,

  * September 2023

# Generalized Second Law for Non-minimally Coupled Matter Theories

Prateksh Dhivakar1 and Krishna Jalan2,3 1Department of Physics, Indian
Institute of Technology Kanpur, Kalyanpur, Kanpur 208016, India  2The
Institute of Mathematical Sciences, 4th Cross Street, C.I.T. Campus, Taramani,
Chennai 600113, India  3Homi Bhabha National Institute, Training School
Complex, Anushakti Nagar, Mumbai 400094, India<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We prove the generalized second law (GSL) for higher curvature gravity
theories when the matter sector is non-minimally coupled. The validity of our
proof is in the regime of linearized fluctuations about equilibrium black
holes, which is the same regime as considered in the previous proofs by Wall
and Sarkar. These proofs were provided in different gravity theories – for
instance, Lovelock theory and higher curvature gravity – but the matter sector
was always taken to be minimally coupled. In this article, we describe how to
generalize the proof of linearized semi-classical GSL when the matter sector
comes with non-minimal couplings. The proof proceeds by suitably evaluating
the matter path integral in the stress tensor expectation value by treating
the higher derivative couplings in an effective field theory setting. We use
the recently established result of the linearized second law for such
theories.

Keywords: generalized second law (GSL), non-minimal, effective field theory
(EFT)

##  1 Introduction

General Relativity (GR) is considered incomplete due to the presence of
singularities [1], which are expected to be resolved by a quantum theory of
gravity. At energies significantly below the Planck scale, any UV complete
theory of gravity simplifies to Einstein’s general relativity with suppressed
higher derivative terms [2, 3]. Consequently, we can treat gravity as an
effective field theory (EFT) and concentrate on the primary quantum
corrections at these energy levels [4]. Black holes, as solutions of
Einstein’s theory, provide a useful context for studying the quantum nature of
gravity. In this framework, black holes exhibit thermodynamic properties, such
as well-defined notions of energy and temperature [5, 6, 7]. The black hole’s
entropy is linked to its horizon’s area [8, 9], and the area theorem ensures
that entropy never decreases, akin to the second law of black hole
thermodynamics.

Due to the presence of higher derivative terms, there’s a question about
whether black hole thermodynamics holds beyond general relativity. In the
presence of higher derivative interactions, the area law typically fails [10,
11, 12, 13]. Iyer and Wald, [14, 15], introduced a major generalization for
diffeomorphism invariant theories of gravity, where black hole entropy is
derived from the integral of the Noether charge associated with the horizon’s
generating Killing isometry. While Wald entropy satisfies the first law of
black hole mechanics, it suffers from JKM ambiguities for non-dynamical
horizons [16, 17]. As a result, except for exceptional cases like
$f(\mathcal{R})$ theories, where $\mathcal{R}$ is the Ricci scalar and $f$ is
a polynomial, Wald entropy doesn’t generally adhere to the second law of black
hole mechanics, which dictates that entropy should increase at each stage of
evolution

To prove the second law of black hole mechanics, one common approach is to
consider an equilibrium black hole with a Killing horizon being slightly
perturbed by external matter. As the black hole eventually returns to
equilibrium, an entropy functional should be constructed to increase
throughout this dynamical process. This can be achieved by extending the Wald
entropy to dynamic situations while addressing the associated JKM ambiguities
to ensure compliance with the second law. This approach has been used in
previous studies [18, 19, 20, 21, 22, 23, 24].

In [22, 23, 24], building on [20], an ultra-local version of the second law of
black hole thermodynamics was established for arbitrary diffeomorphism
invariant theories of gravity. This was accomplished by introducing a local
“entropy current” with a non-negative divergence on the black hole’s dynamic
horizon. The time component of the entropy current was a combination of the
Wald entropy and uniquely resolved JKM ambiguities. The spatial components of
the current facilitated the redistribution of entropy across different spatial
sections of the horizon. In previous works [18, 20, 22, 23], it was assumed
that the matter sector of the theory was minimally coupled and satisfied the
null energy condition (NEC), which contributed to entropy production and the
establishment of the second law. The minimal coupling prescription that we
work with is the following: given any Lorentz invariant expression involving
the matter fields in flat Minkowski space $\eta_{ab}$, the minimally coupled
expression is given by taking the flat metric $\eta_{ab}$ to the curved space
metric $g_{ab}$ and the partial derivative $\partial_{m}$ to the covariant
derivative $D_{m}$.

It is well known that if one considers a matter sector non-minimally coupled
to gravity, it can violate NEC [25, 26, 27, 28]. The analysis of [24] lifts
this crucial assumption, establishing a linearized second law for arbitrary
matter couplings to gravity. The current work aims to extend the analysis of
[24] by treating matter fields quantum mechanically, thereby establishing a
Generalized Second Law (GSL).

The GSL states that the total entropy of the matter fields outside the horizon
$S_{\text{out}}$ and the entropy of the black hole $S_{H}$ always increases in
a dynamical evolution. Thus, the proof of the GSL proceeds by showing that the
generalized entropy $S_{\text{gen}}=S_{H}+S_{\text{out}}$ always increases.

In the semi-classical setting that we are interested in, one expands the
metric in powers of the Planck mass, or equivalently, an $\hbar$ expansion
[29, 30, 18]:

$g_{ab}=g^{(0)}_{ab}+g^{({\frac{1}{2}})}_{ab}+g^{(1)}_{ab}+\mathcal{O}(\hbar^{\frac{3}{2}})\,.$
(1)

The zeroth-order term denotes the background metric, and the half-order term
denotes the fluctuations due to quantized gravitons. The first-order term
arises from the gravitational field due to quantized matter fields. Since we
are considering linearized fluctuations, we will ignore all terms higher than
$\mathcal{O}(\hbar)$. As detailed in [30], we are interested in a regime where
the metric fluctuations are solely driven by the quantum fluctuations of
matter fields. This is the most interesting case because if we consider
classical fluctuations, they far outweigh the quantum fluctuations. Unlike the
previous proofs of GSL in different kinds of gravity theories, we consider the
matter sector to be non-minimally coupled. We were able to establish a
statement of linearized semi-classical GSL for the non-minimally coupled
quantum field in an EFT sense (cf. Section 3.2). In this EFT treatment, we
assume a clear separation of scale between the minimal and non-minimal
sectors, which results in the splitting of the minimal and non-minimal parts
of the stress-energy tensor. We find that the non-minimal sector, treated
perturbatively over the minimal sector, should only contribute to the horizon
entropy, whereas the minimal sector is the dominant contribution to
$S_{\text{out}}$ at the linear order. With this simplification, we can make a
statement of GSL for the non-minimally coupled matter by suitably building up
on the previous works of Sarkar and Wall [18, 20] and using the boost-weight
analysis and entropy current structure of [23, 24]. We should stress that this
analysis does not incorporate the graviton contributions. We would like to
mention that the GSL was considered for a particular example of non-minimal
coupling in [31], and it has also been considered for specific examples of
pure gravity theories in [32].

Recently, in [33], the authors prove a statement of linearized GSL for higher
curvature gravity (including graviton contributions), generalizing the works
in [34, 35] (CPLW, CPW). Following up on the work of Leutheusser and Liu [36,
37], Witten in [38] first showed that there is a change in the nature of the
von Neumann algebra associated to local observables in the black hole exterior
in AdS spacetimes with holographic boundary duals. This change in the nature
of local observable algebra allows for associating a notion of density matrix
and a (renormalized) von Neumann entropy with the black hole state, which is
well-defined up to a state-independent additive constant. Later in [34, 35],
this idea was generalized to black holes in asymptotically flat spacetimes,
such that the entropy of any semi-classical state of the transformed algebra
agrees with the generalized entropy, up to an additive constant that is
independent of the choice of state. In [33], the authors claim a
generalization of the results of [34, 35] to an arbitrary diffeomorphism
invariant theory of gravity. However, the monotonicity of the generalized
entropy, which is essential to the proof of GSL, was a statement in the
boundary theory in [35], and it also does not generalize to out-of-equilibrium
horizon cuts, unlike the case in [30] where Wall proves that for any horizon
cut, the statement of GSL is given by $\derivative{S_{\text{gen}}}{v}\geq
0\,$. The main novelty of our proof of GSL is that one does not need the input
from the boundary theory, and we also have a statement for arbitrary out-of-
equilibrium cuts.

