Generalized Entropy in Higher Curvature Gravity And Entropy of Algebra of
Observables

Mohd Ali<EMAIL_ADDRESS>Vardarajan
<EMAIL_ADDRESS>
The Indian Institute of Science Education and Research (IISER),
Pune, India - 411008.

Abstract

Recently, Chandrasekaran, Penington and Witten (CPW) have shown that the
generalized entropy of the Schwarzschild black hole at the bifurcation surface
equals the entropy of an extended von Neumann algebra of quantum observables
in the black hole exterior, in semiclassical Einstein gravity. They also
derive a version of the Generalized Second law. We generalize these results to
a static black hole in an arbitrary diffeomorphism invariant theory of
gravity. Thus, a version of the Generalized second law for an arbitrary
diffeomorphism invariant theory of gravity follows.

##  I Introduction

Generalized entropy in Einstein gravity was introduced by Bekenstein in order
that the second law of thermodynamics be valid near black holes [1], [2]. He
suggested the Generalized Second Law (GSL) holds, namely that the generalized
entropy increases under future evolution along the black hole horizon. The
generalized entropy for a quantum black hole coupled to matter, in the
semiclassical $G\to 0$ limit was defined to be

$S_{gen}=<\frac{A}{4\hbar G}>+S_{QFT},$ (I.1)

where $A$ is the black hole horizon area and $S_{QFT}$ is the entanglement
entropy of the quantum fields in the black hole exterior. As has been pointed
out, both the terms in (I.1) are individually ultraviolet (UV) divergent (the
first term due to loop effects which renormalize $G$ and the second term,
entanglement entropy, which is UV divergent), but the sum is UV finite
[3]—[10]. The GSL for Einstein gravity was proved by Wall [34]. In an
arbitrary diffeomorphism invariant theory of gravity, we can analogously
define the generalized entropy for a black hole with quantum fields to be the
sum of its horizon entropy $S$ in that theory and the entanglement entropy of
quantum fields in the black hole exterior,

$S_{gen}=S+S_{QFT}.$ (I.2)

A candidate for the horizon entropy for a stationary black hole is Wald
entropy [14], [17]. The Wald entropy is ambiguous for a non-stationary black
hole — these ambiguities were first discussed in [12]. A linearized GSL
(ignoring gravitons) was proved for Lovelock gravity in [11].

In an interesting recent development, the generalized entropy in Einstein
gravity appeared as the entropy of a von Neumann algebra of observables in the
black hole exterior where the black hole is the Schwarzschild solution in
Einstein gravity [24]. Leutheusser and Liu [25] (see also [26] ) studied the
holographic boundary operator algebra of the CFT dual to gravity in the
asymptotically anti-de Sitter (AdS) black hole spacetime. They found a
emergent Type III1 von Neumann algebra for single trace operators in the large
$N$ limit of the boundary CFT (see also [27]). Later, Witten [33] showed that
by enlarging the set of operators to include the boundary Hamiltonian and
enlarging the Hilbert space to include a degree of freedom corresponding to a
boundary time shift, the algebra becomes a Type II∞ von Neumann algebra. In
[23], Chandrasekaran, Longo, Penington and Witten discuss how this
construction can be generalized to asymptotically flat black holes. Starting
with quantum fields in the exterior of a Schwarzschild black hole, by
including the ADM Hamiltonian in the set of operators and the time shift
degree of freedom in the Hilbert space, the algebra of these operators in the
bulk spacetime is Type II∞. A Type II von Neumann algebra (unlike a Type III
algebra) has a notion of a (renormalized) trace, density matrix and
corresponding von Neumann entropy associated with the density matrix (for a
review of von Neumann algebras and their classification, see [29]). In [24],
Chandrasekaran, Penington and Witten (CPW) showed that the entropy of
semiclassical states in the boundary algebra is equal to the generalized
entropy of the black hole at the bifurcation surface. This has a bulk
interpretation — in semiclassical gravity in the $G\to 0$ limit, the
generalized entropy is the entropy of the algebra of operators in the black
hole exterior. Finally, CPW use the monotonicity of the entropy of the algebra
under trace preserving inclusions to prove a version of the Generalized Second
Law (GSL) showing monotonicity of generalized entropy between early and late
times.

In this paper, we study the generalization of these results to an arbitrary
diffeomorphism invariant theory of gravity by also including gravitons. We
first write the black hole entropy in such a theory at an arbitrary horizon
cut, which is the Wald entropy [14], [17] with an extra term representing an
ambiguity in the Wald entropy for a non-stationary black hole [12]. We work in
semiclassical gravity. We consider a static (therefore stationary) black hole
that is slightly perturbed due to infalling quantum matter and gravitons. In
the limit when the cut $v\to\infty$, the perturbed black hole approaches a
stationary black hole ($v$ is the affine parameter along the null generator of
the horizon). We compute the entropy at $v\to\infty$ minus the entropy at the
bifurcation surface up to quadratic order in the perturbation, and we take
into account the contribution due to gravitons to the stress-energy tensor. It
is possible to fix the ambiguity in the Wald entropy in such a way that this
difference of entropies takes a simplified form proportional to the $vv$
component of the stress-energy tensor. Generalizing the computations of CPW
[24], we find the difference of _generalized_ entropies at $v\to\infty$ and at
the bifurcation surface to be proportional to the relative entropy, which is
non-negative. By computing the entropy of the extended von Neumann algebra of
the black hole exterior [33], we show that the entropy of the algebra is the
generalized entropy at the bifurcation surface just as in [24] for Einstein
gravity. All the above constructions can be done for asymptotically flat
static black holes [23]. Finally, we discuss the monotonicity result of CPW
[24] who show that monotonicity of relative entropy under trace preserving
inclusions can be used to argue that the generalized entropy at late times is
more than that at early times. For this, we need asymptotically AdS black
holes with a holographic dual, but modulo this change, the monotonicity result
of CPW goes through for the generalized entropy of a black hole in an
arbitrary diffeomorphism invariant theory of gravity.

In section II, we discuss the difference of entropies at $v\to\infty$ and at
the bifurcation surface for a slightly perturbed black hole. We use boost
arguments which we summarize in section II.1 to simplify this difference of
entropies. By expanding the Raychaudhuri equation order by order in the
perturbation parameter, we compute the change in entropies to quadratic order
in section II.4 both without graviton contributions to the stress-energy
tensor, and with the graviton contribution included. In section III, we
discuss the entropy of the algebra of operators in the black hole exterior. We
first summarize salient results from earlier papers of Witten [33],
Chandrasekaran, Longo, Penington and Witten [23] and CPW [24] who discussed
how the entropy of the algebra was related to the generalized entropy in
Einstein gravity. We then generalize these results to an arbitrary
diffeomorphism invariant theory of gravity. We conclude in section IV with a
discussion.

##  II Entropy Change in Higher Curvature Theory

Figure 1: Accretion of matter across the horizon in an asymptotically flat
black hole.

In what follows and the rest of the paper, we work in units where $G=1$.
Consider an entropy function for a black hole horizon in an arbitrary
diffeomorphism invariant theory of gravity with matter,

$S=\frac{1}{4}\int\rho\sqrt{h}d^{D-2}x$ (II.3)

where $\rho=1+\rho_{w}+\Omega$, where $h$ is the induced metric on the $D-2$
dimensional transverse cut on the horizon and $1+\rho_{w}$ is the Wald local
entropy density [14], [17]. As is well-known, the Wald entropy is
unambiguously defined for a stationary black hole, but suffers from
ambiguities when evaluated on a non-stationary black hole. These ambiguities
were pointed out by Jacobson, Kang and Myers (JKM) [12] and by Iyer and Wald
[17]. $\Omega$ is a correction to Wald entropy density representing this JKM
ambiguity, such that it vanishes for a stationary solution. We are interested
in a black hole spacetime with a regular bifurcation surface $\mathcal{B}$,
which is slightly perturbed from stationarity by throwing some quantum matter.
Let $v$ be an affine parameter along the null generator of the future horizon,
such that $v=0$ is the bifurcation surface as shown in Figure 1. Then, the
entropy at an arbitrary horizon cut (given by $v=constant$) is

$S[v]=\frac{1}{4}\int_{v}\rho\sqrt{h}d^{D-2}x$ (II.4)

where the subscript $v$ in the integral indicates that the integral is over
the transverse space at fixed $v$ on the horizon. We can compute the change in
the entropy along the horizon,

$\frac{dS}{dv}=\frac{1}{4}\int_{v}\sqrt{h}d^{D-2}x\Big{(}\frac{d\rho}{dv}+\theta\rho\Big{)}$
(II.5)

where expansion $\theta\equiv\frac{1}{\sqrt{h}}\frac{d\sqrt{h}}{dv}$. To
compute change in the entropy from $v=0$ to $v\rightarrow\infty$, we can
integrate both the sides with respect to $v$. This yields

$\Delta
S=\frac{1}{4}\int_{0}^{\infty}dv\int_{v}\sqrt{h}d^{D-2}x\Big{(}\frac{d\rho}{dv}+\theta\rho\Big{)}.$
(II.6)

Here, $\Delta S=S(\infty)-S(0)$. Using integration by parts,

$\Delta
S=\frac{1}{4}\int_{v}\Big{\\{}v\sqrt{h}\Big{(}\frac{d\rho}{dv}+\theta\rho\Big{)}\Big{\\}}\Big{|}_{v=0}^{v\rightarrow\infty}d^{D-2}x\hskip
5.69054pt-\frac{1}{4}\int_{0}^{\infty}dv\int\sqrt{h}d^{D-2}xv\Big{\\{}\frac{d^{2}\rho}{dv^{2}}+\frac{d\theta}{dv}\rho+2\frac{d\rho}{dv}\theta+\theta^{2}\rho\Big{\\}}.$
(II.7)

We will assume that $\sqrt{h}\Big{(}\frac{d\rho}{dv}+\theta\rho\Big{)}$ goes
to zero faster than $\frac{1}{v}$, therefore the first term in the above
equation is identically zero and we are left with

$\Delta
S=-\frac{1}{4}\int_{0}^{\infty}dv\int\sqrt{h}d^{D-2}xv\Big{\\{}\frac{d^{2}\rho}{dv^{2}}+\frac{d\theta}{dv}\rho+2\frac{d\rho}{dv}\theta+\theta^{2}\rho\Big{\\}}.$
(II.8)

To compute $\Delta S$ order by order, we will now consider the metric
perturbation sourced by a stress-energy tensor of order $\epsilon$, i.e,
$<T_{vv}>\hskip 5.69054pt\sim O(\epsilon)$. We will also assume that the
perturbation is about a stationary black hole background and at late times,
the black hole will again settle down into a stationary state. The
perturbation expansion we are interested in is

$g_{\mu\nu}=g^{(0)}_{\mu\nu}+\epsilon^{\frac{1}{2}}g^{(\frac{1}{2})}_{\mu\nu}+\epsilon
g^{(1)}_{\mu\nu}+O(\epsilon^{\frac{3}{2}}),$ (II.9)

where the zeroth order term corresponds to the stationary black hole solution
with regular bifurcation surface, the $\sqrt{\epsilon}$ term is due to
quantized graviton fluctuations, and the $\epsilon$ term is due to the
gravitational field of matter or gravitons. We can think of $\epsilon$ as
$\hbar$. We want to emphasize that $\Omega$ vanishes at order
$\sqrt{\epsilon}$ at the bifurcation surface [19], a fact which will be useful
later in the calculations.

