BRX-TH-6690 Semi-classical thermodynamics of quantum extremal surfaces in Jackiw-Teitelboim gravity Juan F. Pedraza, 1,2 Andrew Svesko, 1 Watse Sybesma 3 and Manus R. Visser 4 1 Department of Physics and Astronomy, University College London, London, WC1E 6BT, United Kingdom 2 Martin Fisher School of Physics, Brandeis University Waltham MA 02453, USA 3 Science Institute, University of Iceland Dunhaga 3, 107 Reykjav´ ık, Iceland 4 Department of Theoretical Physics, University of Geneva 24 quai Ernest-Ansermet, 1211 Gen` eve 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 (AdS 2 ) 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. arXiv:2107.10358v1 [hep-th] 21 Jul 2021
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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
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 = Lõ, 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=
õ
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 orõ 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
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
(õ
Lr∗
)∂r∗φ
]=
8cG
3
( µ
4L2+ tv(v)
)sinh2
(õ
Lr∗
), (2.58)
– 16 –
∂r∗
[sinh2
(õ
Lr∗
)∂r∗φ
]=
8cG
3
( µ
4L2+ tu(u)
)sinh2
(õ
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
(õ
Lr∗
)+Gc
3
(1 +
4τL2
µ
)(1 +
õ
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
(õ
Lr∗
)+Gc
3
(1 +
õ
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
(õ
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
(õ
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 +
õ
2Lu , ξ(3)
v = cv −√µ
2Lv ,
ξ(4)u = cu −
õ
2Lu , ξ(4)
v = cv +
õ
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 ∓
õ
2Lu+ ... , lim
r∗→−∞ξ(5)v = c′v −Kv ±
õ
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
(õ
LVK
),
ξ(4)UK
= cu , ξ(4)VK
= cv ,
ξ(5)UK
= kUK− log
[−√µ
LUK +KUK
], ξ
(5)VK
= kVK− log
[õ
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
eõu/L
)(1 +KVK
e−√µv/L
)]+ k ,
= −1
2log
4µ(1 + µUKVK
L2
)2
− log
[(−√µUK
L+KUK
)(õVK
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
(õ
Lr∗
)+Gc
3
(1 +
õ
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
õ
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
(õ
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
+cõ
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(õLVK +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=
õ
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
(õ
2L(vB − uB)
)]− 1
2log
[1
(√µδ/L)2(
√µδB/L)2
],
Ku = −√µuB
L, Kv =
õvB
L,
(3.18)
such that the von Neumann entropy SvN in static conformal coordinates (v, u) takes the form
SBvN =
c
12log
[1
µsinh2
(õ
2L(v − u)
)]+
c
12log
[1
µsinh2
(õ
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 = Lõ
(1− µ
L2UKVK
1 + µL2UKVK
)≈ rH
(1 +
2ε2
9e−
2õ
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
õ) +
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
(õ
2`(v − u)
)+Gc
3
(1 +
õ
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
õ
L(−2u+ vB + uB) +
ε
2sinh
[õ
L(−2u+ vB + uB)
]− 4ε
3
Lõ
u− uBsinh2
[õ
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) +
Lõ
log
(√ε
2
),
uQES ≈1
2(uB + vB)− L
õ
log
(√ε
2
).
(3.42)
Equivalently, in Kruskal coordinates
õ
LV QES
K ≈ 1
2
(−V B
K
UBK
)1/2√ε ,
õ
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 , % =
Lõ
√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)
[Lõ
cosh
(õ
Lα
)− r/L√
r2
L2 − µcosh
(õ
L(t− t0)
) ∂t
+ L
√r2
L2− µ sinh
(õ
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):
õ
Lu = log
[e√µt0/L + eκu+
√µ(t0+α)/L
e√µα/L + eκu
],
õ
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
(õ
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 = Lõ 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
(õ
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
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 =
õ
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) =
õ
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 =cõ
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 =
õ
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→∞ = −
õL
2(1− 2 log(2
õ)) . (4.85)
The semi-classical Smarr law for the black hole can thus be simplified to
2Mφr +Mc =
õ
2πL(Sφr + Sc) +
cΛ
12πΘcζ,α→∞ , (4.86)
with Sφr =φr√µ
4G and Sc = c12 + c
6 log(2õ). 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) =
õ
2πLδ
(φ0 + φH
4G− cχH
6
)−√µ
8πGLδφ0 +
(µφr
16πGL+
√µφ0
8πGL
)δG
G
− (Mφr +Mc)δL
L+
(Mc −
cõ
24πL−
cõ
12πLlog(2
õ)
)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 =õ
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-
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) = 2Lõ
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.
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
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
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
[1õ
sinh
(−√µr∗B
L
)(õ
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
)
+Lõ
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 =
õ/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 =cõ
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
eõu/2L
)(eõv/2L +KVK
e−√µv/2L
)]. (C.54)
The derivative with respect to the tortoise coordinate r∗ is
∂r∗ξ(5)u,v =
õ
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
eõv/L
− 1
1 +KUKeõu/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,