We conclude this section by briefly summarizing the different sections. In
Section 2, we briefly summarize the proof of classical second law (CSL),
introducing the idea of horizon-adapted coordinates, boost-weight analysis,
and the entropy current structure. These form some key ingredients in our
proof for the GSL as well. Readers familiar with these ideas can directly jump
to Section 3 which is the key section of the paper. In Section 3, we first
summarize previous attempts at proving GSL in different gravity theories. This
sets the stage to explain our setup and how our proof compares to the previous
literature. In this section, we also describe our EFT treatment of the non-
minimal sector and give a proof of the linearized semi-classical GSL in the
context of non-minimally coupled matter sector. Finally, in Section 4, we
discuss the relevance of our work, mentioning some issues to be addressed and
pointing out some open questions for future work.

##  2 Brief review of the classical second law

In this section, we will recap the basic setup of the proof of linearized
second law [20, 22, 23, 24]. We begin by explaining our coordinate choice for
the black hole spacetime. Then, we detail the notion of stationarity in our
spacetime, allowing us to define the fluctuating quantities about equilibrium
solutions. These fluctuating quantities can be classified according to how
they transform under the symmetries of the stationary spacetime. One can then
argue for the linearized second law from the structures of these fluctuating
quantities. We will state the main facts we will use, and we refer [23] for
further details.

###  2.1 Horizon adapted coordinates

The choice of coordinates for our black hole spacetime is given by $\\{v,\ r,\
x^{A}\\}$ where $(A=1,\ldots,d-2)$ such that the horizon is located at $r=0$.
The coordinate $v$ is chosen along the null generators of the horizon
$\partial_{v}$. On each constant $v$ slice, we shoot off a set of spatial
tangents $\partial_{A}$ in the $d-2$ spatial directions. These spatial
tangents $\partial_{A}$ will span the coordinates $x^{A}$. Thus, the horizon
will be spanned by $\\{v,x^{A}\\}$. The spacetime away from the horizon will
be spanned by a set of null geodesics $\partial_{r}$ emanating from the
horizon, making appropriate angles with $\partial_{A}$ and $\partial_{v}$.
This results in the gauge choice of [23, 24], namely

$\differential s^{2}=g_{ab}\differential x^{a}\differential x^{b},\mbox{\quad
where\quad}g_{rr}=g_{vA}\big{|}_{r=0}=g_{vv}\big{|}_{r=0}=g_{rA}=\partial_{r}g_{vv}\big{|}_{r=0}=0\,,g_{rv}=1\,.$
(2)

With this, the near-horizon metric for a generic dynamical black hole can
always be cast in the form given by [39]

$\differential s^{2}=2\differential v\,\differential
r-r^{2}X(r,v,x^{C})\differential v^{2}+2r\omega_{A}(r,v,x^{C})\differential
v\,\differential x^{A}+h_{AB}(r,v,x^{C})\differential x^{A}\,\differential
x^{B}\,.$ (3)

We assume a form of the Zeroth law that enables us to express the metric in
this form. A Zeroth law for arbitrary pure gravity diffeomorphism invariant
theories was established in [40] following upon initial work in [41].

Here $X,\omega_{A},h_{AB}$ are general functions of coordinates. This gauge
choice does not completely fix the choice of coordinates on constant $v$ and
$r$ slices as there are residual symmetries in the gauge of (3) of the form

$\begin{split}v\to\tilde{v}&=vf_{1}(x^{i})+f_{2}(x^{i})\,,\ \text{with
appropriate redefinition of}~{}r\,,\\\
x^{i}\to\tilde{x}^{i}&=g^{i}(x^{j})\,.\end{split}$ (4)

We work within a regime where we are perturbatively close to the equilibrium
solution. For this, we must construct an equilibrium solution in our gauge
(3). We will use the residual symmetries of (4) to facilitate this. Let us
consider a simple scaling symmetry of the form

$(r,v)\mapsto\left(\lambda r,\frac{v}{\lambda}\right)\,.$ (5)

This is called “boost transformation” and is a particular case of (4) when
$f_{1}$ is a constant and $f_{2}=0$. Since equilibrium black holes must have a
Killing vector that becomes null on the horizon, we can use (5) to fix our
stationary background metric. As (5) is generated by

$\xi=v\partial_{v}-r\partial_{r}\,,$ (6)

a stationary black hole with metric $g_{ab}^{\text{(eq)}}$ is given by

$\differential s^{2}=2\differential v\differential
r-r^{2}X(rv,x^{C})\differential v^{2}+2r\omega_{A}(rv,x^{C})\differential
v\differential x^{A}+h_{AB}(rv,x^{C})\differential x^{A}\differential x^{B},$
(7)

where the coefficients $X,\omega_{A},h_{AB}$ are now functions of the products
of the coordinates $r,v$. See A for a demonstration of how the standard
Schwarzschild and Kerr solutions can be brought to this gauge. The metric is
invariant under the scaling (5). This is said to be the boost symmetry of the
horizon, and (6) is a Killing vector of the background spacetime. The horizon
$r=0$ is generated by (6) and is a Killing horizon of the spacetime.
$\xi^{\mu}$ of (6) becomes null on the horizon $r=0$. For our construction, it
is enough to consider a Killing horizon. We will assume that the event horizon
of the equilibrium black hole is a Killing horizon. We want to describe the
dynamics only in a perturbative sense, up to the linearized order, in which
case we decompose the metric $g_{ab}$ in (3) using (7) as

$g_{ab}=g^{\text{(eq)}}_{ab}(rv,x^{C})+\delta g_{ab}(r,v,x^{C}),$ (8)

where $g^{\text{(eq)}}_{ab}$ is the equilibrium metric as in (7), and $\delta
g_{ab}$ is the fluctuation in the metric at the linearized order in $\hbar$ or
$\epsilon$.

Once we have the notion of the background plus fluctuation, we can quantify
non-equilibrium structures based on their “boost weight”. The boost weight is
defined according to how covariant tensors/structures transform under the
boost transformation (5):

$\mathcal{T}\to\bar{\mathcal{T}}=\lambda^{w}\mathcal{T}\,,~{}\text{under}~{}\left\\{r\to\bar{r}=\lambda
r,\,v\to\bar{v}=\frac{v}{\lambda}\right\\}~{}~{}\implies~{}~{}\text{boost
weight of $\mathcal{T}$ is $w$}\,.$ (9)

We have a non-equilibrium quantity if $w>0$ for a covariant tensor. This can
be understood as follows. Suppose if $\mathcal{T}$ was of the form

$\mathcal{T}\sim(\partial_{r})^{k_{r}}(\partial_{v})^{k_{v}}\mathcal{A}\,,$
(10)

where $\mathcal{A}$ is constructed out of $X,\omega_{A},h_{AB}$ and
$\nabla_{A}$ (the covariant derivative compatible with induced metric
$h_{AB}$) only. When we evaluate $\mathcal{A}$ on (8), it breaks up into an
equilibrium contribution and a fluctuating contribution. The equilibrium
contribution
$\mathcal{A}(r,v,x^{A})\big{|}_{\text{eqbrm}}\sim\mathcal{A}(rv,x^{A})$. If we
take $k_{v}>k_{r}$, i.e., $w>0$, then the equilibrium contribution of (10) has
$k_{v}-k_{r}$ factors of $r$. This evaluates to zero on the horizon $r=0$.
Thus, only the fluctuating part of (10) remains:

$\mathcal{T}\big{|}_{r=0}\sim(\partial_{r})^{k_{r}}(\partial_{v})^{k_{v}}\mathcal{A}\sim\epsilon(\partial_{r})^{k_{r}}(\partial_{v})^{k_{v}}\delta\mathcal{A}\sim\mathcal{O}(\epsilon)\,.$
(11)

Now suppose if we consider a product of two such terms, then they become
$\mathcal{O}(\epsilon^{2})$ and we neglect them up to the linearized order we
are working with:

$(\partial^{m_{1}}_{r}\partial^{m_{1}+k_{1}}_{v})\mathcal{A}_{1}\,(\partial^{m_{2}}_{r}\partial^{m_{2}+k_{2}}_{v})\mathcal{A}_{2}\sim\mathcal{O}(\epsilon^{2})\,.$
(12)

Thus, when we have positive boost weight quantities, they are non-trivial only
for non-equilibrium configurations. Likewise, we neglect the terms, which are
in the form of a product of two positive boost weight quantities at the
linearized approximation we are working with.

###  2.2 Review of the second law within the setup of the entropy current

The entropy of a dynamical black hole in arbitrary diffeomorphism invariant
theories is given as [18, 20, 22, 23]

$S_{\text{tot},v}=\frac{1}{4\hbar
G}\int_{H_{v}}\differential^{d-2}x\,\sqrt{h}\,(1+s^{\text{HD}}_{\text{wald}}+s_{\text{cor}})\,,$
(13)

where, $h_{AB}$ is the induced metric on the space-like $(d-2)$ dimensional
cross-section of the horizon $H_{v}$, (the subscript $v$ indicates we are
looking at some constant $v$ slice) and $h$ denotes its determinant. The
factor of 1 contribution arises from the two-derivative Einstein-Hilbert term
in the gravity Lagrangian and it is the standard Area term,
$s^{\text{HD}}_{\text{wald}}$ is the entropy density contribution arising only
from the higher-derivative terms in the gravity action, and $s_{\text{cor}}$
is the possible correction to the entropy density due to non-equilibrium
effects, which includes JKM ambiguities. It is not contained in the Wald
entropy density $s_{\text{wald}}=1+s^{\text{HD}}_{\text{wald}}$. Clearly, in
the equilibrium limit, this contribution should vanish, i.e.,
$s_{\text{cor}}\big{|}_{\text{eqbrm}}\to 0$.

Define $\rho=s_{\text{wald}}+s_{\text{cor}}$. As we perturb the horizon, the
entropy changes and the rate of change is given by

$\derivative{S_{\text{tot},v}}{v}=\frac{1}{4\hbar
G}\int_{H_{v}}\differential^{d-2}x\,\sqrt{h}\,\left[\derivative{\rho}{v}+\theta\rho\right]\,,$
(14)

where $\theta$ is the expansion parameter of the null generators and is given
by

$\theta=\frac{1}{\sqrt{h}}\partial_{v}\sqrt{h}\,.$ (15)

The total change can then be obtained as

$\Delta S_{\text{tot}}=\frac{1}{4\hbar G}\int_{H}\differential
v\,\differential^{d-2}x\,\sqrt{h}\,\left(\derivative{\rho}{v}+\theta\rho\right)\,.$
(16)

We now define the “generalized expansion parameter” $\Theta$ as the rate at
which the entropy density of an infinitesimal region of the horizon changes,
i.e.,

$\Theta\coloneqq\derivative{\rho}{v}+\theta\rho\,,$ (17)

which for the Einstein-Hilbert term reduces to the usual expansion parameter
$\theta$ that goes into the Raychaudhuri equation [42]. It will be assumed
that the dynamical horizon will eventually reach a stationary configuration
such that $\Theta(v\to\infty)=0$. Thus, the classical second law follows if we
can show $\Delta S_{\text{tot}}>0$. A way to show this is to prove $\Theta>0$,
which is a stronger statement than $\Delta S_{\text{tot}}>0$. This follows if
we prove $\partial_{v}\Theta<0$ and then use $\Theta(v\to\infty)=0$ to infer
$\Theta>0$ for every finite $v$.

Now $\partial_{v}\Theta$ is given by

$\begin{split}\partial_{v}\Theta&=\partial^{2}_{v}(s^{\text{HD}}_{\text{wald}}+s_{\text{cor}})+\partial_{v}\theta(1+s^{\text{HD}}_{\text{wald}}+s_{\text{cor}})+\theta\partial_{v}\rho\\\
&=\partial_{v}\theta+\partial_{v}\left(\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}(s^{\text{HD}}_{\text{wald}}+s_{\text{cor}})\right)\right)+\mathcal{O}(\epsilon^{2})\\\
&=-\mathcal{R}_{vv}+\partial_{v}\left(\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}(s^{\text{HD}}_{\text{wald}}+s_{\text{cor}})\right)\right)+\mathcal{O}(\epsilon^{2})\,.\end{split}$
(18)

In the second equality, we have used the boost weight prescription of (12) to
get rid of quadratic terms like $\theta\partial_{v}\rho$ and other terms that
arise as the result of expressing them in the above form. In the final
equality, we have used the linearized Raychaudhuri equation to write
$\partial_{v}\theta=-\mathcal{R}_{vv}+\mathcal{O}(\epsilon^{2})$. Since the
Lagrangian is given by the Einstein-Hilbert (Ricci scalar) term in addition to
higher derivative terms and a matter sector (non-minimally coupled matter plus
a minimally coupled sector that satisfies the NEC), the equations of motion
projected onto $r=0$ (the horizon) in our gauge (3) are given by

$\mathcal{R}_{vv}+E^{\text{HD}}_{vv}+T^{\text{(non-
min)}}_{vv}=T^{\text{(min)}}_{vv}\,.$ (19)

Here $E_{vv}=\mathcal{R}_{vv}+E^{\text{HD}}_{vv}$ denotes the contribution of
pure gravity terms, and $T_{vv}$ denotes the contribution of matter terms
coupled to gravity tensors. Substituting (19) in (18), we get

$\partial_{v}\Theta=E^{\text{HD}}_{vv}+T^{\text{(non-
min)}}_{vv}+\partial_{v}\left(\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}(s^{\text{HD}}_{\text{wald}}+s_{\text{cor}})\right)\right)-T^{\text{(min)}}_{vv}+\mathcal{O}(\epsilon^{2})\,.$
(20)

From (20), it is clear that if one can show that the equations of motion
projected onto the horizon $r=0$ in our gauge (3) satisfy

$E^{\text{HD}}_{vv}+T^{\text{(non-
min)}}_{vv}=-\partial_{v}\left(\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}(s^{\text{HD}}_{\text{wald}}+s_{\text{cor}})\right)\right)+\mathcal{O}(\epsilon^{2})\,,$
(21)

then we have

$\partial_{v}\Theta=-T^{\text{(min)}}_{vv}+\mathcal{O}(\epsilon^{2})<0\,,$
(22)

because of null energy condition $T^{\text{(min)}}_{vv}>0$. It is important to
note that this inequality sign in the context of linearized approximation
necessarily implies that $T^{\text{(min)}}_{vv}$ cannot be
$\mathcal{O}(\epsilon)$.

This is because one can switch the sign of $\epsilon$ to violate the
inequality. Thus, we have $T^{(\text{min})}_{vv}\sim\mathcal{O}(\epsilon^{2})$
and we are essentially proving
$\partial_{v}\Theta\sim\mathcal{O}(\epsilon^{2})$. An intuitive way to
understand this is that there are no terms in the linear order that can make
$\partial_{v}\Theta>0$. Such terms might occur in the quadratic order, and one
must carefully analyze them like [43]. We will not have anything to say about
the non-linear order in our paper.