###  II.1 Boost Argument

We now use boost arguments first used in [21] and later in [13], [15]. The
metric near any null hypersurface and therefore near the event horizon can be
given in Gaussian null coordinates as

$ds^{2}=2dvdu-u^{2}X(u,v,x^{k})dv^{2}+2u\omega_{i}(u,v,x^{k})dvdx^{i}+h_{ij}(u,v,x^{k})dx^{i}dx^{j}$
(II.10)

where $v$ is an affine parameter along the null generator of the horizon,
$x^{i}$ corresponds to coordinates on the $D-2$ transverse surface (cut) and
$u$ is chosen in a way that $\partial_{v}.\partial_{u}=1$ and
$\partial_{u}.\partial_{i}=0$. In this coordinate system, $u=0$ is the future
horizon and $u=0,v=0$ corresponds to the bifurcation surface $\mathcal{B}$.
These coordinates may not cover the entire spacetime, but the near-horizon
region of any dynamical black hole spacetime can always be written in this
form. Now, the black hole spacetime we consider in this paper is a static
(therefore stationary) black hole spacetime which is a solution in a
diffeomorphism invariant theory of gravity. Then, the black hole event horizon
is a Killing horizon [14]. First consider the case where this horizon is a
Killing horizon with respect to the boost field
$\xi=v\partial_{v}-u\partial_{u}$ (see [13]). This is true for any stationary
black hole spacetime which, near the horizon, looks like a Rindler spacetime,
hence the terminology ‘boost field’. The near-horizon metric of this
stationary black hole will then be boost invariant, i.e the Lie derivative of
the metric $\mathcal{L}_{\xi}g_{\mu\nu}=0$. Then, the near-horizon metric
(II.10) is of the form

$ds^{2}_{st}=2dvdu-u^{2}X(uv,x^{k})dv^{2}+h_{ij}(uv,x^{k})dx^{i}dx^{j}$
(II.11)

Here, $\omega_{i}=0$ since the spacetime is static. This is the most general
form of a static spacetime with a Killing horizon near the horizon. It can
easily be seen from (II.10) that along the horizon, any non-zero tensor $A$,
which is constructed out of metric components can always be written as
$A=\partial^{n}_{v}\partial^{m}_{u}B$, where $m,n$ are integers and $B$ is
constructed out of metric components $X,\omega_{i},h_{ij}$ and their
derivatives with respect to $\nabla_{i}$ . Then, we can associate a boost
weight with these tensors as boost weight $=$ $\\#v$ index $-$ $\\#u$ index.
Furthermore, we can write the schematic form for the $vv$ component of any
2-tensor $A_{vv}$ constructed from metric components as

$A_{vv}=\tilde{X}\partial_{v}^{2}Y+C\partial_{v}A\partial_{v}B.$ (II.12)

Here, $\tilde{X},Y,C,A,B$ have boost weight zero and are constructed out of
metric components. Now for the stationary black hole spacetime, the above
equation reduces down to

$A_{vv}|_{st}=u^{2}\tilde{X}\partial_{uv}^{2}Y+u^{2}C\partial_{uv}A\partial_{uv}B.$
(II.13)

This is because the stationary black hole has a Killing symmetry which reduces
on the horizon to a scaling symmetry under $u\to pu$ and $v\to v/p$. Thus, the
metric components in the stationary case only depend on $uv$ at the horizon.
This implies that the $vv$ component of any 2-tensor $A_{vv}$ constructed from
metric components in a stationary black hole spacetime vanishes at the future
horizon i.e at $u=0$.

$A_{vv}|_{st}^{u=0}=0.$ (II.14)

Now, the $vv$ component of the equation of motion for any higher curvature
theory takes the following form,

$R_{vv}+H_{vv}=8\pi T_{vv}$ (II.15)

where $H_{vv}$ corresponds to a higher curvature contribution to the equation
of motion. Using (II.13) for the stationary black hole $R_{vv}=0$ and
$H_{vv}=0$, this implies that

$T_{vv}|^{u=0}=0$ (II.16)

for any _classical_ matter stress-energy tensor. Furthermore, whenever a $v$
derivative acts on the stationary background metric component, it gives a
factor of $u$ as well, since the metric component depends on $v$ only through
$uv$. Hence such a term will vanish at the future horizon $u=0$. Therefore,
from (II.12), the $vv$ component of any 2- tensor $A_{vv}$ _linear_ in the
metric perturbation at the future horizon $u=0$ can always be written in the
following form,

$A_{vv}|^{u=0}=\partial_{v}^{2}\zeta,$ (II.17)

where $\zeta$ has boost weight zero and is constructed from the background
metric and the linear perturbation over stationarity.

###  II.2 Semi-classical gravity equations

Following Chandrasekaran, Penington and Witten [24] and Wall [34], we will
work in semi-classical gravity where the expectation value of the matter
stress energy tensor is a source term in the gravity equations. Now let us
look at the semiclassical equations of motion. The $vv$ component is

$R_{vv}+H_{vv}=8\pi<T_{vv}>$ (II.18)

Writing this order by order in $\epsilon$, we get:

$\displaystyle\epsilon^{0}:\hskip 71.13188ptR_{vv}+H_{vv}=0$ (II.19)
$\displaystyle\epsilon^{\frac{1}{2}}:\hskip
62.59605ptR^{(\frac{1}{2})}_{vv}+H^{(\frac{1}{2})}_{vv}=0$ (II.20)
$\displaystyle\epsilon^{1}:\hskip
14.22636ptR^{(1)}_{vv}+H^{(1)}_{vv}=8\pi<T^{Q}_{vv}>.$ (II.21)

where $T^{Q}_{vv}=T^{M}_{vv}+t_{vv}$, $<T^{M}_{vv}>\hskip 5.69054pt\sim
O(\epsilon)$ is a matter stress energy tensor and $<t_{vv}>\hskip
5.69054pt\sim O(\epsilon)$, the pseudo-stress-energy tensor of the graviton.
Further, $R^{(\frac{1}{2})}_{vv}$, $H^{(\frac{1}{2})}_{vv}$ are linear in
$g_{\mu\nu}^{(\frac{1}{2})}$ perturbation, and $R^{(1)}_{vv}$, $H^{(1)}_{vv}$
are linear in $g_{\mu\nu}^{(1)}$ perturbation.

###  II.3 Raychaudhuri equation order by order

As is well-known, the Raychaudhuri equation (II.20) plays a key role in the
proof of the second law and we will use it later in our computation. The
Raychaudhuri equation is given by,

$\frac{d\theta}{dv}=-\Big{\\{}\frac{\theta^{2}}{D-2}+\sigma^{\alpha\beta}\sigma_{\alpha\beta}+R_{vv}\Big{\\}}$
(II.22)

Now, if we write it order by order in $\epsilon$, we get,

$\displaystyle\epsilon^{0}:\hskip 202.01456pt\frac{d\theta^{(0)}}{dv}=0$
(II.23) $\displaystyle\epsilon^{\frac{1}{2}}:\hskip
173.5618pt\frac{d\theta^{(\frac{1}{2})}}{dv}=-R^{(\frac{1}{2})}_{vv}$ (II.24)
$\displaystyle\epsilon^{1}:\hskip
5.69054pt\frac{d\theta^{(1)}}{dv}=-\Big{\\{}\frac{\theta^{(\frac{1}{2})}\theta^{(\frac{1}{2})}}{D-2}+\sigma^{\alpha\beta}_{(\frac{1}{2})}\sigma_{\alpha\beta}^{(\frac{1}{2})}+R^{(1)}_{vv}+R^{(\frac{1}{2},\frac{1}{2})}_{vv}\Big{\\}}$
(II.25)

Further using (II.21), (II.25) can be written as

$\frac{d\theta^{(1)}}{dv}=-\Big{\\{}\frac{\theta^{(\frac{1}{2})}\theta^{(\frac{1}{2})}}{D-2}+\sigma^{\alpha\beta}_{(\frac{1}{2})}\sigma_{\alpha\beta}^{(\frac{1}{2})}+8\pi<T^{Q}_{vv}>-H^{(1)}_{vv}+R^{(\frac{1}{2},\frac{1}{2})}_{vv}\Big{\\}}.$
(II.26)

Furthermore, if we compute $R_{vv}^{(\frac{1}{2},\frac{1}{2})}$ in TT
(transverse traceless) gauge at the horizon 333The perturbative expansion of
the Ricci tensor to quadratic order can be found in [16]., we will get,

$R^{(\frac{1}{2},\frac{1}{2})}_{vv}\underset{TT}{=}-\frac{1}{4}\frac{dg^{ij}_{(\frac{1}{2})}}{dv}\frac{dg_{ij}^{(\frac{1}{2})}}{dv}+\frac{1}{2}\frac{d}{dv}\Big{(}g^{ij}_{(\frac{1}{2})}\frac{dg_{ij}^{(\frac{1}{2})}}{dv}\Big{)}.$
(II.27)

Now using the fact that
$\frac{1}{2}\frac{dg_{ij}^{(\frac{1}{2})}}{dv}=\sigma^{(\frac{1}{2})}_{ij}+\frac{1}{D-2}g^{(0)}_{ij}\theta^{(\frac{1}{2})}$
[34] ($i,j$ are transverse coordinate indices), we can write equation (II.27)
as

$R^{(\frac{1}{2},\frac{1}{2})}_{vv}\underset{TT}{=}-\Big{(}\frac{\theta^{(\frac{1}{2})}\theta^{(\frac{1}{2})}}{D-2}+\sigma^{\alpha\beta}_{(\frac{1}{2})}\sigma_{\alpha\beta}^{(\frac{1}{2})}\Big{)}+\frac{1}{4}\frac{d^{2}}{dv^{2}}\Big{(}g^{ij}_{(\frac{1}{2})}g_{ij}^{(\frac{1}{2})}\Big{)}$
(II.28)

which yields

$\frac{d\theta^{(1)}}{dv}=-\Big{\\{}8\pi<T^{Q}_{vv}>-H^{(1)}_{vv}+\frac{1}{4}\frac{d^{2}}{dv^{2}}\Big{(}g^{ij}_{(\frac{1}{2})}g_{ij}^{(\frac{1}{2})}\Big{)}\Big{\\}}.$
(II.29)

(II.23) follows from the fact that the background solution is stationary. The
other equations are obtained by expanding the Raychaudhuri equation order by
order.

