aainstitutetext: School of Physical Sciences, National Institute of Science
Education and Research, HBNI, Bhubaneswar, Khurda 752050, Odisha,
Indiabbinstitutetext: Department of Physics, Indian Institute of Technology
Kanpur, Kalyanpur, Kanpur 208016, India

# An entropy current and the second law in higher derivative theories of
gravity

Sayantani Bhattacharyya b , Prateksh Dhivakar a , Anirban Dinda b , Nilay
Kundu a , Milan Patra a and Shuvayu Roy<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We construct a proof of the second law of thermodynamics in an arbitrary
diffeomorphism invariant theory of gravity working within the approximation of
linearized dynamical fluctuations around stationary black holes. We achieve
this by establishing the existence of an entropy current defined on the
horizon of the dynamically perturbed black hole in such theories. By
construction, this entropy current has non-negative divergence, suggestive of
a mechanism for the dynamical black hole to approach a final equilibrium
configuration via entropy production as well as the spatial flow of it on the
null horizon. This enables us to argue for the second law in its strongest
possible form, which has a manifest locality at each space-time point. We
explicitly check that the form of the entropy current that we construct in
this paper exactly matches with previously reported expressions computed
considering specific four derivative theories of higher curvature gravity.
Using the same set up we also provide an alternative proof of the physical
process version of the first law applicable to arbitrary higher derivative
theories of gravity.

###### Keywords:

Black hole entropy, Entropy current, Second law of black hole thermodynamics,
Diffeomorphism invariant theories of gravity, Higher derivative theories of
gravity.

##  1 Introduction and summary

In Einstein’s classical theory of gravity, black holes are interesting
solutions that behave like thermodynamic objects Hawking:1971tu ;
Bardeen:1973gs ; Bekenstein:1973ur ; Hawking:1974sw  and one can associate
notions like temperature and entropy to them. It is believed that there is
also a consistent statistical picture of black hole thermodynamics. This
allows us to consider black holes as basically a thermodynamic ensemble of
micro-states in an underlying quantum theory of gravity.

Any UV complete theory of gravity, in its low energy limit, will typically
generate several higher derivative corrections to the classical two derivative
theory of Einstein gravity. Black hole solutions should continue to exist in
these theories, at least when the higher derivative coupling is treated
perturbatively. The fact that black holes are indeed a collection of a large
number of micro-states will remain so even with the higher derivative
corrections being added to the two derivative gravity action. Therefore, they
should also maintain the laws of thermodynamics provided we could correctly
identify the thermodynamic properties with the geometric properties of black
holes. This identification is well-known in two-derivative Einstein gravity.
However, in higher derivative theories, we still do not have a complete
understanding of how to do it for dynamical black holes.

More precisely, we know that in two-derivative theories of gravity, the area
of a time-slice of the horizon plays the role of entropy of the black hole
that satisfies both the first and the second law of thermodynamics. In
particular, the second law follows from the area increase theorem for black
holes Hawking:1973uf ; waldbook . However, when one considers higher
derivative theories of gravity, the same horizon area does not work as the
definition of black hole entropy. We need to modify it to account for the
higher derivative corrections. In (PhysRevD.48.R3427, ; Iyer:1994ys, ) a
geometrical notion of black hole entropy, called the Wald entropy, was
provided for an arbitrary diffeomorphism invariant theory of gravity
(including theories with higher derivative corrections to Einstein’s theory)
in such a way that the first law of thermodynamics is satisfied. However, once
a dynamical black hole solution is considered in general higher derivative
theories of gravity, Wald entropy’s construction becomes ambiguous, meaning a
bunch of terms could be added to Wald entropy without affecting the first law
Jacobson:1993xs ; Jacobson:1993vj ; Jacobson:1995uq . These are generally
known as the JKM ambiguities in the literature, and they are non-zero only for
dynamical configurations.

The concept of an equilibrium black hole configuration in a theory of gravity
is associated with a space-time metric admitting a Killing vector that becomes
null on the event horizon. Such a geometric configuration is also known as the
stationary black hole solution. As we know, the first law of thermodynamics is
essentially a comparison between the black hole parameters (like mass, charge,
etc.) for two slightly different but nearby stationary configurations. The
second law of thermodynamics, however, necessarily refers to dynamics. It
states that the black hole entropy compared between final and initial
stationary configurations (not necessarily nearby) can never decrease, i.e.,
entropy is always produced in every dynamical process. Although the Wald
entropy satisfies the first law by construction, there is no clear proof that
it will obey the second law in any arbitrary diffeomorphism invariant theory
of gravity. In other words, unlike the two-derivative theory, we still do not
have a completely satisfactory geometric construction of entropy that
satisfies both the first and the second law of thermodynamics for dynamical
black hole solutions in higher derivative theories of gravity.

Over the years, various attempts have been made to understand if indeed there
is a notion of the second law for black hole thermodynamics in higher
derivative theories of gravity, Iyer:1994ys ; Jacobson:1993xs ;
Jacobson:1993vj ; Jacobson:1995uq ; Sarkar:2013swa ; Bhattacharjee:2015yaa ;
Wall:2015raa ; Bhattacharjee:2015qaa ; Wall:2011hj 111See the recent reviews
of black hole thermodynamics in higher derivative theories of gravity
Wall:2018ydq ; Sarkar:2019xfd  and references therein, for a detailed
discussion on this topic.. A natural approach is to start from Wald entropy as
the equilibrium definition of black hole entropy and explore the possibilities
of extending it to dynamical situations such that the second law is satisfied.
It essentially means fixing the JKM ambiguities related to the Wald entropy
for dynamical configurations by imposing the second law as a principle. This
has so far been the basic theme of various approaches in this context.

In any such attempt, one would, however, naturally have to devise an algorithm
to analyze dynamical black hole space-times, which by itself is very difficult
to execute even in general relativity. Therefore we need to adapt a
perturbative approach around the exactly known stationary solutions. A
standard procedure, which we will also follow in our analysis in this paper,
is to consider some stationary black hole space-time which then gets slightly
perturbed due to an external source and then finally settles down to another
stationary configuration. The amplitude of the dynamical fluctuation is small
enough in the sense that we could always perform a perturbative expansion in
its amplitude and work with only the linearized term in this expansion,
ignoring all non-linear terms. We will call this approximation the linearized
amplitude expansion222One should note that with this linearized amplitude
expansion, we cannot study violent time-dependent processes like black hole
mergers. Although we would have nothing to say about those situations within
this paper’s premise, see Liko:2007vi ; Sarkar:2010xp ; Chatterjee:2013daa
where a similar problem has been studied with exciting conclusions.

##### The context and the backdrop of our current work:

Our analysis in this paper is significantly motivated and based on the
important results that were reported recently in Wall:2015raa  and
subsequently in a follow-up paper Bhattacharya:2019qal . Therefore, to better
understand the context of this current paper, it is useful and informative
that we discuss them a priori333For a detailed discussion the reader is
requested to see section $2$ of Bhattacharya:2019qal  where a comprehensive
review of Wall:2015raa  is given..

Working within the approximation of linearized amplitude expansion of small
dynamical perturbation to a stationary black hole configuration, in
Wall:2015raa , it was shown that these out-of-equilibrium JKM ambiguities of
Wald entropy could uniquely be fixed up to a particular order by demanding
that it must satisfy the linearized version of the second law. Additionally,
in Wall:2015raa , the statement of the second law was satisfied in a more
robust sense: not only the difference between the total Wald entropy of the
final and the initial equilibrium configuration was non-negative; instead, the
entropy was monotonically increasing at every instant of “time” throughout the
evolution from the initial to the final stationary point. This work, thereby,
provides an explicit construction of an entropy function - an out-of-
equilibrium extension of the Wald entropy - satisfying the second law.

The general strategy that was followed in Wall:2015raa  was to show that the
“time” derivative of this entropy function is non-negative at every instant
along the evolution444The reason we are putting “time” within the quotation
mark is the fact that on horizon rather than having a ‘time’ coordinate as the
component of a time-like vector, we will have a parameter running along the
curve generated by a null vector, the generator of the horizon. The horizon
being a null surface, we cannot actually have such a time like coordinate in
true sense.. This way, the basic working principle essentially resembles the
same that is responsible for the area increase theorem in Einstein’s theory of
gravity. After choosing a particular metric gauge suitable for the case at
hand and working within the approximation of linearized amplitude expansion, a
specific ‘time-time’ component of the rank-two equation of motion was studied.
More precisely, a particular sub-class of the residual symmetry transformation
of the chosen metric gauge (named as the “boost transformation”) was used to
classify how different quantities transform with definite weights (termed as
the “boost-weight”) under that symmetry of the stationary background.
Eventually, using this classification based on boost-weights, a specific
expression of that “time-time” component of the equation of motion was
obtained, maintaining the second law up to linearized order in amplitude.
Finally, it was argued that the zero boost-weight part of that expression
determines the equilibrium Wald entropy. In contrast, the higher boost sector
fixes the JKM ambiguities of the entropy (which is the same as the out-of-
equilibrium part of it).