In the above proof, the structure of equations of motions projected onto the
null horizon in (21) played a crucial role. Consider a Lagrangian which is a
diffeomorphism invariant theory of gravity of the following form

$\mathcal{L}=\mathcal{L}(g_{ab},\mathcal{R}_{abcd},D_{e}\mathcal{R}_{abcd},\dots,\Phi,D_{a}\Phi,D_{(a}D_{b)}\Phi,\dots,F_{ab},D_{c}F_{ab},\dots)\,,$
(23)

where the Lagrangian can have a complicated dependence of the metric $g_{ab}$,
scalar $\Phi$, and $U(1)$ gauge fields $A_{a}$, with field strength tensor
$F_{ab}$. If one can show that the equations of motion projected onto the null
horizon $r=0$ in our gauge (3) take the form

$E^{\text{HD}}_{vv}+T^{\text{(non-
min)}}_{vv}=\partial_{v}\left(\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}\mathcal{J}^{v}\right)+\nabla_{A}\mathcal{J}^{A}\right)+\mathcal{O}(\epsilon^{2})\,,$
(24)

then (21) holds, and thus, a linearized second law holds for black holes in
the theory we are considering. (24) has been established for the Lagrangian of
the form (23) in [23, 24]. Using constraints from the boost symmetry (5) of
the horizon and diffeomorphism invariance of the Lagrangian, [23] showed that
$E_{vv}$ of (19) has the structure of (24). The input of diffeomorphism
invariance can be relaxed as this structure was also established for Chern-
Simons theories of gravity, which are diffeomorphism invariant up to total
derivatives only [44]. The analysis follows by carefully working out how
different covariant tensors, built out of the Lagrangian, transform under the
boost transformation (5). For $T_{vv}^{\text{(non-min)}}$, we can invoke the
analysis of [24] to show that it also has the structure of (24). This proves
the linearized classical second law. Here, $\mathcal{J}^{v}$ and
$\mathcal{J}^{A}$ are considered as components of a $d-1$ “vector” on the
horizon known as entropy current. $\mathcal{J}^{v}$ is given by

$\mathcal{J}^{v}=1+s^{\text{HD}}_{\text{wald}}+s_{\text{cor}}\,,$ (25)

which is the equilibrium Wald entropy density plus the associated JKM
ambiguities, which are now fixed as we are just rewriting the equations of
motion. We put vector in quotes because as the horizon is null, this is not a
covariant vector of the full space-time. If we integrate $\mathcal{J}^{v}$
over the cross-section of the horizon, we get the total entropy (13).
$\mathcal{J}^{A}$ is a non-equilibrium quantity, and it can be understood as
quantifying the redistribution of entropy across the spatial cross-section of
the constant $v$ slices. This interpretation holds because the increase of the
expansion parameter $\Theta>0$ implies

$\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}\mathcal{J}^{v}\right)+\nabla_{A}\mathcal{J}^{A}>0\,.$
(26)

We have the result that the divergence of an entropy current with components
$\mathcal{J}^{v}$ and $\mathcal{J}^{A}$ is positive. This is a stronger ultra-
local version of the linearized second law when compared to the integrated
version of the second law considered in [20], i.e., in (16). This is because
the divergence of entropy current not only quantifies an increase in entropy
along the “time” direction but also quantifies a redistribution along the
spatial directions in such a way that the total entropy always increases
(because there is ultra-local entropy production). Clearly, the presence of
$\mathcal{J}^{A}$ would not contribute to (14) if we assume compact horizons,
whence the total derivative term drops out of the spatial integral. Thus, (24)
and (21) are equivalent. At this point, one might raise an objection that the
construction is heavily reliant on the horizon adapted coordinates. However,
one can show that the entropy current structure is covariant under affine
reparametrizations of the null generators [45]. This completes our review of
the linearized classical second law.

##  3 The Generalized Second Law

In this section, we will give a proof of the GSL when the matter is non-
minimally coupled to gravity. We work in an EFT sense, which will be
elaborated in Section 3.2. This also requires defining the expectation value
of the stress-energy tensor, which involves the first-order quantum correction
to the classical theory and will be treated similarly as in [46]. We then give
a proof of the linearized semi-classical GSL in Section 3.2.2.

###  3.1 Summary of GSL in other gravity theories

We briefly summarize [18, 20], which formed the primary motivation for our
work. Sarkar and Wall (SW) [18] constructed a proof for the (integrated form
of) linearized classical second law and the linearized generalized second law
for the semi-classical, minimally-coupled matter sector in the context of
Lovelock theories. This was later extended to an arbitrary diffeomorphism
invariant theory of gravity with minimally coupled matter by Wall [20]. The
setup considered has been detailed in Section 2: a stationary black hole is
perturbed slightly and it settles down to some equilibrium solution. This
allows us to use the linearized approximation in the dynamics. The central
idea was to argue that (13) with the JKM ambiguities fixed, resulted in a
linearized version of the second law.

The semi-classical equation of motion for the metric is given by

$E_{ab}=8\pi G_{N}\expectationvalue{T^{\text{(full)}}_{ab}}\,,$ (27)

where $E_{ab}$ will be considered to be a c-number whereas the stress-energy
tensor $T^{\text{(full)}}_{ab}$ is treated as a quantum operator. The precise
meaning of $\expectationvalue{T^{\text{(full)}}_{ab}}$ will be described in
Section 3.2.1, [46, 30]. The state with respect to which the expectation value
is calculated is chosen to be the generic semi-classical vacuum state
corresponding to the background equilibrium solution $g_{ab}^{(0)}$. The semi-
classical equation in (27) is justified because we will be working in the
linearized weak-field approximation limit, where the semi-classical equation
in (27) is obtained from the full quantum operator equation by taking
expectation values of the $\mathcal{O}(\hbar)$ part of the metric, [29]. In
[18], $T^{\text{(full)}}_{ab}$ received contribution just from the minimally
coupled matter sector, but for a generic non-minimal matter theory it would
include the non-minimal terms as well. Then, if we substitute (27) instead of
(19) in (18), we have for first-order changes

$\derivative{\Theta}{v}\approx-8\pi
G_{N}\expectationvalue{T^{\text{(full)}}_{vv}}\,.$ (28)

Now, the claim is that in any theory of gravity, if (28) is obeyed, then this
will imply the linearized semi-classical GSL [18, 20]. This can be seen as
follows. We first integrate (28) once in the transverse directions $x^{A}$ and
twice along the null generator $\partial_{v}$ to get

$\dfrac{\hbar}{2\pi}\left(S_{\text{tot}}(\infty)-S_{\text{tot}}(v^{\prime})\right)=\int\differential^{d-2}x\sqrt{h}\int_{v>v^{\prime}}\differential
v\,(v-v^{\prime})\expectationvalue{T^{\text{(full)}}_{vv}}\,.$ (29)

[18, 20] showed that this would imply a semi-classical GSL for matter
minimally coupled to gravity.

In both works above, certain assumptions were made on the algebra of
observables on the horizon. These assumptions were first detailed in Wall’s
earlier work on proving GSL for Einstein’s theory with fairly generic fields
minimally coupled to gravity and for arbitrary horizon slices, [30]. All the
above proofs required some (as in [18]) or all (as in [30]) of the four axioms
that the algebra of observables restricted to the horizon must satisfy,
namely, Determinism (the horizon algebra completely specifies all the
information falling across the horizon such that together with the information
available at the future null infinity, all the information outside the event
horizon is completely determined), Ultralocality (the degrees of freedom on
distinct horizon generators should be understood as independent systems),
Local Lorentz Invariance (there exists an infinite dimensional group of
symmetries, associated with the horizon algebra, which correspond to the
affine transformations of each horizon generator), and Stability (the
generator of null translations should be non-negative).

