# The Entropy of Dynamical Black Holes

Stefan Hollands<EMAIL_ADDRESS>Institute for Theoretical
Physics
Leipzig University
Brüderstrasse 16, D-04103 Leipzig  Max-Planck-Institute MiS,
Inselstrasse 22, D-04103 Leipzig  Robert M. Wald<EMAIL_ADDRESS>Victor G.
Zhang<EMAIL_ADDRESS>Enrico Fermi Institute and Department of
Physics
University of Chicago
933 E. 56th St., Chicago, IL 60637

###### Abstract

We propose a new formula for the entropy of a dynamical black hole—valid to
leading order for perturbations off of a stationary black hole background—in
an arbitrary classical diffeomorphism covariant Lagrangian theory of gravity
in $n$ dimensions. In stationary eras, this formula agrees with the usual
Noether charge formula, but in nonstationary eras, we obtain a nontrivial
correction term. In particular, in general relativity, our formula for the
entropy of a dynamical black hole differs from the standard Bekenstein–Hawking
formula $A/4$ by a term involving the integral of the expansion of the null
generators of the horizon. We show that, to leading perturbative order, our
dynamical entropy in general relativity is equal to $1/4$ of the area of the
apparent horizon. Our formula for entropy in a general theory of gravity is
obtained from the requirement that a “local physical process version” of the
first law of black hole thermodynamics hold for perturbations of a stationary
black hole. It follows immediately that for first order perturbations sourced
by external matter that satisfies the null energy condition, our entropy obeys
the second law of black hole thermodynamics. For vacuum perturbations, the
leading order change in entropy occurs at second order in perturbation theory,
and the second law is obeyed at leading order if and only if the “modified
canonical energy flux” is positive (as is the case in general relativity but
presumably would not hold in more general theories of gravity). Our formula
for the entropy of a dynamical black hole differs from a formula proposed
independently by Dong and by Wall. We obtain the general relationship between
their formula and ours. We then consider the generalized second law in
semiclassical gravity for first order perturbations of a stationary black
hole. We show that the validity of the quantum null energy condition (QNEC) on
a Killing horizon is equivalent to the generalized second law using our notion
of black hole entropy but using a modified notion of von Neumann entropy for
matter. On the other hand, the generalized second law for the Dong–Wall
entropy is equivalent to an integrated version of QNEC, using the unmodified
von Neumann entropy for the entropy of matter.

††preprint: APS/123-QED

##  I Introduction

The discoveries that black holes obey laws of thermodynamics [1], radiate
thermally [2], and should be assigned an entropy [3] provide some of the
deepest insights we presently have on the fundamental nature of black holes
within a quantum theory of gravity. A formula for the entropy of a stationary
black hole in an arbitrary theory of gravity obtained from a diffeomorphism
covariant Lagrangian was obtained in [4]. However, the derivation of [4]
required evaluation of the entropy on the bifurcation surface, $\mathcal{B}$,
of the black hole, thus restricting the validity of the formula for entropy to
stationary black holes and their linear perturbations, evaluated at the “time”
represented by $\mathcal{B}$. It is of considerable interest to obtain an
expression for the entropy of a non-stationary black hole in a general theory
of gravity at a “time” represented by an arbitrary cross-section
$\mathcal{C}$, since this would allow one to investigate whether, classically,
black hole entropy satisfies a second law (i.e., whether it is non-decreasing
with time) and whether, semiclassically, black hole entropy satisfies a
generalized second law (i.e., whether the sum of black hole entropy and a
matter contribution to entropy is non-decreasing). A formula for dynamical
black hole entropy in a general theory of gravity on an arbitrary cross-
section $\mathcal{C}$ was proposed in [4]. However, that proposed dynamical
entropy formula is not field redefinition invariant, and the authors retracted
their proposal in a “note in proof” in the published version of [4].
Subsequently, proposed prescriptions for dynamical black hole entropy were
given independently by Dong [5] and Wall [6]. The Dong and Wall entropy
formulas agree in the cases where both have been evaluated and are expected to
correspond generally, although this has not been proven. We will refer to
their proposal for dynamical black hole entropy as the Dong–Wall entropy.

The main aim of this paper is to apply a new strategy to the definition of
dynamical black hole entropy, based upon the validity of a local, “physical
process version” of the first law of black hole mechanics. Since this law is
valid only at leading nontrivial order in perturbation theory about a
stationary black hole, our resulting definition of dynamical black hole
entropy, $S[\mathcal{C}]$, on an arbitrary cross-section, $\mathcal{C}$, of
the black hole horizon is intended to be applied only at leading nontrivial
order in perturbation theory about a stationary black hole. Our formula does
not agree with the Dong–Wall entropy, but, as we will elucidate in section VI,
there is a close relationship between our formula and theirs. Furthermore, in
the case of general relativity, we obtain a nontrivial dynamical correction to
the Bekenstein–Hawking entropy formula; namely we obtain

$S[\mathcal{C}]=\frac{A[\mathcal{C}]}{4}-\frac{1}{4}\int_{\mathcal{C}}V\vartheta$
(1)

where $A[\mathcal{C}]$ is the area of the cross-section $\mathcal{C}$ (so
$A[\mathcal{C}]/4$ is the usual Bekenstein–Hawking entropy), $V$ is an affine
parameter of the null generators of the horizon (with $V=0$ corresponding to
the bifurcation surface $\mathcal{B}$), and $\vartheta$ is the expansion of
these generators with respect to this affine parametrization.

In the remainder of this introductory section, we shall give an outline of our
strategy that leads to our new dynamical entropy formula and explain some of
the key features of our formula. To begin, we recall that the derivation of
[4] of the entropy of a stationary black hole follows from a “fundamental
identity” (see (30) below) that holds in an arbitrary theory of gravity
obtained from a diffeomorphism covariant Lagrangian. For source-free
perturbations of a stationary black hole whose event horizon is a Killing
horizon, the fundamental identity reduces to

$d[\delta\textbf{Q}[\xi]-\xi\cdot\bm{\theta}(\phi,\delta\phi)]=0.$ (2)

Here $\xi^{a}$ denotes the horizon Killing field, $\textbf{Q}[\xi]$ is the
Noether charge associated with $\xi$, $\bm{\theta}$ denotes the symplectic
potential, and $\phi$ collectively denotes all of the dynamical fields (i.e.,
the metric together with any matter fields appearing in the Lagrangian).
Integration of this equation over a spacelike hypersurface $\Sigma$ that
extends from the bifurcation surface $\mathcal{B}$ of the black hole to
infinity yields

$\int_{\mathcal{B}}\delta\textbf{Q}[\xi]=\delta M-\Omega_{H}\delta J.$ (3)

Here the right side is the boundary term from infinity resulting from
integration of (2) and it involves the mass, $M$, and angular momentum, $J$,
at spatial infinity (with $\Omega_{H}$ being angular velocity of the horizon,
which enters the expression when one expresses $\xi^{a}$ in terms of
asymptotic time translations and rotations). The left side is the boundary
term at $\mathcal{B}$; very importantly, there is no contribution to this term
from $\xi\cdot\bm{\theta}$ because $\xi^{a}=0$ on $\mathcal{B}$. We define

$S\equiv\frac{2\pi}{\kappa}\int_{\mathcal{B}}\textbf{Q}[\xi]$ (4)

where $\kappa$ denotes the surface gravity of the black hole (so that
$\kappa/2\pi$ is the Hawking temperature of the black hole). It can be shown
that $S$ is a local, geometrical quantity [4]. We thereby obtain the first law
of black hole mechanics for arbitrary source-free perturbations of a
stationary black hole

$\frac{\kappa}{2\pi}\delta S=\delta M-\Omega_{H}\delta J.$ (5)

This formula gives $S$ the interpretation of representing the entropy of the
black hole, valid to first order for perturbations of a stationary black hole.

However, as already mentioned above, (4) has the deficiency that the horizon
boundary term is evaluated at the bifurcation surface, $\mathcal{B}$, and,
thus, $S$ has the interpretation of representing the entropy,
$S[\mathcal{B}]$, of the black hole only at the “time” corresponding to
$\mathcal{B}$. This deficiency is not of importance for a stationary black
hole, since the entropy should be “time independent,” i.e., independent of the
cross-section of the horizon of the black hole at which it is evaluated.
However, for a non-stationary black hole, the entropy should evolve with time,
and one would like to have an expression for the entropy, $S[\mathcal{C}]$, on
an arbitrary cross-section, $\mathcal{C}$.

An expression for $S[\mathcal{C}]$ that would satisfy the first law of black
hole mechanics would be obtained if we could define $S[\mathcal{C}]$ so that

$\delta
S[\mathcal{C}]=\frac{2\pi}{\kappa}\int_{\mathcal{C}}\delta\textbf{Q}[\xi]-\xi\cdot\bm{\theta}(\phi,\delta\phi).$
(6)

If, on the event horizon $\mathcal{H}$, the pullback,
$\underline{\bm{\theta}}$, of the symplectic potential $\bm{\theta}$ were of
the form $\underline{\bm{\theta}}=\delta\textbf{B}_{\mathcal{H}}$ for some
quantity $\textbf{B}_{\mathcal{H}}$ defined on $\mathcal{H}$, then the desired
quantity $S[\mathcal{C}]$ could be defined by

$S[\mathcal{C}]=\frac{2\pi}{\kappa}\int_{\mathcal{C}}\left(\textbf{Q}[\xi]-\xi\cdot\textbf{B}_{\mathcal{H}}\right).$
(7)

However, it is not possible for $\underline{\bm{\theta}}$ to be of the form of
a total variation $\delta\textbf{B}_{\mathcal{H}}$ on $\mathcal{H}$ in general
because such a form would imply that the symplectic current flux through the
horizon would vanish identically (since the symplectic current is the
antisymmetrized second variation of $\bm{\theta}$), which is not the case.
Indeed, it was for this reason that [4] defined the entropy via (4) only on
the bifurcation surface $\mathcal{B}$.

A similar issue arises for the definition of ADM mass, $M$, at spatial
infinity with respect to an asymptotic time translation $t^{a}$. One wishes to
define $M$ so that it is the Hamiltonian conjugate to $t^{a}$, i.e., so that
it generates asymptotic time translations on phase space. This leads to the
requirement that under asymptotically flat perturbations of any asymptotically
flat background solution, we have [4]

$\delta M=\int_{\infty}\delta\textbf{Q}[t]-t\cdot\bm{\theta}(\phi,\delta\phi)$
(8)

where “$\infty$” denotes that the integral is taken over a sphere that
approaches spatial infinity. However, in this case, as one approaches spatial
infinity, the symplectic current goes to zero sufficiently rapidly that the
symplectic current flux vanishes. Thus, there is no obstruction to finding a
quantity $\textbf{B}_{\infty}$ such that to leading asymptotic order

$\bm{\theta}=\delta\textbf{B}_{\infty}$ (9)

and such a $\textbf{B}_{\infty}$ can be explicitly found [4]. One may then
define the ADM mass as

$M=\int_{\infty}\left(\textbf{Q}[t]-t\cdot\textbf{B}_{\infty}\right)$ (10)

and, thus, there is no difficulty in defining a quantity $M$ that satisfies
the desired relation (8).

The problem of defining dynamical black hole entropy is much more analogous to
the problem of defining the Bondi mass, $M_{\rm B}$, at null infinity
$\mathcal{I}^{+}$. Here, the symplectic current flux does not, in general,
vanish at null infinity, so an analog of (9) cannot hold in general.
Nevertheless, it was found in [7] that one could find another symplectic
potential $\bm{\theta}^{\prime}(\phi,\delta\phi)$ for the pullback of the
symplectic current to $\mathcal{I}^{+}$ with the property that
$\bm{\theta}^{\prime}=0$ in a stationary background $\phi$. The authors of [7]
then defined $\textbf{B}_{\mathcal{I}^{+}}$ by

$\underline{\bm{\theta}}-\bm{\theta}^{\prime}=\delta\textbf{B}_{\mathcal{I}^{+}}$
(11)

where $\underline{\bm{\theta}}$ denotes the pullback of $\bm{\theta}$ to
$\mathcal{I}^{+}$. They then defined the Bondi mass relative to an asymptotic
time translation $t^{a}$ on a cross-section, $\mathcal{C}$, of
${\mathcal{I}^{+}}$ by

$M_{\rm
B}[\mathcal{C}]=\int_{\mathcal{C}}\left(\textbf{Q}[t]-t\cdot\textbf{B}_{\mathcal{I}^{+}}\right).$
(12)

In this case, $M_{\rm B}$ does not satisfy (8) in general—since, in general,
there is no solution to this equation—but it does satisfy (8) for
perturbations off of a stationary background spacetime.

In this paper, we will apply the strategy of [7] to the definition of
dynamical black hole entropy. We will prove that for first order perturbations
of a stationary black hole, there exists a quantity $\textbf{B}_{\mathcal{H}}$
defined on the black hole horizon that satisfies111Note that this is
equivalent to the Wald–Zoupas requirement for the existence of a
$\bm{\theta}^{\prime}$ that satisfies (11) and vanishes for a stationary
background, since for a non-stationary background, one could define
$\bm{\theta}^{\prime}=\underline{\bm{\theta}}-\delta\textbf{B}_{\mathcal{H}}$.

$\underline{\bm{\theta}}~{}\hat{=}~{}\delta\textbf{B}_{\mathcal{H}}$ (13)

where $\underline{\bm{\theta}}$ denotes the pullback of $\bm{\theta}$ to
${\mathcal{H}}$ and “$~{}\hat{=}~{}$” denotes that the equality holds only on
the horizon $\mathcal{H}$. This enables us to define dynamical black hole
entropy via (7). For first order vacuum perturbations of stationary black
hole, it follows immediately from integration of (2) over a hypersurface that
extends from an arbitrary cross-section $\mathcal{C}$ of the horizon to
spatial infinity that

$\frac{\kappa}{2\pi}\delta S[\mathcal{C}]=\delta
M-\Omega_{{\mathcal{H}}}\delta J,$ (14)

i.e., the first law of black hole mechanics holds for an arbitrary cross-
section. However, it also follows that $\delta S[\mathcal{C}]=\delta
S[\mathcal{B}]$, i.e., for vacuum perturbations of a stationary black hole,
the entropy is “time independent” to first order and is equal to the entropy
of [4]. In order to obtain a nontrivial time variation of black hole entropy,
we must either work to second order in perturbation theory in the vacuum case
or allow an external stress-energy, $\delta T_{ab}$, to be present in the
first order perturbation. As will be discussed in subsection V.2, the results
in the vacuum case at second order are closely analogous to the first order
results with an external stress-energy, with the perturbed matter stress-
energy $\delta T_{ab}$ replaced by the second order “modified canonical
energy” of the gravitational field. For simplicity, we describe only the first
order results with an external stress-energy in the discussion below.

When an external stress-energy $\delta T_{ab}$ is present, the fundamental
identity (2) on the black hole horizon becomes

$d[\delta\textbf{Q}[\xi]-\xi\cdot\bm{\theta}(\phi,\delta\phi)]=-\xi^{a}\delta\bm{\textbf{C}}_{a}$
(15)

where, restoring the form indices, $\delta\bm{\textbf{C}}_{a}$ is given by

$\delta{{C}}_{aa_{1}\cdots a_{n-1}}=\delta T_{ae}{{\epsilon}^{e}}_{a_{1}\cdots
a_{n-1}}$ (16)

where ${\epsilon}_{a_{1}\cdots a_{n}}$ is the spacetime volume $n$-form. The
change in the dynamical entropy with “time” (i.e., its dependence on the
cross-section $\mathcal{C}$) can be seen by integrating (15) over the portion
of the horizon, $\mathcal{H}_{12}$, bounded by cross-sections
$\mathcal{C}_{1}$ and $\mathcal{C}_{2}$, where we take $\mathcal{C}_{2}$ to
lie to the future of $\mathcal{C}_{1}$. Using
$\underline{\bm{\theta}}=\delta\textbf{B}_{\mathcal{H}}$, we obtain

$\frac{\kappa}{2\pi}\left(\delta S[\mathcal{C}_{2}]-\delta
S[\mathcal{C}_{1}]\right)=\int_{\mathcal{H}_{12}}-\xi^{a}\delta\bm{\textbf{C}}_{a}=\int_{\mathcal{H}_{12}}\delta
T_{ab}\xi^{a}k^{b}\sqrt{h}dVd^{n-2}x$ (17)

where $k^{a}$ is the tangent to the affinely parametrized generators of the
horizon. Thus, we obtain

$\frac{\kappa}{2\pi}\Delta\delta S=\Delta\delta{E}$ (18)

where $\Delta\delta S$ is the first order entropy difference between “times”
$\mathcal{C}_{2}$ and $\mathcal{C}_{1}$, and $\Delta\delta{E}$ is the first
order matter energy flux (relative to the horizon Killing field $\xi^{a}$)
through the horizon between these “times.” Since $\kappa/2\pi$ is the
temperature of the black hole, this has exactly the form of the “physical
process” version of the first law of black hole mechanics [8]. Equation (18)
is valid for first order perturbations with matter sources off of a stationary
black hole in an arbitrary theory of gravity obtained from a diffeomorphism
covariant Lagrangian.

If the matter stress-energy satisfies the null energy condition, then $\delta
T_{ab}\xi^{a}k^{b}\geq 0$. It then follows immediately from (18) that our
definition of entropy satisfies the classical second law $\Delta\delta S\geq
0$ for first order perturbations. For source free first order perturbations,
we have $\Delta\delta S=0$, so the leading order change in entropy occurs at
second order. As previously mentioned, we will show in subsection V.2 that the
entropy is non-decreasing with time at second order if and only if the
modified canonical energy is non-negative at second order. This is the case in
general relativity but would not be expected to hold in more general theories
of gravity.

For the case of general relativity, the term
$-\xi\cdot\textbf{B}_{\mathcal{H}}$ appearing in (7) gives rise to the
“dynamical correction term” to the Bekenstein–Hawking entropy given in (1).
This correction term makes $S[\mathcal{C}]$ smaller than the
Bekenstein–Hawking entropy. It follows immediately from (18) that, to first
order, our entropy increases only when matter crosses the horizon. This
contrasts with the Bekenstein–Hawking entropy, which increases before matter
is thrown into the black hole since the event horizon is teleologically
defined and moves outward in anticipation of matter being thrown in at a later
time. We show in appendix A that, to first order, our entropy (1) is, in fact,
the area of the apparent horizon corresponding to “time” $\mathcal{C}$, and
thus can be determined locally, without knowledge of the future behavior of
the spacetime.

As previously mentioned, our formula for dynamical black hole entropy differs
from the Dong–Wall formula. However, we show in section VI that there is a
close relationship between our notion of entropy and the Dong–Wall entropy
$S_{\rm DW}$. If, for simplicity, we consider the entropy $S[V]$ evaluated on
a cross-section of constant affine time $V$, then we have

$S[V]=S_{\rm DW}[V]-V\frac{d}{dV}S_{\rm DW}[V].$ (19)

In particular, we have

$\frac{d}{dV}S=-\frac{d^{2}}{dV^{2}}S_{\rm DW}.$ (20)

As we shall see in section VII, the fact that the first derivative of our
entropy is related to the second derivative of the Dong–Wall entropy will play
an important role in the analysis of the generalized second law and its
relationship to the quantum null energy condition (QNEC). We will show that
with our notion of entropy, the generalized second law is equivalent to QNEC,
but with a dynamical correction to the von Neumann entropy. On the other hand,
with the Dong–Wall notion of entropy, the generalized second law is equivalent
to an integrated version of QNEC, using the unmodified notion of von Neumann
entropy of matter.