In essence, the importance of the work Wall:2015raa  lies in the fact that it
not only provides an algorithm to fix the JKM ambiguities but also to
reproduce the Wald entropy of the stationary configuration itself. As a check
of its consistency, this entropy function defined locally in time should
reduce to the known expression of Wald entropy for stationary configurations
once the dynamics are switched off. However, in Bhattacharya:2019qal , a
subtle technical issue of this construction regarding its implementation was
pointed out. Actually, this constructive algorithm of Wall:2015raa  stumbles
upon a road-block at the very leading order in amplitude expansion. The zero
boost-weight sector of the “time-time” component of the equations of motion
doesn’t acquire the desired form, which is responsible for reproducing the
Wald entropy of the stationary black holes. However, this should not be a
problem since, for stationary black holes, we already know that Wald’s
construction works. In Wall:2015raa , this trickier set of terms were dealt
with by taking recourse to the physical process version of the first law
Jacobson:1995uq ; Gao:2001ut ; Amsel:2007mh ; Bhattacharjee:2014eea ;
Chakraborty:2017kob ; Chatterjee:2011wj ; Kolekar:2012tq . It was argued that
if the physical process version of the first law is to be valid, then things
should work out nicely to reproduce the correct expression for the Wald
entropy at equilibrium.

The main goal of the analysis carried out in Bhattacharya:2019qal  was to
perform a brute force calculation focussing on only four derivative theories
of gravity to check if the arguments presented in Wall:2015raa  are indeed
true. By explicitly working out the “time-time” component of the equations of
motion for those theories, it was shown that there are terms, in that zero
boost-weight sector of it, which cannot be cast in the form required to
identify them as the Wald entropy density. However, it was also noticed that
the structure of every such anomalous term could be rearranged as the spatial
divergence of a specific spatial vector, named the entropy current. In
Wall:2015raa  the appearance of such additional terms in the particular
components of the equations of motion with the specific structures mentioned
above was not paid its due attention, but the physical process version of the
first law was called for the rescue.

The physical process version of the first law is actually formulated along the
same lines as the linearized version of the second law555It should be noted
that although the necessary formalism to address both of them is along the
same lines, the implications of them are pretty different. One needs to fix
the JKM ambiguities to prove a linearized version of the second law; however,
they have vanishing contribution for the first law.. A stationary black hole
is driven out of its equilibrium state by a small stress tensor associated
with some matter falling in the black hole from asymptotic infinity, and
finally, it settles down to another stationary configuration. A relation
between the changes in the parameters (like entropy, mass, etc.) of the two
equilibrium black holes follows accordingly. The presence of the additional
anomalous terms in the form of a spatial divergence of a spatial current,
which was pointed out in Bhattacharya:2019qal , didn’t affect the statement of
the physical process version of the first law. This is because, in the
physical process version of the first law, one compares the total entropy
integrated over the spatial slices of the horizon, and thereby the entire
spatial divergence terms drop out. In turn, this also justifies the arguments
presented in Wall:2015raa  that, once the dynamical evolution of a black hole
between two equilibrium configurations satisfies the physical process version
of the first law, the zero boost-weight sector of the “time-time” component of
the equations of motion reproduces the correct expression of the Wald entropy
for stationary black holes.

Although in Wall:2015raa  there was a temporal locality in the version of the
second law, that the entropy was considered as an integrated expression over
the spatial slices of the horizon (for the reasons mentioned above) signifies
that there was no locality in the spatial directions. In other words, all the
statements that were made in Wall:2015raa  involved an entropy function that
was local in time but could only be associated to each globally complete
spatial section of the horizon but not to any infinitesimal elements of it. On
the other hand, the analysis of Bhattacharya:2019qal  explicitly showed that,
although only for specific four derivative theories of gravity, introducing
the notion of an entropy current, the second law for black holes could be
formulated in its most potent possible form, i.e., in an ultra-local version
which is local in both time and space. In fact, this statement has been recast
even in a more robust sense as follows: in a general four derivative theory of
gravity, working up to linear order in the amplitude of the dynamical
fluctuations, the spatial components of the entropy current has to be
accounted for if one aims to prove an ultra-local version of the second law
(as described above).

The entropy current, constructed in Bhattacharya:2019qal , is a $(d-1)$-vector
defined on the horizon, with coordinates running along with the ‘time’ and the
spatial coordinates that span the horizon. The ‘time’ component of the entropy
current gives us the instantaneous entropy density, whereas the spatial
components of the same account for a flow of entropy across different local
segments of the spatial slice of the horizon. The existence of a non-negative
divergence of this entropy current is definitely the statement that entropy is
being produced locally at every point of the dynamical space-time. However,
due to the non-zero spatial components of the entropy current, it also
suggests that there is another simultaneous mechanism by which redistribution
of entropy is taking place between different adjacent spatial regions666The
existence of such an entropy current is well known in theories where there is
no dynamical gravity, e.g., in fluid dynamics Bhattacharyya:2008xc ;
Bhattacharyya:2012nq ; Bhattacharyya:2013lha ; Bhattacharyya:2014bha .. This
physical picture highlighting the significance of having an entropy current or
the idea of an ultra-locality for the second law is undoubtedly consistent in
Einstein’s theory of gravity. It is well known that the area increase theorem
in general relativity is a local statement that works for every instance of
time as well as at each spatial point of a time slice of the horizon. It is
therefore not at all absurd to expect that a similar physical picture of the
same kind would prevail even when higher derivative theories of gravity are
considered, at least for dynamical fluctuations with small enough amplitudes.
Actually, the results of Bhattacharya:2019qal  very strongly favour such an
interpretation, at least for the theories explicitly studied there.

It is, therefore, of utmost importance to know whether the lessons that we
have learned so far in the discussions above are applicable much more
generally and to other generic theories of gravity as well. In this present
work, our primary goal is to address this question. Rather than focussing on
specific theories as was done in Bhattacharya:2019qal , here in this work, we
will be considering a general diffeomorphism invariant theory of gravity.
However, our methodology will still be the same. For example, we will
extensively use the same boost-symmetry of the horizon geometry to constrain
the possible structural form of the “time-time” component of the equations of
motion.

Additionally, we will also restrict ourselves only to those dynamical
situations where the approximation of a small amplitude of dynamical
perturbation is good enough. In other words, a linearized expansion in the
amplitude of the non-stationary fluctuations is allowed. Interestingly, our
final result turns out to be an affirmative answer to the question we posed
above: in an arbitrary diffeomorphism invariant theory of gravity,
construction of an entropy current to account for the ultra-local version of
the second law of thermodynamics, at least for dynamical processes with small
amplitudes, is indeed possible.

It is worth highlighting that one crucial aspect of the construction of an
entropy current in this paper has been to prove a linearized version of the
second law without ever invoking the physical process version of the first
law. It has been instrumental in enabling us to maintain strict locality
throughout our work. As we have mentioned before, this is in contrast to
Wall:2015raa , where the physical process version of the first law was used to
take care of the zero boost-weight sector in specific components of equations
of motion, and hence the spatial locality was lost. On the other hand, we have
been able to design an independent proof of the first law’s physical process
version, using the same basic formalism that was formulated to prove a
linearized version of the second law.

Before we proceed, let us also mention that in the literature, the concept of
an entropy current in the context of dynamical black holes has already been
introduced; see Guedens:2011dy ; Chapman:2012my ; Eling:2012xa ;
Dandekar:2019hyc ; Saha:2020zho . Although they share the common primary goal
in terms of constructing a local entropy current, the methodology and the
working principles that we are using in our paper are very different from
theirs. For example, in Guedens:2011dy  the context was to interpret the field
equations of gravity as an equation of state, the primary motivation behind
Chapman:2012my ; Eling:2012xa  has mainly been the celebrated fluid gravity
correspondence Bhattacharyya:2008jc . On the other hand, a certain duality
between membrane dynamics and black holes called the membrane-gravity duality
had been the driving principle for Dandekar:2019hyc ; Saha:2020zho .

The outline of this paper is as follows. We start in §1.1 with making a
precise statement of the problem at hand and a summary of the main result of
our paper. In §2 we will then discuss the basic set up of our construction.
This will include an extensive discussion of various basic concepts which play
an important role in our analysis. After introducing the coordinate system
that we will work with, the idea of a boost transformation and the weights of
different quantities under this specific symmetry of the horizon geometry in
our metric gauge will be explained. We will also discuss how these are related
to the concepts of stationarity and a slight deviation from it.

Furthermore, an important identity relating the equations of motion and the
Noether charges under diffeomorphism invariance will be established. We will
end this section with a discussion of the strategy that will be followed. In
the next section, i.e., §3, we will present detailed proof of how to extract
out the entropy current. This will be the central technical part of our paper.
In the following section §4, we will show that the expression of the entropy
current obtained from the abstract proof in this paper exactly reproduces the
results known in four derivative theories of gravity which were reported in
Bhattacharya:2019qal . Next, in §5 we use the same technical set up and
present an independent proof of the physical process version of the first law
for any arbitrary diffeomorphism invariant theory of gravity. Finally, in §6,
we conclude with a brief discussion of the consequences of our results and
possible future directions. We have also summarised the notations,
conventions, and useful definitions in Appendix-(A) for the convenience of the
reader. The other appendices provide useful technical details for our
computations.

###  1.1 Statement of the problem and summary of the final result

With the motivations and the context of our paper being discussed so far, in
this sub-section, we will write down a more precise and explicit statement of
the problem at hand as well as the final result that we have been able to
achieve. This will help us organize our presentation in the following sections
in a much more coherent manner.