###  II.4 Entropy change due to accretion of quantum matter across the horizon

In this subsection, we will compute the order-by-order change in the entropy
due to the accretion of quantum matter and gravitons across the horizon. We
will assume that the background black hole solution is stationary, as well as
the final state of the black hole at late times. Also, we assume the
perturbation will fall off fast enough, so that all boundary terms at late
times vanish 444This implies that at late times all the perturbations would
have either crossed the horizon or gone to asymptotic infinity. For AdS black
hole spacetime with reflecting boundary conditions, all perturbations will
cross the horizon. Also, we dynamically impose the gauge (II.10) at all times.
. To compute the entropy change order by order, first we will write the
perturbative expansion of entropy density as

$\rho=\rho^{(0)}+\epsilon^{\frac{1}{2}}\rho^{(\frac{1}{2})}+\epsilon\rho^{(1)}+O(\epsilon^{\frac{3}{2}})$
(II.30)

Now, we have all the tools to compute the change in entropy. First we will do
the change in entropy computation in the absence of the graviton. Then we will
do the computation in which we will include gravitons.

####  II.4.1 Entropy change without graviton contribution

When there is no graviton, all the terms with $g_{\mu\nu}^{(\frac{1}{2})}$
perturbation will go away in all of the above equations. Also, the stress-
energy tensor will have only the matter contribution i.e,
$<T^{Q}_{vv}>=<T^{M}_{vv}>$, which we will take to be $O(\epsilon)$. The
background solution is a stationary black hole, with a Killing horizon and
regular bifurcation sphere. For the stationary black hole, the expansion
coefficient of the horizon is zero and the entropy density will be independent
of the chosen horizon cut. Therefore

$\epsilon^{0}:\hskip 14.22636pt\Delta S^{(0)}=0$ (II.31)

Now, we will compute the change in the entropy due to accretion of matter by
the stationary black hole, which takes it away from stationarity. As we have
already mentioned, the black hole will settle down into a new stationary state
at late times. Now using (II.8) and (II.29) with the fact that there is no
$\rho^{(\frac{1}{2})}$ and $\Theta^{\frac{1}{2}}$ we will get

$\epsilon:\hskip 14.22636pt\Delta\delta
S^{(1)}=-\frac{1}{4}\int_{0}^{\infty}dv\int
d^{D-2}x\sqrt{h}v\Big{\\{}\frac{d^{2}\rho^{(1)}}{dv^{2}}+\frac{d\theta^{(1)}}{dv}\rho^{(0)}\Big{\\}}.$
(II.32)

Here, $\delta$ is the perturbation away from stationarity. Since
$\rho^{(0)}=1+\rho^{(0)}_{w}$,

$\frac{d\theta^{(1)}}{dv}\rho^{(0)}=\frac{d\theta^{(1)}}{dv}+\frac{d\theta^{(1)}}{dv}\rho^{(0)}_{w}=-8\pi<T_{vv}^{Q}>+H_{vv}^{(1)}-R_{vv}^{(1)}\rho^{(0)}_{w}.$
(II.33)

The equation (II.33) is obtained using (II.25),(II.29) and the fact that there
is no $g^{(\frac{1}{2})}$ perturbation, which will make
$\frac{d\theta^{(1)}}{dv}=-R_{vv}^{(1)}$. Further, we used the equation of
motion to rewrite $R_{vv}^{(1)}$ in terms of the stress energy tensor. Putting
the equation (II.33) in the equation (II.32), we get

$\epsilon:\hskip 14.22636pt\Delta\delta
S^{(1)}=-\frac{1}{4}\int_{0}^{\infty}dv\int
d^{D-2}x\sqrt{h}v\Big{\\{}\Big{(}\frac{d^{2}\rho^{(1)}}{dv^{2}}-R^{(1)}_{vv}\rho^{(0)}_{w}+H^{(1)}_{vv}\Big{)}-8\pi<T^{Q}_{vv}>\Big{\\}}.$
(II.34)

We note that
$\Big{(}\frac{d^{2}\rho^{(1)}}{dv^{2}}-R^{(1)}_{vv}\rho_{w}^{(0)}+H^{(1)}_{vv}\Big{)}$
is constructed out of background metric components and the perturbation and is
linear in the perturbation 555We can replace the ordinary derivatives with
respect to $v$ in the first term with covariant derivatives in the gauge we
are in.. Therefore, using boost arguments we can write
$\Big{(}\frac{d^{2}\rho^{(1)}}{dv^{2}}-R^{(1)}_{vv}\rho_{w}^{(0)}+H^{(1)}_{vv}\Big{)}=\partial^{2}_{v}\zeta_{(1)}$,
which will yield

$\Delta\delta S^{(1)}=-\frac{1}{4}\int_{0}^{\infty}dv\int
d^{D-2}x\sqrt{h}v\Big{\\{}\partial^{2}_{v}\zeta_{(1)}-8\pi<T^{Q}_{vv}>\Big{\\}}.$
(II.35)

We can simplify the above term using integration by parts,

$\Delta\delta S^{(1)}=2\pi\int_{0}^{\infty}dv\int
d^{D-2}x\sqrt{h}v<T^{Q}_{vv}>+\frac{1}{4}\int
d^{D-2}x\sqrt{h}\int_{0}^{\infty}dv\partial_{v}\zeta_{(1)}\\\ -\frac{1}{4}\int
d^{D-2}x\sqrt{h}\Big{(}v\partial_{v}\zeta_{(1)}\Big{)}\Big{|}_{v=0}^{v\rightarrow\infty}.$
(II.36)

Now we assume fall-off conditions at late times i.e, all perturbations and
their derivatives should fall fast enough such that this boundary term goes to
zero at late times. The contribution from the last term in (II.36) also
vanishes at $v=0$. Let us recall that we are in the gauge (II.10) in which the
horizon is always at $u=0$. Then we will get

$\Delta\delta S^{(1)}=2\pi\int_{0}^{\infty}dv\int
d^{D-2}x\sqrt{h}v<T^{Q}_{vv}>-\frac{1}{4}\int
d^{D-2}x\sqrt{h}\zeta_{(1)}\Big{|}_{v=0}.$ (II.37)

We can get rid of the second term by assuming that we can fix the JKM
ambiguity in such a way that

$\frac{d^{2}\rho^{(1)}}{dv^{2}}-R^{(1)}_{vv}\rho_{w}^{(0)}+H^{(1)}_{vv}=\partial^{2}_{v}\zeta_{(1)}=0$
(II.38)

everywhere on the horizon. This will get rid of the term which is giving rise
to the second term in (II.37). This is because then $\zeta_{(1)}(v)=av+b$,
where $a$ and $b$ are only functions of transverse coordinate. $\zeta_{(1)}$
is constructed out of the background metric and the linear perturbation in the
gauge (II.10). We have to further impose the fall-off conditions on the
perturbation i.e., the perturbation and its derivatives with respect to $v$
must go to zero at late times. Thus $\zeta_{(1)}(v)=0$. There is no
contradiction with the fact that $\Omega$ vanishes at the bifurcation surface
in linear order. It is shown in [20] that the second term in (II.37) is zero
for $F(R)$ gravity and arbitrary order Lovelock gravity. It is also argued
there that this will be true for an arbitrary diffeomorphic theory at linear
order. Therefore, assuming this,

$\Delta\delta S^{(1)}=2\pi\int_{0}^{\infty}dv\int
d^{D-2}x\sqrt{h}v<T^{Q}_{vv}>$ (II.39)

The above derivation is of course true even when the accreting matter is
classical. For classical matter, imposing the null energy condition i.e
$T^{Q}_{vv}\geq 0$ will give the second law.

####  II.4.2 Entropy change with the graviton contribution included

In this section, we include the graviton contribution, and therefore we will
work with the full perturbation expansion. We will again do an order by order
expansion. Using (II.8) and the fact that background solution is stationary,

$\epsilon^{0}:\hskip 14.22636pt\Delta S^{(0)}=0$ (II.40)

Now, let us compute change in entropy at $\epsilon^{\frac{1}{2}}$ order.
Writing (II.8) at $\epsilon^{\frac{1}{2}}$ will give

$\epsilon^{\frac{1}{2}}:\hskip 14.22636pt\Delta\delta
S^{(\frac{1}{2})}=-\frac{1}{4}\int_{0}^{\infty}dv\int
d^{D-2}x\sqrt{h}v\Big{\\{}\frac{d^{2}\rho^{(\frac{1}{2})}}{dv^{2}}+\frac{d\theta^{(\frac{1}{2})}}{dv}\rho^{(0)}\Big{\\}}.$
(II.41)

Here, $\delta$ corresponds to entropy change due to a perturbation that takes
the solution away from stationarity. Using (II.24), we can write (II.41) as

$\Delta\delta S^{(\frac{1}{2})}=-\frac{1}{4}\int_{0}^{\infty}dv\int
d^{D-2}x\sqrt{h}v\Big{\\{}\frac{d^{2}\rho^{(\frac{1}{2})}}{dv^{2}}-R_{vv}^{(\frac{1}{2})}\rho^{(0)}\Big{\\}}$
(II.42)

Furthermore using the boost argument in (II.17), we can write
$R_{vv}^{(\frac{1}{2})}=\partial_{v}^{2}\zeta_{(\frac{1}{2})}$, where
$\zeta_{(\frac{1}{2})}$ is constructed out of the background metric and linear
perturbation in $g_{\mu\nu}^{(\frac{1}{2})}$ where we work in the gauge
(II.10). Further, $\rho^{(0)}$ is independent of $v$. Therefore, the equation
(II.42) becomes

$\Delta\delta S^{(\frac{1}{2})}=-\frac{1}{4}\int_{0}^{\infty}dv\int
d^{D-2}x\sqrt{h}v\frac{d^{2}}{dv^{2}}\Big{\\{}\rho^{(\frac{1}{2})}-\rho^{(0)}\zeta_{(\frac{1}{2})}\Big{\\}}.$
(II.43)

Using integration by parts,

$\Delta\delta S^{(\frac{1}{2})}=-\frac{1}{4}\int
d^{D-2}x\sqrt{h}v\frac{d}{dv}\Big{\\{}\rho^{(\frac{1}{2})}-\rho^{(0)}\zeta_{(\frac{1}{2})}\Big{\\}}\Big{|}_{v=0}^{v\rightarrow\infty}+\frac{1}{4}\int_{0}^{\infty}dv\int
d^{D-2}x\sqrt{h}\frac{d}{dv}\Big{\\{}\rho^{(\frac{1}{2})}-\rho^{(0)}\zeta_{(\frac{1}{2})}\Big{\\}}.$
(II.44)

Using the fall-off condition on the perturbation at late times, the first term
vanishes. Therefore, we get

$\Delta\delta S^{(\frac{1}{2})}=\frac{1}{4}\int_{0}^{\infty}dv\int
d^{D-2}x\sqrt{h}\frac{d}{dv}\Big{\\{}\rho^{(\frac{1}{2})}-\rho^{(0)}\zeta_{(\frac{1}{2})}\Big{\\}}.$
(II.45)