In section II, we give the necessary background material on the Lagrangian
formalism that we will be using throughout the paper and we also define our
tetrad choice and specify our gauge conditions on the horizon. In section III,
we prove that for perturbations of a stationary black hole whose event horizon
is a bifurcate Killing horizon, there exists a quantity
$\textbf{B}_{\mathcal{H}}$ such that
$\underline{\bm{\theta}}~{}\hat{=}~{}\delta\textbf{B}_{\mathcal{H}}$. This
enables us to give a general definition of dynamical black hole entropy and
elucidate its properties in section IV. We derive the physical process version
of the first law of black hole mechanics in section V for both external matter
perturbations and vacuum perturbations and discuss its implications for the
second law. The relationship of our notion of entropy with the Dong–Wall
entropy is analyzed in section VI. The generalized second law and its
relationship to QNEC is analyzed in section VII for both our notion of entropy
and the Dong–Wall notion.

We will generally follow the conventions of [9]. As we have already done
above, we will use boldface to denote differential forms when the form indices
are not explicitly written. Pullbacks of differential forms to the event
horizon $\mathcal{H}$ will be denoted by underlining them as in (13) above.
The covariant spacetime volume element is written as $\epsilon_{a_{1}\dots
a_{n}}$, and our orientation conventions for the event horizon and horizon
cross-sections are given by (47) and (48) below. We will use the notation
“$~{}\hat{=}~{}$” to denote that an equality that holds only when both sides
are restricted (but not necessarily pulled back) to $\mathcal{H}$, although we
will often simply use “$=$” for equations involving quantities that are only
defined on $\mathcal{H}$, where no confusion can arise. We also set
$\hbar=1=G$.

##  II Lagrangian Formalism, Killing Horizons, Tetrad Choice, and Gauge
Conditions

In this section, we review some basic definitions and constructions in the
Lagrangian formalism, we specify a tetrad choice on the horizon, and we impose
some gauge conditions on the horizon.

###  II.1  Lagrangians, Noether Charge, and the Fundamental Identity

We consider an arbitrary diffeomorphism covariant theory of gravity in
$n$-dimensions derived from a Lagrangian $n$-form L of the form

$\displaystyle\textbf{L}={}\textbf{L}(g_{ab},R_{abcd},\nabla_{a_{1}}R_{bcde},\dots,\nabla_{(a_{1}}\cdots\nabla_{a_{m})}R_{bcde},\psi,\dots,\nabla_{(b_{1}}\cdots\nabla_{a_{p})}\psi)$
(21)

where $g_{ab}$ is the spacetime metric, $\nabla$ denotes the derivative
operator associated with $g_{ab}$, $R_{abcd}$ denotes the curvature of
$g_{ab}$, and $\psi$ denotes any matter fields included in the Lagrangian. We
shall assume that $\psi$ is a tensor field or fields; spinor fields can be
treated via the approach developed in [10]. We collectively refer to the
dynamical fields as $\phi=(g_{ab},\psi)$. It was shown in [4] that any
diffeomorphism covariant Lagrangian can be put in the form (21). Beginning in
section III, we will restrict consideration to the case where matter fields,
$\psi$, are not present in the Lagrangian, but we allow their presence for
now.

The first variation of the Lagrangian can always be expressed in the form

$\displaystyle\delta\textbf{L}={}$
$\displaystyle\textbf{E}\delta\phi+d\bm{\theta}$ (22)

where E is locally constructed out of $\phi$ and its derivatives and
$\bm{\theta}(\phi,\delta\phi)$ is locally constructed out of $\phi$,
$\delta\phi$, and their derivatives and is linear in $\delta\phi$. In (22),
contraction of the tensor indices of $\phi$ with the corresponding dual tensor
indices of E is understood. The Euler–Lagrange equations of motion obtained
from L are $\textbf{E}=0$. The $(n-1)$-form $\bm{\theta}$ plays the role of a
symplectic potential [4] in that the symplectic current $(n-1)$-form is
obtained from it via222Here we assume that the field variations
$\delta_{1}\phi$ and $\delta_{2}\phi$ arise from a two-parameter variation
$\phi(\lambda_{1},\lambda_{2})$ and thus commute. Note that (23) uses the
opposite sign convention of [4], but the same sign convention as [11].

$\displaystyle\bm{\omega}(\phi,\delta_{1}\phi,\delta_{2}\phi)={}$
$\displaystyle\delta_{1}\bm{\theta}(\phi,\delta_{2}\phi)-\delta_{2}\bm{\theta}(\phi,\delta_{1}\phi).$
(23)

The symplectic form $W$ is obtained by integrating the symplectic current over
a Cauchy surface, $\Sigma$, of the spacetime

$\displaystyle W_{\Sigma}(\phi,\delta_{1}\phi,\delta_{2}\phi)={}$
$\displaystyle\int_{\Sigma}\bm{\omega}(\phi,\delta_{1}\phi,\delta_{2}\phi)\,.$
(24)

Let $\chi^{a}$ be an arbitrary vector field on the spacetime $\mathcal{M}$.
Since $\chi^{a}$ is the infinitesimal generator of a diffeomorphism, i.e., a
local symmetry of L, there is an associated Noether current $(n-1)$-form J
defined by

$\displaystyle\textbf{J}={}$
$\displaystyle\bm{\theta}(\phi,\mathscr{L}_{\chi}\phi)-\chi\cdot\textbf{L}.$
(25)

where $\mathscr{L}_{\chi}$ denotes the Lie derivative with respect to
$\chi^{a}$ and we use the notation “$\cdot$” to denote the contraction of a
vector field into the first index of a differential form. This definition of J
holds for any field configuration $\phi$, i.e., $\phi$ need not be a solution
to the field equations, $\textbf{E}=0$. It can be shown that the Noether
current can be written in the form [12]

$\displaystyle\textbf{J}={}$ $\displaystyle
d\textbf{Q}[\chi]+\chi^{a}\bm{\textbf{C}}_{a}\,.$ (26)

Here, the $(n-2)$-form Q is referred to as the “Noether charge” and the dual
vector valued $(n-1)$-form $\bm{\textbf{C}}_{a}$ vanishes when the equations
of motion are satisfied and is referred to as the “constraints” of the theory
associated with diffeomorphism invariance. The Noether charge takes the
general form [4]

$\displaystyle\textbf{Q}[\chi]={}$
$\displaystyle\textbf{W}_{c}(\phi)\chi^{c}+\textbf{X}^{cd}(\phi)\nabla_{[c}\chi_{d]}+\textbf{Y}(\phi,\mathscr{L}_{\chi}\phi)+d\textbf{Z}(\phi,\chi)$
(27)

where $\textbf{W}_{c}$, $\textbf{X}^{cd}$, Y, and Z are covariant quantities,
with $\textbf{W}_{c}$ and $\textbf{X}^{cd}$ locally constructed from $\phi$,
and Y and Z locally constructed from the indicated fields, with Y linear in
$\mathscr{L}_{\chi}\phi$ and Z linear in $\chi$. For the case where the matter
field $\psi$ is a single tensor field $\tensor{A}{{}^{a_{1}\cdots
a_{k}}_{b_{1}\cdots b_{l}}}$ of type $(k,l)$, the constraints take the
explicit form [13]

$\displaystyle{{C}}_{aa_{1}\cdots a_{n-1}}$ $\displaystyle={}$
$\displaystyle{\epsilon}_{ca_{1}\cdots
a_{n-1}}\left[2\tensor{({E}_{G})}{{}^{c}_{a}}-\sum\tensor{A}{{}^{{d_{1}}\cdots{d_{k}}}_{b_{1}\cdots
a\cdots b_{l}}}\tensor{({E}_{M})}{{}_{d_{1}\cdots d_{k}}^{b_{1}\cdots c\cdots
b_{l}}}+\sum\tensor{A}{{}^{d_{1}\cdots c\cdots d_{k}}_{b_{1}\cdots
b_{l}}}\tensor{({E}_{M})}{{}_{d_{1}\cdots a\cdots d_{s}}^{b_{1}\cdots
b_{l}}}\right],$ (28)

where ${\epsilon}_{a_{1}\cdots a_{n}}$ is the spacetime volume form,
$({E}_{G})_{ab}=0$ are the equations of motion for $g^{ab}$ (obtained by
varying the Lagrangian with respect to $g^{ab}$) and
$\tensor{({E}_{M})}{{}_{a_{1}\cdots a_{k}}^{b_{1}\cdots b_{l}}}$ are the
equations of motion for the matter field $\tensor{A}{{}^{a_{1}\cdots
a_{k}}_{b_{1}\cdots b_{l}}}$. The summations in (28) run over the possible
index positions of $a$ and $c$. The generalization of (28) to more than one
matter field is straightforward.

Taking the first variation of (25) (taking the vector field $\chi^{a}$ to be
fixed) and using (22) and (23), we obtain [4]

$\displaystyle\delta\textbf{J}={}$
$\displaystyle-\chi\cdot(\textbf{E}\delta\phi)+\bm{\omega}(\phi;\delta\phi,\mathscr{L}_{\chi}\phi)+d[\chi\cdot\bm{\theta}(\phi,\delta\phi)],$
(29)

Equating this to the first variation of (26), we obtain

$\displaystyle\bm{\omega}(\phi,\delta\phi,\mathscr{L}_{\chi}\phi)={}$
$\displaystyle\chi\cdot(\textbf{E}\delta\phi)+\chi^{a}\delta\bm{\textbf{C}}_{a}+d[\delta\textbf{Q}-\chi\cdot\bm{\theta}(\phi,\delta\phi)].$
(30)

Following Hollands and Wald [11], we shall refer to this equation as the
fundamental identity. The fundamental identity holds for arbitrary $\phi$ and
$\delta\phi$, i.e., they need not satisfy the equations of motion or
linearized equations of motion. For the case where $\chi^{a}$ is a Killing
field of the background $\phi$ so that $\mathscr{L}_{\chi}\phi=0$, a second
variation of (30) yields

$\displaystyle\begin{split}\bm{\omega}(\phi,\delta\phi,\mathscr{L}_{\chi}\delta\phi)={}&\chi\cdot(\delta\textbf{E}\delta\phi)+\chi\cdot(\textbf{E}\delta^{2}\phi)+\chi^{a}\delta^{2}\bm{\textbf{C}}_{a}+d[\delta^{2}\textbf{Q}-\chi\cdot\delta\bm{\theta}(\phi,\delta\phi)].\end{split}$
(31)

###  II.2 Tetrad Choice on a Bifurcate Killing Horizon

We are interested in this paper in first and second order perturbations of
spacetimes that contain a black hole whose event horizon is a bifurcate
Killing horizon. By definition, a Killing horizon $\mathcal{H}$ is a null
surface to which a Killing field $\xi^{a}$ is normal. Thus, $\xi^{a}$ is
tangent to the null geodesic generators of $\mathcal{H}$. The surface gravity,
$\kappa$, of $\mathcal{H}$ is defined by

$\xi^{b}\nabla_{b}\xi^{a}=\kappa\xi^{a}$ (32)

so $\kappa$ measures the failure of Killing parametrization of the null
generators to be an affine parametrization. It is necessary for $\kappa$ to be
constant in order that there not be a parallelly propagated curvature
singularity on $\mathcal{H}$ [14]. We shall consider here only the case where
$\kappa$ is constant and $\kappa\neq 0$, in which case $\mathcal{H}$
corresponds to a bifurcate Killing horizon [15].

We wish to use a tetrad basis on $\mathcal{H}$ to the future of the
bifurcation surface $\mathcal{B}$ that is Lie transported by the Killing
field. One way of doing this would be to introduce Gaussian null coordinates,
using the Killing parameter $v$ as one of the coordinates (see, e.g., [16]).
However, the introduction of Gaussian null coordinates requires an arbitrary
choice of a cross-section, $\mathcal{C}$, of $\mathcal{H}$, and this would
raise questions in our subsequent analysis of the extent to which our
constructions depend upon our choice of $\mathcal{C}$. Instead, we will
introduce a tetrad that is constructed only from $\xi^{a}$—since our
constructions will, in any case, depend upon $\xi^{a}$.

We do so by introducing a vector field $N^{a}$ on $\mathcal{H}$ that satisfies

$\displaystyle\nabla_{a}\xi_{b}$ $\displaystyle~{}\hat{=}~{}$ $\displaystyle
2\kappa N_{[a}\xi_{b]}$ (33) $\displaystyle N^{a}N_{a}$
$\displaystyle~{}\hat{=}~{}$ $\displaystyle 0$ (34)

where $~{}\hat{=}~{}$ denotes equality on $\mathcal{H}$. That there exists a
unique $N^{a}$ satisfying these conditions can be seen as follows: From the
hypersurface orthogonality of $\xi^{a}$ on $\mathcal{H}$ and Frobenius’s
theorem, there exists a vector field $w^{a}$ on $\mathcal{H}$ such that

$\nabla_{[a}\xi_{b]}~{}\hat{=}~{}2w_{[a}\xi_{b]}$ (35)

Since $\xi^{a}$ is a Killing field, we have
$\nabla_{[a}\xi_{b]}=\nabla_{a}\xi_{b}$ and it follows from (32) that

$\xi^{a}w_{a}~{}\hat{=}~{}\kappa$ (36)

so, in particular, $w_{a}\neq 0$. We define

$N^{a}~{}\hat{=}~{}(\xi^{b}w_{b})^{-1}w^{a}+c\xi^{a}=\frac{1}{\kappa}w^{a}+c\xi^{a}$
(37)

with

$c=\frac{1}{2(\xi^{b}w_{b})^{2}}w^{a}w_{a}=\frac{1}{2\kappa^{2}}w^{a}w_{a}$
(38)

Then it is easily seen that $N^{a}$ satisfies (33) and (34). Uniqueness
follows from the fact that (33) uniquely determines $N^{a}$ up to addition of
a multiple of $\xi^{a}$ at each point and (34) then fixes that multiple. Note
that if we contract (33) with $\xi^{a}$ and use (32), we find that
$N^{a}\xi_{a}~{}\hat{=}~{}1$. Thus, $N^{a}$ is a past-directed null vector.
Note also that $N^{a}$ must be invariant under the action of the isometries
generated by $\xi^{a}$, since this action preserves (33) and (34) but $N^{a}$
is unique. Thus, we have $\mathscr{L}_{\xi}N^{a}~{}\hat{=}~{}0$.

It is useful to complete $\xi^{a}$ and $N^{a}$ to a basis of the tangent space
on $\mathcal{H}$ by introducing vectors $s_{i}^{a}$ ($i=1,\cdots,n-2$) on
$\mathcal{H}$ such that $s_{i}^{a}\xi_{a}=0=s_{i}^{a}N_{a}$,
$s_{i}^{a}s_{ja}=\delta_{ij}$ and
$\mathscr{L}_{\xi}s_{i}^{a}\Big{|}_{\mathcal{H}}=0$. This can be done by
choosing $s_{i}^{a}$ on a cross-section333Note that there is no reason why
there need exist a cross-section to which $N^{a}$ is orthogonal, so, unlike a
Gaussian null coordinate basis, $s_{i}^{a}$ is not assumed to be tangent to
the cross-section. It may not be possible to make global choices of
$s_{i}^{a}$ on a cross-section, in which case our construction would have to
be done in patches. Different choices of $s_{i}^{a}$ will merely differ by
rotations in the $(n-2)$-plane orthogonal to $\xi^{a}$ and $N^{a}$, and it
will be manifest that our results below will not depend upon the choice of
$s_{i}^{a}$. so as to satisfy these properties and then Lie-transporting
$s_{i}^{a}$ along $\xi^{a}$.

It is important to understand the behavior of our tetrad
$\\{\xi^{a},N^{a},s_{i}^{a}\\}$ as one approaches the bifurcation surface
$\mathcal{B}$. This is most conveniently analyzed by introducing Gaussian null
coordinates based on a choice of affine parameter $V$ of the null generators
of $\mathcal{H}$ (see, e.g., [17])—with $V=0$ corresponding to the bifurcation
surface $\mathcal{B}$—since such coordinates are smooth at $\mathcal{B}$.
Since we require $V=0$ at $\mathcal{B}$, the choice of $V$ is unique up to
$V\to fV$, where $f$ may vary from generator to generator but is constant on
each generator. The results below hold for any choice of $V$. By (32), the
relationship between Killing parameter $v$ and affine parameter $V$ is

$V=e^{\kappa v}$ (39)

where the (generator dependent) scaling freedom of $V$ corresponds to the
(generator dependent) freedom in the choice of origin of $v$. Thus, we have

$\xi^{a}=\kappa e^{\kappa v}k^{a}=\kappa Vk^{a}$ (40)

where $k^{a}$ is tangent to the affinely parametrized geodesics and thus is
smooth and nonvanishing on $\mathcal{B}$. Thus, we see that ${\xi^{a}=O(V)}$
at $V=0$. Similarly, $N^{a}=O(V^{-1})$ at $V=0$, i.e., the components of
$N^{a}$ in Gaussian null coordinates blow up as $1/V$ as $V\to 0$. Finally,
the tetrad vectors $s_{i}^{a}$ smoothly extend to $\mathcal{B}$. To see this,
we note that

$\mathscr{L}_{k}s_{i}^{a}=\frac{1}{\kappa}e^{-\kappa
v}\mathscr{L}_{\xi}s_{i}^{a}+e^{-\kappa v}\xi^{a}s_{i}^{b}\nabla_{b}v=\kappa
k^{a}s_{i}^{b}\nabla_{b}v$ (41)

Since ${\mathscr{L}}_{\xi}[\kappa s_{i}^{b}\nabla_{b}v]=0$, the right side is
smooth, which implies that $s_{i}^{a}$ smoothly extends to $\mathcal{B}$.