We will work with the most general diffeomorphism invariant theory of gravity
in $d$ space-time dimensions with coordinates denoted by $x^{\mu}$, with the
action given by777We are using a convention such that the Newton’s constant
$G=1/4$, such that the area of the horizon gives the entropy of a black hole
in Einstein’s gravity.

$I={1\over 4\pi}\int d^{d}x\,\sqrt{-g}\,\left(L_{grav}+L_{mat}\right)\,.$ (1)

In eq.(1), $L_{grav}$ is the Lagrangian for the degrees of freedom
corresponding to the gravity part888One can argue that, see section $2$ of
Iyer:1994ys , any diffeomorphism invariant Lagrangian will have the specific
functional dependence as mentioned here.,

$L_{grav}=L_{grav}(g_{\alpha\beta},\,R_{\alpha\beta\rho\sigma},\,D_{\alpha_{1}}R_{\alpha\beta\rho\sigma},\cdots)\,.$
(2)

Moreover, $L_{mat}$ is the Lagrangian for the matter fields present in the
theory999All possible couplings between the matter and the gravity sector are
contained within the Lagrangian $L_{mat}$. It is also possible that $L_{mat}$
has to contain higher spin fields to make the entire higher derivative theory
consistent with causality concerning processes such as graviton scattering, as
was pointed out in Camanho:2014apa .. In this paper, we will be focussing on
the gravity part of the Lagrangian $L_{grav}$. For the matter sector, all that
we will need is to consider that it gives us a stress tensor satisfying the
null energy condition. In what follows, we will make this more precise.

We will first make the following choice of gauge for the space-time metric
with coordinates $x^{\mu}=\\{r,\,v,\,x^{i}\\}$, for $i=1,\,\cdots,\,(d-2)$,
that represents a dynamical black hole solution with a regular event horizon
at $r=0$,

$\begin{split}ds^{2}=2\,dv\,dr-
g_{vv}(r,v,x)~{}dv^{2}+2g_{vi}(r,v,x)\,dv\,dx^{i}+h_{ij}(r,v,x)\,dx^{i}\,dx^{j}\,,\\\
\text{such
that}~{}g_{rv}=1\,,~{}g_{rr}=0\,,~{}g_{ri}=0\,,~{}g_{vv}\big{|}_{r=0}=0\,,~{}\partial_{r}g_{vv}\big{|}_{r=0}=0\,,~{}g_{vi}\big{|}_{r=0}=0\,.\end{split}$
(3)

The null hyper-surface of the event horizon, which is spanned by the
coordinates $\\{v,\,x^{i}\\}$, will be denoted by $\cal H$. The constant
$v$-slices of the horizon will be represented by ${\cal H}_{v}$. The
coordinate $v$ is an affine parameter corresponding to the null generator
$\chi^{\mu}\partial_{\mu}\equiv\partial_{v}$ of the horizon101010For the
reasons mentioned in footnote-$4$, we have been loosely calling the parameter
$v$ as the “time” coordinate, although it is actually a null direction.. In
§2.1 we will explain in more detail the choice of our coordinates.

This paper will only focus on scenarios where the dynamics can be treated as
small fluctuations around a stationary background. In operational terms, the
full space-time metric can be decomposed as follows

$g_{\mu\nu}=g_{\mu\nu}^{eq}+\epsilon\,\delta g_{\mu\nu}\,,$ (4)

where $g_{\mu\nu}^{eq}$ is the metric for the stationary black hole space-
time, and $\epsilon$ is a parameter denoting the amplitude of small
fluctuations around that stationary background. The smallness of the
fluctuation allows us to perform a perturbative expansion in the amplitude and
work only up to linear order in that amplitude expansion, i.e., neglecting
terms with $\mathcal{O}(\epsilon^{2})$ and higher orders.

As we have emphasized before, in this current work, the basic idea behind
constructing a proof of the linearized version of the second law is to
constrain the structure of a particular component of the equations of motion,
namely the $vv$-component of it denoted by ${\cal E}_{vv}$. It should be noted
that by ${\cal E}_{\mu\nu}$ we are denoting the equations of motion which
follow from varying the complete Lagrangian, including both the gravitational
and the matter sector, with respect to the space-time metric. The part of
${\cal E}_{\mu\nu}$ that is derived only from the gravitational part of the
Lagrangian will be denoted by $E_{\mu\nu}$. Therefore, we will have the
following relation

${\cal E}_{\mu\nu}=E_{\mu\nu}+T_{\mu\nu}\,,$ (5)

where $T_{\mu\nu}$ is the stress-energy tensor obtained from the matter part
of the Lagrangian 111111We are working with the following definition of the
stress tensor. While obtaining the equations of motion ${\cal E}_{\mu\nu}$ by
varying the total Lagrangian with respect to the fluctuations in the metric,
there will be one type of terms involving only the metric components and their
derivatives. We denote them as $E_{\mu\nu}$. However, in ${\cal E}_{\mu\nu}$,
there will be another type of terms that involve both the metric and the
matter fields. They will all be included in $T_{\mu\nu}$.. In §2.4 we have
discussed in detail why the structure of this particular component of the
equations of motion is so crucial to validate the second law121212Remembering
that we are writing the components of equations of motion, we will always have
${\cal E}_{vv}=0$ when thinking of it in an on-shell manner. However, this is
not how we would consider ${\cal E}_{vv}$; rather, we would like to emphasize
that our purpose here is to constrain the off-shell structure of ${\cal
E}_{vv}$..

The primary content of our analysis in this paper can be precisely summarised
as the following statement:

Using arguments based on diffeomorphism invariance and a certain boost
symmetry of the horizon geometry to classify possible structures that can
appear in ${\cal E}_{vv}$, when evaluated on the horizon (i.e. at $r=0$), we
have been able to show that

$\begin{split}{\cal
E}_{vv}\big{|}_{r=0}=&\,\partial_{v}\bigg{[}{1\over\sqrt{h}}\partial_{v}\left({\sqrt{h}}\,{\cal
J}^{v}\right)+{1\over\sqrt{h}}\partial_{i}\left({\sqrt{h}}\,{\cal
J}^{i}\right)\bigg{]}+T_{vv}+\mathcal{O}(\epsilon^{2})\\\
=&\,\partial_{v}\bigg{[}{1\over\sqrt{h}}\partial_{v}\left({\sqrt{h}}\,{\cal
J}^{v}\right)+\nabla_{i}{\cal
J}^{i}\bigg{]}+T_{vv}+\mathcal{O}(\epsilon^{2})\,,\end{split}$ (6)

where $h$ is the determinant of the induced spatial metric, i.e. $h_{ij}$, on
the co-dimension two constant $v$-slices of the horizon, and $\nabla_{i}$ is
the covariant derivative compatible with $h_{ij}$. Also, ${\cal J}^{v}$ and
${\cal J}^{i}$ will get contribution only from the gravitational part of the
Lagrangian and eq.(6) can equivalently be stated as

$\begin{split}E_{vv}\big{|}_{r=0}=&\,\partial_{v}\bigg{[}{1\over\sqrt{h}}\partial_{v}\left({\sqrt{h}}\,{\cal
J}^{v}\right)+\nabla_{i}{\cal
J}^{i}\bigg{]}+\mathcal{O}(\epsilon^{2})\,,\end{split}$ (7)

which follows from eq.(5).

Furthermore, we will only focus on situations where the dynamics is initiated
due to some perturbation coming from a matter stress tensor such that it
satisfies the null energy condition. In our metric gauge this becomes
$T_{vv}\geq 0$. Using this and also the fact that for on-shell processes the
RHS of eq.(6) should vanish, we immediately obtain

$\partial_{v}\bigg{[}{1\over\sqrt{h}}\partial_{v}\left({\sqrt{h}}~{}{\cal
J}^{v}\right)+\nabla_{i}{\cal J}^{i}\bigg{]}\leq 0\,.$ (8)

Additionally, since we also require that the dynamics settles down to some
stationary black hole configuration at late future as $v\rightarrow\infty$,
where $\partial_{v}$ becomes proportional to the Killing direction, we
get131313The reason for this is the following: by construction ${\cal J}^{i}$
includes at least one $\partial_{v}$-derivative and when evaluated on a
stationary metric all the $\partial_{v}$’s should vanish.