In [33], the authors arrive at (29) by crucially using the structure of
$E_{vv}$ in (21) proved for arbitrary diffeomorphism invariant theories of
gravity in [23, 24] following up on the works of [18, 20, 47, 48]. For
instance, (II.38) of [33] follows crucially from equation (5.7) of [23]. They
then proceed to generalize the constructions of [34, 35] for arbitrary
diffeomorphism invariant theories of gravity by suitably generalizing the
notions of canonical energy in the covariant phase space formalism of [14,
15]. However, as mentioned in the Introduction (cf. Section 1), the proof of
GSL does not generalize to out-of-equilibrium cuts as the notion of the
crossed product type II von Neumann algebra, and hence an associated entropy,
does not exist corresponding to such arbitrary horizon cuts.

###  3.2 The setup and a proof

In this section, we will explain in detail the setup in which we prove the
GSL. We start with the following action

$\begin{split}\mathcal{S}&=\mathcal{S}_{\text{g}}[g_{ab},\mathcal{R}_{abcd},D_{m}\mathcal{R}_{abcd},D_{(m}D_{n)}\mathcal{R}_{abcd},\ldots]+\mathcal{S}_{\text{min}}[g_{ab},\Phi,D_{m}\Phi]\\\
&+\mathcal{S}_{\text{nm}}[g_{ab},\mathcal{R}_{abcd},D_{m}\mathcal{R}_{abcd},D_{(m}D_{n)}\mathcal{R}_{abcd},\ldots,\Phi,D_{m}\Phi,D_{(m}D_{n)}\Phi,\ldots]\,,\end{split}$
(30)

where $\mathcal{S}_{\text{g}}$ is just the gravitational part of the action
which is an arbitrary function of the $\mathcal{R}_{abcd}$ and covariant
derivatives acting on it, $\mathcal{S}_{\text{min}}$ represents the minimally
coupled matter sector (containing terms only up to two derivatives in the
field), and $S_{\text{nm}}$ represents the non-minimal interaction of the
gravity and matter fields, and we assume that it has to have terms with at
least four derivatives in them.

We want to show that GSL holds to the linear order in $\hbar$ corrections
within the EFT approximation of [43]. We will consider the Lagrangian to be a
formal sum of different terms with increasing number of derivatives in the
fields. Each term comes with a suitable factor of some UV length scale $\ell$,
more precisely, a term with $n+2$ derivatives will be multiplied by
$\ell^{n}$. The terms with two or fewer derivatives are understood as the
usual Einstein-Hilbert term with a minimally coupled matter sector, besides
some cosmological constant. The validity of EFT depends on the fact that the
terms with an increasing number of derivatives become increasingly less
significant. We can achieve this by restricting to spacetimes varying over
some characteristic length/time scale $L$ with $\frac{\ell}{L}\ll 1$. The
scale $L$ could be considered a lower bound on the size of the final
equilibrium black hole state and any perturbation scale away from the
equilibrium, [43].

We will further consider the semi-classical regime where we can control the
physical effects as a perturbative expansion in $\hbar$ and $\frac{\ell}{L}$.
We will aim to argue for the GSL to linear order in $\hbar$ from (28). The
non-trivial part of the proof would be to develop a proposal that can evaluate
the expectation value of the matter stress tensor when the matter is non-
minimally coupled. One cannot straight-away use the four axioms of [30] to
deal with the stress tensor expectation value because non-minimal matter
contributes to the horizon entropy. Our proposal in Section 3.2.1 will deal
with this issue in such a way that we can argue for the linearized GSL from
(29). The higher order corrections to this equation in $\hbar$ come from the
renormalization theory, but these are safely neglected in our analysis, which
treats backreaction only at the leading order in $\hbar$. This is along the
same lines as previous analyses of Wall in [29, 20], and Sarkar and Wall in
[18].

Further, we stress that $\hbar\sim\epsilon$, which follows from the fact that
we treat the metric fluctuations as semi-classical, i.e., the size of the
quantum backreaction is comparable to the dynamical fluctuations in the
classical background. This is, in fact, the most interesting regime where the
validity of GSL should be carefully checked. If we consider classical
fluctuations, they far outweigh any quantum fluctuation. Thus, any violation
of GSL is apparent only when the size of the classical fluctuations is
comparable to the quantum fluctuations [30]. Thus, we consider the variation
of the metric fluctuations along the horizon in the semi-classical case to be
comparable to that of the classical fluctuations.

####  3.2.1 Stress tensor expectation value: a proposal

In this subsection, we will describe a prescription for computing the
expectation value of the stress tensor for the complete non-minimal theory by
treating the non-minimal part as a small perturbation over the minimal theory,
where we know how to compute the expectation values of operators, in
particular the stress tensor, up to the one-loop level.111The theory truncated
at one-loop level has all terms of the complete quantum theory to
$\mathcal{O}(\hbar)$, and it is in this sense the first order quantum
correction to the classical theory, [46]. For a discussion on the issues
related to semi-classical Einstein equation see [49].

The basic object we need to analyze is the generating functional of the full
theory given by

$\mathcal{Z}[J]=\int\mathscr{D}\Phi\,\exp[\dfrac{\iota}{\hbar}(\mathcal{S}_{\text{g}}+\mathcal{S}_{\text{min}}+\mathcal{S}_{\text{nm}})+\iota\int\differential^{D}x\,J(x)\Phi(x)]\,,$
(31)

taken over the space of fields $\Phi$ with suitable measure. Since the
$\mathcal{S}_{\text{g}}$ term does not depend on $\Phi$, it will contribute as
an overall multiplicative factor to the generating functional, which we denote
by $\mathcal{N}_{g}$. Further, we can analyze the non-minimal part of the
action as a perturbation over the minimal theory because
$\mathcal{S}_{\text{nm}}$ is at least $\mathcal{O}((\frac{\ell}{L})^{2})$
suppressed compared to $\mathcal{S}_{\text{min}}$. Thus we have

$\mathcal{Z}[J]=\mathcal{N}_{g}\int\mathscr{D}\Phi\,\exp[\dfrac{\iota}{\hbar}\,\mathcal{S}_{\text{min}}+\iota\int\differential^{D}x\,J(x)\Phi(x)]\exp[\dfrac{\iota\,\ell^{2}}{\hbar\,L^{2}}\mathcal{S}_{\text{non-
min}}]\,,$ (32)

where

$\mathcal{S}_{\text{nm}}=\frac{\ell^{2}}{L^{2}}\mathcal{S}_{\text{non-
min}}\,.$

We now define the “minimal generating function” as

$\mathcal{Z}_{\text{min}}[J]=\int\mathscr{D}\Phi\,\exp[\dfrac{\iota}{\hbar}\,\mathcal{S}_{\text{min}}+\iota\int\differential^{D}x\,J(x)\Phi(x)]\,,$
(33)

and for any operator in the matter sector, $\hat{A}$, defined in the full non-
minimal theory, we define

$\expectationvalue{\hat{A}(\Phi,D_{m}\Phi,\ldots)}\coloneqq\frac{\int\mathscr{D}\Phi\,e^{\frac{\iota}{\hbar}\,\mathcal{S}_{\text{min}}}A(\Phi,D_{m}\Phi,\ldots)}{\int\mathscr{D}\Phi\,e^{\frac{\iota}{\hbar}\,\mathcal{S}_{\text{min}}}}\,.$
(34)

This implies (31) takes the form

$\mathcal{Z}[J]=\mathcal{Z}_{\text{min}}[0]\,\expectationvalue{\exp[\dfrac{\iota\,\ell^{2}}{\hbar\,L^{2}}\,\mathcal{S}_{\text{non-
min}}]}\,.$ (35)

At this point, it is helpful to comment on how we treat the gravity tensors in
the path integral. At the level of the matter path integral, they are
quantities independent of the matter fields. Since we have not imposed any
gravity equations of motion, we can directly evaluate the path integral over
the matter fields by treating gravity tensors as some “background” quantities.
After the path integral, the matter fields couple to gravity through the semi-
classical equation (27). Only at this point, the geometric tensors would
couple to quantum matter fields non-trivially.