The following is an important consequence of the results of the previous
paragraph: Suppose that in a spacetime with a bifurcate Killing horizon we
have a tensor field ${S^{a_{1}\cdots a_{k}}}_{b_{1}\cdots b_{l}}$ that is Lie
derived by $\xi^{a}$ and is smooth on ${\mathcal{H}}\setminus{\mathcal{B}}$.
We can expand ${S^{a_{1}\cdots a_{k}}}_{b_{1}\cdots b_{l}}$ on
${\mathcal{H}}\setminus{\mathcal{B}}$ in the basis
$\\{\xi^{a},N^{a},s_{i}^{a}\\}$. Since both ${S^{a_{1}\cdots
a_{k}}}_{b_{1}\cdots b_{l}}$ and the basis elements are Lie derived by
$\xi^{a}$, the coefficients appearing in the basis expansion must also be Lie
derived by $\xi^{a}$. It follows immediately that if any nonvanishing term in
the basis expansion has strictly more $N^{a}$’s than $\xi^{a}$’s, then
${S^{a_{1}\cdots a_{k}}}_{b_{1}\cdots b_{l}}$ cannot be extended to
$\mathcal{B}$. Indeed, in smooth coordinates covering $\mathcal{B}$, some
components of ${S^{a_{1}\cdots a_{k}}}_{b_{1}\cdots b_{l}}$ will blow up as
$1/V^{p}$, where $p$ is the maximum of the number of $N^{a}$’s minus the
number of $\xi^{a}$’s appearing in any individual nonvanishing term in the
basis expansion. On the other hand, if all of the nonvanishing terms in the
basis expansion of ${S^{a_{1}\cdots a_{k}}}_{b_{1}\cdots b_{l}}$ have at least
as many $\xi^{a}$’s as $N^{a}$’s, then ${S^{a_{1}\cdots a_{k}}}_{b_{1}\cdots
b_{l}}$ can be smoothly extended to $\mathcal{B}$.

###  II.3 Gauge Conditions on Perturbations

In the remainder of this paper, we will be concerned with perturbations,
$\delta g_{ab}$, of a black hole with a bifurcate Killing horizon. Without
loss of generality, we will assume that the true event horizon of the
perturbed spacetime coincides with $\mathcal{H}$. We take $\xi^{a}$ to be
fixed under the variation, i.e., we take $\delta\xi^{a}=0$. We will impose the
following two gauge conditions on $\delta g_{ab}$ at the event horizon:

$\displaystyle\xi^{a}\delta g_{ab}$ $\displaystyle~{}\hat{=}~{}$
$\displaystyle 0$ (42) $\displaystyle\nabla_{a}(\xi^{b}\xi^{c}\delta g_{bc})$
$\displaystyle~{}\hat{=}~{}$ $\displaystyle 0.$ (43)

Both of the above gauge conditions can be imposed without any loss of
generality. This is most easily seen from the fact that they automatically
hold for any perturbation in Gaussian null coordinates based on the Killing
parameter $v$ (see (145) below). However, we will use only the above two gauge
conditions in our analysis below, e.g., we need not assume that the full
Gaussian null gauge conditions are imposed.

Equation (42) states that $\xi^{a}$ remains the null normal to $\mathcal{H}$
under perturbations, since the condition implies that $\xi^{a}$ remains null
and orthogonal to any vector tangent to $\mathcal{H}$. Equation (43) implies
that

$\delta\left(\xi^{b}\nabla_{b}\xi^{a}\right)=\xi^{b}\delta{\Gamma^{a}}_{bc}\xi^{c}~{}\hat{=}~{}-\frac{1}{2}\xi^{b}\xi^{c}\nabla^{a}\delta
g_{bc}~{}\hat{=}~{}-\frac{1}{2}\nabla^{a}\left(\xi^{b}\xi^{c}\delta
g_{bc}\right)~{}\hat{=}~{}0$ (44)

where (42) was used to get the second and third equalities. Comparing with
(32), we see that the second condition is equivalent to

$\delta\kappa=0$ (45)

i.e., the surface gravity of $\xi^{a}$ on $\mathcal{H}$ does not change under
the perturbation.

If matter fields $\psi$ are present in the Lagrangian, then additional
conditions on their perturbations, $\delta\psi$, may have to be imposed in
order for the results of the next section to hold. In particular, if an
electromagnetic field is present, we must impose the condition $\xi^{a}\delta
A_{a}~{}\hat{=}~{}0$, which can always be achieved without loss of generality
by an electromagnetic gauge transformation. However, we have not investigated
whether, in all cases, these additional conditions on matter fields are always
gauge conditions (as opposed to physical restrictions on the perturbation). In
order not to deal further with this issue, for the remainder of this paper, we
shall restrict consideration to the case where the metric is the only
dynamical field in the Lagrangian, i.e., no additional dynamical fields $\psi$
are present. However, it should be possible to extend the analysis to include
additional dynamical fields,444 Very recently, Wall and Yan [18] have extended
the definition of the Dong–Wall entropy to Proca fields without imposing
additional restrictions on the Proca field. provided that, when necessary,
suitable conditions analogous to (42) and (43) are imposed on their
perturbations.

##  III The Pullback of the Symplectic Potential is a Total Variation for
Perturbations of a Stationary Black Hole

In this section, we consider perturbations of a stationary black hole whose
event horizon is a bifurcate Killing horizon. We assume that the perturbations
are everywhere smooth and satisfy the gauge conditions (42) and (43). However,
we need not assume that the background spacetime metric $g_{ab}$, is a
solution to the field equations, nor do we need to assume that the perturbed
metric, $\delta g_{ab}$ is a solution to the linearized field equations. We
will prove that there exists a quantity $\textbf{B}_{\mathcal{H}}$ defined on
the black hole horizon that is locally and covariantly constructed from the
metric and555The vector field $N^{a}$ on $\mathcal{H}$ will also appear in our
expression for $\textbf{B}_{\mathcal{H}}$. However, $N^{a}$ on $\mathcal{H}$
is locally and covariantly determined from the metric and $\xi^{a}$ via (33)
and (34). $\xi^{a}$ such that the pullback, $\underline{\bm{\theta}}$, to
$\mathcal{H}$ of the symplectic potential $\bm{\theta}(g_{ab},\delta g_{ab})$
is given by

$\underline{\bm{\theta}}~{}\hat{=}~{}\delta\textbf{B}_{\mathcal{H}}.$ (46)

As explained in the Introduction, (46) will enable us to define dynamical
black hole entropy, and we shall then do so in the next section.

The horizon $\mathcal{H}$ is an $(n-1)$-dimensional null surface. Since the
pullback of the spacetime metric to $\mathcal{H}$ is degenerate, it does not
induce a unique volume form on $\mathcal{H}$. However, we can define a volume
form $\bm{\epsilon}^{(n-1)}_{a_{1}\cdots a_{n-1}}$ on $\mathcal{H}$ that is
Lie derived by $\xi^{a}$ in the background spacetime by

$-n\xi_{[a_{1}}{\epsilon}^{(n-1)}_{a_{2}\cdots
a_{n}]}={\epsilon}_{a_{1}a_{2}\cdots a_{n}}$ (47)

where ${\epsilon}_{a_{1}a_{2}\cdots a_{n}}$ is the spacetime volume form. It
is also convenient to define an $(n-2)$-form ${\epsilon}^{(n-2)}_{a_{1}\cdots
a_{n-2}}$ on $\mathcal{H}$ by

${\epsilon}^{(n-2)}_{a_{1}\cdots
a_{n-2}}=\xi^{c}{\epsilon}^{(n-1)}_{ca_{1}\cdots a_{n-2}}$ (48)

The pullback of $\bm{\epsilon}^{(n-2)}$ to any cross-section, $\mathcal{C}$,
of $\mathcal{H}$ will then agree with the volume element on $\mathcal{C}$
obtained from the nondegenerate metric on $\mathcal{C}$ obtained from the
pullback of the spacetime metric.

The symplectic potential $\bm{\theta}$ is an $(n-1)$-form on spacetime, so its
pullback $\underline{\bm{\theta}}$ to $\mathcal{H}$ must be proportional to
${\epsilon}^{(n-1)}_{a_{1}\cdots a_{n-1}}$. It follows from (47) that we have

$\underline{\bm{\theta}}~{}\hat{=}~{}\alpha^{a}\xi_{a}\bm{\epsilon}^{(n-1)}$
(49)

where we have defined

$\alpha^{a}\equiv\frac{1}{(n-1)!}{\theta}_{a_{1}\cdots
a_{n-1}}{\epsilon}^{aa_{1}\cdots a_{n-1}}.$ (50)

Our results on the form of $\underline{\bm{\theta}}$ will be a direct
consequence of the following theorem.

######  Theorem 1.

Let $(\mathcal{M},g_{ab})$ be an n-dimensional spacetime with a bifurcate
Killing horizon $\mathcal{H}$ with horizon Killing field $\xi^{a}$. Let
$\delta g_{ab}$ be an arbitrary smooth perturbation of $g_{ab}$ satisfying the
gauge conditions (42) and (43). Let $\alpha^{a}$ be a vector field on
$\mathcal{M}$ of the form

$\displaystyle\alpha^{a}={}$ $\displaystyle\sum_{i=0}^{k}T_{(i)}^{ab_{1}\cdots
b_{i}cd}\nabla_{(b_{1}}\cdots\nabla_{b_{i})}\delta g_{cd},$ (51)

where the the tensors $T_{(i)}^{ab_{1}\cdots b_{i}cd}=T_{(i)}^{a(b_{1}\cdots
b_{i})(cd)}$ are smooth and are locally and covariantly constructed from the
metric, curvature, and covariant derivatives of the curvature. Then on the
horizon, the scalar function $\alpha_{c}\xi^{c}$ takes the form

$\displaystyle\alpha_{c}\xi^{c}~{}\hat{=}~{}{}$
$\displaystyle\sum_{i=0}^{k-1}\tilde{T}_{(i)}^{b_{1}\cdots
b_{i}cd}\nabla_{(b_{1}}\cdots\nabla_{b_{i})}\mathscr{L}_{\xi}\delta g_{cd}$
(52)

where the tensors $\tilde{T}_{(i)}^{b_{1}\cdots b_{i}cd}$ are smooth on
$\mathcal{H}$ and are locally and covariantly constructed from the metric,
curvature, and its derivatives as well as from $\xi^{a}$ and $N^{a}$ (see (33)
and (34)), with $\xi^{a}$ and $N^{a}$ appearing only algebraically.

###### Proof.

To begin, we focus on the tensor field $T_{(k)}^{ab_{1}\cdots b_{i}cd}$ on
$\mathcal{H}$ appearing in the highest derivative term in the expansion (51)
of $\alpha^{a}$. Our aim is to show that $\xi_{a}T_{(k)}^{ab_{1}\cdots
b_{i}cd}\nabla_{(b_{1}}\cdots\nabla_{b_{k})}\delta g_{cd}$ can be written in
the form $\tilde{T}_{(k-1)}^{b_{1}\cdots
b_{k-1}cd}\nabla_{(b_{1}}\cdots\nabla_{b_{k-1})}{\mathscr{L}}_{\xi}\delta
g_{cd}$ plus lower derivative terms of essentially the same character as
appeared in the original expression for $\xi_{a}\alpha^{a}$. This will enable
us to make an inductive argument to prove the theorem.

By the hypothesis of the theorem, the tensor field $T_{(k)}^{ab_{1}\cdots
b_{i}cd}$ is smooth on $\mathcal{H}$, including at the bifurcation surface
$\mathcal{B}$. Since $\xi^{a}$ is a Killing field of the background spacetime,
$T_{(k)}^{ab_{1}\cdots b_{k}cd}$ is Lie derived by $\xi^{a}$. Therefore, as
shown in the last paragraph of subsection II.2, if we expand it in the basis
$\\{\xi^{a},N^{a},s_{i}^{a}\\}$, the only nonvanishing terms in the basis
expansion will have at least as many factors of $\xi^{a}$ as $N^{a}$. If we
then contract $\alpha^{a}$ with $\xi^{a}$, we will eliminate one factor of
$N^{a}$ from each term in the basis expansion, so each nonvanishing term will
have at least one more factor of $\xi^{a}$ than $N^{a}$. More explicitly, the
basis expansion of $\xi_{a}T_{(k)}^{ab_{1}\cdots b_{i}cd}$ takes the form

$\displaystyle\begin{split}\xi_{a}T_{(k)}^{ab_{1}\cdots
b_{k}cd}~{}\hat{=}~{}{}&c_{1}\xi^{b_{1}}\xi^{b_{2}}\cdots\xi^{b_{k}}\xi^{c}\xi^{d}\\\
{}&+c_{2}N^{(b_{1}}\xi^{b_{2}}\cdots\xi^{b_{k})}\xi^{c}\xi^{d}+c_{3}\xi^{b_{1}}\xi^{b_{2}}\cdots\xi^{b_{k}}N^{(c}\xi^{d)}\\\
{}&+c_{4}s_{i_{1}}^{(b_{1}}\xi^{b_{2}}\cdots\xi^{b_{k})}\xi^{c}\xi^{d}+c_{5}\xi^{b_{1}}\xi^{b_{2}}\cdots\xi^{b_{k}}s_{i}^{(c}\xi^{d)}\\\
{}&+\cdots\\\ {}&+c_{6}s_{i_{1}}^{(b_{1}}s_{i_{2}}^{b_{2}}\cdots
s_{i_{{k-1}}}^{b_{k-1}}\xi^{b_{k})}s_{j_{1}}^{(c}s_{j_{2}}^{d)}+c_{7}s_{i_{1}}^{b_{1}}s_{i_{2}}^{b_{2}}\cdots
s_{i_{k}}^{b_{k}}\xi^{(c}s_{j}^{d)},\end{split}$ (53)

where $c_{1},c_{2},\dots,c_{7}$ are Lie derived by $\xi^{a}$. Here, the first
term in the basis expansion is the term with all $\xi^{a}$’s. In the second
line, we replace one of the $\xi^{a}$’s with an $N^{a}$. Since
$T_{(k)}^{ab_{1}\cdots b_{i}cd}$ is symmetric in $b_{1}\dots b_{k}$ as well as
in $cd$, there are two distinct terms that we can get via such a replacement,
namely, one term where the $N^{a}$ replaces a $\xi^{a}$ on a $b_{i}$ index and
one term where it replaces a $\xi^{a}$ on a $c$ or $d$ index. In the third
line, we similarly replace a $\xi^{a}$ by an $s_{i}^{a}$. (Since $i$ ranges
from $1$ to $n-2$, there are $n-2$ distinct expressions of the form appearing
in the third line.) The “$\dots$” in the fourth line represents all of the
terms we get by continuing to replace $\xi^{a}$’s by $N^{a}$’s and
$s_{i}^{a}$’s, thereby continually decreasing the number of factors of
$\xi^{a}$. The key point is that each term must have at least one more
$\xi^{a}$ than $N^{a}$. Thus, the final two terms in the basis expansion—shown
on the fifth line—have $(k+1)$ factors of $s_{i}^{a}$ and one factor of
$\xi^{a}$.

We now consider each term in the basis expansion separately. Since at least
one $\xi^{a}$ appears in each term in the basis expansion, there are three
possible cases, which we will consider separately:

  * •

Case (1): There are no $\xi^{b_{i}}$ factors, i.e., there are no factors of
$\xi^{a}$ that contract into $\nabla_{(b_{1}}\cdots\nabla_{b_{k})}$ in (51).
Furthermore, there is only one $\xi^{c}$ or $\xi^{d}$ factor, i.e., only one
index of $\delta g_{cd}$ contracts into factor of $\xi^{a}$.

  * •

Case (2): As in case (1), there are no $\xi^{b_{i}}$ factors. However, there
are $\xi^{c}$ and $\xi^{d}$ factors, i.e., both indices of $\delta g_{cd}$
contract into factors of $\xi^{a}$.

  * •

Case (3): There is at least one $\xi^{b_{i}}$ factor, i.e., there is at least
one contraction of $\xi^{a}$ with a derivative index in (51).

Case (1) is the most straightforward to analyze. Since only one factor of
$\xi^{a}$ occurs in this case, the only terms in the basis expansion (53) that
meet the criterion of case (1) take the form of the very last term in (53).
Thus, to analyze this case, we need to consider the quantity

$s_{i_{1}}^{b_{1}}s_{i_{2}}^{b_{2}}\cdots
s_{i_{k}}^{b_{k}}\xi^{c}s_{j}^{d}\nabla_{(b_{1}}\cdots\nabla_{b_{k})}\delta
g_{cd}$ (54)

where we dropped the symmetrization over $cd$ in the basis term since $\delta
g_{cd}$ is symmetric in $c$ and $d$. We now move $\xi^{c}$, through all of the
derivatives $\nabla_{(b_{1}}\cdots\nabla_{b_{k})}$ to write it as

$s_{i_{1}}^{b_{1}}s_{i_{2}}^{b_{2}}\cdots
s_{i_{k}}^{b_{k}}\xi^{c}s_{j}^{d}\nabla_{(b_{1}}\cdots\nabla_{b_{k})}\delta
g_{cd}=s_{i_{1}}^{b_{1}}s_{i_{2}}^{b_{2}}\cdots
s_{i_{k}}^{b_{k}}s_{j}^{d}\nabla_{(b_{1}}\cdots\nabla_{b_{k})}(\delta
g_{cd}\xi^{c})+{\rm lower\,derivative\,terms}$ (55)

where the “lower derivative terms” have one or more derivatives
$\nabla_{b_{i}}$ acting only on $\xi^{c}$ and thus have no more than $k-1$
derivatives remaining to act on $\delta g_{cd}$. However, by our gauge
condition (42), we have $\delta g_{cd}\xi^{c}~{}\hat{=}~{}0$. Since
derivatives of this quantity in directions tangential to the horizon must also
vanish, we see that the terms occurring in case (1) are equivalent to an
expression with strictly fewer than $k$ derivatives acting on $\delta g_{cd}$.

The analysis of case (2) is similar to case (1). We move both $\xi^{c}$ and
$\xi^{d}$ through $\nabla_{(b_{1}}\cdots\nabla_{b_{k})}$ to obtain a term
where all $k$ derivatives act on $\delta g_{cd}\xi^{c}\xi^{d}$ plus “lower
derivative terms,” i.e., terms where at least one derivative acts on $\xi^{c}$
or $\xi^{d}$ and thus no more than $k-1$ derivatives act on $\delta g_{cd}$.
However, the term $\nabla_{(b_{1}}\cdots\nabla_{b_{k})}(\delta
g_{cd}\xi^{c}\xi^{d})$ can have at most one factor $N^{b_{i}}$ contracting
into it, since there are precisely two factors of $\xi^{a}$ in the terms that
arise in case (2) and we must have strictly more factors of $\xi^{a}$ than
$N^{a}$. Therefore, in the term $\nabla_{(b_{1}}\cdots\nabla_{b_{k})}(\delta
g_{cd}\xi^{c}\xi^{d})$, there is at most one derivative of $\delta
g_{cd}\xi^{c}\xi^{d}$ that is taken in a direction that is not tangential to
the horizon. Consequently, this term vanishes by our gauge condition
$\nabla_{b}(\delta g_{cd}\xi^{c}\xi^{d})~{}\hat{=}~{}0$. Therefore, we are
left only with the “lower derivative terms,” i.e., terms with $k-1$ or fewer
derivatives acting on $\delta g_{cd}$.