$\bigg{[}{1\over\sqrt{h}}\partial_{v}\left({\sqrt{h}}~{}{\cal
J}^{v}\right)+\nabla_{i}{\cal J}^{i}\bigg{]}\rightarrow 0~{}~{}\text{
as}~{}~{}v\rightarrow\infty\,.$ (9)

Therefore, from eq.(8) and eq.(9) it follows that

$\bigg{[}{1\over\sqrt{h}}\partial_{v}\left({\sqrt{h}}~{}{\cal
J}^{v}\right)+\nabla_{i}{\cal J}^{i}\bigg{]}\geq 0~{}~{}\text{for all finite
$v$}\,.$ (10)

From eq.(10), we can also conclude that our procedure of arriving at eq.(7)
enables us to construct a local entropy current on the horizon with components
${\cal J}^{v}$ and ${\cal J}^{i}$. Consequently, we can immediately identify
its $v$-component given by ${\cal J}^{v}$ as the local entropy density, which
not only reproduces the equilibrium expression of Wald entropy but also
determines the out-of-equilibrium extension of it in the form of fixing the
JKM ambiguities. On the other hand, the spatial components, i.e., ${\cal
J}^{i}$, are identified with the entropy current density signifying a possible
spatial flow of entropy locally at each point of the constant $v$-slices of
the horizon. Also, we can see that with these identifications, eq.(10)
signifies a local statement of entropy production at each space-time point on
the horizon. Interestingly, we have also been able to show that when the
stationary limit is taken by switching off the dynamical perturbations, ${\cal
J}^{v}$ reduces to the expression of Wald entropy known for equilibrium black
holes and ${\cal J}^{i}$ identically vanishes141414This actually justifies
eq.(9) above. It is worth highlighting that obtaining eq.(7), therefore, not
only justifies a consistent out-of-equilibrium extension of the Wald entropy
but also provides us an algorithm to obtain ${\cal J}^{v}$ and ${\cal J}^{i}$
as entirely geometric quantities in the sense that they are constructed solely
out of the metric components and their derivatives. However, an important
point worth emphasizing is the fact that the choice of our gauge for the
space-time metric, eq.(3), plays a vital role in the definitions of ${\cal
J}^{v}$ and ${\cal J}^{i}$ via eq.(6) and eq.(7). Consequently, we must admit
that our construction of the local entropy current is not a covariant one, and
it relies on the choice of the constant $v$-slices of the horizon.

Finally, we conclude our introduction with the following comments regarding
the novelty of the results reported in this paper. The key result of this
paper is to prove that eq.(7) is valid for arbitrary diffeomorphism invariant
theories of gravity, up to linearized order in the amplitude of the non-
stationary perturbations. However, we must note that the rest of the arguments
connected to the linearized version of the second law have been used before,
e.g., see Wall:2015raa . Even in the case of two derivative Einstein gravity,
the area increase theorem for the second law of black hole mechanics is very
similar. On the other hand, a case by case and explicit computation of
$E_{vv}$ in Bhattacharya:2019qal  has already revealed the appearance of the
terms involving ${\cal J}^{i}$ in eq.(7), but only for four derivative
theories of gravity. In this context, the originality of our results,
therefore, lies in the fact that we have been able to justify the ultra-local
version of the second law via the entropy current on the horizon with an
analysis that is based on general principles like Noether charge for
diffeomorphism invariance and most importantly it makes a statement applicable
to any higher derivative theory of gravity.

##  2 Basic setup and key conceptual elements

In this section, we will explain the basic setup of our problem. We will
introduce a coordinate system adapted to the horizon. The choice of this
coordinate system is akin to choosing a gauge for the metric, and we will work
with this throughout this paper. Subsequently, we will also elaborate on the
approximations that will be used. A specific symmetry, called the boost-
invariance, will be introduced. By knowing how different terms transform under
this boost transformation, we will see how any generic tensor quantity, built
out of metric functions and their derivatives, in a theory of gravity can be
constrained. After we have discussed various elements of the basic setup
151515We shall be very brief in explaining our setup here. For details, we
refer the readers to (Wall:2015raa, ), (Bhattacharyya:2016xfs, ),
(Bhattacharya:2019qal, )., we will also describe the main strategy, which will
be followed in the subsequent section to prove the existence of an entropy
current for a general diffeomorphism invariant theory of gravity.

###  2.1 The coordinate system adapted to the horizon

We are considering a classical theory of gravity, and black holes are specific
solutions described by a space-time that is endowed with a horizon - a null
hyper-surface separating the black hole singularity from the asymptotic
infinity. Our goal is to choose a coordinate system to describe a space-time
with dynamical black holes in this subsection. We will work in $d$-dimensional
space-times, and let us suppose that on the horizon, a co-dimension one hyper-
surface, $\partial_{v}$ is the null generator with $v$ being an affine
parameter. We will choose $v$ to be one of the coordinates along the horizon.
Let $\\{x^{i}\\}$, for $i=1,2,\cdots,(d-2),$ be the other $(d-2)$ spatial
coordinates along the horizon. Thus, $\\{v,x^{i}\\}$ together construct a
coordinate system on the horizon. Next, we consider null geodesics generated
by $\partial_{r}$, emanating out of the horizon. We further demand that these
geodesics, generated by $\partial_{r}$, make an angle such that, on the
horizon, the inner product of $\partial_{r}$ and $\partial_{v}$ is one.
Additionally, the inner product of $\partial_{r}$ and $\partial_{i}$ vanishes
for every $i$. The origin of the $r$ coordinate is chosen to lie on the
horizon, i.e., $r=0$ will always give the horizon’s location. Finally, we also
restrict the parameter $r$ to be an affine parameter along these geodesics,
denoting the coordinate away from the horizon. The full $d$ dimensional
coordinates will be denoted by $x^{\mu}\,=\,\\{r,v,x^{i}\\}$. In this
coordinate system, the metric takes the following form

$\begin{split}ds^{2}=2\,dv\,dr-r^{2}\,X(r,v,x)~{}dv^{2}+2r\,\omega_{i}(r,v,x)\,dv\,dx^{i}+h_{ij}(r,v,x)\,dx^{i}\,dx^{j}\,.\end{split}$
(11)

The detailed arguments justifying that the most general form of the metric
describing a black hole space-time can always be written as in eq.(11) are
given in Appendix-A of (Bhattacharyya:2016xfs, ) 161616The particular form of
the metric in which we have extracted out an explicit factor of $r^{2}$ in
$g_{vv}$ and a factor of $r$ in $g_{vi}$ components of the metric has
previously been used in Bhattacharya:2019qal .. It is interesting to note that
the metric written in eq.(11) should not be considered as the global
coordinates. However, the space-time sufficiently close to the horizon will
always obtain the form mentioned above. It can be explicitly checked that for
Schwarzschild black holes the full space-time can be written in the form given
in eq.(11). On the other hand, for rotating Kerr black holes the near-horizon
metric can be written in the form of eq.(11) but these coordinates cannot be
used as global coordinates. For the arguments used in this paper it is
sufficient that the local patch near the horizon can always be described by
the metric in eq.(11).

However, one should note that the metric structure given in eq.(11) does not
completely fix the coordinates on constant $r$ and $v$ slices. The following
two types of coordinate redefinitions are still allowed, which preserve the
gauge choice in eq.(11):

  1. 1.

We are allowed to do a coordinate transformation as

$v\rightarrow\tilde{v}=f_{1}(x^{i})\,v+f_{2}(x^{i}),$ (12)

along with an appropriate redefinition of the coordinate $r$ such that the
form of the metric in eq.(11) remains invariant. Here $f_{1}$ and $f_{2}$ are
arbitrary functions of the $x^{i}$ coordinates. This coordinate transform
essentially redefines the constant $v$-slicing of the horizon and hence, will
be important in determining the form of the entropy current. We will actually
consider a special sub-class of this residual freedom of coordinate
transformation in the following sub-section extensively for our analysis in
this paper.

  2. 2.

We can also perform a relabelling of the null generators of the horizon by

$x^{i}\rightarrow\tilde{x}^{i}=g^{i}(x^{j})\,,$ (13)

which also keeps our metric gauge in eq.(11) intact. Additionally, this does
not change the constant $v$-slicing of the horizon. It rather mixes only the
$x^{i}$ coordinates among themselves. Therefore, it represents the
diffeomorphism invariance for the metric component $h_{ij}$, the induced
metric on a co-dimension two constant $r$ and $v$ hyper-surface. This would,
for example, enable us to convert all partial derivatives with respect to
$x_{i}$ coordinates to covariant derivatives $\nabla_{i}$ compatible with the
metric $h_{ij}$.

We are interested in obtaining an expression for the entropy density and
entropy current density, both of which are defined on the horizon (i.e., on
the $r=0$ hyper-surface). Hence, the entropy density and entropy current
should be a scalar and a vector, respectively, under the diffeomorphism that
mixes only the $x^{i}$ coordinates among themselves as in eq.(13). Also, in
our chosen gauge eq.(11), both are constructed out of the metric functions
$X$, $\omega_{i}$, $h_{ij}$ and their derivatives. Therefore, in our attempt
to construct an entropy current for a diffeomorphism invariant theory of
gravity, we identify the following quantities as our basic building blocks
listed in Table-1.

1. |  The metric coefficients: a scalar function $X(r,v,x)$, a vector function $\omega_{i}(r,v,x)$ and a tensor function $h_{ij}(r,v,x)$.
---|---
2. |  Three differential operators: $\\{\partial_{r}$, $\partial_{v}$, $\nabla_{i}\\}$ acting on $X(r,v,x),\,\omega_{i}(r,v,x)$ and $h_{ij}(r,v,x)$.
Table 1: The basic building blocks

One should note that in the list given in Table-1, the scalar, vector, and
tensor properties of different quantities are determined with respect to the
spatial diffeomorphism that mixes only the $x^{i}$ coordinates among
themselves.

###  2.2 Stationarity and small deviation from it by dynamical fluctuations

As we have mentioned before, our analysis in this paper is limited to the
perturbative approximation of working only up to linear order in the amplitude
of fluctuations around a stationary black hole configuration. In other words,
we should be able to decompose the full metric as a sum of a stationary metric
and a dynamical fluctuation with a small amplitude around it 171717We only
consider fluctuations that preserve our metric gauge eq.((11)). Working with
the horizon adapted coordinates $\\{r,\,v,\,x\\}$ described above in §2.1, in
this subsection, our aim is to make these notions more precise, namely what it
means for space-time to be stationary and also what a non-stationary dynamical
perturbation is.