Integrating and using fall-off condition gives

$\Delta\delta S^{(\frac{1}{2})}=-\frac{1}{4}\int
d^{D-2}x\sqrt{h}\Big{\\{}\rho^{(\frac{1}{2})}-\rho^{(0)}\zeta_{(\frac{1}{2})}\Big{\\}}\Big{|}_{v=0}.$
(II.46)

Using the fact that $\theta=\frac{d}{dv}\log{(\sqrt{h})}$ and (II.24), we can
write $\delta\log{(\sqrt{h})}=-\zeta_{(\frac{1}{2})}$. Therefore, we can write
(II.46) as

$\Delta\delta
S^{(\frac{1}{2})}=-\frac{1}{4}\int_{\mathcal{B}}d^{D-2}x\sqrt{h}\Big{\\{}\rho^{(\frac{1}{2})}+\delta\log{(\sqrt{h})}\rho^{(0)}\Big{\\}}.$
(II.47)

Using (II.3), it is straightforward to verify that

$\Delta\delta S^{(\frac{1}{2})}=-\delta
S^{(\frac{1}{2})}\Big{|}_{\mathcal{B}}$ (II.48)

$\mathcal{B}$ is the bifurcation surface. Now, we use Theorem $6.1$ in Iyer
and Wald (IW) [17], i.e $(\delta
S=\delta\mathcal{E}-\Omega_{H}\delta\mathcal{J})\Big{|}_{\mathcal{B}}$, where
$\mathcal{E}$ is the canonical energy and $\mathcal{J}$ is the canonical
angular momentum of the black hole in the covariant phase space formalism
666Since we have a static black hole, angular velocity at the horizon is
zero.. This was proved by IW at the bifurcation surface for any non-stationary
perturbation satisfying the linearized equation of motion777IW’s first law at
the bifurcation surface $\mathcal{B}$ is unaffected by the JKM ambiguity.. Now
in our case, there is no stress-energy tensor at $\epsilon^{(\frac{1}{2})}$
order. This implies

$\Delta\delta S^{(\frac{1}{2})}=-\delta
S^{(\frac{1}{2})}\Big{|}_{\mathcal{B}}=0$ (II.49)

Hence, $\delta S^{(\frac{1}{2})}(0)=\delta S^{(\frac{1}{2})}(\infty)=0$.
Therefore, from (II.49), if $\delta S^{(\frac{1}{2})}$ is non-zero at any cut,
then for some range of $v$ entropy will definitely decrease. This violates the
second law. The only way for the second law to be true is to assume that
$\delta S^{(\frac{1}{2})}$ will vanish at arbitrary cut. It was shown
explicitly by the authors in [19, 20] that

$\mathcal{R}_{vv}\equiv\frac{d^{2}\rho}{dv^{2}}-R_{vv}\rho_{w}^{(0)}+H_{vv}$
(II.50)

vanishes for $F(R)$ theory and Lovelock theory of arbitrary order, at the
linear order in perturbation theory about the stationary black hole
(perturbation can be non-stationary). Using (II.20), it can can be checked
that the term in curly brackets in (II.41) is the same as
$\mathcal{R}_{vv}^{(\frac{1}{2})}$. The authors in [19, 20] also argued that
this relation may be true for an arbitrary theory of gravity with an
appropriate definition of local entropy density. Vanishing of $\delta
S^{(\frac{1}{2})}$ in general will yield,

$\frac{d^{2}\rho^{(\frac{1}{2})}}{dv^{2}}=-\rho^{(0)}\partial_{v}\theta_{(\frac{1}{2})}$
(II.51)

which after integration and using the boundary condition that the perturbation
vanishes at late times will give
$\rho^{(\frac{1}{2})}=\rho^{(0)}\zeta_{(\frac{1}{2})}$.
Now, let us compute the $\epsilon$ order change in entropy, writing (II.8) to
the $\epsilon$ order. We get

$\Delta\delta S^{(1)}=-\frac{1}{4}\int_{0}^{\infty}dv\int
v\Big{\\{}\frac{d^{2}\rho^{(1)}}{dv^{2}}+\rho^{(0)}\frac{d\theta^{(1)}}{dv}+\frac{d\theta^{(\frac{1}{2})}}{dv}\rho^{(\frac{1}{2})}+2\theta^{(\frac{1}{2})}\frac{d\rho^{(\frac{1}{2})}}{dv}+\rho^{(0)}\theta^{(\frac{1}{2})}\theta^{(\frac{1}{2})}\Big{\\}}\sqrt{h}d^{D-2}x.$
(II.52)

Using (II.21), (II.25) and (II.50), the first two terms in the above
expression can be written as

$\frac{d^{2}\rho^{(1)}}{dv^{2}}+\rho^{(0)}\frac{d\theta^{(1)}}{dv}=-8\pi<T^{Q}_{vv}>+\mathcal{R}^{(1)}_{vv}-\frac{1}{4}\frac{d^{2}}{dv^{2}}\Big{(}g^{ij}_{(\frac{1}{2})}g_{ij}^{(\frac{1}{2})}\rho^{(0)}\Big{)}.$
(II.53)

Using (II.50), (II.17) and the fact that $H^{(1)}_{vv}$ and $R^{(1)}_{vv}$ are
constructed out of background metric components and the perturbation and are
linear in $g^{(1)}_{\mu\nu}$ perturbation in the gauge (II.10),
$\mathcal{R}^{(1)}_{vv}$ can be written as
$\mathcal{R}^{(1)}_{vv}=\partial_{v}^{2}\zeta_{(1)}$. This yields

$\frac{d^{2}\rho^{(1)}}{dv^{2}}+\rho^{(0)}\frac{d\theta^{(1)}}{dv}=-8\pi<T^{Q}_{vv}>+\partial_{v}^{2}\zeta^{\prime}_{(1)}.$
(II.54)

where,
$\zeta^{\prime}_{(1)}=\zeta_{(1)}-\frac{1}{4}\Big{(}g^{ij}_{(\frac{1}{2})}g_{ij}^{(\frac{1}{2})}\rho^{(0)}\Big{)}$.Putting
the above equation in (II.52) we get,

$-\frac{1}{4}\int_{0}^{\infty}dv\int
v\sqrt{h}d^{D-2}x\Big{(}\frac{d^{2}\rho^{(1)}}{dv^{2}}+\rho^{(0)}\frac{d\theta^{(1)}}{dv}\Big{)}=2\pi\int_{0}^{\infty}dv\int
v\sqrt{h}d^{D-2}x<T^{Q}_{vv}>\\\
-\frac{1}{4}\int\sqrt{h}\zeta^{\prime}_{(1)}d^{D-2}x\Big{|}_{v=0}.$ (II.55)

We get the above equation using integration by parts in the
$\partial_{v}^{2}\zeta^{\prime}_{(1)}$ integral and the fact that the term at
$v\rightarrow\infty$ will vanish due to the fall-off condition. Let us
consider the rest of the terms in (II.52), we will call it
$A(\frac{1}{2},\frac{1}{2})$,

$A(\frac{1}{2},\frac{1}{2})=-\frac{1}{4}\int_{0}^{\infty}dv\int
v\Big{\\{}\frac{d\theta^{(\frac{1}{2})}}{dv}\rho^{(\frac{1}{2})}+2\theta^{(\frac{1}{2})}\frac{d\rho^{(\frac{1}{2})}}{dv}+\rho^{(0)}\theta^{(\frac{1}{2})}\theta^{(\frac{1}{2})}\Big{\\}}\sqrt{h}d^{D-2}x.$
(II.56)

After integrating (II.51) once, we get
$\frac{d\rho^{(\frac{1}{2})}}{dv}=-\rho^{(0)}\theta_{(\frac{1}{2})}$. Putting
this in (II.56) will yield

$A(\frac{1}{2},\frac{1}{2})=-\frac{1}{4}\int_{0}^{\infty}dv\int
v\Big{\\{}\frac{d\theta^{(\frac{1}{2})}}{dv}\rho^{(\frac{1}{2})}+\theta^{(\frac{1}{2})}\frac{d\rho^{(\frac{1}{2})}}{dv}\Big{\\}}\sqrt{h}d^{D-2}x$
(II.57)

which can be further simplified using integration by parts and using fall-off
conditions as $v\to\infty$,

$A(\frac{1}{2},\frac{1}{2})=\frac{1}{4}\int_{0}^{\infty}dv\int\theta^{(\frac{1}{2})}\rho^{(\frac{1}{2})}\sqrt{h}d^{D-2}x.$
(II.58)

Using $\frac{d\rho^{(\frac{1}{2})}}{dv}=-\rho^{(0)}\theta_{(\frac{1}{2})}$ in
(II.58), we get

$A(\frac{1}{2},\frac{1}{2})=-\frac{1}{8}\int_{0}^{\infty}dv\int\frac{d}{dv}\Big{(}\frac{(\rho^{(\frac{1}{2})})^{2}}{\rho^{(0)}}\Big{)}\sqrt{h}d^{D-2}x.$
(II.59)

After integrating and using fall-off at late times, we will get

$A(\frac{1}{2},\frac{1}{2})=\frac{1}{8}\int\Big{(}\frac{(\rho^{(\frac{1}{2})})^{2}}{\rho^{(0)}}\Big{)}\sqrt{h}\Big{|}_{v=0}d^{D-2}x$
(II.60)

This quantity is thus manifestly positive. From (II.55) and (II.60), we get

$\Delta\delta S^{(1)}=2\pi\int_{0}^{\infty}dv\int
v\sqrt{h}d^{D-2}x<T^{Q}_{vv}>-\frac{1}{4}\int\sqrt{h}\zeta^{\prime}_{(1)}d^{D-2}x\Big{|}_{v=0}+A(\frac{1}{2},\frac{1}{2}).$
(II.61)

We know that the Wald entropy has JKM ambiguities when the metric is not
stationary. That was a motivation for putting $\Omega$ as the correction to
the Wald entropy in the definition of local entropy density. We now fix
$\Omega$ such that the last two terms in (II.61) vanish. These terms are
anyway zero for a stationary black hole, as can be seen from the expression
for $A(\frac{1}{2},\frac{1}{2})$ in (II.60) and $\mathcal{R}^{(1)}_{vv}$ is
zero for a stationary black hole from results in the subsection on the boost
argument 888For a stationary black hole, $v$ derivatives of $\rho$ are zero
and $R_{vv}$ and $H_{vv}$ are zero for a stationary metric as discussed using
boost arguments.. So $\Omega$ will be non-zero only when the metric is not
stationary. For the cases when it is possible to set these two terms to zero
by a choice of $\Omega$, we will get

$\Delta\delta S^{(1)}=2\pi\int_{0}^{\infty}dv\int
v\sqrt{h}d^{D-2}x<T^{Q}_{vv}>$ (II.62)

Some of the ambiguities in Wald’s entropy were fixed for some class of
theories at linear order in perturbation theory[21]. Moreover, this entropy
was shown to be equal to holographic entanglement entropy computed by Dong
[22].It is also pointed out in [21] that considering the second law at linear
order does not fix the ambiguities at higher order. Therefore these results
are not in contradiction with our computation. One can also get rid of the
second and third term in (II.61), by restricting the perturbation to a special
class of perturbations which vanishes at $v=0$. One physical case in which
such perturbation can be realised is when matter falls after $v=0$.