Finally, to treat case (3), we must analyze terms of the form

$\xi^{b_{i}}\nabla_{b_{1}}\cdots\nabla_{b_{i}}\cdots\nabla_{b_{k}}\delta
g_{cd}.$ (56)

To do so, we first bring $\xi^{b_{i}}$ through
$\nabla_{b_{1}}\cdots\nabla_{b_{i-1}}$ to rewrite this term as

$\xi^{b_{i}}\nabla_{b_{1}}\cdots\nabla_{b_{i}}\cdots\nabla_{b_{k}}\delta
g_{cd}=\nabla_{b_{1}}\cdots\nabla_{b_{i-1}}\left[\xi^{b_{i}}\nabla_{b_{i}}\left(\nabla_{b_{i+1}}\cdots\nabla_{b_{k}}\delta
g_{cd}\right)\right]+{\rm lower\,derivative\,terms}.$ (57)

We then write

$\xi^{b_{i}}\nabla_{b_{i}}\left(\nabla_{b_{i+1}}\cdots\nabla_{b_{k}}\delta
g_{cd}\right)={\mathscr{L}}_{\xi}\left(\nabla_{b_{i+1}}\cdots\nabla_{b_{k}}\delta
g_{cd}\right)+{\rm lower\,derivative\,terms}.$ (58)

Finally, we use the fact that since $\xi^{a}$ is a Killing field of the
background metric, ${\mathscr{L}}_{\xi}$ commutes with the background
derivative operator $\nabla_{b}$, so we have

${\mathscr{L}}_{\xi}\left(\nabla_{b_{i+1}}\cdots\nabla_{b_{k}}\delta
g_{cd}\right)=\nabla_{b_{i+1}}\cdots\nabla_{b_{k}}{\mathscr{L}}_{\xi}\delta
g_{cd}.$ (59)

Putting this all together, we have shown that

$\xi^{b_{i}}\nabla_{b_{1}}\cdots\nabla_{b_{i}}\cdots\nabla_{b_{k}}\delta
g_{cd}=\nabla_{b_{1}}\cdots\nabla_{b_{i-1}}\nabla_{b_{i+1}}\cdots\nabla_{b_{k}}{\mathscr{L}}_{\xi}\delta
g_{cd}+{\rm lower\,derivative\,terms}.$ (60)

Combining all of the results of the analyses of cases (1), (2), and (3) above,
we see that the highest derivative term in $\xi_{a}\alpha^{a}$ can be written
in the form

$\xi_{a}T_{(k)}^{ab_{1}\cdots
b_{k}cd}\nabla_{(b_{1}}\cdots\nabla_{b_{k})}\delta
g_{cd}~{}\hat{=}~{}\tilde{T}_{(k-1)}^{b_{1}\cdots
b_{k-1}cd}\nabla_{(b_{1}}\cdots\nabla_{b_{k-1})}{\mathscr{L}}_{\xi}\delta
g_{cd}+\sum_{i=0}^{k-1}{T^{\prime}}_{(i)}^{b_{1}\cdots
b_{i}cd}\nabla_{(b_{1}}\cdots\nabla_{b_{i})}\delta g_{cd}$ (61)

Here $\tilde{T}_{(k-1)}^{b_{1}\cdots b_{k-1}cd}$ arises entirely from case
(3), wherein one $\xi^{b}$ in terms in the basis expansion of
$\xi_{a}T_{(k)}^{ab_{1}\cdots b_{k}cd}$ has been combined with a $\nabla_{b}$
and converted to a ${\mathscr{L}}_{\xi}$. It follows that each term in basis
expansion of $\tilde{T}_{(k-1)}^{b_{1}\cdots b_{k-1}cd}$ has at least as many
$\xi^{a}$’s as $N^{a}$’s and thus is smooth on $\mathcal{H}$. The components
of $\tilde{T}_{(k-1)}^{b_{1}\cdots b_{k-1}cd}$ are obtained algebraically from
$\xi_{a}T_{(k)}^{ab_{1}\cdots b_{k}cd}$ together with the basis elements
$N^{a}$, $\xi^{a}$ and $\\{s_{i}^{a}\\}$. However, it is clear that
$\tilde{T}_{(k-1)}^{b_{1}\cdots b_{k-1}cd}$ does not depend on the choice of
$\\{s_{i}^{a}\\}$, as can be seen from the fact that we could have worked with
the projector

$q^{ab}=g^{ab}-2\xi^{(a}N^{b)}$ (62)

rather than choosing the basis $\\{s_{i}^{a}\\}$. Since
$\xi_{a}T_{(k)}^{ab_{1}\cdots b_{k}cd}$ is locally and covariantly constructed
from the metric, curvature, and its derivatives, we see that
$\tilde{T}_{(k-1)}^{b_{1}\cdots b_{k-1}cd}$ is locally and covariantly
constructed from the metric, curvature, and its derivatives together with
$\xi^{a}$ and $N^{a}$, with only algebraic dependence on $\xi^{a}$ and
$N^{a}$.

On the other hand ${T^{\prime}}_{(i)}^{b_{1}\cdots b_{i}cd}$ is algebraically
constructed from not only $\xi_{a}T_{(k)}^{ab_{1}\cdots b_{k}cd}$, $\xi^{a}$,
and $N^{a}$ but also from derivatives of $\xi^{a}$. However, we can use the
Killing field identity

$\nabla_{a}\nabla_{b}\xi_{c}=R_{cbad}\xi^{d}$ (63)

to eliminate second derivatives of $\xi^{a}$ in terms of $\xi^{a}$ and
curvature. By successively using this identity, we can rewrite all factors
involving an arbitrary number of derivatives of $\xi^{a}$ in terms of
curvature, derivatives of curvature, $\xi^{a}$, and $\nabla_{b}\xi^{a}$. We
may then use

$\nabla_{b}\xi_{a}~{}\hat{=}~{}2\kappa N_{[b}\xi_{a]}$ (64)

to eliminate the first derivative of $\xi^{a}$ in terms of $\xi^{a}$ and
$N^{a}$. In this way, we see that ${T^{\prime}}_{(i)}^{b_{1}\cdots b_{i}cd}$
is locally and covariantly constructed from the metric, curvature, derivatives
of curvature, $\xi^{a}$ and $N^{a}$, with only algebraic dependence on
$\xi^{a}$ and $N^{a}$.

Next, we note that since $\xi^{a}$ is a Killing field of the background
spacetime, we have

$\displaystyle\tilde{T}_{(k-1)}^{b_{1}\cdots
b_{k-1}cd}\nabla_{(b_{1}}\cdots\nabla_{b_{k-1})}{\mathscr{L}}_{\xi}\delta
g_{cd}={}$
$\displaystyle{\mathscr{L}}_{\xi}\left[\tilde{T}_{(k-1)}^{b_{1}\cdots
b_{k-1}cd}\nabla_{(b_{1}}\cdots\nabla_{b_{k-1})}\delta g_{cd}\right]$ (65)
$\displaystyle={}$
$\displaystyle\xi^{a}\nabla_{a}\left[\tilde{T}_{(k-1)}^{b_{1}\cdots
b_{k-1}cd}\nabla_{(b_{1}}\cdots\nabla_{b_{k-1})}\delta g_{cd}\right].$ (66)

Thus, we may rewrite (61) as

$\sum_{i=0}^{k-1}{T^{\prime}}_{(i)}^{b_{1}\cdots
b_{i}cd}\nabla_{(b_{1}}\cdots\nabla_{b_{i})}\delta
g_{cd}=\xi_{a}T_{(k)}^{ab_{1}\cdots
b_{k}cd}\nabla_{(b_{1}}\cdots\nabla_{b_{k})}\delta
g_{cd}-\xi^{a}\nabla_{a}\left[\tilde{T}_{(k-1)}^{b_{1}\cdots
b_{k-1}cd}\nabla_{(b_{1}}\cdots\nabla_{b_{k-1})}\delta g_{cd}\right].$ (67)

The right side of (67) is smooth and vanishes on the bifurcation surface on
account of the explicit factor of $\xi^{a}$ in both terms. Thus, the left side
of (67) has exactly the same properties as $\xi_{a}\alpha^{a}$ except that (i)
there now is additional algebraic dependence on $\xi^{a}$ and $N^{a}$ and (ii)
the highest derivative term has only $k-1$ derivatives of $\delta g_{cd}$. The
first point is of no consequence for any of the arguments that led from (53)
to (61). The second point enables us to prove the theorem by the following
inductive argument.

We take as an inductive hypothesis in $k$ that a slight strengthening of the
statement of the theorem holds, namely, we allow the tensors appearing in (51)
to depend algebraically on $\xi^{a}$ and $N^{a}$ as well as on the metric,
curvature, and covariant derivatives of the curvature. We assume that the
inductive hypothesis holds for $k=p-1$. For $k=p$, the above arguments show
that the highest derivative term in $\xi_{a}\alpha^{a}$ takes the form (61)
and that the lower derivative terms are such that the $k=p-1$ hypothesis can
be applied to them. Therefore, the inductive hypothesis for $k=p-1$ implies
the inductive hypothesis for $k=p$, and all that remains to show is that the
inductive hypothesis holds for $k=0$. However, this is just a simple special
case of cases (1) and (2) in the above proof of the proposition, and it
follows immediately from our gauge conditions (42) and (43) that
$\xi_{a}\alpha^{a}=0$ when $k=0$. Thus, our inductive hypothesis holds for
$k=0$ and the proof is completed.

∎

The main result of this section can now be obtained as follows. For
$\alpha^{a}$ defined by (50), the general form of $\bm{\theta}$ given by eq.
(23) of [4] shows that (51) holds with $k=m+1$, where $m$ is the maximum
number of derivatives of the curvature appearing in the Lagrangian (21).
Therefore, by (49) and the above theorem, we have

$\underline{\bm{\theta}}~{}\hat{=}~{}\bm{\epsilon}^{(n-1)}\sum_{i=0}^{m}\tilde{T}_{(i)}^{b_{1}\cdots
b_{i}cd}\nabla_{(b_{1}}\cdots\nabla_{b_{i})}\mathscr{L}_{\xi}\delta g_{cd}.$
(68)

Define $\textbf{B}_{\mathcal{H}}$ on $\mathcal{H}$ by

$\textbf{B}_{\mathcal{H}}=\bm{\epsilon}^{(n-1)}\sum_{i=0}^{m}\tilde{T}_{(i)}^{b_{1}\cdots
b_{i}cd}\nabla_{(b_{1}}\cdots\nabla_{b_{i})}\mathscr{L}_{\xi}g_{cd}.$ (69)

Then we have

$\underline{\bm{\theta}}~{}\hat{=}~{}\delta\textbf{B}_{\mathcal{H}}.$ (70)

Namely, when we take a first variation of $\textbf{B}_{\mathcal{H}}$, we will
get a factor of $\mathscr{L}_{\xi}g_{cd}=0$ appearing in all terms except the
terms where the variation acts on this factor. However, these terms manifestly
yield $\underline{\bm{\theta}}$.

The quantities $\tilde{T}_{(i)}^{b_{1}\cdots b_{i}cd}$ needed to obtain
$\textbf{B}_{\mathcal{H}}$ can be computed by following the steps used in the
proof of the theorem. We will compute $\textbf{B}_{\mathcal{H}}$ explicitly
for general relativity and for theories with Lagrangians that are quadratic in
the curvature tensor in the next section. We will also discuss the ambiguities
in the definition of $\textbf{B}_{\mathcal{H}}$ in the next section.

##  IV Dynamical Black Hole Entropy

###  IV.1 General Definition of Dynamical Black Hole Entropy

In the previous section, we have proven that for a stationary black hole whose
event horizon, ${\mathcal{H}}$, is a bifurcate Killing horizon there exists an
$(n-1)$-form $\textbf{B}_{\mathcal{H}}$ on ${\mathcal{H}}$ with the property
that (70) holds for arbitrary perturbations. Following the motivation given in
the Introduction, we define the entropy $(n-2)$-form S on ${\mathcal{H}}$ by

$\displaystyle\textbf{S}\equiv{}$
$\displaystyle\frac{2\pi}{\kappa}(\textbf{Q}[\xi]-\xi\cdot\textbf{B}_{\mathcal{H}}).$
(71)

We define the entropy, $S[\mathcal{C}]$, of the black hole at a “time”
represented by an arbitrary cross-section $\mathcal{C}$ by

$\displaystyle S[\mathcal{C}]\equiv{}$
$\displaystyle\int_{\mathcal{C}}\textbf{S}=\frac{2\pi}{\kappa}\int_{\mathcal{C}}(\textbf{Q}[\xi]-\xi\cdot\textbf{B}_{\mathcal{H}}).$
(72)

Since the definition of S does not depend on a choice of cross-section and S
is smooth on ${\mathcal{H}}$, it follows immediately that $S[\mathcal{C}]$
varies continuously with the cross-section $\mathcal{C}$ in the sense defined
in [19]. We now consider the ambiguities in our definition of $S[\mathcal{C}]$
for stationary black holes, for first order perturbations of stationary black
holes, and for second and higher order deviations from a stationary black
hole.

For the unperturbed stationary black hole, we have
$\textbf{B}_{\mathcal{H}}=0$ by (69), so we have

$S[\mathcal{C}]=\frac{2\pi}{\kappa}\int_{\mathcal{C}}\textbf{Q}[\xi].$ (73)

Thus, the only possible ambiguities in $S[\mathcal{C}]$ for stationary black
holes arise from ambiguities in the definition of Q, which are given by [4]

$\displaystyle\textbf{Q}[\xi]\rightarrow{}$
$\displaystyle\textbf{Q}[\xi]+\xi\cdot\bm{\mu}(\phi)+\textbf{Y}(\phi,\mathscr{L}_{\xi}\phi)+d\textbf{Z}(\phi,\xi).$
(74)

Here the first two terms result from the ambiguity in $\bm{\theta}$

$\displaystyle\bm{\theta}(\phi,\delta\phi)\rightarrow{}$
$\displaystyle\bm{\theta}(\phi,\delta\phi)+\delta\bm{\mu}(\phi)+d\textbf{Y}(\phi,\delta\phi)$
(75)

arising from adding an exact term $d\bm{\mu}$ to the Lagrangian and an exact
term $d\textbf{Y}$ to $\bm{\theta}$ (see [4]). Since the $(n-1)$-form
$\bm{\mu}$ is locally and covariantly constructed from the metric, we have
$\mathscr{L}_{\xi}\bm{\mu}=0$ in the stationary background. Since $\bm{\mu}$
smoothly extends to $\mathcal{B}$, it then follows from the type of argument
given at the end of subsection II.2 that the pullback of $\bm{\mu}$ to
${\mathcal{H}}$ vanishes in the stationary background. The Y term in (74)
vanishes in the stationary background since $\mathscr{L}_{\xi}\phi=0$.
Finally, $d\textbf{Z}$ integrates to zero over any cross-section
$\mathcal{C}$. Thus, $S[\mathcal{C}]$ is completely unambiguous in the
stationary background. Furthermore, if the equations of motion hold in the
stationary background, then $d\textbf{Q}=\textbf{J}$ by (26). However, (25)
implies that the pullback of J to ${\mathcal{H}}$ vanishes, because
$\bm{\theta}(\phi,\mathscr{L}_{\xi}\phi)=0$ (since $\xi^{a}$ is a Killing
field) and the pullback of $\xi\cdot\textbf{L}$ to ${\mathcal{H}}$ vanishes
(since $\xi^{a}$ is tangent to ${\mathcal{H}}$). Thus, we find that the
pullback of $d\textbf{Q}$ to ${\mathcal{H}}$ vanishes. This implies that
$S[\mathcal{C}]$ does not depend on $\mathcal{C}$ for a stationary black hole,
i.e., the entropy is “time independent.” In particular, we may evaluate
$S[\mathcal{C}]$ on the bifurcation surface $\mathcal{B}$, in which case it
reduces to the definition given in [20, 4]. Thus, our definition (72) reduces
to the standard expression for entropy for stationary black holes on any
cross-section $\mathcal{C}$.

For first order perturbations of a stationary black hole, we have

$\delta
S[\mathcal{C}]=\frac{2\pi}{\kappa}\int_{\mathcal{C}}(\delta\textbf{Q}[\xi]-\xi\cdot\bm{\theta})$
(76)

where (70) was used to replace $\delta\textbf{B}_{\mathcal{H}}$ by
$\bm{\theta}$. The only possible ambiguities in this expression result from
the ambiguities (74) and (75). Thus, the ambiguity in $\delta\textbf{S}$ is

$\displaystyle\delta\textbf{S}\rightarrow{}$
$\displaystyle\delta\textbf{S}+\frac{2\pi}{\kappa}\left[\xi\cdot\delta\bm{\mu}+\textbf{Y}(\phi,\mathscr{L}_{\xi}\delta\phi)+d\delta\textbf{Z}(\phi,\xi)-\xi\cdot\delta\bm{\mu}-\xi\cdot
d\textbf{Y}(\phi,\delta\phi)\right]$ (77) $\displaystyle={}$
$\displaystyle\delta\textbf{S}+\frac{2\pi}{\kappa}\left[\mathscr{L}_{\xi}\textbf{Y}(\phi,\delta\phi)+d\delta\textbf{Z}(\phi,\xi)-\xi\cdot
d\textbf{Y}(\phi,\delta\phi)\right]$ (78) $\displaystyle={}$
$\displaystyle\delta\textbf{S}+\frac{2\pi}{\kappa}\left[d(\xi\cdot\textbf{Y})(\phi,\delta\phi)+d\delta\textbf{Z}(\phi,\xi)\right]$
(79)

where the general identity

$\displaystyle\mathscr{L}_{\xi}\bm{\alpha}=\xi\cdot
d\bm{\alpha}+d(\xi\cdot\bm{\alpha})$ (80)

for any form $\bm{\alpha}$ was used in the last line. The exact terms in (79)
integrate to zero on any cross-section. Thus, for arbitrary first order
perturbations, $\delta S[\mathcal{C}]$ is entirely unambiguous. As we will see
explicitly below, in the case of general relativity our formula for $\delta
S[\mathcal{C}]$ does not agree with the Bekenstein–Hawking entropy formula in
non-stationary eras.

For second order perturbations, there will, in general, be substantial
additional ambiguities resulting from ambiguities in defining
$\textbf{B}_{\mathcal{H}}$ at second order. As we shall now explain, there are
two sources of these ambiguities.

The first source has to do with the essential presence of $\xi^{a}$ in the
formula for $\textbf{B}_{\mathcal{H}}$. At first order, the perturbed metric
enters the formula for $\textbf{B}_{\mathcal{H}}$ only via the factor
$\mathscr{L}_{\xi}\delta g_{ab}$, so all of the other quantities in that
formula take their background values. Since we require $\delta\xi^{a}=0$,
$\xi^{a}$ is rigidly fixed on spacetime to be the horizon Killing field of the
background metric $g_{ab}$. Similarly, $N^{a}$ on $\mathcal{H}$ is rigidly
fixed by (33) and (34). However, the varied metric will not, in general, admit
any Killing fields at all, and, at second and higher orders in perturbation
theory, the formula for $\textbf{B}_{\mathcal{H}}$ will depend upon how
$\xi^{a}$ is chosen. We can (and will) require as a gauge choice that
$\xi^{a}$ is a fixed vector field on spacetime, but unless $g_{ab}$ is rigidly
tied to $\xi^{a}$, the formula for $\textbf{B}_{\mathcal{H}}$ will have a
gauge dependence with regard to the metric. In order to avoid this, we must
provide a prescription to rigidly fix $\xi^{a}$ in terms of the metric in a
situation where the metric no longer possesses a horizon Killing field.
Without loss of generality, we may assume that the true event horizon of the
perturbed spacetime coincides with $\mathcal{H}$, that $\xi^{a}$ remains
normal to ${\mathcal{H}}$, and that $\xi^{b}\nabla_{b}\xi^{a}=\kappa\xi^{a}$
on ${\mathcal{H}}$. This rigidly fixes $\xi^{a}$ on $\mathcal{H}$ up to a
choice of cross-section $\mathcal{B}$ of $\mathcal{H}$ in the varied spacetime
corresponding to the bifurcation surface. However, since as many as $m+1$
derivatives of $\xi^{a}$ occur in (69), we must also give a construction of
$\xi^{a}$ off of $\mathcal{H}$ at least up to this order for the case of a
metric that does not possess a horizon Killing field. Once we have determined
$\xi^{a}$, we may define $N^{a}$ on $\mathcal{H}$ by (35) and the first
equalities of (37) and (38).