It is well known that for a stationary black hole, we can always find one
Killing vector such that it is null on the event horizon, a co-dimension one
null hyper-surface. In other words, the event horizon becomes a Killing
horizon for that Killing vector181818However, for a static black hole, we have
a stronger requirement, i.e., the Killing vector, which becomes null on the
event horizon, is the time translations at the asymptotic infinity.. With the
knowledge of this, we can learn about what specific restrictions does the
requirement of stationarity imposes on the space-time metric given in eq.(11).

As was discussed before, we consider a special sub-class of the residual
coordinate transformation freedom mentioned in eq.(12) with
$f_{1}(x^{i})=\lambda$, an arbitrary constant parameter and $f_{2}(x^{i})=0$.
To be more precise, we get the following rescaling of the $v$ and $r$
coordinate

$r\rightarrow\tilde{r}=\lambda~{}r,~{}~{}v\rightarrow\tilde{v}={v\over\lambda}\,,$
(14)

with $\lambda$ being some arbitrary constant. Following (Wall:2015raa, ) and
(Bhattacharya:2019qal, ), we will call this scaling transformation as the
boost transformation in the rest of this paper. The infinitesimal version of
this transformation is generated by the following vector 191919The norm of
$\xi$ is given by $\xi^{\mu}\xi^{\nu}g_{\mu\nu}=-2rv-r^{2}v^{2}X(r,v,x)$. So,
$\xi$ becomes null at the horizon located at $r=0$, where it is proportional
to the null generator ($\partial_{v}$) of the horizon.

$\xi=\xi^{\mu}\partial_{\mu}=v\partial_{v}-r\partial_{r}\,.$ (15)

Under this coordinate transformation, the metric eq.(11) remains almost
invariant apart from the scaling of the arguments ($r$ and $v$) of the metric
functions $X$, $\omega_{i}$ and $h_{ij}$,

$ds^{2}=2d\tilde{v}~{}d\tilde{r}-\tilde{r}^{2}\tilde{X}(\tilde{r},\tilde{v},x)~{}d\tilde{v}^{2}+2\tilde{r}~{}\tilde{\omega}_{i}(\tilde{r},\tilde{v},x)~{}d\tilde{v}~{}dx^{i}+\tilde{h}_{ij}(\tilde{r},\tilde{v},x)dx^{i}~{}dx^{j}\,,$
(16)

where
$\tilde{X}(\tilde{r},\tilde{v},x)=X\left({\tilde{r}\over\lambda},\lambda\tilde{v},x\right),~{}\tilde{\omega}_{i}(\tilde{r},\tilde{v},x)=\omega_{i}\left({\tilde{r}\over\lambda},\lambda\tilde{v},x\right),~{}\tilde{h}_{ij}(\tilde{r},\tilde{v},x)=h_{ij}\left({\tilde{r}\over\lambda},\lambda\tilde{v},x\right)$.
It is then obvious that the metric remains invariant if all the metric
functions $X$, $\omega_{i}$ and $h_{ij}$ depend on the coordinates $r$ and $v$
only through their product $rv$. More precisely, when evaluated on the
horizon, the vector field $\xi$, which reduces to the null generator of the
horizon, will be a Killing vector for any space-time with the following metric

$ds^{2}=2\,dv\,dr-r^{2}\,X(rv,x)\,dv^{2}+2\,r\,\omega_{i}(rv,x)\,dv\,dx^{i}+h_{ij}(rv,x)\,dx^{i}\,dx^{j}\,.$
(17)

Therefore, we learn that in our horizon adapted coordinates
$\\{r,\,v,\,x^{i}\\}$ any metric of the form given above in eq.(17) describes
a stationary space-time202020It might seem strange to see that the stationary
metric in eq.(17) has $v$ dependent components. As explained in detail in
Appendix-A of (Bhattacharyya:2016xfs, ), we should remember that $v$ is an
affine parameter along $\partial_{v}$, the null generator of the horizon.
However, the metric functions for a stationary black hole should be
independent of the Killing coordinate. The latter is generated along the
Killing vector $(v\partial_{v}-r\partial_{r})$. This is not exactly equal to,
but rather proportional to the $\partial_{v}$, even at the horizon $r=0$.. In
(Bhattacharya:2019qal, ) (see Appendix-A in it), it was shown that up to
possible coordinate transformations, this is the most general form of
stationary space-time with a Killing horizon.

Having understood the most general form of any stationary metric, our next job
will be to quantify the departure from stationarity due to a dynamical
perturbation with small amplitude. We remember that our basic building blocks
are the metric functions $X$, $\omega_{i}$ and $h_{ij}$ and their
$\partial_{r}$, $\partial_{v}$ and $\nabla_{i}$ derivatives given in Table-1.
Let us now consider a generic covariant tensor with an expression of the form

${\cal A}\sim(\partial_{r})^{m_{r}}(\partial_{v})^{m_{v}}{\cal B}\,,$ (18)

where ${\cal B}$ collectively denotes the metric coefficients:
$X,\,\omega_{i},\,h_{ij}$ or any other covariant tensor constructed out of
them using the action of $\nabla_{i}$ only 212121It should be noted that the
possible appearances of $\partial_{r}$ and $\partial_{v}$ in $\cal A$ are
explicitly shown in eq.(18), i.e. ${\cal B}$ does not contain any factors
involving $\partial_{r}$ or $\partial_{v}$ acting on
$X,\,\omega_{i},\,h_{ij}$. To avoid clutter of indices, we have also
suppressed the components of the tensors ${\cal A}$ and ${\cal B}$.. If we
evaluate ${\cal B}$ for any equilibrium or stationary space-time configuration
given by the metric eq.(17), it will be a function of the product $rv$ and
$x$, i.e. ${\cal B}(r,v,x)\,|_{equil.}\sim{\cal B}(rv,x)$. Therefore, it can
be argued that whenever $m_{v}>m_{r}$, the expression of ${\cal A}$ given in
eq.(18) will vanish on the horizon ($r=0$) as it will have $(m_{v}-m_{r})$
factors of $r$. However, this will not be true if we are evaluating ${\cal A}$
for a general dynamical metric where the functional dependence of metric
coefficients ($X,\,\omega_{i},\,h_{ij}$) on $r$ and $v$ is not restricted only
through the product of them. Hence, we conclude that whenever any expression
of the form given in eq.(18) with $m_{v}>m_{r}$ evaluates to a non-zero value,
the corresponding space-time metric is dynamical,

$\begin{split}\text{Equilibrium
configuration}\rightarrow~{}&(\partial_{r})^{m_{r}}(\partial_{v})^{m_{v}}{\cal
B}(rv,x)|_{r=0}=0~{}\text{(for $m_{v}>m_{r}$)}\,,\\\ \text{Non-equilibrium
configuration}\rightarrow~{}&(\partial_{r})^{m_{r}}(\partial_{v})^{m_{v}}{\cal
B}(r,v,x)|_{r=0}\neq 0~{}\text{(for $m_{v}>m_{r}$)}\,.\end{split}$ (19)

We will use this as a criterion for stationary and dynamical space-times.

To explain the linearity of dynamical fluctuations to a stationary
configurations, we decompose the full space-time metric $g_{\mu\nu}$ as a sum
of two parts

$g_{\mu\nu}=g_{\mu\nu}^{eq}+\epsilon\,\delta g_{\mu\nu}\,,$ (20)

where $g_{\mu\nu}^{eq}$ is a stationary metric and $\delta g_{\mu\nu}$
captures the dynamics away from equilibrium. The small parameter $\epsilon$
denotes the amplitude of the dynamical perturbation. In the coordinate system
that we are working with, such a decomposition will imply that all the metric
functions could also be written accordingly as a sum of two contributions

$\begin{split}X=&\,X^{eq}(rv,x)+\epsilon\,\delta X(r,v,x)\,,\\\
\omega_{i}=&\,\omega^{eq}_{i}(rv,x)+\epsilon\,\delta\omega_{i}(r,v,x)\,,\\\
h_{ij}=&\,h_{ij}^{eq}(rv,x)+\epsilon\,\delta h_{ij}(r,v,x)\,.\end{split}$ (21)

Furthermore, as a consequence, we can also argue that for any ${\cal B}$, some
covariant expression constructed out of the metric functions and their
$\nabla_{i}$ derivative(s), can always be written as

${\cal B}(r,v,x)={\cal B}^{eq}(rv,x)+\epsilon\,\delta{\cal B}(r,v,x)\,.$ (22)

Following our arguments leading to eq.(19), it is obvious that if we operate
$\left(\partial_{r}^{m}\partial_{v}^{m+k}\right)$ (for $k>0$) on such a ${\cal
B}$ given in eq.(22) and evaluate it on the horizon, it is only $\delta{\cal
B}$ that will contribute, rendering the expression linear in dynamics

${\cal A}\,|_{r=0}\sim\left(\partial_{r}^{m}\partial_{v}^{m+k}\right){\cal
B}|_{r=0}=\epsilon\,\left(\partial_{r}^{m}\partial_{v}^{m+k}\right)\delta{\cal
B}|_{r=0}\,\sim\,{\mathcal{O}}(\epsilon)\,,~{}~{}~{}(\text{for $k>0$}).$ (23)