##  III The Entropy of Algebra And Generalized Entropy in Higher Curvature
Theory

In this section, we utilize the algebraic approach to quantum field theory,
specifically the constructions of Chandrasekaran, Longo, Penington, and Witten
(CLPW) in [23], and Chandrasekaran, Penington, and Witten (CPW) in [24], to
study black holes in higher curvature theories. We have employed their
construction to prove a version of the GSL (Generalized Second Law) for an
arbitrary diffeomorphism invariant theory of gravity, with certain appropriate
assumptions. The setup that we are interested in involves both the
asymptotically flat and asymptotically AdS stationary black hole solutions in
an arbitrary diffeomorphism invariant theory. Throughout this section we
follow the notation of CPW. In this section, we will first review the work of
CPW and then generalize it to our case.

###  III.1 Brief review of the recent papers

The series of recent papers by CLPW, Witten and CPW [23, 33, 24] have helped
to understand the generalized entropy introduced by Bekenstein [1] better.
They have addressed the question of why the generalized entropy is well-
defined, whereas the gravity contribution and the quantum field contribution
in the generalized entropy are not well-defined separately. CPW showed for an
eternal black hole that is either asymptotically flat and asymptotically AdS,
that there is an entropy associated with the von Neumann algebra of quantum
fields (including the graviton) in the black hole left or right exterior,
together with either the left or right ADM/boundary Hamiltonian. Further, they
showed that this algebra entropy is the generalized entropy at the bifurcation
surface of the black hole. They have also discussed the monotonicity of the
generalized entropy of asymptotically AdS black holes by using techniques from
von Neumann algebras.

We first provide a brief review of the salient features of these constructions
and then generalize them to our case of black holes in arbitrary
diffeomorphism invariant theories of gravity.

Figure 2: Maximally extended stationary black hole with Cauchy surface $S$.
The bulk algebra of the left and the right exterior region is
$\mathcal{A}_{\ell,0}$ and $\mathcal{A}_{r,0}$ respectively.

Let $M$ be the asymptotically flat, maximally extended Schwarzschild black
hole in Einstein’s theory or the maximally extended AdS-Schwarzschild black
hole. We consider quantum fields in this spacetime, including gravitons. The
left and right exterior regions of $M$ will be denoted by $\ell$ and $r$
respectively, while $L$ and $R$ will be used to denote left and right spatial
infinity. Let $\mathcal{H}_{0}$ be the Hilbert space of this theory that we
get by quantizing the fields and the local algebra of observables of the left
and right exterior region be $\mathcal{A}_{\ell,0}$ and $\mathcal{A}_{r,0}$
respectively as shown in Figure 2. It is well known that algebras
$\mathcal{A}_{\ell,0}$ and $\mathcal{A}_{r,0}$ are Type III1 factors (their
centers are trivial) [28, 29, 30]999The algebra of operators in quantum field
theory in a causal wedge is always a von Neumann algebra of Type III [25].
When the center is trivial it is a Type III1 algebra. . Moreover,
$\mathcal{A}_{\ell,0}$ and $\mathcal{A}_{r,0}$ are each other’s commutants
(i.e all the operators of $\mathcal{A}_{\ell,0}$ commute with all the
operators in $\mathcal{A}_{r,0}$).
The spacetime is stationary and equipped with a time translation Killing field
$V$. $V$ is future directed in the right exterior region and past-directed in
the left exterior region. Due to background diffeomorphism invariance, one can
define a conserved quantity $\hat{h}$ associated with the time translation
vector field $V$. Let $S$ be the bulk Cauchy surface going from the spatial
infinity of the right exterior region to the spatial infinity of the left
exterior region, through the bifurcation surface as shown in Figure 2. Then
$\hat{h}$ can be defined as

$\hat{h}=\int_{S}d\Sigma^{\mu}V^{\nu}T_{\mu\nu}.$ (III.63)

Here, $T_{\mu\nu}$ is the stress-energy tensor of the bulk
fields101010$T_{\mu\nu}$ includes the contribution from the pseudo-stress
tensor of gravitons.. In Tomita-Takesaki theory of the quantum fields in the
black hole exterior, $\beta\hat{h}$ is the modular Hamiltonian associated with
the Hartle-Hawking state $|\Psi_{HH}>$ of the black hole and $\beta$ is the
Hawking temperature111111The modular operator is defined as
$\Delta=\exp{(-H_{mod})}$ and for $\lvert\Psi_{HH}\rangle$,
$H_{mod}=\beta\hat{h}$.[31, 32]. It is well known that $\hat{h}$ in Einstein’s
gravity is the difference between the right ADM Hamiltonian $H_{R}$ and the
left ADM Hamiltonian $H_{L}$ i.e. $\hat{h}=H_{R}-H_{L}$.

CLPW and CPW now extend the Type III1 algebra $\mathcal{A}_{\ell,0}$ and
$\mathcal{A}_{r,0}$ by including one more operator $h_{L}$ with
$\mathcal{A}_{\ell,0}$ and $h_{R}$ similarly for the right algebra. This
extended (crossed product) algebra acts on an extended Hilbert space
$\mathcal{H}=\mathcal{H}_{0}\otimes L^{2}(\mathbb{R})$ where the extra degree
of freedom that has been introduced is the time-shift (the sum of the times in
the left and the right exteriors). The extended crossed product right algebra
is denoted $\mathcal{A}_{r}=\mathcal{A}_{r,0}\rtimes\mathbb{R}_{h}$ and
similarly for the left algebra. Here,

$\displaystyle h_{L}=H_{L}-M_{0}$ $\displaystyle h_{R}=H_{R}-M_{0}$ (III.64)

$M_{0}$ is the ADM mass of some reference black hole. CPW work in a micro-
canonical ensemble i.e. an energy eigenstate centered around some energy
$M_{0}$ (mass of the reference black hole)121212CPW explicitly do the
microcanonical ensemble construction in the boundary CFT using a thermofield
double state.. The algebra of observables for the right exterior region is
studied in a semi-classical limit i.e. $G\rightarrow 0$. In this limit, the
ADM masses $H_{R}$ and $H_{R}$ diverge because the black hole mass $M_{0}$
(Schwarzschild radius divided by $2G$) diverges. So, CPW work with the non-
divergent subtracted Hamiltonians $h_{L}$ and $h_{R}$.

We can also write $\hat{h}=h_{r}-h_{\ell}$ where

$h_{r}=\int_{S_{1}}d\Sigma^{\mu}V^{\nu}T_{\mu\nu}$ (III.65)

$h_{\ell}=-\int_{S_{2}}d\Sigma^{\mu}V^{\nu}T_{\mu\nu}$ (III.66)

Figure 3: This figure depicts the split of Cauchy surface $S$ into union of
the red Cauchy surface $S_{1}$ in the right exterior and the green Cauchy
surface $S_{2}$ in the left exterior.

where $S_{1}$ and $S_{2}$ are the right exterior and the left exterior part of
the Cauchy surface $S$ as shown in Figure 3. As pointed out by CPLW, $h_{r}$
and $h_{\ell}$ have divergent fluctuations. Thus, such a splitting is not
true, strictly speaking [33], but in the extended algebra of Witten, the
modular operator has a factorization into a product of operators in the left
and right algebra.

$\hat{h}=h_{r}-h_{\ell}=h_{R}-h_{L}$ (III.67)

Further,

$h_{R}=\hat{h}+h_{L}\equiv\frac{h_{\psi}}{\beta}+x$ (III.68)

where $h_{\Psi}=H_{mod}$ is the modular Hamiltonian for the Hartle-Hawking
state $\lvert\Psi\rangle=\lvert\Psi_{HH}\rangle$.
$\mathcal{A}_{r}=\mathcal{A}_{r,0}\rtimes\mathbb{R}_{h}$ is the crossed
product algebra of the algebra $\mathcal{A}_{r,0}$ by the modular group for
the cyclic separating vector $\lvert\Psi\rangle$. As discussed in [33], the
algebra $\mathcal{A}_{r}$ is then a von Neumann algebra of Type II∞. As
explained in [33], unlike a Type III algebra, a Type II algebra has a notion
of trace, density matrix and entropy.

An operator in $\mathcal{A}_{r}$ which we will denote as $\hat{a}$ has the
form $\hat{a}=ae^{({ish_{\psi}/\beta})}\otimes e^{(isx)}$, where
$a\in\mathcal{A}_{r,0}$ . The states on which this operator acts has the form
$\lvert\hat{\Psi}\rangle=\lvert\Psi\rangle\otimes g(x)\in\mathcal{H}$ where
$\lvert\Psi\rangle\in\mathcal{H}_{0}$ and $g(x)\in L^{2}(\mathbb{R})$. The
most generic operator in $\mathcal{A}_{r}$ can be written as

$\hat{a}=\int_{-\infty}^{\infty}a(s)e^{is(x+\hat{h})}ds$ (III.69)

where $a(s)\in\mathcal{A}_{r,0}$. Similarly, we can write the most general
state as

$\lvert\hat{\Psi}\rangle=\int dxf(x)\lvert\Psi\rangle\lvert x\rangle$ (III.70)

As we have already mentioned, a Type II algebra has a trace, which is a
positive linear functional on operators in the algebra satisfying
$tr[\hat{a}\hat{b}]=tr[\hat{b}\hat{a}]$. The trace of an operator
$\hat{a}\in\mathcal{A}_{r}$ can be denoted as [24],

$tr[\hat{a}]=\int_{-\infty}^{\infty}dxe^{\beta
x}\langle\hat{\Psi}\rvert\hat{a}\lvert\hat{\Psi}\rangle$ (III.71)

where the expectation value
$\langle\hat{\Psi}\rvert\hat{a}\lvert\hat{\Psi}\rangle$ is in general some
non-trivial function of $x$. This trace is not the same as a standard trace on
Hilbert space $\mathcal{H}$ but one can think of it more as a renormalized
trace. Also, $tr$ is defined up to a scaling, which follows from the fact that
the algebra $\mathcal{A}_{r}$ has a one-parameter family of outer
automorphisms $x\rightarrow x+c$, which just rescales $tr$ by $e^{c}$. This
rescaling has the physical interpretation of shifting the renormalization
constant $M_{0}$ to $M_{0}-c$. Now, using the trace, we can define the density
matrix for the algebra $\mathcal{A}_{r}$. For any given state
$\lvert\hat{\Phi}\rangle\in\mathcal{H}$, the density matrix
$\rho_{\hat{\Phi}}\in\mathcal{A}_{r}$ is defined by

$tr[\rho_{\hat{\Phi}}\hat{a}]=\langle\hat{\Phi}\rvert\hat{a}\lvert\hat{\Phi}\rangle\hskip
14.22636pt\forall\in\mathcal{A}_{r}$ (III.72)