The basic idea of the calculation is to mimic the interaction theory analysis
for perturbative QFT. Ultimately, it boils down to calculating things with
respect to the minimal theory in the sense that the weight factor in the path
integral is defined only via the minimal part. Our EFT expansion is based on
the parameter

$\lambda=\dfrac{\ell^{2}}{\hbar L^{2}}\ll 1\,.$ (36)

This parameter clarifies why we can safely neglect renormalization effects on
the couplings. They tend to be suppressed because, at one loop, we have
$\lambda\hbar\ll\lambda\ll 1$. We then define a quantity which we call the
effective action for the quantum matter fields as

$\mathcal{W}_{\text{min}}=-\iota\hbar\,\log\mathcal{Z}_{\text{min}}[0]\equiv-\iota\hbar\,\log\mathcal{Z}_{\text{min}}\,,$
(37)

and

$\mathcal{W}=-\iota\hbar\log\mathcal{Z}[0]=-\iota\hbar\log\left[\mathcal{Z}_{\text{min}}\expectationvalue{\exp[\iota\lambda\,\mathcal{S}_{\text{non-
min}}]}\right]\,,$ (38)

such that

$\dfrac{2}{\sqrt{-g}}\functionalderivative{\mathcal{W}_{\text{min}}}{g_{ab}}\coloneqq\expectationvalue{T_{\text{min}}^{ab}}=-\frac{2\iota\,\hbar}{\sqrt{-g}\mathcal{Z}_{\text{min}}}\functionalderivative{\mathcal{Z}_{\text{min}}}{g_{ab}}\mbox{\quad
and\quad}\dfrac{2}{\sqrt{-g}}\functionalderivative{\mathcal{W}}{g_{ab}}\coloneqq\expectationvalue{T^{ab}}\,.$
(39)

So the expectation value of the stress tensor is defined to be

$\expectationvalue{T^{ab}}=\expectationvalue{T^{ab}_{\text{min}}}-\dfrac{2\iota}{\sqrt{-g}}\dfrac{\hbar}{\expectationvalue{\exp[\iota\lambda\,\mathcal{S}_{\text{non-
min}}]}}\functionalderivative{g_{ab}}\expectationvalue{\exp[\iota\lambda\,\mathcal{S}_{\text{non-
min}}]}\,.$ (40)

We emphasize that [46] gives us a prescription for defining the expectation
value of the minimal sector only. The above expectation value for a generic
non-minimal coupling is our prescription based on EFT, where we crucially
required that $\mathcal{S}_{\text{non-min}}$ be suppressed compared to
$\mathcal{S}_{\text{min}}$. To evaluate the final step, we first note that the
expectation value is defined with respect to the minimal theory in (34). We
can simplify the path integral by doing a saddle point approximation. This
amounts to setting the field $\Phi$ in $\mathcal{S}_{\text{non-min}}$ as the
field $\Phi_{\text{min}}^{\text{sol}}$ which satisfies the classical minimal
equation of motion, i.e.,

$D_{a}D^{a}\Phi_{\text{min}}^{\text{sol}}+m^{2}\Phi_{\text{min}}^{\text{sol}}=0\,.$
(41)

This saddle-point approximation is valid because the minimal part of the
action has only up to second-order derivatives in the field. Thus, we have

$\expectationvalue{\exp[\iota\lambda\,\mathcal{S}_{\text{non-
min}}]}=\exp[\iota\lambda\,\mathcal{S}_{\text{non-
min}}[\Phi=\Phi^{\text{sol}}_{\text{min}}]]\,.$ (42)

We stress the point that we were able to arrive at the above equation because
$\mathcal{S}_{\text{non-min}}$ is suppressed compared to
$\mathcal{S}_{\text{min}}$ in an EFT sense (36). Using (42) in (40), we get

$\expectationvalue{T^{ab}}=\expectationvalue{T^{ab}_{\text{min}}}+\dfrac{2\lambda\,\hbar}{\sqrt{-g}}\functionalderivative{g_{ab}}\mathcal{S}_{\text{non-
min}}[\Phi=\Phi^{\text{sol}}_{\text{min}}]=\expectationvalue{T^{ab}_{\text{min}}}+\dfrac{\ell^{2}}{L^{2}}\,T^{ab}_{\text{non-
min}}[\Phi=\Phi^{\text{sol}}_{\text{min}}]\,.$ (43)

For our proof, we would be interested only in the $vv$-component of the stress
tensor, using the fact that the terms are either proportional to $g_{vv}$, or
have positive boost weight, and with our gauge choice (7) it is easy to see
that

$\expectationvalue{T_{vv}^{\text{(min)}}}\big{|}_{r=0}\sim\mathcal{O}(\hbar)\,.$
(44)

We are finally left with

$\expectationvalue{T_{vv}}\big{|}_{r=0}=\expectationvalue{T_{vv}^{\text{(min)}}}\big{|}_{r=0}+\dfrac{\ell^{2}}{L^{2}}\,T^{\text{(non-
min)}}_{vv}[\Phi=\Phi^{\text{sol}}_{\text{min}}]\big{|}_{r=0}+\mathcal{O}(\hbar^{2})\,.$
(45)

####  3.2.2 Generalized second law: a proof

Using (45), the $vv$ component of the semi-classical equation of motion in
(27) becomes

$E_{vv}\big{|}_{r=0}=8\pi
G_{N}\left[\expectationvalue{T_{vv}^{\text{(min)}}}+\dfrac{\ell^{2}}{L^{2}}T_{vv}^{\text{(non-
min)}}(\Phi_{\text{min}}^{\text{sol}},D_{a}\Phi_{\text{min}}^{\text{sol}},\ldots)\right]_{r=0}\,.$
(46)

Here $E_{vv}$ gets contribution only from $\mathcal{S}_{\text{g}}$ of the
action (30) while $T_{vv}^{\text{(non-min)}}$ is the classical equation of
motion obtained from $\mathcal{S}_{\text{non-min}}$. We now invoke the
$E_{vv}$ structure at $r=0$ in (24) [23, 24]. Thus, (46) becomes

$\partial_{v}\Theta_{\text{tot}}=8\pi
G_{N}\,\expectationvalue{T_{vv}^{\text{(min)}}}\,,$ (47)

where $\Theta_{\text{tot}}$ is given by

$\Theta_{\text{tot}}=\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}\,\left[\mathcal{J}^{v}+\dfrac{\ell^{2}}{L^{2}}\mathcal{J}^{v}_{\text{non-
min}}\right]\right)+\dfrac{1}{\sqrt{h}}\partial_{A}\left(\sqrt{h}\,\left[\mathcal{J}^{A}+\dfrac{\ell^{2}}{L^{2}}\mathcal{J}^{A}_{\text{non-
min}}\right]\right)\,.$ (48)

Here $\mathcal{J}^{v}_{\text{non-min}}$ and $\mathcal{J}^{A}_{\text{non-min}}$
are given by [24]

$-T^{(\text{non-
min})}_{vv}\big{|}_{r=0}=\partial_{v}\left(\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}\,\mathcal{J}^{v}_{\text{non-
min}}\right)+\dfrac{1}{\sqrt{h}}\partial_{A}\left(\sqrt{h}\,\mathcal{J}^{A}_{\text{non-
min}}\right)\right)+\mathcal{O}(\hbar^{2})\,,$ (49)

and $\mathcal{J}^{v}$ and $\mathcal{J}^{A}$ are defined through [23]

$E_{vv}\big{|}_{r=0}=\partial_{v}\left(\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}\,\mathcal{J}^{v}\right)+\dfrac{1}{\sqrt{h}}\partial_{A}\left(\sqrt{h}\,\mathcal{J}^{A}\right)\right)+\mathcal{O}(\hbar^{2})\,.$
(50)

(47) implies a linearized GSL following the analysis of [30, 18]. The way one
argues is as follows. In (47), we have split the minimal and non-minimal parts
of the stress tensor. The part of the stress tensor that sits inside the
expectation value is the minimal term, and the path integral is weighted with
the minimal action. The matter fields of the full non-minimal theory might not
always satisfy the Stability axiom of [30]. It is reasonable to assume that
the algebra satisfies Determinism, Ultra-locality, and Local Lorentz symmetry.
We assume that given such an algebra, we can consistently restrict the algebra
to the minimal sector of the theory. This algebra is taken to satisfy the
stability axiom as shown in [30]. Once we have this restricted “minimal”
algebra of observables on the horizon, we can borrow the setup of section 5 of
[18] to argue for the GSL.