Now, let us consider an expression of the form

${\cal A}_{1}\sim\left(\partial_{r}^{m_{1}}\partial_{v}^{m_{1}+k_{1}}{\cal
B}_{1}\right)\left(\partial_{r}^{m_{2}}\partial_{v}^{m_{2}+k_{2}}{\cal
B}_{2}\right)\,,~{}~{}~{}(\text{for $k_{1},k_{2}>0$}),$ (24)

where both ${\cal B}_{1}$ and ${\cal B}_{2}$ are two different but arbitrary
covariant tensor expressions admitting similar decomposition as ${\cal B}$ in
eq.(22). Following the similar set of arguments presented above, we could now
see that both of the factors in the RHS of eq.(24) will have non-zero
contributions only when ${\cal B}_{1}$ and ${\cal B}_{2}$ are replaced by
$\delta{\cal B}_{1}$ and $\delta{\cal B}_{2}$ respectively,

$\begin{split}{\cal
A}_{1}\,|_{r=0}\sim\left(\partial_{r}^{m_{1}}\partial_{v}^{m_{1}+k_{1}}\right){\cal
B}_{1}\,\left(\partial_{r}^{m_{2}}\partial_{v}^{m_{2}+k_{2}}\right){\cal
B}_{2}\,|_{r=0}\,\sim\,{\mathcal{O}}(\epsilon^{2})\,,~{}~{}~{}(\text{for
$k_{1},k_{2}>0$})\,,\end{split}$ (25)

leading us to the conclusion that the expression ${\cal A}_{1}$ will be non-
linear (quadratic) in the amplitude of dynamical fluctuations.

We will summarise the main lessons that we have learnt regarding the
classification of generic tensor structures built out of the metric
coefficient functions and their derivatives, in terms of stationarity and
dynamical fluctuations (linear or non-linear) around it in Table-2

1. |  Any expression that has more $\partial_{v}$-derivatives than $\partial_{r}$-derivatives could be non-zero solely because of dynamics.
---|---
2. |  The expression will be linear in the amplitude of dynamics when the extra $\partial_{v}$ derivatives (i.e. those $\partial_{v}$’s which are not compensated or paired with $\partial_{r}$’s) are all acting on a single covariant expression of the form of ${\cal B}$, eq.(23).
3. |  Whenever the extra $\partial_{v}$ derivatives are distributed between different factors, the resulting expression (e.g. ${\cal A}_{1}$ in eq.(25)) would be non-linear in the amplitude of dynamics.
Table 2: Distribution of $\partial_{v}$-derivatives and non-linearities in
amplitude expansion.

###  2.3 Killing symmetry and boost weight of quantities

As we have mentioned in the introduction, we would be analyzing the equation
of motion for the metric, and our goal is to write its most general form in
terms of our building blocks given in Table-1. The equation of motion in a
theory of gravity is a rank two tensor with respect to the full set of
diffeomorphism that mixes all the space-time coordinates. In this subsection,
we shall briefly describe how we can relate a particular component of a tensor
(when evaluated on the horizon) with the number and distribution of
$\partial_{r}$ and $\partial_{v}$ derivatives acting on different factors.

In our horizon adapted coordinate system $\\{r,\,v,\,x\\}$, the most general
stationary black hole solution as in eq.(17), admits a Killing vector

$\xi=\xi^{\mu}\partial_{\mu}=(v\partial_{v}-r\partial_{r})\,.$ (26)

As was mentioned in the previous subsection (see eq.(15)), this is also the
generator of the boost transformation eq.(14), or more precisely of the
infinitesimal version of it.

In order to study small dynamical fluctuations away from a stationary black
hole configuration, it will be useful to see how various quantities transform
under this scaling or boost transformation that generates the Killing symmetry
of the stationary background. Furthermore, since we are viewing every quantity
to be built out of the basic building blocks (Table-1), it is sufficient to
know the transformation property of them.

Let us define the power of $\lambda$ by which any quantity transform under the
boost transformation eq.(14), as the boost weight of that quantity

${\cal A}\rightarrow\widetilde{\cal A}=\lambda^{w}{\cal
A}\,,~{}~{}\text{under}~{}~{}\\{r\rightarrow\tilde{r}=\lambda~{}r,~{}v\rightarrow\tilde{v}=\lambda^{-1}v\\}~{}~{}\Rightarrow~{}~{}\text{boost
weight of ${\cal A}$ is $w$}\,.$ (27)

The first thing we should note following this definition, is that the
coordinates $r$ and $v$ have boost weights $+1$ and $-1$ respectively.
Similarly all the metric functions: $X$, $\omega_{i}$ and $h_{ij}$ have zero
boost weight and only $\partial_{v}$ and $\partial_{r}$ are the operators that
transform non-trivially under eq.(14)

$\partial_{r}\rightarrow\partial_{\tilde{r}}=\lambda^{-1}\,\partial_{r}\,,~{}~{}\partial_{v}\rightarrow\partial_{\tilde{v}}=\lambda\,\partial_{v}\,.$
(28)

Hence, the operators $\partial_{r}$ and $\partial_{v}$ have boost weights $-1$
and $+1$ respectively, and the operator $\nabla_{i}$ has zero boost weight
when operated on $X$, $\omega_{i}$ and $h_{ij}$.

In other words, since $\xi$ given in eq.(26) generates Killing symmetry of the
stationary background, the Lie derivative with respect to $\xi$ (denoted as
${\cal L}_{\xi}$) when operated on any covariant tensor that is constructed
out of the stationary metric eq.(17), will vanish. Therefore, a non-zero value
of this ${\cal L}_{\xi}$ operator acting on any covariant tensor built out of
metric components, is indicative of non-stationary or out-of-equilibrium
dynamics. The action of ${\cal L}_{\xi}$ on any covariant tensor
$S_{\alpha_{1}\alpha_{2}\cdots\alpha_{k}}$ is given by

$\begin{split}{\cal
L}_{\xi}S_{\alpha_{1}\alpha_{2}\cdots\alpha_{k}}=\,&\xi^{\beta}\partial_{\beta}S_{\alpha_{1}\alpha_{2}\cdots\alpha_{k}}+\left(\partial_{\alpha_{1}}\xi^{\beta}\right)S_{\beta\alpha_{2}\cdots\alpha_{k}}+\left(\partial_{\alpha_{2}}\xi^{\beta}\right)S_{\alpha_{1}\beta\alpha_{3}\cdots\alpha_{k}}+\cdots\\\
&+\left(\partial_{\alpha_{k}}\xi^{\beta}\right)S_{\alpha_{1}\alpha_{2}\cdots\beta}\,,\end{split}$
(29)

and when we evaluate this in our coordinate system, eq.(11), with
$\xi=(v\partial_{v}-r\partial_{r})$, we will obtain the following general form

$\begin{split}{\cal
L}_{\xi}S_{\alpha_{1}\alpha_{2}\cdots\alpha_{k}}=\,\left[w+(v\partial_{v}-r\partial_{r})\right]S_{\alpha_{1}\alpha_{2}\cdots\alpha_{k}}\,\end{split}$
(30)

where we identify the quantity $w$ as the boost weight of the covariant tensor
$S_{\alpha_{1}\alpha_{2}\cdots\alpha_{k}}$ as

$\begin{split}w\,\equiv\,&\text{net boost weight of a covariant tensor (with
all indices lowered)}\\\ =\,&\text{number of lower $v$ indices - number of
lower $r$ indices.}\end{split}$ (31)

A more detailed explanation regarding the idea of boost weight of different
quantities is provided in Appendix-(B). Following eq.(31), we can check that
under ${\cal L}_{\xi}$, the operators $\partial_{v}$ and $\partial_{r}$
transform like the $v$ and $r$ components of a lower-indexed vector
$\partial_{\mu}$ and therefore have boost weights $+1$ and $-1$ respectively,
as expected. We also learn that if we evaluate a particular component of a
covariant tensor on the horizon $r=0$, the number of lower $v$ indices
corresponds to the number of $\partial_{v}$’s (and the number of lower $r$
indices corresponds to the number of $\partial_{r}$’s) acting on metric
functions ($X$, $\omega_{i}$, and $h_{ij}$) with boost weight zero222222If we
allow ourselves to go away from the horizon, positive boost weights could also
be absorbed by factors of $r$. For example, the $vv$ component of the metric,
$g_{vv}$, which is one example of a rank-2 covariant tensor component with
boost weight $2$, has a factor of $r^{2}$ multiplying a boost-invariant
function $X$. The same is true for $g_{vi}$ and $g_{ij}$ the $vi$ and $ij$
components of the metric respectively. In fact, this is why all the metric
functions ($X$, $\omega_{i}$ and $h_{ij}$) are of zero boost weights..

Before we proceed, let us summarise the main points of this section in
Table-3, which we are going to use in the rest of the paper.