Moreover $\rho_{\hat{\Phi}}$ exists and is unique, which follows from the fact
that the trace is non-degenerate. If we are able to define a density matrix,
then we can also define the entropy of the algebra, by

$S(\hat{\Phi})_{\mathbb{A}_{r}}=-tr[\rho_{\hat{\Phi}}\log{\rho_{\hat{\Phi}}}]=-\langle\hat{\Phi}\rvert\log{\rho_{\hat{\Phi}}}\lvert\hat{\Phi}\rangle$
(III.73)

As emphasized by CPW, $S(\hat{\Phi})$ should not be thought of as the
entanglement entropy of $\mathcal{A}_{r}$ but is a renormalized entropy. Also,
because of the ambiguity in the definition of trace (trace is defined up to a
scaling), it is only entropy differences that are unambiguous, not the entropy
itself. CPW now work with a semi-classical state, i.e the state with
fluctuation in timeshift $p$, $\Delta p\sim O(\varepsilon)$, where
$\varepsilon$ is some parameter much smaller than unity. Then, with $x=h_{L}$,
$\Delta x\sim O(\frac{1}{\varepsilon})$. CPW consider the AdS-Schwarzschild
black hole and write down the semi-classical state in the _boundary_ CFT.
However, it could have equally been defined in the bulk and we will assume
that the formulae below correspond to the equivalent bulk statement. The
general form of such a state is,

$\lvert\hat{\Phi}\rangle=\int_{\infty}^{\infty}\varepsilon^{\frac{1}{2}}g(\varepsilon
x)\lvert\Phi\rangle\lvert x\rangle\hskip 8.53581pt\textrm{where}\hskip
14.22636pt\lvert\Phi\rangle\in\mathcal{H},\hskip 5.69054ptg(x)\in
L^{2}(\mathbb{R})$ (III.74)

It is shown by CPW that the density matrix $\rho_{\hat{\Phi}}$ for the state
$\lvert\hat{\Phi}\rangle$ is approximately 131313The expression is only valid
up to corrections suppressed by $O(\varepsilon)$ terms. given by

$\rho_{\hat{\Phi}}\approx\varepsilon\bar{g}(\varepsilon h_{R})e^{-\beta
x}\Delta_{\Phi|\Psi}g(\varepsilon h_{R})$ (III.75)

where $\Delta_{\Phi|\Psi}=e^{-h_{\Psi|\Phi}}$ is a relative modular operator
and $h_{\Psi|\Phi}$ is the relative modular Hamiltonian 141414$\Psi$ is the
Hartle Hawking state and $\Phi$ is any arbitrary state of quantum fields in
the black hole spacetime.. The relative modular Hamiltonian $h_{\Phi|\Psi}$ is
defined such that $h_{\Psi|\Psi}=h_{\Psi}$. As we have already mentioned, the
Type II algebra modular operator factorizes, i.e
$\Delta_{\hat{\Phi}|\hat{\Psi}}=\rho_{\hat{\Psi}}\rho^{\prime-1}_{\hat{\Phi}}$
(where prime denotes the element of the commutant of the algebra
$\mathcal{A}_{r}$)[23]. Putting (III.75) in (III.73) yields

$S(\hat{\Phi})_{\mathcal{A}_{r}}=\langle\hat{\Phi}\rvert\beta
h_{R}\lvert\hat{\Phi}\rangle-\langle\hat{\Phi}\rvert
h_{\Psi|\Phi}\lvert\hat{\Phi}\rangle-\langle\hat{\Phi}\rvert\log{(\varepsilon|g(\varepsilon
h_{R})|^{2})}\lvert\hat{\Phi}\rangle+O(\varepsilon)$ (III.76)

By definition, the second term in the above equation is the relative entropy,

$S_{rel}(\Phi||\Psi)=-\langle\hat{\Phi}\rvert
h_{\Psi|\Phi}\lvert\hat{\Phi}\rangle$ (III.77)

###  III.2 Generalization To Higher Curvature Gravity

We now generalize the construction of the previous sub-section to an arbitrary
diffeomorphism invariant theory of gravity. We note that some of the
constructions in the previous sub-section such as the semi-classical state
were done by CPW for the AdS-Schwarzschild black hole in the _boundary_ CFT.
But we can analogously define such a semi-classical state in the bulk using
the same construction. In fact, only in the sections on monotonicity of
generalized entropy in CPW, are results in the boundary theory crucially used.
Therefore, except while discussing monotonicity of the generalized entropy, we
can confine our analysis to the bulk, and we can even work with an
asymptotically flat black hole, as discussed by CPLW. So, in our case, $M$ is
the $(3+1)$ dimensional, asymptotically flat, maximally extended static
(therefore stationary) black hole solution in an arbitrary diffeomorphism
invariant theory of gravity. Therefore, its horizon is a Killing horizon. Now,
the expectation value of the energy-momentum tensor of the quantum fields
$\langle{\Phi}\rvert T_{\mu\nu}\lvert{\Phi}\rangle$ is covariantly conserved
as a consequence of invariance of the action under background diffeomorphisms
(see appendix (VI)). The equation (III.63) will thus define a conserved
quantity even in the arbitrary theory of gravity if $V^{\mu}$ is a Killing
vector. Let us define the 1-form $J=T_{\mu\nu}V^{\nu}dx^{\mu}$ where $V$ is
the timelike Killing vector of $M$. Then, divergence of $J_{\mu}$ is zero,
i.e., $J^{\mu}$ is covariantly conserved. This implies $d*J=0$, where $*$ is
the Hodge dual. Since the spacetime does not have non-trivial topology, this
implies $*J=dQ$ for some 2 form $Q$. Also, notice that the integral of $*J$
over the 3 dimensional Cauchy surface is $\hat{h}$. Since $*J=dQ$, this
reduces to an integral over the codimension 2 surface which is the boundary of
the Cauchy slice. Therefore we can write $\hat{h}=H_{R}-H_{L}$ where $H_{R}$
and $H_{L}$ are codimension 2 integrals at right and left spatial infinity
respectively. We note that the canonical energy $\mathcal{E}$ in the covariant
phase space formalism is given by (III.63) apart from a surface term ambiguity
(see appendix of IW [17]).

$\mathcal{E}=\int_{\Sigma}J+\text{Surface
term}=\int_{\Sigma}d\Sigma^{\mu}T_{\mu\nu}V^{\nu}+\text{Surface term}$
(III.78)

So we can think of $H_{R}$ and $H_{L}$ as being the right and left canonical
energy, respectively, apart from ambiguities in the canonical energy 151515As
discussed in the Appendix of IW [17], the ambiguity in the canonical energy is
the sum of two surface terms, one of which vanishes for common matter theories
in a background spacetime. There is an ambiguity due to the second surface
term which is a function of the background metric, Killing vector, matter
fields and their derivatives. In what follows, we ignore this ambiguity.. We
will call them right and left Hamiltonian. Although the results of IW are only
for classical gravity and matter, and we are interested in quantum fields, the
IW equations can be understood as expectation values in semiclassical gravity
as discussed in appendix (VI). Now, $\hat{h}$ is the modular Hamiltonian
corresponding to the Hartle-Hawking state as before. This follows from the
analysis of Sewell [32] for any metric of the following form:

$ds^{2}=A(t^{2}-w^{2},y)(-dt^{2}+dw^{2})+B(t^{2}-w^{2},y)d\sigma^{2}(y).$
(III.79)

The Schwarzschild spacetime in Kruskal coordinates is of this form. We will
assume that our static black hole solution also has this form (i.e., we assume
the existence of Kruskal-like coordinates).

Now we can proceed by defining $h_{L}$ and $h_{R}$ as defined in (III.64).
Following the argument in the previous section that including gravity changes
the algebra to Type II, we can split $\hat{h}$ as done in (III.65) and
(III.66). Further, we can straightforwardly obtain the equation in (III.67)
and (III.68). The only difference is now $h_{R}$ in (III.68) is the
renormalized Hamiltonian in the higher curvature theory which generates the
time translation on the boundary of the right exterior region. Afterwards, the
construction of the crossed -product algebra (extended algebra) and other
constructions like defining the trace and entropy will analogously go through
as done in the previous section. We will work with a semi-classical state as
defined in (III.74). Therefore we can define the entropy of the algebra in the
right exterior region by the same formula (III.76) i.e.

$S(\hat{\Phi})_{\mathcal{A}_{r}}=\langle\hat{\Phi}\rvert\beta
h_{R}\lvert\hat{\Phi}\rangle-\langle\hat{\Phi}\rvert
h_{\Psi|\Phi}\lvert\hat{\Phi}\rangle-\langle\hat{\Phi}\rvert\log{(\varepsilon|g(\varepsilon
h_{R})|^{2})}\lvert\hat{\Phi}\rangle+O(\varepsilon)$ (III.80)

where $S_{rel}(\Phi||\Psi)=-\langle\hat{\Phi}\rvert
h_{\Psi|\Phi}\lvert\hat{\Phi}\rangle$ is relative entropy as defined earlier.
So the form of the equation (III.76) remains intact —the only change is that
$h_{R}$ is the renormalized canonical energy in higher curvature theory and
$h_{\Phi|\Psi}$ is the relative modular Hamiltonian in that particular theory.
In our case of interest, the black hole settles down to a stationary state at
very late times. This is plausible since at late times all the flux of matter
would either have crossed the horizon or would have escaped through future
null infinity. So, at late times we will not be able to distinguish between
$\lvert\Psi\rangle$ and $\lvert\Phi\rangle$. We get

$S_{bulk}(\infty)_{\Phi}=S_{bulk}(\infty)_{\Psi}=S_{bulk}(b)_{\Psi}$ (III.81)

Now let us analyze $h_{r}$, using (III.65) and the fact that the deformation
of Cauchy surfaces $S$ does not affect the conserved quantity $\hat{h}$. We
deform $S_{1}$ such that $S^{\prime}_{1}=\mathcal{H}^{+}\cup\mathcal{I}^{+}$,
where $\mathcal{H}^{+}$ is the future horizon and $\mathcal{I}^{+}$ is future
null infinity 161616In the case of an asymptotically AdS black hole, the
deformed Cauchy surface is just $\mathcal{H}^{+}$. Therefore

$\beta
h_{r}^{\mathcal{H}^{+}}=\beta(h_{r}-h_{r}^{\mathcal{I^{+}}})=\int_{0}^{\infty}dv\int_{\mathcal{H}^{+}}d^{D-2}x\sqrt{h}vT_{vv}$
(III.82)

where $\beta h_{r}^{\mathcal{I^{+}}}$ is the time translation generator at
future null infinity and $\beta h_{r}^{\mathcal{H}^{+}}$ is the boost
generator on the horizon. The second equality in the above equation can be
obtained using (III.65) and the fact that $h_{r}^{\mathcal{I^{+}}}$ is just
the integral of the stress tensor supported at future null infinity. Let us
define a one-sided modular operator (boost operator) at arbitrary cut
$v=v_{*}$ ( which is the $D-2$ dimensional transverse surface) at the horizon.
It is well known that the density matrix $(\rho_{r})_{HH}$ of the Hartle
Hawking state in the region $r$ is thermal with respect to [31, 34]