Using the axioms of local Lorentz symmetry and stability, we invoke the
Bisongano-Wichmann theorem [50], which states that the vacuum state restricted
to the Rindler wedge is thermal with respect to the boost energy. Then, we use
Sewell’s generalization [51] to the algebra of observables on the horizon when
the vacuum state is restricted to the horizon cut above the bifurcation
surface. The vacuum then remains thermal for any cut $v>v^{\prime}$ by virtue
of the local Lorentz symmetry (supertranslations in this case). One can then
use the monotonicity property of the relative entropy to argue for a GSL [30].
We note that this analysis assumes a suitable renormalization scheme, which
maybe argued for based on Wall’s original formalism [30] and the recent works
in [34, 35, 33].

##  4 Discussion

Here, we summarize what we have been able to achieve so far. We considered
arbitrary non-minimally coupled diffeomorphism invariant matter-gravity
theories and established a linearized GSL for dynamical black holes in such
theories. The non-trivial physics of non-minimally coupled theories is that it
violates the Null Energy Condition. It is also unknown if they satisfy the
Averaged Null Energy Condition (ANEC) [52]. Thus, one cannot impose the
stability axiom of [30] to help us in the proof. We got around this thorny
issue by treating the higher derivative non-minimally coupled terms in an EFT
sense. This allowed us to directly evaluate the matter path integral for the
expectation value of the stress tensor. In this perturbative regime, it is
clear that the non-minimal couplings only contribute to the horizon entropy
via (47). With our formalism, it is not possible to treat the $S_{\text{QFT}}$
corrections to the generalized entropy from non-minimal terms since they are
parametrically suppressed. (45) implies that $S_{\text{out}}$ receives
contributions only from the minimal sector. Our proof relied on some basic
assumptions about the algebra of observables as in [30, 18]. It also suggests
that the validity of the linearized GSL necessitates the perturbative EFT
treatment of the higher derivative terms. We can then incorporate higher-order
contributions. This is also in line with the recent observations of [43],
where the second law at the quadratic order in perturbations was valid only in
the EFT sense.

At this point, it is helpful to comment on how our proof is related to or
different from the proofs of [53, 30, 18, 20, 33]. [30] building on [53]
worked in a regime where the matter fields across the horizon are rapidly
fluctuating. However, the proof of [30] was restricted to Einstein’s gravity,
with matter fields minimally coupled to gravity. Our proof works in the regime
of linearized fluctuations around stationary black holes, which is the same
regime considered in [18, 20]. The proof outlined in [20] building on [18] was
established for arbitrary pure gravity diffeomorphism invariant theories but
with a minimally coupled matter sector only. We have been able to extend the
proof of the GSL in [20] for arbitrary diffeomorphism invariant theories of
gravity with matter fields non-minimally coupled to gravity, provided we are
working in an EFT using the proof of the linearized second law worked out in
[22, 23, 24]. Further, unlike the case of [33], we have made a stronger
statement of GSL in that we can show that the generalized entropy is
increasing for every horizon cut, using the entropy current structure detailed
in [22, 23, 24]. Our proof crucially uses the advantages of working in the
horizon adapted coordinates (2) and the resulting entropy current structure
for the $E_{vv}$ in (50). This significantly simplifies the analysis of the
$S_{\text{out}}$, the entanglement entropy of the matter fields outside the
horizon. The entropy current structure (50) indicates that in (46), the non-
minimally coupled matter fields at one loop continue to contribute to the
entropy of the horizon $S_{H}$ as expected classically. This was possible
because of the EFT regime we worked with in the above analysis. This completes
our proof of the linearized GSL for a matter sector that can be non-minimally
coupled.

We now highlight some future directions. Our analysis does not consider the
graviton contribution. [33] have taken the gravitons into account and it would
be interesting to see how these subtleties fit into our perturbative EFT
analysis. We also assume a sharp suppression of scale between minimal and non-
minimal sectors, making our analysis not the most generic treatment of the
non-minimal sector. Two crucial open issues remain: first, what happens when
the non-minimal contribution is comparable to the minimal sector? In
particular, we do not know how to treat all possible non-minimal couplings
that could appear at two derivatives in the EFT treatment; second, our
analysis only holds in the perturbative setup, and we do not yet understand
how to incorporate any non-perturbative effects up to the linear order.

To conclude the discussion, we mention that the horizon entropy in our context
is coarse-grained as we do not know its microstate description. It would be
interesting to see if there is a holographic entanglement entropy analog of
the monotonic quantity, as proposed in Section 3.2.2,222This is in
anticipation of an analogous calculation of the holographic entanglement
entropy of Dong [54] which agrees with the quantity satisfying the linearized
semi-classical GSL in [20]. that satisfies the linearized semi-classical GSL
for higher curvature gravity theory with non-minimally coupled matter sector,
and check whether the horizon will turn out to be the quantum extremal surface
(QES), [55], in that context. We do not expect the latter to hold; otherwise,
it would correspond to saying that the horizon entropy is fine-grained in our
setup. However, this is not the case as here entropy is derived from Wald-like
prescription [15] which in turn arises from $\delta M=T\delta S$, where
$\delta M$ is given by the energy integral on the surface at infinity and thus
represents the collective energy of the spacetime. It is an extensive
quantity, and in this sense, the horizon entropy is coarse-grained. This could
have some interesting connections to other recent works that rely on the QES
formalism [56, 57].

We would like to thank Sayantani Bhattacharyya, Nilay Kundu, Alok Laddha,
Sudipta Sarkar, Amitabh Virmani, Aron Wall, and Zihan Yan for helpful
discussions at various stages of the project. Also, we thank A P Balachandran,
Jyotirmoy Bhattacharya, Nilay Kundu, Sudipta Sarkar, Suneeta Vardarajan and
Amitabh Virmani for several useful comments on the draft. This research was
supported in part by the International Centre for Theoretical Sciences (ICTS)
for the program “Nonperturbative and Numerical Approaches to Quantum Gravity,
String Theory and Holography” (code: ICTS/numstrings-2022/8). PD duly
acknowledges the Council of Scientific and Industrial Research (CSIR), New
Delhi, for financial assistance through the Senior Research Fellowship (SRF)
scheme. KJ would like to thank the Department of Atomic Energy, Govt. of
India, for financial support.

##  Appendix A Examples of solutions in the horizon adapted coordinates

This appendix will show how one can obtain the standard Schwarzschild and Kerr
solutions in the horizon adapted coordinates of (7). Since we are using
geodesics to rule our coordinates, the coordinate patch is valid only until
the geodesics form a caustic. This necessarily means that the coordinate patch
is valid only locally near the dynamical horizon. Our setup is akin to the
Gaussian Null Coordinates discussed in Section 2.1 of [58]. To expound on this
point, we briefly demonstrate the following: we will show that the
Schwarzschild metric can be bought to the gauge given by (7) such that the
metric is valid for arbitrary values of $r$ away from the horizon. However, in
case of the Kerr metric we bring it to the gauge given by (7) close to the
horizon in a perturbative expansion only.