1. |  We have chosen a special coordinate system adapted to the dynamical null horizon. The coordinates on the horizon are $v$ (parameter along the null generator) and $\\{x^{i}\\}$ (all the spatial coordinates), whereas the coordinate away from the horizon is denoted by $r$, see eq.(11). The coordinates $v$ and $r$ are also affinely parametrized.
---|---
2. |  In this coordinate system, if we evaluate any component of a covariant tensor restricted on the horizon, schematically, it will have a structure with some number of $\partial_{r}$, $\partial_{v}$ and $\nabla_{i}$ operators acting on the functions appearing in the metric eq.(11):($X$, $\omega_{i}$, and $h_{ij}$) or product of such structures.
3. |  For an expression, we defined the boost weight $w$ of it to be the difference between the number of $\partial_{v}$’s and $\partial_{r}$’s, and then we have argued that for a tensor component, this is the same as the difference between the number of lower $v$ (or upper $r$) and the lower $r$ (or upper $v$) indices, see eq.(27) and eq.(31).
4. |  Furthermore, the boost weight of a tensor component is nothing but the multiplicative factor it will have under the action of Lie derivative ${\cal L}_{\xi}$ along the vector $\xi\left(=v\partial_{v}-r\partial_{r}\right)$, the general Killing vector field whenever the metric has a Killing horizon, see eq.(30).
5. |  Finally, any expression with positive boost weight is non-zero only when evaluated on a dynamical, non-stationary metric. It is linear in the amplitude of the dynamical fluctuation provided all the excess $\partial_{v}$’s are acting on a single function, and it is not a product of more than one expression, each having positive boost weight, see eq.(19), eq.(23) and eq.(25).
Table 3: Summary of the important lessons so far that will serve as guidelines
for later computations.

###  2.4 Significance of the structure of certain components of the equations
of motion for a proof of the second law

Until now, we have summarized the basic concepts in relevance to our work. In
the following, we will briefly review the arguments required to justify a
local version of the second law of thermodynamics for linearized dynamical
perturbations around a stationary black hole configuration. As we have
mentioned before, to achieve this, we will be looking at the structure of a
particular component of the equations of motion, to be more specific, the
‘$vv$’-component of it, i.e., $E_{vv}$. Thereby, we will also explain how
significant the role of the structure of $E_{vv}$ is to establish the second
law.

The notion of entropy for stationary black holes in arbitrary diffeomorphism
invariant theory of gravity is associated with the Wald entropy defined as

$S_{W}=-2\pi\int_{{\cal
H}_{v}}d^{d-2}x\,\sqrt{h}\,{E^{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}}_{R}}\epsilon_{\alpha_{1}\alpha_{2}}\epsilon_{\alpha_{3}\alpha_{4}}\,=\int_{{\cal
H}_{v}}d^{d-2}x\,\sqrt{h}\,\left(1+s^{\text{HD}}_{w}\right),$ (32)

where $E^{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}}_{R}$ is given by eq.(70),
$\epsilon_{\alpha_{1}\alpha_{2}}$ are the bi-normal to the constant $v$-slices
of the horizon, $\sqrt{h}$ is the determinant of the induced metric $h_{ij}$
on the horizon and $s^{\text{HD}}_{w}$ is the contribution to the Wald entropy
density from the higher derivative part of the Lagrangian leaving the
Einstein-Hilbert part aside. This definition of the Wald entropy is, by
construction, consistent with the first law of thermodynamics. From the last
expression on the RHS above it is clear that for Einstein’s gravity $S_{W}$
reduces to the well known formula of computing area of the spatial slices of
the horizon.

Starting from the Wald entropy formula in eq.(32), one can then write down an
entropy function which is the out-of-equilibrium extension of it as the
following

$S_{\text{total}}=S_{W}+\int_{{\cal
H}_{v}}d^{d-2}x\,\sqrt{h}\,s_{\text{cor}}=\int_{{\cal
H}_{v}}d^{d-2}x\,\sqrt{h}\,\left(1+s^{\text{HD}}_{w}+s_{\text{cor}}\right)\,,$
(33)

where $s_{\text{cor}}$ are the corrections to the Wald entropy density due to
non-stationary dynamics, and it certainly includes the JKM ambiguities that we
mentioned before. Once the dynamics is switched off, $s_{\text{cor}}$ vanishes
and therefore $S_{\text{total}}$ reduces to $S_{W}$, i.e.

$s_{\text{cor}}\big{|}_{equilibrium}=0.$

The main goal of Wall:2015raa  was to constrain the form of $s_{\text{cor}}$,
and hence to fix the JKM ambiguities, by requiring $S_{\text{total}}$ to
satisfy a linearized version of the second law. More precisely, the strategy
would be to determine $s_{\text{cor}}$ such that
$\partial_{v}S_{\text{total}}\geq 0$. One defines a local expansion parameter
$\vartheta$, 232323The corresponding quantity in Einstein’s gravity, say
$\vartheta_{E}$, is the expansion parameter for the null congruence
$\partial_{v}$. The evolution of this parameter $\vartheta$ with $v$ is
governed by the Raychaudhuri equation and this plays a crucial role in proving
the area increase theorem in general relativity. as follows

${\partial S_{\text{total}}\over\partial v}\equiv\int_{{\cal
H}_{v}}d^{d-2}x\,\sqrt{h}\,\vartheta\,.$ (34)

Now, one way to prove $\partial_{v}S_{\text{total}}\geq 0$ is to show that
$\vartheta\geq 0$ for all $v\geq 0$. The trick one uses for this is to argue
that $\vartheta$ is a monotonically decreasing function of $v$ (i.e.
$\partial_{v}\vartheta\leq 0$), along with the restriction that at asymptotic
future it approaches to zero, meaning $\vartheta\rightarrow 0$ for
$v\rightarrow\infty$. This is actually motivated by the physical situation
that we are examining here. After getting slightly perturbed by a small
dynamical fluctuation, the initial stationary black hole solution finally
settles down to another stationary black hole solution in the future.

Now calculating $\partial_{v}\vartheta$ in our chosen metric gauge by using
eq.(34) as the definition of $\vartheta$ and $S_{\text{total}}$ from eq.(33),
we obtain242424We have used here the fact that
$\partial_{v}\vartheta_{E}=-R_{vv}+\mathcal{O}(\epsilon^{2})$, where
$\vartheta_{E}$ is the corresponding quantity for $\vartheta$ when computed
from just the Einstein-Hilbert term in the action.

$\partial_{v}\vartheta=-R_{vv}+\partial_{v}\left({1\over\sqrt{h}}\partial_{v}\left(\sqrt{h}\left(s^{\text{HD}}_{w}+s_{\text{cor}}\right)\,\right)\right)+\mathcal{O}(\epsilon^{2})\,.$
(35)

Next, we should remember that the $vv$-component of the equation of motion
$E_{\mu\nu}$ can be written as

$-E_{vv}=R_{vv}+E^{\text{HD}}_{vv}=T_{vv}\,$ (36)

where $E_{vv}$ is obtained from just the gravity part of the
Lagrangian252525The minus sign on the LHS is there to make it consistent with
the convention used in eq.(5). With this choice of convention, for Einstein
gravity we get  $E_{vv}=-R_{vv}=-T_{vv},$ and this also defines the sign
convention of the stress tensor such that the null energy condition works out
as $T_{vv}\geq 0$. We should also remember that while writing the
$vv$-component of the equations of motion in the form of eq.(36), we have
evaluated it on the horizon which is located at $r=0$. For example, from our
choice of metric given in eq.(11) it is clear that $g_{vv}|_{r=0}=0$. ,
$E^{\text{HD}}_{vv}$ only contains the contribution from the higher derivative
part of the gravity Lagrangian excluding the Einstein-Hilbert term, $T_{vv}$
is the $vv$-component of the stress energy tensor $T_{\mu\nu}$ coming from the
matter sector. Substituting this for $R_{vv}$ in eq.(35) we get

$\partial_{v}\vartheta=E^{\text{HD}}_{vv}+\partial_{v}\left({1\over\sqrt{h}}\partial_{v}\left(\sqrt{h}\left(s^{\text{HD}}_{w}+s_{\text{cor}}\right)\,\right)\right)-T_{vv}+\mathcal{O}(\epsilon^{2})\,.$
(37)

Furthermore, we also assume that the stress-energy tensor, for the matter
part, satisfies the null energy condition, which, when translated to our
choice of metric gauge and $\xi^{\mu}$ eq.(26), takes the form of $T_{vv}\geq
0$. Thus from eq.(37) we see that the imposition of null energy condition on
the stress-energy tensor for the matter perturbation actually contributes in
favor of making $\partial_{v}\vartheta\leq 0$. In that sense, with the
assumption of null energy condition, $T_{vv}$ trivially drops out from our
analysis.

We are considering dynamical perturbations to a stationary black hole
configuration such that the metric of the equilibrium background gets
corrected at $\mathcal{O}(\epsilon)$, see eq.(20). Consequently, we must also
take into consideration the fact that terms linear in the amplitude of the
dynamical fluctuation can come with both signs depending on the sign of
$\epsilon$. On the other hand, on equilibrium configurations, i.e.
$\epsilon\rightarrow 0$, we must get $\partial_{v}\vartheta\rightarrow 0$.
Therefore, the only way in which $\partial_{v}\vartheta\leq 0$ can be met
while working up to linear order $\mathcal{O}(\epsilon)$, is to show that

$\partial_{v}\vartheta=\mathcal{O}(\epsilon^{2})\,,$ (38)

or in other words, $\partial_{v}\vartheta$ should vanish up to linear order in
the amplitude expansion.

