutions for the dilaton φ. The semi-classical gravitational equations of motion (2.14) now

become

T φµν + 〈Tχµν〉+ Tmµν = 0 . (E.4)

such that in conformal gauge the equation of motion (2.20) changes into

e2ρ∂±(e−2ρ∂±φ) = −8πG〈Tχ±±〉 − 8πGTm±± , (E.5)

while (2.21) remains unchanged since Tm±∓ = 0.

For convenience, below we will work in Poincare coordinates y+, y− = V,U,

ds2 = −e2ρdV dU , e2ρ =4L2

(V − U)2. (E.6)

We will also consider, when we are near equilibrium, that the system is in the Hartle-Hawking

vacuum state, for which the expectation values of the χ stress-energy tensor take the following

form in Poincare coordinates (see Appendix A)

〈Tχ±±〉 = 0 , 〈Tχ±∓〉 =c

12π

1

(y+ − y−)2. (E.7)

To study more realistic evolution as matter is dropped in during some short time duration,

one should not restrict to the Hartle-Hawking state. One way this assumption shows up

is that we do not take into account backreaction effects on χ which directly enter into the

generalized entropy. Namely, while the black hole is settling towards its new equilibrium it is

reasonable to expect a mismatch between left- and right-moving modes, which is reflected in

χ. We explicitly assume that h(y−) is mild enough to keep the error under control. A more

careful analysis was presented in [64]. Here we will only focus on the system when it is near

equilibrium where our assumption about being in the Hartle-Hawking state holds.

The ++ component of (E.5) together with (2.21) yield

φ =4L2

y− − y+h(y−) +

cG

3− 2L2h′(y−) , (E.8)

here h(y−) is some “constant” of integration and h′(y−) = dhdy− . Substituting this into the

−− component of (E.5), where Tm−− 6= 0, we find

2L2h′′′(y−) = 8πGTm−− . (E.9)

We may solve this differential equation explicitly for Tm−−(y+, y−), however, we won’t need

to know the exact form of the solution for the derivation of the generalized second law.

– 91 –

Importantly, carrying out an identical quasinormal mode analysis as in [64], at late times,

where y− → y+B , one has h(y−) ∼ (y+

B − y−) where y+B denotes the null coordinate along the

future Killing horizon associated to the quantum extremal surface.

As in the classical case, we have dy−

dλ = (y+B − y−)2. Using that the auxiliary field χ(5),

for example, in Poincare coordinates is simply42

χ(5) = −1

2log

[4L2

(y− − y+)2

]− log

[(−√µ

Ly− + k−

)(√µ

Ly+ + k+

)]+ c , (E.10)

where c, k± are real constants, we find that thedSgen

dλ is simply

dSgen

dλ= (y+

B − y−)2(∂−Sgen) = (y+

B − y−)2

1

4G

(− 4L2

(y− − y+)2h+

4L2

(y− + y+)h′ − 2L2h′′

)

− c

6

1

y− − y++

√µ

L

1(k− −

√µL y−) . (E.11)

Upon using h(y−) ∼ (y+B − y−), we find at late times

dSgen

dλ vanishes.

Consider next the second derivative of the generalized entropy. We have

d2Sgen

dλ2= 16e−2ρ∂−

(e−2ρ∂−Sgen

)=

4e−4ρ

G

([∂2−φ− 2(∂−φ)(∂−ρ)] +

2Gc

3[−∂2

−χ+ 2(∂−ρ)(∂−χ)]

).

(E.12)

Invoking the gravitational equations of motion (E.5) and (2.21), with 〈Tχ−−〉 = 0 in the

Hartle-Hawking vacuum, yields

d2Sgen

dλ2= −4e−4ρ

G

(8πGTm

−− +2Gc

3(∂−χ)2

)≤ 0 , (E.13)

where the inequality follows from the classical null energy condition. Note that the second

derivative calculation did not make use of the explicit form of χ or φ. We also see that had

one considered the Boulware state, where 〈Tχ±±〉 6= 0, the second derivative of Sgen is not

guaranteed to satisfy (E.13).

Lastly, using thatdSgen

dλ → 0 at equilibrium and (E.13), it follows

dSgen

dλ≥ 0 , (E.14)

such that the generalized entropy increases monotonically along the future Killing horizon

associated with the quantum extremal surface, as in the case of the black hole Killing horizon

[64]. This completes our heuristic derivation. A more careful analysis would solve the entire

problem without assuming we are in the Hartle-Hawking state, such that we study the entire

evolution of the system. This would be interesting and we save it for future work.

42This solution follows straightforwardly from a nearly identical analysis presented in Section 2, where now

we use the transformation between static coordinates (v, u) and Poincare coordinates (V,U).

– 92 –

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Documents

Semi-classical thermodynamics of quantum extremal surfaces