### Schwarzschild solution

Consider the Eddington-Finkelstein Patch of the Schwarzschild solution

$ds^{2}=2\differential v\,\differential
r-\left(1-\dfrac{2M}{r}\right)\differential
v^{2}+r^{2}\differential\Omega^{2}\,.$ (51)

$v,r$ are not affine and $r=0$ is not the horizon. Thus, do the following
transformations successively:

$v=4M\,\log\lambda~{}~{}\rightarrow~{}~{}\tilde{r}=r-2M~{}~{}\rightarrow~{}~{}\tilde{r}=\dfrac{\lambda\,\rho}{4M}\,.$
(52)

We can thus bring the metric to the desired gauge (7) where $\lambda,\rho$ are
affine and $\rho=0$ is the horizon:

$ds^{2}=2\differential\lambda\,\differential\rho+\rho^{2}\left(\dfrac{2}{\lambda\rho}-\left(1-\dfrac{8M^{2}}{\lambda\rho+8M^{2}}\right)\dfrac{16M^{2}}{(\lambda\rho)^{2}}\right)\differential\lambda^{2}+\left(\dfrac{\lambda\rho}{4M}+2M\right)^{2}\differential\Omega^{2}\,.$
(53)

It is important to note that this metric is valid for arbitrary values of
$\rho$ away from the horizon. Near the horizon $\rho=0$, this metric takes the
desired form as,

$ds^{2}=2\differential\lambda\,\differential\rho+\rho^{2}\left(\dfrac{1}{4}-\dfrac{\lambda\rho}{32M}+\dfrac{(\lambda\rho)^{2}}{256M^{2}}-\mathcal{O}[(\lambda\rho)^{3}]\right)\,\differential\lambda^{2}+\left(\dfrac{\lambda\rho}{4M}+2M\right)^{2}\differential\Omega^{2}\,.$
(54)

### Kerr solution

Consider the Eddington-Finkelstein patch of the Kerr solution

$\begin{split}ds^{2}=2\differential v\,\differential
r-\left(1-\dfrac{2Mr}{(r^{2}+a^{2}\cos^{2}\theta)}\right)\differential
v^{2}-\dfrac{4Mar\sin^{2}\theta}{r^{2}+a^{2}\cos^{2}\theta}\differential
v\,\differential\phi+(r^{2}+a^{2}\cos^{2}\theta)\,\differential\theta^{2}\\\
-2a\sin^{2}\theta\,\differential
r\,\differential\phi+\dfrac{\sin^{2}\theta}{r^{2}+a^{2}\cos^{2}\theta}[(r^{2}+a^{2})^{2}-(r^{2}+a^{2}-2Mr)a^{2}\sin^{2}\theta]\,\differential\phi^{2}\,.\end{split}$
(55)

Following the arguments of Appendix A of [59], we can bring the Kerr metric
(55) to our gauge. First, go to a co-rotating frame

$\phi=\varphi+\dfrac{a}{r^{2}_{+}+a^{2}}v$ (56)

Re-write the metric with respect to the following null geodesics

$\begin{split}n&=-\left(\dfrac{a^{2}\sin^{2}\theta}{2r^{2}_{+}+2a^{2}\cos^{2}\theta}\right)\dfrac{\partial}{\partial
v}-\left(\dfrac{r^{2}_{+}+a^{2}}{r^{2}_{+}+a^{2}\cos^{2}\theta}\right)\dfrac{\partial}{\partial
r}-\left(\dfrac{a}{r^{2}_{+}+a^{2}}\right)\dfrac{\partial}{\partial\phi}\,,\\\
l&=\dfrac{\partial}{\partial v}\,.\end{split}$ (57)

Consider a family of null geodesics with $n$ as the tangent field crossing the
horizon $H$ at the point $(v,\theta,\phi)$. Parametrize them using an affine
$\rho$ such that $\rho=0$ is $H$. Thus, we can perturbatively construct the
geodesic coordinates:

$X^{\alpha}_{(v,\theta,\phi)}(\rho)\approx
X^{\alpha}|_{\rho=0}+\rho\left.\dfrac{d\,X^{\alpha}}{d\,\rho}\right|_{\rho=0}+\dfrac{\rho^{2}}{2}\left.\dfrac{dX^{\alpha}}{d\rho}\right|_{\rho=0}+\mathcal{O}(\rho^{3})\,,$
(58) $X^{\alpha}|_{H}=(v,r_{+},\theta,\phi)\,,$ (59)
$\left.\dfrac{dX^{\alpha}}{d\rho}\right|_{H}=n^{\alpha}=\left(-\dfrac{a^{2}\sin^{2}\theta}{2r^{2}_{+}+2a^{2}\cos^{2}\theta},-\dfrac{r^{2}_{+}+a^{2}}{r^{2}_{+}+a^{2}\cos^{2}\theta},0,-\dfrac{a}{r^{2}_{+}+a^{2}}\right)\,,$
(60)
$n^{\beta}D_{\beta}n^{\alpha}=0\implies\left.\dfrac{d^{2}X^{\alpha}}{d\rho^{2}}\right|_{H}=-\Gamma^{\alpha}_{\beta\gamma}n^{\beta}n^{\gamma}\,.$
(61)

Thus, we have the transformation from $(v,r,\theta,\phi)$ to
$(v,\rho,\theta,\phi)$. The Kerr metric can thus be schematically written as

$g_{\alpha\beta}\approx
g^{(0)}_{\alpha\beta}+\rho\,g^{(1)}_{\alpha\beta}+\dfrac{\rho^{2}}{2}\,g^{(2)}_{\alpha\beta}+\mathcal{O}(\rho^{3})\,.$
(62)

Once we reach (A.11) of [59], we simply do the following transformations:

$\rho=\kappa\,u\,\lambda,~{}~{}~{}v=\dfrac{1}{\kappa}\,\log\lambda,~{}~{}~{}\text{where
}~{}~{}~{}\kappa=\dfrac{\Delta^{\prime}}{2\chi_{o}}\,,$ (63)

where $u,\lambda$ are the new affine coordinates we define, and other
quantities are defined as in [59]. This will result in the Kerr metric in our
gauge (7):

$\begin{split}ds^{2}&=2\differential\lambda\,\differential
u+\dfrac{u^{2}}{2}g^{(2)}_{vv}\,\differential\lambda^{2}+u\,(2g^{(1)}_{v\theta}+\kappa
u\lambda\,g^{(2)}_{v\theta})\,\differential\lambda\,\differential\theta\\\
&+u\,(2g^{(1)}_{v\phi}+\kappa
u\lambda)\,\differential\lambda\,\differential\phi+\left(g^{(0)}_{\theta\theta}+\kappa
u\lambda g^{(1)}_{\theta\theta}+\dfrac{(\kappa
u\lambda)^{2}}{2}g^{(2)}_{\theta\theta}\right)\,\differential\theta^{2}\\\
&+\left(g^{(0)}_{\phi\phi}+\kappa u\lambda g^{(1)}_{\phi\phi}+\dfrac{(\kappa
u\lambda)^{2}}{2}g^{(2)}_{\phi\phi}\right)\,\differential\phi^{2}\\\
&+\left(\kappa u\lambda g^{(1)}_{\theta\phi}+\dfrac{(\kappa
u\lambda)^{2}}{2}g^{(2)}_{\theta\phi}\right)\,\differential\theta\,\differential\phi+\dots\,.\end{split}$
(64)

From the above analysis, we can see that the Kerr metric in our gauge (7) is
valid only close to the horizon.

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