A simple and natural way to rigidly tie $g_{ab}$ to $\xi^{a}$ in a
neighborhood of $\mathcal{H}$ would be to introduce Gaussian null coordinates
(GNCs) on $\mathcal{H}$ adapted to a choice of affine parameter $V$ on the
null geodesics of $\mathcal{H}$, with $V=0$ corresponding to $\mathcal{B}$. In
these coordinates, the metric takes the form

$ds^{2}=2dV(d\rho-\rho\beta_{A}dx^{A}-\tfrac{1}{2}\rho^{2}\alpha
dV)+\gamma_{AB}dx^{A}dx^{B},$ (81)

where $x^{A}$ are coordinates on a cut of constant $V$ and $\rho$ is an affine
parameter along the null geodesics transverse to $\mathcal{H}$ that are
orthogonal to these cuts. We may then define $\xi^{a}$ in a neighborhood of
$\mathcal{H}$ by

$\xi^{a}=\kappa\left[V\left(\frac{\partial}{\partial
V}\right)^{a}-\rho\left(\frac{\partial}{\partial\rho}\right)^{a}\right],$ (82)

and we may then apply a suitable diffeomorphism to $g_{ab},\xi^{a}$ such that
$\xi^{a}$ is the same in each spacetime666In other words, we identify
different spacetimes by identifying points near $\mathcal{H}$ with the same
coordinates $(V,\rho,x^{A})$..

For the case of a metric with a horizon Killing field with surface gravity
$\kappa$ and bifurcation surface $\mathcal{B}$, this definition reproduces the
horizon Killing field [11], but this formula is well defined in a neighborhood
of $\mathcal{H}$ for an arbitrary metric. In addition, the metric in this
gauge satisfies conditions that yield (42) and (43) for perturbations about a
metric with a horizon Killing field. However, there is a rescaling freedom
$V\to f(x^{A})V$ in the choice of $V$. Although this rescaling freedom does
not affect the definition (82) of $\xi^{a}$ in the case where the metric
possesses a horizon Killing field with surface gravity $\kappa$ and
bifurcation surface $\mathcal{B}$, it will affect the definition of $\xi^{a}$
in the general case. Thus, even with this choice of prescription, we have
ambiguities in the definition of $\xi^{a}$ and corresponding ambiguities in
the definition of $\textbf{B}_{\mathcal{H}}$. Of course, other prescriptions
for defining $\xi^{a}$ off of $\mathcal{H}$ would also be possible. In
appendix C we will give an alternative construction of $\xi^{a}$.

It is not necessarily a bad thing that $\textbf{B}_{\mathcal{H}}$—and thereby
our definition of entropy—depends on a choice of “time translation” $\xi^{a}$.
It is generally recognized that the notion of “energy” should be defined as
conjugate to a notion of time translations. Since “energy” and “entropy”
appear together in the first law of thermodynamics for near-equilibrium
systems, it seems reasonable that the notion of “entropy” should similarly
depend on a notion of time translations. Nevertheless, the ambiguities in
defining a notion of “time translation” gives rise to ambiguities in our
definition of entropy beginning at second order in perturbation theory about a
black hole whose event horizon is a Killing horizon.

The second source of ambiguities in the formula for entropy at second order
arises from the fact that $\textbf{B}_{\mathcal{H}}$ was constructed so as to
satisfy (70), which is a condition that applies only to first order
perturbations. Therefore, we have freedom to add to $\textbf{B}_{\mathcal{H}}$
terms that vanish identically for first order perturbations about a background
spacetime with a Killing horizon but do not vanish at second order.
Nevertheless, based upon the construction of $\textbf{B}_{\mathcal{H}}$ given
in the previous section, it seems natural to impose the following conditions
on $\textbf{B}_{\mathcal{H}}$:

  1. 1.

We require $\textbf{B}_{\mathcal{H}}$ to be locally and covariantly
constructed out of the metric, the curvature tensor, derivatives of the
curvature tensor, and the vector fields $\xi^{a}$, $N^{a}$ and their covariant
derivatives777Note that $N^{a}$ on ${\mathcal{H}}$ itself is locally and
covariantly constructed from the metric, $\xi^{a}$, and the first derivative
of $\xi^{a}$, since it is uniquely determined in a local manner by (35) and
the first equalities of (37) and (38). It would be consistent with our
construction of $\textbf{B}_{\mathcal{H}}$ in section III at first order to
require that no derivatives of $N^{a}$ appear in the expression for
$\textbf{B}_{\mathcal{H}}$..

  2. 2.

We require that each term in the expression for $\textbf{B}_{\mathcal{H}}$
contains a factor of $\mathscr{L}_{\xi}g_{ab}$ (or covariant derivatives of
this quantity). In particular, we have $\textbf{B}_{\mathcal{H}}=0$ in the
stationary background.888It would be consistent with our construction of
$\textbf{B}_{\mathcal{H}}$ in section III at first order to require that there
be no more than $m$ covariant derivatives of $\mathscr{L}_{\xi}g_{ab}$.

  3. 3.

We require that (70) holds for arbitrary perturbations of a stationary
background.

  4. 4.

We require that the total number of derivatives appearing in
$\textbf{B}_{\mathcal{H}}$ cannot exceed the total number of derivatives in
$\bm{\theta}$, which is one fewer than the total number of derivatives
appearing in the Lagrangian. (Here, a factor of the curvature tensor counts as
“two derivatives” in the Lagrangian and in $\textbf{B}_{\mathcal{H}}$. The
factor of $\mathscr{L}_{\xi}g_{ab}$ in $\textbf{B}_{\mathcal{H}}$ counts as
“one derivative.” However, the appearance of $N^{a}$ does not count towards
the derivative limitation in $\textbf{B}_{\mathcal{H}}$—despite its being
determined by the derivative of $\xi^{a}$—since its presence can also arise
from the purely algebraic basis expansion.)

  5. 5.

We require that $\textbf{B}_{\mathcal{H}}$ is invariant under a rescaling of
$\xi^{a}$ by a constant, $\xi^{a}\to c\xi^{a}$ and the corresponding rescaling
$\kappa\to c\kappa$.

These conditions are quite restrictive for theories with a relatively low
total number of derivatives in the Lagrangian. In particular, they uniquely
determine $\textbf{B}_{\mathcal{H}}$ in general relativity. However, by the
time one reaches curvature cubed theories (i.e., a total of 6 derivatives in
the Lagrangian), it is not difficult to construct terms—such as
${\epsilon}_{ab_{1}\cdots
b_{n-1}}R_{pq}N^{p}N^{q}{R^{a}}_{c}\xi^{c}g^{mn}\mathscr{L}_{\xi}g_{mn}$—that
could be added to $\textbf{B}_{\mathcal{H}}$ so as satisfy all of the above
conditions and are nonvanishing at second order. It is possible that one could
impose additional conditions on $\textbf{B}_{\mathcal{H}}$ so as to further
restrict its ambiguities and/or obtain other desired properties of S. One
attractive possibility would be to impose conditions that imply that
$S[\mathcal{C}]$ satisfies a second law (i.e., that it is non-decreasing with
time) at second order (see subsection V.2). However, we do not believe that it
would be possible to do this within our framework.

It is instructive to compare the nature of our framework and its definition
freedom arising at second order with the framework and definition freedom in
the approach of Hollands, Kovacs, and Reall (HKR) [17]. As will be discussed
further in appendix C.3, HKR take an effective field theory approach, wherein
the Lagrangian is expanded about the general relativity Lagrangian in powers
of a small parameter $\ell$ and one works only to some finite order in $\ell$.
The HKR entropy-current $(n-2)$-form $\textbf{S}_{\rm HKR}$ is constructed so
as to agree with the Dong–Wall entropy (see section VI) rather than our
entropy current to first order in perturbation theory about a black hole with
a bifurcate Killing horizon. The HKR construction is aimed at providing an
entropy satisfying the 2nd law to second order in perturbation theory [17], or
even non-perturbatively in a recent extension [21]. But it need not be
covariant in the metric [22]. As discussed further in appendix C.3, the lack
of a covariant HKR entropy current $(n-2)$-form means that there is no
guarantee that the HKR entropy will be “cross-section continuous,” [19] i.e.,
if we “wiggle” the cut $\mathcal{C}$ into another cut $\mathcal{C}^{\prime}$
that is arbitrarily close to $\mathcal{C}$, there is no guarantee that $S_{\rm
HKR}[\mathcal{C}^{\prime}]$ will be close to $S_{\rm HKR}[\mathcal{C}]$, as
illustrated in Fig. 1. Although we will see in section VI that (our and) the
Dong–Wall entropy is cross-section continuous, it appears that the HKR entropy
will not, in general, satisfy this property. A detailed comparison of the
definition freedom in entropy at second order in our approach and the HKR
approach is given in appendix C.3.

Figure 1: Cross section continuity [19] of $S[\mathcal{C}]$ means that
$S[{\mathcal{C}}^{\prime}]$ approaches $S[{\mathcal{C}}]$ for a cut
${\mathcal{C}}^{\prime}$ tending to a cut $\mathcal{C}$, even if
${\mathcal{C}}^{\prime}$ is very wiggly as illustrated.

Finally, we comment on the definition of S and $S[\mathcal{C}]$ outside of the
perturbative context, i.e., for an arbitrary dynamical black hole. We have
already noted above that our approach to defining entropy is heavily reliant
on the presence of a time translation vector field $\xi^{a}$. In the case of a
stationary black hole whose event horizon is a Killing horizon, the horizon
Killing field $\xi^{a}$ provides a natural notion of “time translations” upon
which to base notions of energy and entropy. For a black hole that is very
nearly stationary—in particular, in first and second order perturbation theory
about a stationary black hole—it also appears to be natural to use the
definition of $\xi^{a}$ given above to play the role of “time translations.”
However, for a black hole that is far from “equilibrium,” it seems quite
artificial to attempt to assign a surface gravity $\kappa$ and bifurcation
surface $\mathcal{B}$ to the event horizon $\mathcal{H}$ so as to construct a
Killing-like vector field $\xi^{a}$. Furthermore, even in ordinary physics
where a notion of time translation is present, the definition of the entropy
of a system that is far from equilibrium is not well defined without making a
choice of coarse-grained observable [23]. Thus, we do not advocate applying
our formulas for S and $S[\mathcal{C}]$ outside of the perturbative context
around a stationary (i.e., “equilibrium”) black hole.

We now evaluate our formula for S and $S[\mathcal{C}]$ for the case of general
relativity and, more generally, for theories whose Lagrangians do not depend
on derivatives of the curvature tensor.

###  IV.2 Dynamical Black Hole Entropy in General Relativity

The Lagrangian for general relativity is

$\displaystyle{L}_{a_{1}\cdots a_{n}}={}$
$\displaystyle\frac{1}{16\pi}{\epsilon}_{a_{1}\cdots a_{n}}R.$ (83)

The symplectic potential $\bm{\theta}$ is [4]

$\displaystyle{\theta}_{a_{1}\cdots a_{n-1}}={}$
$\displaystyle\frac{1}{16\pi}{{\epsilon}^{e}}_{a_{1}\cdots
a_{n-1}}g^{fh}(\nabla_{f}\delta g_{eh}-\nabla_{e}\delta g_{fh})$ (84)

and the Noether charge is [4]

$\displaystyle{Q}_{a_{1}\cdots a_{n-2}}={}$
$\displaystyle-\frac{1}{16\pi}{\epsilon}_{a_{1}\cdots
a_{n-2}cd}\nabla^{c}\xi^{d}.$ (85)

Using the methods employed in section III to obtain $\textbf{B}_{\mathcal{H}}$
from $\bm{\theta}$, we obtain

$\displaystyle{{B}_{\mathcal{H}}}_{a_{1}\cdots a_{n-1}}={}$
$\displaystyle-\frac{1}{16\pi}N^{b}{\epsilon}_{ba_{1}\cdots
a_{n-1}}g^{ef}\mathscr{L}_{\xi}g_{ef}\,.$ (86)

It is easily seen that ${\textbf{B}_{\mathcal{H}}}$ satisfies all of the
requirements enumerated above in subsection IV.1. Furthermore,
${\textbf{B}_{\mathcal{H}}}$ is unique, since it is allowed to contain only
one derivative—which is used up by the required factor of
$\mathscr{L}_{\xi}g_{ef}$—and no terms of the required form can be constructed
that vanish for arbitrary first order perturbations. In terms of the affinely
parametrized tangent $k^{a}=\xi^{a}/(\kappa V)$ (see (40)), we can rewrite
${\textbf{B}_{\mathcal{H}}}$ as

$\displaystyle{{B}_{\mathcal{H}}}_{a_{1}\cdots a_{n-1}}={}$
$\displaystyle-\frac{1}{16\pi}n^{b}{\epsilon}_{ba_{1}\cdots
a_{n-1}}g^{ef}\mathscr{L}_{k}g_{ef}=-\frac{1}{8\pi}n^{b}{\epsilon}_{ba_{1}\cdots
a_{n-1}}\vartheta$ (87)

where $\vartheta$ is the expansion of the null generators of the horizon in
the affine parameterization $V$ and $n^{a}=\kappa VN^{a}$ (so $n^{a}k_{a}=1$).
This agrees with eq. (6.24) in [24].

The entropy $(n-2)$-form (71) in general relativity is

$\displaystyle{S}_{a_{1}\cdots a_{n-2}}={}$
$\displaystyle-\frac{1}{8\kappa}{\epsilon}_{a_{1}\cdots
a_{n-2}cd}\nabla^{c}\xi^{d}-{\epsilon}^{(n-2)}_{a_{1}\cdots
a_{n-2}}\frac{1}{4}V\vartheta$ (88)

where ${\epsilon}^{(n-2)}_{a_{1}\cdots
a_{n-2}}=\xi^{b}N^{c}{\epsilon}_{bca_{1}\cdots a_{n-2}}$ was defined by (47)
and (48) and is such that its pullback to any cross-section $\mathcal{C}$ on
the horizon yields the volume element on $\mathcal{C}$ obtained from the
pullback of the metric to $\mathcal{C}$. For a given cross section
$\mathcal{C}$ of the horizon, the entropy is

$\displaystyle S[\mathcal{C}]={}$
$\displaystyle\frac{A[\mathcal{C}]}{4}-\frac{1}{4}\int_{\mathcal{C}}V\vartheta\,\bm{\epsilon}^{(n-2)}$
(89)

where $A[\mathcal{C}]$ is the ($(n-2)$-dimensional) area of $\mathcal{C}$ and
the integral over $\mathcal{C}$ in the second term is taken with respect to
the natural volume element ${\epsilon}^{(n-2)}_{a_{1}\cdots a_{n-2}}$ on
$\mathcal{C}$. The first term in (89) is the usual Bekenstein–Hawking entropy.
For a stationary black hole, we have $\vartheta=0$. However, for a dynamical
black hole, we have $\vartheta>0$, so $S[\mathcal{C}]$ is smaller than the
Bekenstein–Hawking entropy. We will show in appendix A that for first-order
perturbations, $S[\mathcal{C}]$ in general relativity is, in fact, equal to
the area of the apparent horizon at the “time” corresponding to the cross-
section, $\mathcal{C}$, of the horizon.

Rignon-Bret [25] (see also [26]) has given an analysis of aspects of our
entropy expression (89) as well as an alternative entropy expression based on
the York symplectic potential that is constructed to vanish on any cross
section of a light cone in Minkowski spacetime.

###  IV.3 Entropy Form for $L(R_{abcd})$ Theories

Consider a theory of gravity obtained from a Lagrangian that depends on the
metric and curvature but does not depend on derivatives of the curvature

$\displaystyle{L}_{a_{1}\cdots a_{n}}={}$
$\displaystyle{\epsilon}_{a_{1}\cdots a_{n}}L(g_{ab},R_{abcd}).$ (90)

The symplectic potential $(n-1)$-form can be calculated from the algorithm
given in Lemma 3.1 of [4] or directly from999The sign difference between our
formula and eq. (4.5) of [27] appears to be a result of differences in
conventions. eq. (4.5) of [27]. We obtain,

$\displaystyle{\theta}_{a_{1}\cdots a_{n-1}}={}$ $\displaystyle
2{\epsilon}_{ma_{1}\cdots a_{n-1}}\left[\frac{\partial L}{\partial
R_{mbcd}}\nabla_{d}\delta g_{bc}-\nabla_{d}\left(\frac{\partial L}{\partial
R_{mbcd}}\right)\delta g_{bc}\right].$ (91)

The Noether charge $(n-2)$-form can be obtained either from the Noether
current J via the algorithm given in Lemma 1 of [28] or directly from eq.
(4.6) of [27],

$\displaystyle{Q}_{a_{1}\cdots a_{n-2}}={}$
$\displaystyle{\epsilon}_{cda_{1}\cdots a_{n-2}}\Big{(}-\frac{\partial
L}{\partial R_{cdef}}\nabla_{[e}\xi_{f]}-2\nabla_{f}\Big{(}\frac{\partial
L}{\partial R_{cdef}}\Big{)}\xi_{e}\Big{)}.$ (92)

From the construction in section III, we obtain

$\displaystyle{{B}_{\mathcal{H}}}_{a_{1}\cdots a_{n-1}}={}$ $\displaystyle
2N^{f}{\epsilon}_{fa_{1}\cdots a_{n-1}}\xi_{m}\frac{\partial L}{\partial
R_{mbcd}}N_{d}\mathscr{L}_{\xi}g_{bc}.$ (93)

The entropy $(n-2)$-form is then

$\displaystyle{S}_{a_{1}\cdots a_{n-2}}={}$
$\displaystyle\frac{2\pi}{\kappa}\Big{[}{\epsilon}_{cda_{1}\cdots
a_{n-2}}\Big{(}-\frac{\partial L}{\partial
R_{cdef}}\nabla_{[e}\xi_{f]}-2\nabla_{f}\Big{(}\frac{\partial L}{\partial
R_{cdef}}\Big{)}\xi_{e}\Big{)}+2{\epsilon}^{(n-2)}_{a_{1}\cdots
a_{n-2}}\xi_{m}\frac{\partial L}{\partial
R_{mbcd}}N_{d}\mathscr{L}_{\xi}g_{bc}\Big{]}.$ (94)

We now explicitly evaluate the above quantities for the case of a general
curvature squared Lagrangian, i.e.,

$\displaystyle{L}_{a_{1}\cdots a_{n}}={}$
$\displaystyle{\epsilon}_{a_{1}\cdots
a_{n}}\left(\alpha_{1}R_{abcd}R^{abcd}+\alpha_{2}R_{ab}R^{ab}+\alpha_{3}R^{2}\right),$
(95)

where $\alpha_{1},\alpha_{2},\alpha_{3}$ are arbitrary constants. The
symplectic potential $(n-1)$-form is101010The choice of $\alpha_{1}=1$,
$\alpha_{2}=-4$, $\alpha_{3}=1$ corresponds to Gauss–Bonnet gravity. There is
a typo—previously noted in [29]—in the sign of the second-to-last term in the
formula for $\bm{\theta}$ for Gauss–Bonnet gravity given in eq. (70) of [4].