$K_{r}(v_{*})=\int_{v_{*}}^{\infty}dv\int_{\mathcal{H}^{+}}d^{D-2}x\sqrt{h}(v-v_{*})T_{vv}+K_{r}^{\mathcal{I^{+}}}$
(III.83)

where, $K_{r}^{\mathcal{I^{+}}}=\beta h_{r}^{\mathcal{I^{+}}}$ is the modular
energy at $\mathcal{I^{+}}$, which accounts for energy which goes to
$\mathcal{I^{+}}$ without crossing the horizon. Also notice that
$K_{r}^{\mathcal{I^{+}}}$ is independent of $v_{*}$. When $v_{*}=0$, then the
first term in the above equation will become $\beta h_{r}^{\mathcal{H}^{+}}$
as defined in (III.82), and therefore

$K_{r}(b)=\beta h_{r}=\beta h_{r}^{\mathcal{H}^{+}}+\beta
h_{r}^{\mathcal{I^{+}}}$ (III.84)

where $b$ is the bifurcation surface $v_{*}=0$. We can also define

$K_{r}(\infty)=\lim_{v_{*}\rightarrow\infty}K_{r}(v_{*})$ (III.85)

Using the result from the previous section and equation (III.82), we get

$\langle\Phi\rvert\beta h_{r}^{\mathcal{H^{+}}}\lvert\Phi\rangle=\Delta\delta
S$ (III.86)

If a density matrix were to exist for the algebra $\mathcal{A}_{r,0}$, then
using the definition of modular Hamiltonian for state $\lvert\Psi\rangle$ and
the fact that $\Delta_{\Psi}=\rho_{\Psi}\rho^{\prime-1}_{\Psi}$, we will be
able to write

$\log{\rho_{\Psi}}=-K_{r}(b)+C$ (III.87)

where $C$ is some constant. The density matrix of the Hartle Hawking state in
region $r$ can be written as
$\Psi(\mathcal{H^{+}}\cup\mathcal{I^{+}})=(\rho_{r})_{HH}\otimes\sigma$, which
corresponds to the ground state 171717Hartle Hawking state is a ground state
with respect to the time $v$. $\rho_{HH}$ at $\mathcal{H^{+}}$, product taken
with some arbitrary density matrix defining a faithful state at
$\mathcal{I^{+}}$ [34]. Therefore,

$\langle\Psi\rvert\log{\rho_{\Psi}}\lvert\Psi\rangle=-\langle\Psi\rvert
K_{r}^{\mathcal{I^{+}}}\lvert\Psi\rangle+C$ (III.88)

Here we use the fact that $\langle\Psi\rvert
h_{r}^{\mathcal{H^{+}}}\lvert\Psi\rangle=0$ since it is the Hartle Hawking
state. Further, since
$S_{bulk}(b)_{\Psi}=-\langle\Psi\rvert\log{\rho_{\Psi}}\lvert\Psi\rangle$, we
get $S_{bulk}(b)_{\Psi}=\langle\Psi\rvert
K_{r}^{\mathcal{I^{+}}}\lvert\Psi\rangle-C$.

As mentioned before, it is not strictly true of the algebra
$\mathcal{A}_{r,0}$ that the modular operator factorizes, but by extending the
algebra to $\mathcal{A}_{r}$, it is true that the modular operator factorizes
as $\hat{\Delta}_{\Psi}=\rho_{\hat{\Psi}}\rho^{\prime-1}_{\hat{\Psi}}$ (in the
notation of[33]). We will ignore this detail just for illustrative purposes
following [24] .

$S_{gen}(\infty)-S_{gen}(b)=S(\infty)-S(b)+S_{bulk}(b)_{\Psi}-S_{bulk}(b)_{\Phi}.$
(III.89)

It can be written in terms of the one-sided modular operator,

$S_{gen}(\infty)-S_{gen}(b)=-\langle\Phi\rvert(K_{r}(\infty)-K_{r}(b))\lvert\Phi\rangle+S_{bulk}(b)_{\Psi}-S_{bulk}(b)_{\Phi}.$
(III.90)

Putting $S_{bulk}(b)_{\Psi}$ and $\langle\Phi\rvert K_{r}(b)\lvert\Phi\rangle$
using the equation (III.87) , we get

$S_{gen}(\infty)-S_{gen}(b)=-\langle\Phi\rvert\log{\rho_{\Psi}}\lvert\Phi\rangle+C-\langle\Phi\rvert
K_{r}(\infty)\lvert\Phi\rangle+\langle\Psi\rvert
K_{r}^{\mathcal{I^{+}}}\lvert\Psi\rangle-C-S_{bulk}(b)_{\Phi}$ (III.91)

Now, we use the fact that at late times, every state is indistinguishable from
$\lvert\Psi\rangle$ and $K_{r}(\infty)=K_{r}^{\mathcal{I^{+}}}$. Further,
$K_{r}^{\mathcal{I^{+}}}$ is independent of the cut. The expectation value
$K_{r}^{\mathcal{I^{+}}}$ in state $\lvert\Phi\rangle$ will be equal to its
expectation value in state $\lvert\Psi\rangle$. Therefore, we get

$S_{gen}(\infty)-S_{gen}(b)=-\langle\Phi\rvert\log{\rho_{\Psi}}\lvert\Phi\rangle-
S_{bulk}(b)_{\Phi}$ (III.92)

therefore we got,

$S_{gen}(\infty)-S_{gen}(b)=S_{rel}(\Phi||\Psi)$ (III.93)

As we see, the difference of generalized entropies in (III.93) is manifestly
finite and non-negative. For Einstein gravity, the above expression has been
already obtained by Wall in [34]. The result (III.93) is in an arbitrary
theory of gravity — the difference between generalized entropy at late times
and generalized entropy at the bifurcation surface is relative entropy of the
state of the black hole with respect to the Hartle Hawking state. We now need
to show, as in [24], that the generalized entropy at the bifurcation surface
is the entropy of the algebra. We thus need to show

$S_{gen}(\infty)=\langle\hat{\Phi}\rvert\beta
h_{R}\lvert\hat{\Phi}\rangle-\langle\hat{\Phi}\rvert\log{(\varepsilon|g(\varepsilon
h_{R})|^{2})}\lvert\hat{\Phi}\rangle+\textrm{Const}$ (III.94)

Since both terms in the above equation are only functions of $h_{R}$, and
since we have interpreted $h_{R}$ as the renormalized canonical energy, the
above terms are some distributions of energy in the semi-classical state
$\lvert\hat{\Phi}\rangle$. Also, these terms are independent of the state
$\lvert\Phi\rangle$. To see that, choose $a(s)$ such that

$a(s)=\int e^{-ih^{\prime}_{R}s}f(h^{\prime}_{R})dh^{\prime}_{R}$ (III.95)

where $f(h_{R})$ is some chosen function. Putting the equation (III.95) in the
equation (III.69) and using the fact that $h_{R}=\hat{h_{\Psi}}+x$, will yield
$\hat{a}=f(h_{R})$. Now let us compute the expectation value
$\langle\hat{\Phi}\rvert\beta\hat{a}\lvert\hat{\Phi}\rangle$ for (III.95) with
the semi-classical state $\lvert\hat{\Phi}\rangle$ defined in (III.74). Using
the results (3.25) and (3.26) in CPW [24],

$\langle\hat{\Phi}\rvert\hat{a}\lvert\hat{\Phi}\rangle=\int_{-\infty}^{\infty}dx\int_{-\infty}^{\infty}ds|\varepsilon
g(\varepsilon
x)|^{2}e^{isx}\langle\Psi\rvert\Delta_{\Phi|\Psi}a(s)\lvert\Psi\rangle$
(III.96)

Now put (III.95) in (III.96). Using the fact that $h_{R}=\hat{h_{\Psi}}+x$ and
$\hat{h_{\Psi}}\lvert\Psi\rangle=0$, we get

$\langle\hat{\Phi}\rvert
f(h_{R})\lvert\hat{\Phi}\rangle=\int_{-\infty}^{\infty}dx\int_{-\infty}^{\infty}ds\int_{-\infty}^{\infty}dy|\varepsilon
g(\epsilon
x)|^{2}e^{is(x-y)}f(y)\langle\Psi\rvert\Delta_{\Phi|\Psi}\lvert\Psi\rangle$
(III.97)

By definition, $\langle\Psi\rvert\Delta_{\Phi|\Psi}\lvert\Psi\rangle=1$.
Therefore the above equation is independent of $\lvert\Phi\rangle$, it will
only depend on $f(h_{R})$ and $g(\varepsilon x)$. Therefore, both the terms on
the right-hand side of (III.94) will give the same result either when we
compute them in the state $\lvert\Phi\rangle$ or in the Hartle Hawking state
$\lvert\Psi\rangle$ at late times. Since both terms can be determined from the
late-time behavior of the black holes, the relation (III.94) is plausible.
This is because at late times, all the fields have either fallen across the
horizon or to infinity.
Using equation (III.64), we can write $\langle\hat{\Psi}\rvert\beta
h_{R}\lvert\hat{\Psi}\rangle=\beta(\Delta\mathcal{E})$, where
$\Delta\mathcal{E}$ is the energy difference between the black hole we are
studying and the reference black hole. Since both the black holes are taken in
an equilibrium state, we can apply the first law of black hole mechanics for
two equilibrium configurations in phase space which yields

$\langle\hat{\Psi}\rvert\beta h_{R}\lvert\hat{\Psi}\rangle=\Delta S$ (III.98)

where $\Delta S$ is the difference of entropy of the equilibrium black hole
state we get at late times to the reference black hole. Therefore, the first
term in (III.94) describes the change in $S_{gen}(\infty)$ due to a change in
black hole entropy. At very late times, all the matter would have either
crossed the horizon or would have escaped to null infinity. The second term
should be thought of as the contribution of entropy of fluctuations in black
hole entropy [24]. Finally, we add and subtract the entanglement entropy of
the quantum fields in the Hartle-Hawking state and lump one of the pieces in
the constant in (III.94) using (III.88). This is because at late times, all
the fields have either fallen across the horizon or to infinity.