$\displaystyle\begin{split}{\theta}_{a_{1}\cdots
a_{n-1}}={}&{\epsilon}_{ma_{1}\cdots
a_{n-1}}\Big{\\{}4\alpha_{1}\Big{[}R^{mbcd}\nabla_{d}\delta
g_{bc}-\nabla_{d}R^{mbcd}\delta g_{bc}\Big{]}\\\
{}&+\alpha_{2}\Big{[}2g^{c[m}R^{b]d}\nabla_{d}\delta
g_{bc}+2R^{c[m}\nabla^{b]}\delta g_{bc}+g^{c[b}\nabla^{m]}R\delta
g_{bc}+2\nabla^{[m}R^{b]c}\delta g_{bc}\Big{]}\\\
{}&+2\alpha_{3}\Big{[}Rg^{mc}g^{bd}(\nabla_{d}\delta g_{bc}-\nabla_{c}\delta
g_{bd})-2g^{c[m}\nabla^{b]}R\delta g_{bc}\Big{]}\Big{\\}}.\end{split}$ (96)

The Noether charge $(n-2)$-form is

$\displaystyle\begin{split}{Q}_{a_{1}\cdots
a_{n-2}}={}&-{\epsilon}_{cda_{1}\cdots
a_{n-2}}\Big{[}(2\alpha_{1}R^{cdef}+2\alpha_{2}g^{ce}R^{fd}+2\alpha_{3}Rg^{ce}g^{df})\nabla_{[e}\xi_{f]}\\\
{}&+(4\alpha_{1}\nabla_{f}R^{cdmf}+\alpha_{2}g^{cm}\nabla^{d}R-2\alpha_{2}\nabla^{c}R^{md}+4\alpha_{3}\nabla^{d}Rg^{cm})\xi_{m}\Big{]}.\end{split}$
(97)

The $\textbf{B}_{\mathcal{H}}$ $(n-1)$-form is

$\displaystyle{B}_{\mathcal{H}a_{1}\cdots a_{n-1}}={}$ $\displaystyle
N^{f}{\epsilon}_{fa_{1}\cdots
a_{n-1}}\left(4\alpha_{1}R^{abcd}\xi_{a}N_{d}-\alpha_{2}R^{bc}-\alpha_{2}R_{ad}\xi^{a}N^{d}g^{bc}-2\alpha_{3}Rg^{bc}\right)\mathscr{L}_{\xi}g_{bc}.$
(98)

The entropy $(n-2)$-form is

$\displaystyle\begin{split}{S}_{a_{1}\cdots
a_{n-2}}={}&-\frac{2\pi}{\kappa}\Big{(}{\epsilon}_{cda_{1}\cdots
a_{n-2}}\Big{[}(2\alpha_{1}R^{cdef}+2\alpha_{2}g^{ce}R^{fd}+2\alpha_{3}Rg^{ce}g^{df})\nabla_{[e}\xi_{f]}\\\
{}&+(4\alpha_{1}\nabla_{f}R^{cdmf}+\alpha_{2}g^{cm}\nabla^{d}R-2\alpha_{2}\nabla^{c}R^{md}+4\alpha_{3}\nabla^{d}Rg^{cm})\xi_{m}\Big{]}\\\
{}&-{\epsilon}^{(n-2)}_{a_{1}\cdots
a_{n-2}}(4\alpha_{1}R^{abcd}\xi_{a}N_{d}-\alpha_{2}R^{bc}-\alpha_{2}R_{ad}\xi^{a}N^{d}g^{bc}-2\alpha_{3}Rg^{bc})\mathscr{L}_{\xi}g_{bc}\Big{)}.\end{split}$
(99)

As will be discussed in section VI, this yields a formula for $S[\mathcal{C}]$
that differs from the Dong–Wall entropy in a manner analogous to how our
formula for $S[\mathcal{C}]$ in general relativity differs from the
Bekenstein–Hawking entropy.

##  V The Physical Process First Law and the Second Law

As explained in the Introduction, the main motivation for our new definition
of dynamical black hole entropy is to obtain a local in time version of the
physical process version of the first law of black hole mechanics. In this
section, we derive the physical process version of the first law and discuss
the second law of black hole mechanics. As stated at the end of section II, we
restrict consideration to Lagrangians for which the only dynamical field is
the metric. However, in this context, there are two distinct cases that we
consider.

The first case is where an external stress-energy is present in the first
order perturbation. Here, by an “external stress-energy,” we mean a source
term in the first order gravitational equations arising from matter that is
not represented in the Lagrangian. In this case, as we shall see, there can be
nontrivial time dependence of the dynamical black hole entropy $S$ at first
order when the external matter crosses the horizon. We show that the entropy
change is related to the energy flux by the first law of black hole mechanics.
As an immediate corollary, we see that if the external matter satisfies the
null energy condition, then the second law of black hole mechanics holds,
i.e., the entropy is non-decreasing with time in first order perturbation
theory. These results hold for an arbitrary theory of gravity.

The second case is that of pure vacuum perturbations. In this case, $S$ does
not change with time at first order, so we have to go to second order in
perturbation theory to investigate the time dependence of $S$. We find that at
second order, results similar to the external matter case hold, with the first
order energy flux of external matter replaced by the second order flux of
“modified canonical energy” [11]. The flux of modified canonical energy is
positive in general relativity—it is proportional to the square of the first
order perturbed shear of the horizon—so the second law of black hole mechanics
holds at second order in general relativity. However, the flux of modified
canonical energy would not be expected to be positive in general theories of
gravity, so we would not expect a second law to hold for vacuum perturbations
in general theories.

###  V.1 Perturbations with External Matter Sources

For an arbitrary Lagrangian theory of gravity where the only dynamical field
is the metric, we consider a stationary black hole with horizon Killing field
$\xi^{a}$ that satisfies the vacuum field equations $({E}_{G})_{ab}=0$. We
consider perturbations, $\delta g_{ab}$, of this spacetime that are sourced by
a matter stress-energy tensor, $\delta T_{ab}$, that is conserved,
$\nabla^{a}\delta T_{ab}=0$. Thus, $\delta g_{ab}$ satisfies

$2\delta({E}_{G})_{ab}=\delta T_{ab}.$ (100)

The perturbed matter energy flux $(n-1)$-form e on the horizon is given by

$\delta e_{a_{1}\cdots
a_{n-1}}\equiv-\xi^{b}\delta{T_{b}}^{c}{\epsilon}_{ca_{1}\cdots
a_{n-1}}=\delta T_{bc}\xi^{b}\xi^{c}{\epsilon}^{(n-1)}_{a_{1}\cdots a_{n-1}}$
(101)

where ${\epsilon}^{(n-1)}_{a_{1}\cdots a_{n-1}}$ was defined by (47). We have
the following theorem:

######  Theorem 2 (Physical process version of the first law [8]).

Let $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ be arbitrary cross-sections of the
horizon $\mathcal{H}$ with $\mathcal{C}_{2}$ lying to the future of
$\mathcal{C}_{1}$. Let $\Delta\delta S$ denote the perturbed entropy
difference between the cross-sections

$\Delta\delta S=\delta S[\mathcal{C}_{2}]-\delta S[\mathcal{C}_{1}]$ (102)

and let $\Delta\delta E$ denote the perturbed total energy flux between the
cross-sections

$\Delta\delta
E=\int_{\mathcal{H}_{12}}\delta\textbf{{e}}=-\int_{\mathcal{H}_{12}}\xi^{b}\delta{T_{b}}^{c}{\epsilon}_{ca_{1}\cdots
a_{n-1}}=\int_{\mathcal{H}_{12}}\delta
T_{bc}\xi^{b}\xi^{c}{\epsilon}^{(n-1)}_{a_{1}\cdots a_{n-1}}$ (103)

where ${\mathcal{H}_{12}}$ denotes the portion of $\mathcal{H}$ bounded by
$\mathcal{C}_{1}$ and $\mathcal{C}_{2}$. Then we have

$\displaystyle\frac{\kappa}{2\pi}\Delta\delta S={}$ $\displaystyle\Delta\delta
E.$ (104)

###### Proof.

The pullback to the horizon of the fundamental identity (30) with $\chi^{a}$
taken to be the horizon Killing vector field $\xi^{a}$ yields

$\displaystyle d[\delta\textbf{Q}[\xi]-\xi\cdot\bm{\theta}]={}$
$\displaystyle-\xi^{a}\delta{\textbf{C}}_{a}.$ (105)

Using the definition (71) of S together with (70), we can rewrite the left
side of (105) as

$\displaystyle d[\delta\textbf{Q}[\xi]-\xi\cdot\bm{\theta}]={}$
$\displaystyle\frac{\kappa}{2\pi}d\delta\textbf{S}.$ (106)

Using the general formula (28) with ${E}_{M}=0$ for $\bm{\textbf{C}}_{a}$
together with (100), we can rewrite the right side of (105) as

$-\xi^{a}\delta{\textbf{C}}_{a}=\delta\textbf{e}$ (107)

so we have

$\frac{\kappa}{2\pi}d\delta\textbf{S}=\delta\textbf{e}.$ (108)

Integration of this equation over the region $\mathcal{H}_{12}$ of the horizon
bounded by $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ then immediately yields
(104).

∎

There are two important immediate consequences of Theorem 2. First, if $\delta
T_{bc}$ satisfies the null energy condition (i.e., if $\delta
T_{bc}n^{b}n^{c}\geq 0$ for all null $n^{a}$), then it follows immediately
from (103) that $\Delta\delta E\geq 0$. It then follows immediately from
Theorem 2 that $\Delta\delta S\geq 0$, i.e., $\delta
S[\mathcal{C}_{2}]\geq\delta S[\mathcal{C}_{1}]$ whenever $\mathcal{C}_{2}$
lies to the future of $\mathcal{C}_{1}$. Thus, in an arbitrary theory of
gravity, the second law of black hole mechanics holds to first order for
perturbations of a black hole with external matter satisfying the null energy
condition.

Second, Theorem 2 shows that, to first order, the entropy of a black hole
changes between cross-sections $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ if and
only if there is a net energy flux of matter between $\mathcal{C}_{1}$ and
$\mathcal{C}_{2}$. Thus, the increase in entropy of the black hole has an
entirely local cause. In the case of general relativity, this contrasts
sharply with the properties of the Bekenstein–Hawking entropy, $A/4$. The
event horizon of a black hole is “teleological” in nature, i.e., its location
is determined by properties of the spacetime in the asymptotic future. If one
is going to throw matter into a black hole at a later time, then at early
times the event horizon area will have already slightly increased in
anticipation of the later arrival of the matter. In other words, if one makes
a firm decision to throw a baseball into a black hole, the Bekenstein–Hawking
entropy of the black hole will have increased while the baseball is still in
one’s hand. Our definition of the entropy of a dynamical black hole does not
have this property. Although the $A/4$ term in (89) will have increased while
the baseball is still in one’s hand, the horizon will also be expanding at
that time, so the second term in (89) will contribute negatively. Theorem 2
shows that these two terms cancel at first order, so there is no change in the
black hole entropy while the baseball is still in one’s hand. The entropy of
the black hole changes only when the baseball crosses the horizon.

###  V.2 Vacuum Perturbations

If we have no external matter sources, $\delta T_{ab}=0$, then Theorem 2
states that $\Delta\delta S=0$, i.e., there is no change of black hole entropy
with time at first order. Thus, we must go to second order in perturbation
theory to obtain the leading order dynamical behavior of black hole entropy.
To do so, we use the varied fundamental identity (31) with $\chi^{a}=\xi^{a}$.
Since we assume that the vacuum equations of motion hold, the first three
terms on the right side of (31) vanish111111The first two terms would vanish
in any case under pullback to $\mathcal{H}$.. We obtain121212Here
$\delta^{2}$, the second variation, refers to the second derivative with
respect to a parameter $s$ of a family of metrics $g^{(s)}_{ab}=g_{ab}+s\delta
g_{ab}+\tfrac{1}{2}s^{2}\delta^{2}g_{ab}+O(s^{3})$.

$\bm{\omega}(g,\delta g,\mathscr{L}_{\xi}\delta
g)=d[\delta^{2}\textbf{Q}[g;\xi]-\xi\cdot\delta\bm{\theta}(g,\delta g)]\,.$
(109)

We can rewrite this equation as

$\displaystyle\bm{\omega}(g,\delta g,\mathscr{L}_{\xi}\delta
g)+d[\xi\cdot\delta\bm{\theta}(g,\delta
g)-\xi\cdot\delta^{2}\textbf{B}_{\mathcal{H}}(g))]={}$ $\displaystyle
d[\delta^{2}\textbf{Q}[g;\xi]-\xi\cdot\delta^{2}\textbf{B}_{\mathcal{H}}(g)]=\frac{\kappa}{2\pi}d\delta^{2}\textbf{S}(g).$
(110)

The term $\bm{\omega}(g,\delta g,\mathscr{L}_{\xi}\delta g)$ is known as the
canonical energy $(n-1)$-form [11] of the metric perturbation $\delta g$. We
refer to entire left side of (110) as the modified canonical energy
$(n-1)$-form $\textbf{e}_{G}$,

$\textbf{e}_{G}(g;\delta g,\delta g)\equiv\bm{\omega}(g,\delta
g,\mathscr{L}_{\xi}\delta g)+d[\xi\cdot\delta\bm{\theta}(g,\delta
g)-\xi\cdot\delta^{2}\textbf{B}_{\mathcal{H}}(g))].$ (111)

Note that $\textbf{e}_{G}$ is quadratic in $\delta g_{ab}$ and does not depend
on $\delta^{2}g_{ab}$. The modified canonical energy was introduced in [11]
for the case of general relativity as a means of obtaining a quantity with a
locally positive flux whose total flux through the black hole horizon agrees
with the total canonical energy flux. For the case of general relativity, the
modified canonical energy flux is given by

$\textbf{e}_{G}=\frac{1}{4\pi}(\kappa
V)^{2}\delta\sigma_{cd}\delta\sigma^{cd}\,\bm{\epsilon}^{(n-1)},$ (112)

(see eq. (105) of [11]), where $V$ is an affine parameter, $\sigma_{cd}$ is
the shear of the generators of the horizon with respect to $V$, and we have
used the fact [11] that $\delta\vartheta|_{\mathcal{H}}=0$ for vacuum
perturbations. Although the modified canonical energy was introduced in an
entirely ad hoc way in [11], we now see that it naturally enters the formula
for the local change in dynamical black hole entropy at second order.

From (110) we see that the situation for vacuum perturbations at second order
corresponds to the situation for external matter perturbations at first order,
with the modified canonical energy flux $\textbf{e}_{G}$ replacing the matter
stress-energy flux e. In particular, for vacuum perturbations we have

$\frac{\kappa}{2\pi}\Delta\delta^{2}S=\int_{\mathcal{H}_{12}}\textbf{e}_{G}(g;\delta
g,\delta g).$ (113)

It follows immediately that, in an arbitrary theory of gravity, the second law
of black hole mechanics will hold at second order for vacuum perturbations if
and only if the modified canonical energy flux through the horizon is
positive. This is the case in general relativity. However, it seems unlikely
that this will be the case in more general theories of gravity even if we
allow ourselves to modify the entropy making use of the permissible
ambiguities (see appendix C.3 for further discussion).

##  VI Relationship with the Dong–Wall Entropy

In 2013, Dong [5] proposed a formula for dynamical black hole entropy for
theories obtained from a Lagrangian that depends upon the curvature but not
derivatives of the curvature. In 2015, Wall [6] proposed a formula for
dynamical black hole entropy for general Lagrangian theories, valid to linear
order for perturbations of a stationary black hole. Wall’s approach for
obtaining dynamical black hole entropy does not have any obvious relationship
with Dong’s. Nevertheless, for the case of a Lagrangian that depends upon the
curvature but not its derivatives, Wall’s formula was found to agree [6] with
the linearization of Dong’s formula. In Appendix D, we show that when we apply
the ideas of Wall’s method as elucidated in this section to calculate entropy
in dilaton gravity models, it agrees with the linearization of the entropy
obtained by Dong and Lewkowycz [30] using Dong’s approach. Thus, we believe
that Wall’s entropy agrees with the linearization of Dong’s entropy in
general. We will therefore refer to this entropy as the Dong–Wall entropy and
will denote it by $S_{\rm DW}$. Nevertheless, it should be kept in mind that
general agreement has not been proven. In the event that they do not agree,
the entropy defined in this section would correspond to that proposed by Wall
[6].

In this section, we will give a derivation of the Dong–Wall entropy based on
the fundamental identity (30). We will then obtain a simple, general
relationship between the Dong–Wall entropy and our entropy, valid up to linear
order in perturbations about a stationary black hole. As we shall see, the
relationship between the Dong–Wall entropy and our entropy in general theories
of gravity is very similar to the relationship (89) between the
Bekenstein–Hawking entropy and our entropy in general relativity.