Combining everything, we get

$S(\hat{\Phi})_{\mathcal{A}_{r}}=S_{gen}(b)+\textrm{Const}.$ (III.99)

The equation (III.99) tells us that $S_{gen}(b)$ for black holes at the
bifurcation surface in the arbitrary theory of gravity can be thought of as
the entropy of the algebra $\mathcal{A}_{r}$ modulo a constant. Notice, we
have shown that the generalized entropy at the bifurcation surface is equal to
the entropy of the algebra up to a constant, but we are not making any
statement about entropy at an arbitrary cut of the horizon. In algebraic QFT,
relative entropy is positive. This implies $S_{gen}(\infty)-S_{gen}(b)\geq 0$.
Can we go beyond this result and prove that the generalized entropy is
monotonic? The entropy of the algebra is monotonic under trace-preserving
inclusions [35]. To obtain a GSL (monotonicity of the generalized entropy),
CPW consider an AdS Schwarzschild black hole with a holographic dual CFT.
Then, they have the following clever argument: In the dual CFT, they consider
operator algebras at two different times (early and late times),
$\mathcal{A}_{R,0}$ and $\mathcal{B}_{R,0}$ respectively, separated by a
timescale much larger than the thermal time scale $\beta$. The correlation
functions of operators at these different times factorize into a product of
correlators of early and late times. Thus, the algebra generated by both early
and late time operators is
$\mathcal{C}_{R,0}=\mathcal{A}_{R,0}\otimes\mathcal{B}_{R,0}$. The Hilbert
space factorizes similarly. These algebras are extended similar to what was
done before, to obtain a Type II algebra which has an associated entropy. Now,
consider three different situations: first, the quantum fields at both early
and late times are in an arbitrary state, a second situation where the fields
at early times have fallen into the horizon, so that the state of these fields
is the vacuum times any state of the late time fields, and finally, a
situation where both sets of fields have fallen into the horizon and the state
of the fields in the exterior is the vacuum. CPW argue that the generalized
entropies at these three horizon cuts is the generalized entropy for the
extended algebra $\mathcal{C}_{R}$ for these three different states at a
hypothetical _bifurcation surface_ in the limit of large time gap. From the
property of the monotonicity of the entropy of the algebra under trace
preserving inclusions, it follows that the generalized entropy is increasing
in going from the first to the third situation above. This argument is then a
version of the GSL. We can use these results to prove this version of the GSL
in an arbitrary diffeomorphism invariant theory of gravity if we start with an
asymptotically AdS black hole which has a holographic dual. We can repeat all
the steps in this section for such a black hole. The only thing we need is for
the first law as in the paper of Iyer and Wald [17] to be true in this
situation. Although the statement of the First law is only for asymptotically
flat black holes, the same will be true for an asymptotically AdS black hole
provided the integrals involved in the presymplectic form and the canonical
energy are finite after assuming appropriate fall-offs for the fields. In this
case, the computation of CPW generalizes to these black holes in a higher
curvature theory of gravity and a version of increase of generalized entropy
(comparing the entropy at early and late times) is true. The ambiguity in the
Wald entropy which we fixed in the bifurcation surface in a previous section
was by terms quadratic in the half-order perturbation. This is not affected by
the different states of the quantum fields in the argument of CPW, so their
argument goes over to our case. Can we show a stronger monotonicity result
$\frac{dS_{gen}}{dv}\geq 0$? This is what Wall [34] has done for Einstein
gravity, using an expression for the entropy at any cut of the horizon. Due to
JKM ambiguities in the Wald entropy, this expression will probably need one to
specify the particular theory of gravity.

##  IV Discussion

In the context of Einstein gravity, it was shown by CPW [24] that for the
system of quantum fields in a perturbed Schwarzschild black hole spacetime in
the $G\to 0$ limit with infalling quantum matter across the horizon, the
generalized entropy at the bifurcation surface was equal to the entropy of the
von Neumann algebra of operators in the black hole exterior. This was achieved
by enlarging the operator algebra by including the ADM Hamiltonian, and by
enlarging the Hilbert space. This had the effect of changing the algebra of
operators to a Type II∞ von Neumann algebra, to which we can associate a
notion of trace and entropy. Furthermore, CPW showed that the difference of
the generalized entropy of an arbitrary cut of the horizon, in the limit when
the cut $v\to\infty$ and the generalized entropy at the bifurcation surface
was equal to the relative entropy, and therefore nonnegative. For this, CPW
worked in semiclassical gravity and up to quadratic order in perturbations.
They then obtained a monotonicity result (GSL) for the generalized entropy
from the monotonicity of relative entropy under trace preserving inclusions.
In this paper, we consider quantum fields in a slightly perturbed static black
hole in an arbitrary diffeomorphism invariant theory of gravity in the $G\to
0$ limit. In this set-up, generalized entropy is the sum of Wald entropy and
entanglement entropy of quantum fields in the black hole exterior. We consider
the difference in Wald entropy at infinity and at the bifurcation surface up
to quadratic order in the perturbations and obtain (II.61). Wald entropy has
ambiguities on non-stationary geometries. We fix the ambiguity in order to get
(II.62) which matches the result for Einstein gravity in the paper of CPW and
we obtain a simplified result for the difference of entropies which enables us
to employ the CPW construction. We then consider the difference in
_generalized_ entropies and show that this difference equals the relative
entropy of the state of the quantum fields and the Hartle-Hawking state — it
is thus non-negative. We next consider the von Neumann algebra of the quantum
fields in the black hole exterior, extended to include the Hamiltonian, and an
enlarged Hilbert space as in CPW. Evaluated on the semiclassical states
defined by CPW, we show that the entropy of the von Neumann algebra equals the
generalized entropy of the bifurcation surface. Finally, we see that the
derivation of the increase of generalized entropy by CPW in Einstein gravity
goes through for black holes in an arbitrary gravity theory, provided the
black hole is asymptotically AdS, which has a holographic dual.

Going forward, it would be interesting to not merely restrict to semiclassical
states and see what the entropy of the algebra corresponding to more general
states is. Recently, there has also been an interesting proposal to study the
algebra of operators in general subregions in Einstein gravity with matter in
the $G\to 0$ limit, and it has been shown that the entropy of this algebra
equals the generalized entropy of this subregion up to a constant [36] (see
also the latest papers [37] and [38]). It would be interesting to see how
these results generalize to an arbitrary diffeomorphism invariant theory of
gravity.

##  V Acknowledgements

We would like to thank Sanved Kolekar for useful comments. MA also
acknowledges the Council of Scientific and Industrial Research (CSIR),
Government of India for financial assistance.

##  VI Appendix A

###  VI.1 Quantum Canonical energy in covariant phase space formalism

Consider quantum fields in the stationary black hole background, where the
quantum fields include gravitons. Using diffeomorphism invariance of the
Lagrangian (which is an $n$-form) with respect to the background metric, we
can write

$\frac{\delta L}{\delta\phi}\mathcal{L}_{\xi}\phi=-dJ-\frac{\delta L}{\delta
g_{ab}}\mathcal{L}_{\xi}g_{ab}.$ (VI.100)

Here, $J$ is the Noether current $(n-1)$ form in the covariant phase space
formalism given by equation (49) in IW [17]. $\phi$ here corresponds to all
quantum fields and $g_{ab}$ is the background metric. $\xi$ is a vector field
generating the diffeomorphism and $\mathcal{L}_{\xi}$ denotes the Lie
derivative with respect to $\xi$.

As we have a Lagrangian for quantum fields to begin with, we look at the
expectation value of the relation (VI.100) in an arbitrary state. Since the
terms in the equations are products of operators at the same point, the
expectation values need to be regularized using some procedure. Using a
prescription involving point-splitting and background subtraction, we
regularize these expectation values.

We want to compute the expectation value of (VI.100) in some state
$\lvert\Phi\rangle$. Consider the point-split expression

$\lim_{y\rightarrow x}\langle\Phi\rvert\frac{\delta
L}{\delta\phi}(x)\mathcal{L}_{\xi}\phi(y)\lvert\Phi\rangle=\lim_{y\rightarrow
x}\langle\Phi\rvert\Big{(}-dJ(x,y)-\frac{\delta L}{\delta
g_{ab}}(x)\mathcal{L}_{\xi}g_{ab}(y)\Big{)}\lvert\Phi\rangle$ (VI.101)

From the Schwinger Dyson equation for expectation value in an arbitrary state,
we can compute the left hand side, and in the limit of coincident points, it
is just a state independent divergent term. Therefore if we consider the
difference of the quantity $\Big{(}-dJ(x,y)-\frac{\delta L}{\delta
g}(x)\mathcal{L}_{\xi}g(y)\Big{)}$ in any two states $\lvert\Phi\rangle$ and
$\lvert\Psi\rangle$, then the state independent divergent term will cancel
out. Now we can take the coincidence limit $y\rightarrow x$.

$\langle\Phi\rvert\Big{(}-dJ(x)-\frac{\delta L}{\delta
g_{ab}}(x)\mathcal{L}_{\xi}g_{ab}(x)\Big{)}\lvert\Phi\rangle-\langle\Psi\rvert\Big{(}-dJ(x)-\frac{\delta
L}{\delta g_{ab}}(x)\mathcal{L}_{\xi}g_{ab}(x)\Big{)}\lvert\Psi\rangle=0$
(VI.102)

Now, (VI.102) can be written as

$d\langle\Phi\rvert(J+k.\epsilon)\lvert\Phi\rangle-d\langle\Psi\rvert(J+k.\epsilon)\lvert\Psi\rangle=\nabla_{\mu}(\langle\Phi\rvert
T^{\mu\nu}\lvert\Phi\rangle)\xi_{\nu}-\nabla_{\mu}(\langle\Psi\rvert
T^{\mu\nu}\lvert\Psi\rangle)\xi_{\nu}.$ (VI.103)

Here, we use the fact that $\frac{\delta L}{\delta
g_{\mu\nu}}=T^{\mu\nu}\epsilon$, where $\epsilon$ is the volume form and
$k^{\mu}=T^{\mu\nu}\xi_{\nu}$. Notice that the left hand side in the above
equation is a total derivative for any $\xi$ while the right hand side is not.
The only way that can happen is if $\nabla_{\mu}(\langle\Phi\rvert
T^{\mu\nu}\lvert\Phi\rangle)\xi_{\nu}=\nabla_{\mu}(\langle\Psi\rvert
T^{\mu\nu}\lvert\Psi\rangle)\xi_{\nu}$. Since this has to be true for any two
states and any vector field $\xi^{\mu}$, it can only be if
$\nabla_{\mu}(\langle\Phi\rvert T^{\mu\nu}\lvert\Phi\rangle)$ vanishes for any
state $\lvert\Phi\rangle$ up to a local term independent of state. We may
modify our prescription and get rid of this extra state independent term by
doing background subtraction with respect to some standard state. We do the
background subtraction (VI.102) to make sense of expectation values of
equation (VI.100). Thus the expectation value of the stress energy tensor in
any state is conserved. It has already been extensively discussed in Wald [18]
how to define the expectation value of the stress energy tensor in such a way
that it is conserved, so alternatively, we could use that result. By the same
argument, $d\langle\Phi\rvert(J+k.\epsilon)\lvert\Phi\rangle=0$. The rest of
the analysis of the Appendix of IW [17] follows. Now we can choose $\xi^{\mu}$
to be the Killing field of the static background spacetime. Following IW [17],
we will get

$\big{<}\mathcal{E}\big{>}_{\Phi}=\int_{\Sigma}d\Sigma^{\mu}\big{<}T_{\mu\nu}\big{>}_{\Phi}\xi^{\nu}+\text{Surface
term}$ (VI.104)

where $\mathcal{E}$ is the canonical energy in the covariant phase space
formalism and it is independent of the choice of Cauchy slice.

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