The starting point of our derivation of the Dong–Wall entropy is the formula
(28) for the pullback of the linearized constraints to the horizon, which take
the form

$\displaystyle\xi^{a}\delta{\textbf{C}}_{a}={}$
$\displaystyle-2(\delta{E}_{G})_{ab}\xi^{a}\xi^{b}\bm{\epsilon}^{(n-1)},$
(114)

where $\bm{\epsilon}^{(n-1)}$ was defined by (47). The linearized equations of
motion $(\delta{E}_{G})_{ab}$ take the general form

$\displaystyle\delta{E}_{G}^{ab}={}$
$\displaystyle\sum_{i=0}^{k}U_{(i)}^{abc_{1}\cdots
c_{i}de}\nabla_{(c_{1}}\cdots\nabla_{c_{i})}\delta g_{de},$ (115)

where the tensors $U_{(i)}^{abc_{1}\cdots c_{i}de}=U_{(i)}^{(ab)(c_{1}\cdots
c_{i})(de)}$ are smooth and are locally and covariantly constructed from the
metric, curvature, and covariant derivatives of the curvature. This is closely
analogous to the quantity $\alpha^{a}$ in (51) except that there are two free
indices on $\delta{E}_{G}^{ab}$ instead of the single free index on
$\alpha^{a}$. Both of these free indices on $\delta{E}_{G}^{ab}$ are
contracted into $\xi^{a}$ in (114), as opposed to the single contraction that
occurs in (52). Thus, one might expect that a strengthened form of Theorem 1
should hold for $(\delta{E}_{G})_{ab}\xi^{a}\xi^{b}$. We show in Appendix B
that this is the case. Specifically, we show that
$(\delta{E}_{G})_{ab}\xi^{a}\xi^{b}$ can be written in the form

$\displaystyle(\delta{E}_{G})_{ab}\xi^{a}\xi^{b}~{}\hat{=}~{}{}$
$\displaystyle\sum_{i=0}^{k-2}\tilde{U}_{(i)}^{c_{1}\cdots
c_{i}de}\nabla_{(c_{1}}\cdots\nabla_{c_{i})}(\mathscr{L}^{2}_{\xi}-\kappa\mathscr{L}_{\xi})\delta
g_{de}$ (116)

where the tensors $\tilde{U}_{(i)}^{c_{1}\cdots c_{i}de}$ are smooth on
$\mathcal{H}$ and are locally and covariantly constructed from the metric,
curvature, and its derivatives as well as from $\xi^{a}$, and $N^{a}$ (see
(33) and (34)), with $\xi^{a}$ and $N^{a}$ appearing only algebraically. Thus,
we see that $(\delta{E}_{G})_{ab}\xi^{a}\xi^{b}$ has the form

$(\delta{E}_{G})_{ab}\xi^{a}\xi^{b}~{}\hat{=}~{}\kappa\mathscr{L}_{\xi}P(g,\xi;\delta
g)-\mathscr{L}_{\xi}^{2}P(g,\xi;\delta g)=(\kappa-\mathscr{L}_{\xi})\delta
P(g,\xi;\mathscr{L}_{\xi}g)$ (117)

with

$P(g,\xi;\delta g)=-\sum_{i=0}^{k-2}\tilde{U}_{(i)}^{c_{1}\cdots
c_{i}de}\nabla_{(c_{1}}\cdots\nabla_{c_{i})}\delta g_{de}.$ (118)

We define the Dong–Wall entropy $(n-2)$-form by131313Re-analyzing Wall’s
derivation [6], the authors of [31] have emphasized the fact that the entropy
integrand has a current part, i.e. it is an $(n-2)$-form. However, we have not
been able to find a relationship between their proposed formula and our
formula (119).

$\displaystyle\textbf{S}_{\rm DW}={}$
$\displaystyle\textbf{S}+\frac{4\pi}{\kappa}P(g,\xi;\mathscr{L}_{\xi}g)\bm{\epsilon}^{(n-2)}$
(119)

where our entropy $(n-2)$-form S was defined by (71) and
$\bm{\epsilon}^{(n-2)}$ was defined in (48). It is clear that $\textbf{S}_{\rm
DW}=\textbf{S}$ in the stationary background, wherein both are given by (73).
Since $\delta\textbf{S}$ is uniquely defined for arbitrary first order
perturbations and $\delta P$ is uniquely determined by (117), we see that
$\delta\textbf{S}_{\rm DW}$ is uniquely defined for arbitrary first order
perturbations. However, there will be ambiguities in $\textbf{S}_{\rm DW}$ at
second and higher orders resulting from the ambiguities in S discussed at the
end of subsection IV.1 (see also appendix C) as well as from the additional
ambiguities in defining $P$ at second and higher orders. The Dong–Wall entropy
of a cross-section $\mathcal{C}$ is defined by

$S_{\rm DW}[\mathcal{C}]=\int_{\mathcal{C}}\textbf{S}_{\rm DW}.$ (120)

Since $\textbf{S}_{\rm DW}$ does not depend on a choice of cross-section and
$\textbf{S}_{\rm DW}$ is smooth on ${\mathcal{H}}$, it follows immediately
that $S_{\rm DW}$ is cross-section continuous, as defined in [19].

A simple relationship between $\textbf{S}_{\rm DW}$ and S for first order
perturbations can now be obtained as follows. As in the proof of Theorem 2, we
have

$\displaystyle\frac{\kappa}{2\pi}d\delta\textbf{S}=-\xi^{a}\delta{\textbf{C}}_{a}.$
(121)

Using (114) and (117), we obtain

$\displaystyle\frac{\kappa}{2\pi}d\delta\textbf{S}={}$ $\displaystyle
2(\kappa-\mathscr{L}_{\xi})\left[\delta
P(g,\xi;\mathscr{L}_{\xi}g)\bm{\epsilon}^{(n-1)}\right]\,.$ (122)

Applying the general identity (80) to $\mathscr{L}_{\xi}[\delta
P(g,\xi;\mathscr{L}_{\xi}g)\bm{\epsilon}^{(n-1)}]$, we obtain

$\mathscr{L}_{\xi}[\delta P\bm{\epsilon}^{(n-1)}]=d[\delta
P\xi\cdot\bm{\epsilon}^{(n-1)}]+\xi\cdot d[\delta
P\bm{\epsilon}^{(n-1)}]=d[\delta P\bm{\epsilon}^{(n-2)}]$ (123)

where the second term in the first equality vanishes because
$\bm{\epsilon}^{(n-1)}$ is an $(n-1)$-form on the $(n-1)$-dimensional manifold
$\mathcal{H}$, so its “$d$” vanishes. Thus, bringing the
$\mathscr{L}_{\xi}\delta P$ term on the right side of (122) to the left side
and using (123) and the definition (119) of $\textbf{S}_{\rm DW}$, we obtain

$\frac{\kappa}{2\pi}d\delta\textbf{S}_{\rm DW}=2\kappa\delta
P(g,\xi;\mathscr{L}_{\xi}g)\bm{\epsilon}^{(n-1)}.$ (124)

Canceling the factors of $\kappa$ and contracting both sides with $\xi^{a}$,
we obtain

$\frac{1}{2\pi}\xi\cdot d\delta\textbf{S}_{\rm DW}=2\xi\cdot[\delta
P(g,\xi;\mathscr{L}_{\xi}g)\bm{\epsilon}^{(n-1)}]=2\delta
P(g,\xi;\mathscr{L}_{\xi}g)\bm{\epsilon}^{(n-2)}.$ (125)

By the general identity (80), we have

$\xi\cdot d\delta\textbf{S}_{\rm DW}=\mathscr{L}_{\xi}\delta\textbf{S}_{\rm
DW}-d[\xi\cdot\delta\textbf{S}_{\rm DW}].$ (126)

Thus, we obtain

$4\pi\delta
P(g,\xi;\mathscr{L}_{\xi}g)\bm{\epsilon}^{(n-2)}=\mathscr{L}_{\xi}\delta\textbf{S}_{\rm
DW}-d[\xi\cdot\delta\textbf{S}_{\rm DW}].$ (127)

By the definition (119) of $\textbf{S}_{\rm DW}$, we have

$\delta\textbf{S}=\delta\textbf{S}_{\rm DW}-\frac{4\pi}{\kappa}\delta
P(g,\xi;\mathscr{L}_{\xi}g)\bm{\epsilon}^{(n-2)}$ (128)

and substituting in (127) we have141414Note that we can combine (130) and
(121) into the equation  $8\pi
k^{a}\bm{\textbf{C}}_{a}=d({\mathscr{L}}_{k}\textbf{S}_{\rm DW})$ (129) to
first order, where $\xi^{a}=\kappa Vk^{a}$ and $V$ is an affine parameter
vanishing on $\mathcal{B}$ such that $k^{a}\nabla_{a}V=1$. This is identical
in form to the defining equation (to first order) of the Dong–Wall entropy as
given in [17, eq. 120] if we replace $l^{a}$ there with $k^{a}$, and the
entropy-current $(n-2)$-form $\bm{s}$ defined by [17, eq. 119] with
$\bm{S}_{\rm DW}$, noting that $\bm{F}_{l}$ in [17, eq. 120] vanishes up to
and including first order variations. Also note that $\bm{s}$ is defined only
to first order by [17, eq. 120], and it is shown in [17, Prop. 1] that it can
be modified at second order if necessary so as to be covariant. By contrast,
our derivation automatically delivers a covariant form of $\bm{S}_{\rm DW}$.

$\delta\textbf{S}=\delta\textbf{S}_{\rm
DW}-\frac{1}{\kappa}\mathscr{L}_{\xi}\delta\textbf{S}_{\rm
DW}+\frac{1}{\kappa}d[\xi\cdot\delta\textbf{S}_{\rm DW}].$ (130)

The last term is exact, so it does not contribute when one integrates over a
cross-section $\mathcal{C}$ to get the Dong–Wall entropy, $\delta S_{\rm
DW}[\mathcal{C}]$, of the cross-section. Thus, we obtain the desired general
relationship between our entropy and the Dong–Wall entropy for an arbitrary
cross-section $\mathcal{C}$

$\delta S[\mathcal{C}]=\delta S_{\rm
DW}[\mathcal{C}]-\frac{1}{\kappa}\int_{\mathcal{C}}\mathscr{L}_{\xi}\delta\textbf{S}_{\rm
DW}.$ (131)

It is worth noting that if we multiply (117) by $\bm{\epsilon}^{(n-2)}$ and
use (127) we obtain

$(\delta{E}_{G})_{ab}\xi^{a}\xi^{b}\bm{\epsilon}^{(n-2)}~{}\hat{=}~{}\frac{1}{4\pi}\left(\kappa-\mathscr{L}_{\xi}\right)\left[\mathscr{L}_{\xi}\delta\textbf{S}_{\rm
DW}-d(\xi\cdot\delta\textbf{S}_{\rm DW})\right].$ (132)

Integrating over an arbitrary cross-section $\mathcal{C}$, we obtain

$\int_{\mathcal{C}}(\delta{E}_{G})_{ab}\xi^{a}\xi^{b}\bm{\epsilon}^{(n-2)}=\frac{1}{4\pi}\int_{\mathcal{C}}\left(\kappa-\mathscr{L}_{\xi}\right)\mathscr{L}_{\xi}\delta\textbf{S}_{\rm
DW}.$ (133)

Let $V$ be an affine parameter for null geodesic generators of $\mathcal{H}$
with $V=0$ corresponding to the bifurcation surface. Let $k^{a}$ be the
corresponding affinely parametrized tangent to the null generators. Then
$\xi^{a}=\kappa Vk^{a}$ on $\mathcal{H}$ and on differential forms we have
$\kappa\mathscr{L}_{\xi}-\mathscr{L}_{\xi}^{2}=-\kappa^{2}V^{2}\mathscr{L}^{2}_{k}$
modulo exact terms. We therefore obtain

$\int_{\mathcal{C}}V^{2}(\delta{E}_{G})_{ab}k^{a}k^{b}\bm{\epsilon}^{(n-2)}=-\frac{1}{4\pi}\int_{\mathcal{C}}V^{2}\mathscr{L}^{2}_{k}\delta\textbf{S}_{\rm
DW}.$ (134)

If $\mathcal{C}$ is a cross-section of constant $V$ (which always can be
achieved for any given cross-section using the rescaling freedom of $V$), then
we obtain

$\int_{\mathcal{C}}(\delta{E}_{G})_{ab}k^{a}k^{b}\bm{\epsilon}^{(n-2)}=-\frac{1}{4\pi}\frac{d^{2}}{dV^{2}}\delta
S_{\rm DW}(V).$ (135)

This equation is the defining property151515Note that there is a missing
factor of $1/2\pi$ on the left side of eq. (7) (the first law) in [6]. Putting
this factor back in and taking two derivatives in affine parameter of that
equation, one obtains eq. (8) of [6] with a factor of $2\pi$ on the left side
instead of on the right side. To match with (135), note that the Wall’s
equations of motion tensor $H_{ab}$ (defined in eq. (2) in [6]) is given by
$H_{ab}=2({E}_{G})_{ab}$.  of the entropy given by Wall (see eqs. (7) and (8)
of [6]).

For the case of general relativity (i.e. using the Lagrangian (83)), we have

$(\delta{E}_{G})_{ab}k^{a}k^{b}=\frac{1}{16\pi}\delta
R_{ab}k^{a}k^{b}=-\frac{1}{16\pi}\mathscr{L}_{k}\delta\vartheta$ (136)

where the linearized Raychaudhuri equation was used in the second equality.
Thus, we have

$(\delta{E}_{G})_{ab}\xi^{a}\xi^{b}=-\frac{1}{16\pi}\kappa^{2}V^{2}\mathscr{L}_{k}\delta\vartheta=\frac{1}{16\pi}(\kappa-\mathscr{L}_{\xi})(\kappa
V\delta\vartheta).$ (137)

Comparing with (117), we obtain

$\delta P(g,\xi;\mathscr{L}_{\xi}g)=\frac{1}{16\pi}\kappa V\delta\vartheta.$
(138)

Substituting this result in (119), we find that for general relativity, we
have

$\delta\textbf{S}_{\rm
DW}=\delta\textbf{S}+\frac{1}{4}V\delta\vartheta\bm{\epsilon}^{(n-2)}.$ (139)

Thus, in general relativity the term arising from $P$ in $\textbf{S}_{\rm DW}$
cancels the term arising from $\textbf{B}_{\mathcal{H}}$ in our formula (71)
for S, and we are left with only the contribution from Q, which yields the
Bekenstein–Hawking entropy. Thus, the Dong–Wall entropy equals the
Bekenstein–Hawking entropy in general relativity.

The type of cancellation between the $\textbf{B}_{\mathcal{H}}$ and $P$ terms
that occurs in the calculation of the Dong–Wall entropy in general relativity
does not occur for higher derivative theories. In particular, for the
Lagrangian

$\textbf{L}=R_{ab}R^{ab}\bm{\epsilon},$ (140)

the equations of motion are

$\displaystyle\begin{split}({E}_{G})^{ab}={}&\tensor{R}{{}^{(a}_{cde}}\Big{(}g^{b)d}R^{ec}-R^{b)e}g^{cd}\Big{)}\\\
{}&-\nabla_{d}\nabla_{e}\Big{(}g^{d(a}R^{b)e}-g^{de}R^{ab}-g^{ab}R^{de}+g^{e(a}R^{b)d}\Big{)}-\frac{1}{2}R_{de}R^{de}g^{ab}.\end{split}$
(141)

This can be put in the form

$(\delta{E}_{G})_{ab}\xi^{a}\xi^{b}=\kappa\mathscr{L}_{\xi}P(g,\xi;\delta
g)-\mathscr{L}_{\xi}^{2}P(g,\xi;\delta g)$ (142)

with

$\displaystyle\begin{split}P(g,\xi;\delta g)={}&2\xi^{n}N^{m}\delta
R_{mn}-\frac{1}{2}g^{bd}q^{mn}\nabla_{m}\nabla_{n}\delta
g_{bd}+g^{bd}q^{mn}\nabla_{m}\nabla_{b}\delta g_{nd}\\\
{}&-R_{fmqe}N^{f}\xi^{m}N^{q}\xi^{e}g^{bd}\delta
g_{bd}-g^{bd}q^{mn}R_{fbme}\xi^{e}N^{f}\delta
g_{nd}-q^{mn}\tensor{R}{{}^{f}_{bme}}N^{b}\xi^{e}\delta g_{nf}\\\
{}&-2N^{d}\xi^{f}q^{mn}\nabla_{m}\nabla_{d}\delta
g_{nf}+N^{m}\xi^{f}q^{bd}\nabla_{b}\nabla_{d}\delta
g_{fm}-q^{pn}R_{mnpe}N^{m}\xi^{e}g^{ab}\delta g_{ab}\end{split}$ (143)

where $q^{ab}=g^{ab}-2\xi^{(a}N^{b)}$ is the orthogonal projector to $\xi^{a}$
and $N^{a}$. In this case, it is easily seen that contribution to
$\textbf{S}_{\rm DW}$ from $P$ does not cancel the contribution of
$\textbf{B}_{\mathcal{H}}$ to S (see (98) with $\alpha_{1}=\alpha_{3}=0$ and
$\alpha_{2}=1$).

We now evaluate the perturbed Dong–Wall entropy $\delta S_{\rm
DW}[\mathcal{C}]$ for the Lagrangian (140), evaluated on an arbitrary cross
section $\mathcal{C}$. Taking the Lie derivative of (143), we find, after some
algebra, that $\mathscr{L}_{\xi}P(g,\xi;\delta g)$ can be put in the form

$\displaystyle\begin{split}\mathscr{L}_{\xi}P(g,\xi;\delta
g)~{}\hat{=}~{}{}&2\xi^{n}N^{m}\mathscr{L}_{\xi}\delta
R_{mn}+2q^{mn}\nabla_{m}(\delta R_{nf}\xi^{f})\\\
{}&+\frac{1}{2}q^{mn}\nabla_{m}\nabla_{n}(g^{ab}\mathscr{L}_{\xi}\delta
g_{ab})-q^{pn}R_{mnpe}N^{m}\xi^{e}g^{ab}\mathscr{L}_{\xi}\delta
g_{ab}+g^{ab}\mathscr{L}_{\xi}\delta g_{ab}R_{mn}\xi^{m}N^{n}.\end{split}$
(144)

We now substitute (144) and (99) (with $\alpha_{1}=\alpha_{3}=0$ and
$\alpha_{2}=1$) into (119) and integrate the resulting $(n-2)$-form over
$\mathcal{C}$. The computations needed for this are most readily done using
Gaussian null coordinates based upon the cross-section $\mathcal{C}$ and the
Killing parameterization. The metric in these coordinates takes the form (see
e.g. [32])161616 The Killing Gaussian null coordinates $v,r$ in (145) are
related to the affine Gaussian null coordinates $V,\rho$ of (81) by
$V=e^{\kappa v}$, $\rho=\kappa^{-1}e^{-\kappa v}r$. The quantities $\beta_{A}$
and $\gamma_{AB}$ of (145) are the same quantities that appear in (81), but
$\tilde{\alpha}$ is a different quantity from $\alpha$.

$ds^{2}=2dvdr-2r\tilde{\alpha}dv^{2}-2r\beta_{A}dvdx^{A}+\gamma_{AB}dx^{A}dx^{B}$
(145)

with $\xi^{a}=(\partial/\partial v)^{a}$ and
$\tilde{\alpha}~{}\hat{=}~{}\kappa$. Here $\mathcal{H}$ corresponds to $r=0$
and the cross-section $\mathcal{C}$ of $\mathcal{H}$ corresponds to $v=0$. We
can express $N^{a}$ and $q^{ab}$ on $\mathcal{H}$ in terms of $\xi^{a}$,
$l^{a}\equiv(\partial/\partial r)^{a}$, and $\beta_{a}=\beta_{A}(dx^{A})_{a}$
as follows:

$\displaystyle N^{a}~{}\hat{=}~{}{}$ $\displaystyle
l^{a}+\frac{1}{2\kappa}\beta^{a}-\frac{1}{8\kappa^{2}}\beta^{p}\beta_{p}\xi^{a}$
(146) $\displaystyle q^{ab}~{}\hat{=}~{}{}$
$\displaystyle\gamma^{ab}-\frac{1}{\kappa}\xi^{(a}\beta^{b)}+\frac{1}{4\kappa^{2}}\beta_{p}\beta^{p}\xi^{a}\xi^{b}\,.$
(147)

Using these relations for $N^{a}$ and $q^{ab}$ to rewrite (144), we find

$\displaystyle\begin{split}\frac{4\pi}{\kappa}\mathscr{L}_{\xi}P(g,\xi;\delta
g)\bm{\epsilon}^{(n-2)}={}&2\pi\mathscr{L}_{\xi}\delta\Big{[}\bm{\epsilon}^{(n-2)}\Big{(}4\xi^{m}l^{c}R_{mc}-g^{ab}\mathscr{L}_{\xi}g_{ab}\gamma^{pq}\nabla_{p}l_{q}\Big{)}\Big{]}\\\
{}&+\frac{2\pi}{\kappa}{D}_{q}\Big{(}\bm{\epsilon}^{(n-2)}\Big{[}2\delta
R_{nf}\xi^{f}\gamma^{qn}-\frac{1}{\kappa}\beta^{q}\xi^{m}\xi^{f}\delta
R_{mf}\\\ {}&+\frac{1}{2}D^{q}(g^{ab}\mathscr{L}_{\xi}\delta
g_{ab})-\frac{1}{2}\beta^{q}g^{ab}\mathscr{L}_{\xi}\delta
g_{ab}\Big{]}\Big{)}\,.\end{split}$ (148)

The last two lines are a total spatial divergence, which vanish when
integrated over $\mathcal{C}$. The entropy $(n-2)$-form for the Lagrangian
(140) is given by (99) with $\alpha_{1}=\alpha_{3}=0$ and $\alpha_{2}=1$,
i.e.,

$\displaystyle\begin{split}{S}_{a_{1}\cdots
a_{n-2}}={}&-\frac{2\pi}{\kappa}\Big{(}{\epsilon}_{cda_{1}\cdots
a_{n-2}}\Big{[}2g^{ce}R^{df}\nabla_{[e}\xi_{f]}+(g^{cm}\nabla^{d}R-2\nabla^{c}R^{md})\xi_{m}\Big{]}\\\
{}&+{\epsilon}^{(n-2)}_{a_{1}\cdots
a_{n-2}}(R_{ad}\xi^{a}N^{d}g^{bc}+R^{bc})\mathscr{L}_{\xi}g_{bc}\Big{)}\,.\end{split}$
(149)

Taking a variation of this quantity, pulling it back to $\mathcal{C}$, and
evaluating it in Gaussian null coordinates in a manner similar to what was
done above for $P$, we find, after some algebra, that

$\displaystyle\begin{split}\delta\textbf{S}~{}\hat{=}~{}{}&2\pi\Big{(}\delta\Big{[}\bm{\epsilon}^{(n-2)}\Big{(}4R_{mn}\xi^{m}l^{n}-g^{ab}\mathscr{L}_{\xi}g_{ab}\gamma^{pq}\nabla_{p}l_{q}\Big{)}\Big{]}\\\
{}&-\frac{1}{\kappa}\mathscr{L}_{\xi}\delta\Big{[}\bm{\epsilon}^{(n-2)}\Big{(}4R_{mn}\xi^{m}l^{n}-g^{ab}\mathscr{L}_{\xi}g_{ab}\gamma^{pq}\nabla_{p}l_{q}\Big{)}\Big{]}\\\
{}&-\frac{2}{\kappa}{D}_{m}\Big{[}\bm{\epsilon}^{(n-2)}\delta
R_{dn}\xi^{d}\gamma^{mn}\Big{]}\Big{)}\,.\end{split}$ (150)

Note that the form of this equation—where the second term is minus the Lie
derivative of the first term and the third term is exact—is to be expected in
view of (130). The last line of (150) vanishes when integrated over
$\mathcal{C}$. Adding (148) and (150) and integrating over $\mathcal{C}$, we
find that the Dong–Wall entropy is

$\displaystyle\delta S_{\rm DW}[\mathcal{C}]={}$ $\displaystyle
2\pi\delta\int_{\mathcal{C}}\bm{\epsilon}^{(n-2)}(4R_{mc}\xi^{m}l^{c}-2K\bar{K})$
(151)

where

$\displaystyle K={}$ $\displaystyle\frac{1}{2}g^{ab}\mathscr{L}_{\xi}g_{ab}$
(152) $\displaystyle\bar{K}={}$ $\displaystyle\gamma^{pq}\nabla_{p}l_{q}$
(153)

are the traces of the extrinsic curvatures in the $\xi$ and $l$ directions,
respectively. This agrees with Wall’s formula [6] and with the linearization
of Dong’s formula [5].

Finally, we note that if we integrate (124) between a cross-section
$\mathcal{C}_{1}$ and another cross-section $\mathcal{C}_{2}$ lying to the
future of $\mathcal{C}_{1}$, we obtain

$\delta S_{\rm DW}[\mathcal{C}_{2}]-\delta S_{\rm
DW}[\mathcal{C}_{1}]=4\pi\int_{\mathcal{H}_{12}}\delta P\bm{\epsilon}^{(n-1)}$
(154)

where $\mathcal{H}_{12}$ denotes the portion of $\mathcal{H}$ bounded by
$\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ and we have written $\delta P$ instead
of $\delta P(g,\xi;\mathscr{L}_{\xi}g)$ to make the notation less cumbersome.
Thus, the Dong–Wall entropy will satisfy the second law of black hole
mechanics at first order if and only if $\delta P(x)\geq 0$ for all
$x\in{\mathcal{H}}$. However, from (117), we see that if the matter stress-
energy tensor satisfies the null energy condition, then we have

$-e^{\kappa v}\mathscr{L}_{\xi}\left(e^{-\kappa v}\delta
P\right)=(\kappa-\mathscr{L}_{\xi})\delta
P=(\delta{E}_{G})_{ab}\xi^{a}\xi^{b}=\frac{1}{2}(\delta
T_{ab})\xi^{a}\xi^{b}\geq 0\,$ (155)

where $v$ is the Killing parameter along the null geodesics (see (39)). Thus,
we have

$\mathscr{L}_{\xi}\left(e^{-\kappa v}\delta P\right)\leq 0.$ (156)

For any $x\in{\mathcal{H}}$, let $\gamma$ denote the null geodesic of
$\mathcal{H}$ passing through $x$. Integrating this equation along $\gamma$
with respect to $v$ from $x$ to infinity and making the additional assumption
that $\lim_{v\to\infty}e^{-\kappa v}\delta P=0$, we obtain

$\delta P(x)\geq 0\,.$ (157)

Thus the Dong–Wall entropy also satisfies the second law of black hole
mechanics. However, as with the Bekenstein–Hawking entropy as discussed at the
end of subsection V.1, the Dong–Wall entropy may increase in anticipation of
matter that will be thrown into the black hole at a later time.

##  VII The Generalized second law and QNEC

The generalized second law (GSL) of thermodynamics states that the sum of the
entropy of a black hole and the entropy of matter outside the black hole never
decreases with time. The GSL was originally proposed by Bekenstein [3] in
order to rescue the second law of thermodynamics in view of the fact that the
total entropy of matter in the universe will decrease if matter falls into a
black hole; by assigning an entropy to a black hole, Bekenstein argued that
the total (“generalized”) entropy would be non-decreasing even if the entropy
of matter decreases. Bekenstein’s proposal was originally made within the
context of classical black hole physics, where it is not consistent because,
classically, black holes have vanishing thermodynamic temperature, but they
would have to be assigned a non-zero temperature in order to satisfy the first
law of thermodynamics with a finite entropy. However, shortly thereafter,
Hawking [2] discovered that within the context of semiclassical physics, black
holes radiate thermally at a temperature $T=\kappa/2\pi$. This not only
removed a major inconsistency of Bekenstein’s proposal but it made the GSL the
only viable candidate for a thermodynamic second law, since quantum fields can
violate all local energy conditions and the second law of black hole mechanics
will be violated if there is a flux of negative energy into the black hole—as
occurs during the Hawking radiation process. Thus, matter entropy and black
hole entropy can individually decrease in various circumstances; only their
sum has a possibility of being non-decreasing in all circumstances.

There is considerable circumstantial evidence in support of the GSL. In
particular, gedankenexperiments attempting to violate the GSL fail [33] in a
manner reminiscent of the failure of gedankenexperiments designed to violate
the ordinary second law. Of course, one would like to obtain a precise
formulation and proof of the GSL rather than just having circumstantial
evidence in support of it. However, even in the context of non-gravitational
physics, there is no precise formulation and proof of the ordinary second law
of thermodynamics. Rather, there is a basic argument for the ordinary second
law, which traces back to Boltzmann. This argument involves the introduction
of a coarse-grained observable to define the entropy of matter. In the context
of classical physics, the entropy of a state is defined to be the logarithm of
the volume of phase space corresponding to the value of the coarse-grained
observable in that state. If the entropy of a state is not at its maximum
value, dynamical evolution is very likely to evolve that state to a state of
higher entropy. If the entropy of a state is at its maximum, it is very likely
to remain in a maximum entropy state for much longer than any realistic
observation time. Thus, this argument suggests that if one starts a system in
a “special” (i.e., a low entropy) state, one will observe an entropy increase
until it reaches an “equilibrium” (i.e., maximum entropy) state, at which
point one will observe no further changes. A similar argument applies in the
quantum context, with the phase space volume of the value of the coarse-
grained observable replaced by the dimension of the eigensubspace of the
eigenvalue of the coarse-grained observable [23]. It should be emphasized that
the von Neumann entropy of the quantum state does not play any role in this
argument for justifying the validity of the ordinary second law in the quantum
context [23]. Indeed, for any isolated system in any quantum state, the von
Neumann entropy is constant, but one would still expect to observe an entropy
increase in such a system if one started it in a “special” state.

It is therefore quite remarkable that a precise formulation and argument/proof
of the GSL has been given for semiclassical general relativity, using von
Neumann entropy as the notion of entropy of matter outside the black hole. In
particular, Wall [34] has shown that

$\displaystyle\frac{d}{dV}\left(\frac{A}{4}+S_{\rm vN}\right)\geq{}$
$\displaystyle 0.$ (158)

Here $A(V)/4$ is the Bekenstein–Hawking entropy of the black hole on the
cross-section $\mathcal{C}$ corresponding to affine time $V$ and $S_{\rm
vN}(V)$ is the von Neumann entropy of the quantum field matter outside the
black hole at a “time” corresponding to a spacelike hypersurface extending
from $\mathcal{C}$ to spatial infinity. In fact, $S_{\rm vN}(V)$ is ill
defined on account of the infinite entanglement of the quantum field across
the horizon. However, von Neumann entropy differences between states should be
well defined, and that is what should be needed to make sense of $dS_{\rm
vN}/dV$ in the GSL.

In this section, we will investigate the validity of the GSL for first order
perturbations of a stationary black hole in a general theory of gravity, using
both our definition of entropy, $S[\mathcal{C}]$, and the Dong–Wall
definition, $S_{\rm DW}[\mathcal{C}]$. Our analysis will be based upon the
quantum null energy condition (QNEC) for quantum fields in fixed classical
background, first introduced in [35]. QNEC asserts that if $\mathcal{C}$ is a
cross-section of a null hypersurface $\mathcal{N}$ such that the expansion and
shear of $\mathcal{N}$ vanish on $\mathcal{C}$, then for any point $p$ on
$\mathcal{C}$ we have [35]

$\displaystyle\sqrt{h}\langle T_{ab}\rangle k^{a}k^{b}\geq{}$
$\displaystyle\frac{1}{2\pi}\frac{\delta^{2}}{\delta V(p)^{2}}S_{\rm vN},$
(159)

where $h$ is the determinant of the induced metric on $\mathcal{C}$, $V(y)$ is
the affine parameter (a function of the coordinates in the transverse
direction $y$), and $S_{\rm vN}$ is the von Neumann entropy of the quantum
field “outside” the null hypersurface (where “outside” means on a spacelike
hypersurface whose future Cauchy horizon coincides with $\mathcal{N}$). For
simplicity, we will restrict consderation here to the case where the
deformation of $\mathcal{C}$ is a uniform translation in $V$. Integration of
(159) over $\mathcal{C}$ yields

$\int_{\mathcal{C}}\langle T_{ab}\rangle
k^{a}k^{b}\geq\frac{1}{2\pi}\frac{d^{2}}{dV^{2}}S_{\rm vN}.$ (160)

QNEC has been rigorously proven/formulated in [36] within the general
framework of half-sided modular inclusions and relative entropy, and this
setting is expected to cover the case of a bifurcate Killing horizon in curved
spacetime as considered here. We omit a more precise technical discussion of
the setting and, for purposes of readability, we stick to informal notations
such as (160). In the remainder of this section, we will assume that (160)
holds for an arbitrary cross-section $\mathcal{C}$ of a bifurcate Killing
horizon.

Equation (160) holds in a fixed background spacetime. We assume that for an
unperturbed stationary black hole with a quantum field in a Hartle–Hawking-
like vacuum state, both sides of (160) vanish. We now consider perturbing the
quantum state, taking back-reaction into account by treating $\delta\langle
T_{ab}\rangle$ as an “external matter perturbation” as in subsection V.1. Then
we have

$2\delta({E}_{G})_{ab}k^{a}k^{b}=\delta\langle T_{ab}\rangle k^{a}k^{b}.$
(161)

Consequently, when back-reaction is taken into account, QNEC in the form (160)
yields

$\int_{\mathcal{C}}2\delta({E}_{G})_{ab}k^{a}k^{b}\geq\frac{1}{2\pi}\frac{d^{2}}{dV^{2}}\delta
S_{\rm vN}.$ (162)

The GSL for the Dong–Wall definition of black hole entropy can now be obtained
as follows. Using (135), we obtain

$-\frac{1}{2\pi}\frac{d^{2}}{dV^{2}}\delta S_{\rm
DW}\geq\frac{1}{2\pi}\frac{d^{2}}{dV^{2}}\delta S_{\rm vN}\,,$ (163)

i.e.,

$\frac{d^{2}}{dV^{2}}\left[\delta S_{\rm DW}+\delta S_{\rm vN}\right]\leq
0\,.$ (164)

Integrating this equation from an initial cross-section to $V=\infty$ and
making the additional assumption that $\lim_{V\to\infty}d\delta S_{\rm
DW}/dV=\lim_{V\to\infty}d\delta S_{\rm vN}/dV=0$, we obtain the desired GSL

$\frac{d}{dV}\left[\delta S_{\rm DW}+\delta S_{\rm vN}\right]\geq 0\,.$ (165)

This derivation corresponds closely to Wall’s derivation of the GSL for
general relativity [34], although he based his argument on the monotonicity of
the relative entropy rather than QNEC. The above argument shows that this form
of the GSL corresponds to an integrated form of QNEC.

The GSL for our definition of black hole entropy can be obtained by evaluating
the physical process version of the first law (104) on cross-sections
$\mathcal{C}$ of constant $V$ and differentiating with respect to $V$. We
obtain

$\frac{\kappa}{2\pi}\frac{d}{dV}\delta S=\int_{\mathcal{C}}\delta\langle
T_{ab}\rangle k^{a}\xi^{b}=\kappa V\int_{\mathcal{C}}\delta\langle
T_{ab}\rangle k^{a}k^{b}\geq\frac{\kappa}{2\pi}V\frac{d^{2}}{dV^{2}}\delta
S_{\rm vN}.$ (166)

This relation can be put in a GSL form by defining a modified von Neumann
entropy of matter, $\tilde{S}_{\rm vN}$, by

$\tilde{S}_{\rm vN}\equiv{S}_{\rm vN}-V\frac{d}{dV}{S}_{\rm vN}.$ (167)

Thus, $\tilde{S}_{\rm vN}$ agrees with ${S}_{\rm vN}$ in stationary eras but
has a dynamical correction when ${S}_{\rm vN}$ changes with time. Since

$\frac{d}{dV}\tilde{S}_{\rm vN}=-V\frac{d^{2}}{dV^{2}}{S}_{\rm vN},$ (168)

we see that (166) takes the GSL form171717The idea of defining a modified von
Neumann entropy by (167) so as to be able to write (166) in the form (169) was
suggested to us by Jon Sorce. It would be interesting to see if this
definition of modified von Neumann entropy can be motivated by additional,
independent arguments. In this regard, Igor Klebanov has pointed out to us
that the expression $V\frac{d}{dV}{S}_{\rm vN}$ is formally equal to the
monotonic $c$-function for $2$-dimensional CFTs defined by [37]. The lightray
algebras of a Killing horizon have connections with a chiral CFT, so this
might be more than a coincidence.

$\frac{d}{dV}\left[\delta S+\delta\tilde{S}_{\rm vN}\right]\geq 0\,.$ (169)

This version of the GSL directly corresponds to QNEC, rather than an
integrated form of QNEC.

###### Acknowledgements.

We thank Tom Faulkner, Igor Klebanov, Harvey Reall, and Jon Sorce for
discussions. SH and RMW thank the Erwin Schrödinger Institut in Vienna, where
part of this work has been completed, for its hospitality and support. We
thank Thomas Endler from MPI-MiS for creating Fig. 1 based on our suggestions
and for giving us permission to include it in this publication. SH is grateful
to the Max-Planck Society for supporting the collaboration between MPI-MiS and
Leipzig U., grant Proj. Bez. M.FE.A.MATN0003. The research of RMW and VGZ was
supported in part by NSF grant PHY-2105878 and Templeton Foundation grant
62845 to the University of Chicago.

##  Appendix A $S$ is the Area of the Apparent Horizon to First Order in
General Relativity

The apparent horizon on an arbitrary Cauchy surface $\Sigma$ is the boundary
of the region containing outer trapped surfaces that lie within $\Sigma$. The
apparent horizon is a marginally outer trapped surface, i.e., its outgoing
null expansion vanishes. If the null energy condition holds and if weak cosmic
censorship is valid, then an apparent horizon must lie within a black hole, so
apparent horizons are very useful for determining the presence of a black hole
given only data on $\Sigma$. For a stationary black hole, the apparent horizon
on any Cauchy surface $\Sigma$ is given by the intersection of $\Sigma$ with
the event horizon $\mathcal{H}$.

For a nonstationary black hole, the location of an apparent horizon will
depend on the choice of Cauchy surface, and the area of an apparent horizon at
a “time” (i.e., choice of Cauchy surface) corresponding to a cross-section
$\mathcal{C}$ of the horizon will not, in general, be a meaningful concept.
However, for first order perturbations of a stationary black hole with
bifurcate Killing horizon, the notion of the area of an apparent horizon at a
time corresponding to $\mathcal{C}$ is a well defined concept. To see this, we
note that, as proven in section 2.1 of [11], for a stationary black hole with
bifurcate Killing horizon, any horizon cross-section $\mathcal{C}$ is a
strictly stably outermost marginally trapped surface in the terminology of
[38]. This implies that given any function, $f$, on $\mathcal{C}$, there
exists a unique function, $\psi$ on $\mathcal{C}$ such that if we deform
$\mathcal{C}$ infinitesimally by $\psi l^{a}$ (with $l^{a}$ a fixed vector
field tranverse to the horizon), then to first order, the outgoing expansion
of $\delta\mathcal{C}$ will be $f$. Now suppose we perturb the black hole so