As a result it is obvious from eq.(37) that eq.(38) will be satisfied if the
following holds true

$E^{\text{HD}}_{vv}=-\,\partial_{v}\left({1\over\sqrt{h}}\partial_{v}\left(\sqrt{h}\left(s^{\text{HD}}_{w}+s_{\text{cor}}\right)\,\right)\right)+\mathcal{O}(\epsilon^{2})\,.$
(39)

This relation in eq.(39) highlights the significance of $E^{\text{HD}}_{vv}$
in arriving at a proof of the linearized version of the second law presented
in the form of eq.(38) .

At this point, it is worth making a comment regarding the dependance of
$T_{vv}$ on the order of $\epsilon$ and its significance for eq.(38). As
mentioned before in §2.2 (see eq.(20)), the correction to the stationary
background metric due to dynamical perturbations is of
$\mathcal{O}(\epsilon)$. Let the matter fields that induce the perturbation
are of order $\epsilon$. Now, following our assumptions the stress tensor due
to the matter perturbation satisfies null energy condition $T_{vv}\geq 0$, and
vanishes in the equilibrium limit $\epsilon\rightarrow 0$. As a consequence,
based on general arguments given above for $\partial_{v}\vartheta$, we learn
that $T_{vv}\sim\mathcal{O}(\epsilon^{2})$; see section-(2.1) and Appendix-B
of Bhattacharya:2019qal  for a detailed discussion on this point. Therefore,
up to our working precision of $\mathcal{O}(\epsilon)$, $T_{vv}$ can be
dropped from the RHS of eq.(37) and the equality in eq.(38) is justified once
the restriction on $E_{vv}$ in eq.(39) is met 262626We can instead have a
situation, possibly a more physically appealing one, where the corrections to
the space-time geometry are generated due to backreactions from a
$\mathcal{O}(\epsilon^{2})$ stress tensor representing the non-stationary
perturbation. The metric which is sourced by this stress tensor would then
receive its first correction at $\mathcal{O}(\epsilon^{2})$ and not at
$\mathcal{O}(\epsilon)$. The corrections to the metric at
$\mathcal{O}(\epsilon)$ would trivially vanish. In other words, the dynamical
fluctuations of the metric will get non-trivial contributions starting at
$\mathcal{O}(\epsilon^{2})$. This is certainly true for minimal coupling of
the matter sector, which we will be considering in this paper. However, as it
has been discussed in section-(2.1) of Bhattacharya:2019qal , even in that
situation, the statement made in eq.(39) would be valid. In that case eq.(38)
would become trivial at $\mathcal{O}(\epsilon)$. The first non-trivial
contribution for this would occur at $\mathcal{O}(\epsilon^{2})$ as the
following: $\partial_{v}\vartheta|_{\mathcal{O}(\epsilon^{2})}\leq 0$..

In other words, we can view eq.(39) as a constraint on the off-shell structure
of $E_{vv}$, which can be summarised as follows:

In any diffeomorphism invariant theory of gravity, working in the chosen
metric gauge eq.(11) and up to linear order in the amplitude expansion, if we
can show that the structure of the $vv$-component of equations of motion can
always be written as the RHS of eq.(39), we will be able to construct a proof
of the linearized version of the second law by satisfying eq.(38).

In view of the statement made above, we are now convinced that the primary
goal of this current paper is indeed justified. Once we are able to show that
in any arbitrary higher derivative theory of gravity $E_{vv}$ can always be
written as eq.(7), which we write here again for convenience

$\begin{split}E_{vv}\big{|}_{r=0}=&\,\partial_{v}\bigg{[}{1\over\sqrt{h}}\partial_{v}\left({\sqrt{h}}\,{\cal
J}^{v}\right)+\nabla_{i}{\cal
J}^{i}\bigg{]}+\mathcal{O}(\epsilon^{2})\,.\end{split}$ (40)

Once this is compared with eq.(39), one can straightforwardly obtain the
components of entropy current (i.e. ${\cal J}^{v}$ and ${\cal J}^{i}$).
Furthermore, we are also convinced that when the equilibrium limit is taken
${\cal J}^{v}$ should reproduce the equilibrium Wald entropy
density272727Since $E_{vv}$ also includes the contributions from the Einstein-
Hilbert term, in ${\cal J}^{v}$ we get an extra additive factor of one in
addition to $s^{\text{HD}}_{w}$. This extra factor of one is actually the Area
of the horizon piece.

${\cal J}^{v}\big{|}_{equilibrium}=(1+s^{\text{HD}}_{w})\,.$ (41)

###  2.5 An important identity relating the equations of motion with the
Noether charge for diffeomorphism

In the previous sub-sections, we have established the importance of a certain
desired structure of the $vv$-component of the equations of motion, denoted by
$E_{vv}$, for our work. In the following, we will write down a key equation
that will help us construct the entropy current from $E_{vv}$ by relating the
latter to the Noether charge under diffeomorphism. This relation is, in fact,
an identity that is true for any diffeomorphism invariant theory of gravity.
We will follow the construction of Noether currents and charges corresponding
to the diffeomorphism invariance in such a theory of gravity as prescribed in
Iyer:1994ys 282828Before we proceed, let us emphasize that this is a well-
known identity in gravitational theories and here we are just stating and
reproducing it for the sake of completeness. We refer the readers to
Appendix-(C) where a review of Iyer:1994ys  is provided in component
notation..

In a diffeomorphism invariant theory of gravity with an action
$S=\int\sqrt{-g}\,L$, the variation of the Lagrangian due to an arbitrary
variation $\delta g_{\mu\nu}$ of the metric, is given by

$\begin{split}\delta\left[\sqrt{-g}~{}L\right]&=\sqrt{-g}\,E^{\mu\nu}\,\delta
g_{\mu\nu}+\sqrt{-g}\,D_{\mu}\Theta^{\mu}[\delta g]\,,\end{split}$ (42)

where $E^{\mu\nu}$ is the equation of motion, $\Theta^{\mu}[\delta g]$ is the
total derivative term generated in the variation of the Lagrangian and
following our convention in this paper, $D_{\mu}$ is the covariant derivative
with respect to the metric $g_{\mu\nu}$.

Next we consider that $\delta g_{\mu\nu}$ is produced due to an infinitesimal
coordinate transformation generating the diffeomorphism $x^{\mu}\rightarrow
x^{\mu}+\zeta^{\mu}$,

$\delta g_{\mu\nu}=D_{\mu}\zeta_{\nu}+D_{\nu}\zeta_{\mu},$ (43)

for which we obtain

$\delta\left[\sqrt{-g}\,L\right]=\sqrt{-g}\,D_{\mu}\left(\zeta^{\mu}L\right).$
(44)

Substituting this back in eq.(42) and cancelling the $\sqrt{-g}$ factor on
both sides we get

$\begin{split}D_{\mu}\left(\zeta^{\mu}L-2E^{\mu\nu}\zeta_{\nu}-\Theta^{\mu}[\partial\zeta]\right)=-2\zeta_{\nu}D_{\mu}E^{\mu\nu}\,.\end{split}$
(45)

Now we make a choice for $\zeta^{\mu}$ such that it is non-zero only in a
small region ${\cal R}$. With this, if one integrates the both sides of
eq.(45) over full space-time, the LHS will vanish since it would integrate to
a pure boundary term at infinity where $\zeta$ vanishes. Hence, we obtain

$\int_{\text{full space-time}}\,\zeta_{\nu}\,D_{\mu}E^{\mu\nu}=0\,,$ (46)

which is true for any $\zeta$ as long as it is has a non-zero support over a
finite region $\mathcal{R}$. Therefore, we see that the following relation
identically holds

$D_{\mu}E^{\mu\nu}=0\,.$ (47)

We should note that this is essentially the Bianchi identity corresponding to
any diffeomorphism invariant theory.

Let us now substitute the identity eq.(47) in eq.(45) and allow $\zeta^{\mu}$
to be any vector without any restriction on its support. We obtain that the
vector $\zeta^{\mu}L-2E^{\mu\nu}\zeta_{\nu}-\Theta^{\mu}(\partial\zeta)$ is
identically conserved on any geometry for any $\zeta$

$D_{\mu}\left(\zeta^{\mu}L-2E^{\mu\nu}\zeta_{\nu}-\Theta^{\mu}[\partial\zeta]\right)=0\,.$
(48)

Next, we use the result that an identically conserved vector can itself be
always written as the divergence of an arbitrary antisymmetric rank-$2$
tensor. Hence one can argue that locally it is always true that

$\begin{split}\zeta^{\mu}L-2E^{\mu\nu}\zeta_{\nu}-\Theta^{\mu}=-D_{\nu}Q^{\mu\nu}\,,\end{split}$
(49)

for some arbitrary antisymmetric tensor $Q^{\mu\nu}$292929The negative sign on
the RHS of eq.(49) is due to our convention and could be understood as a part
of the definition of $Q^{\mu\nu}$ through this equation. In Appendix-(C), see
eq.(178), where we have derived this equation while reviewing Iyer:1994ys ..

We can contract the free index on both sides of eq.(49) with another
$\zeta_{\mu}$ to get

$\begin{split}&\zeta^{2}L-\Theta^{\mu}\zeta_{\mu}+\zeta_{\mu}\,D_{\nu}Q^{\mu\nu}=2\,\zeta^{\mu}\zeta^{\nu}E_{\mu\nu}\,.\end{split}$
(50)

Although eq.(50) is true for any $\zeta^{\mu}$, for our purpose, we will
choose $\zeta^{\mu}$ to be $\xi=v\partial_{v}-r\partial_{r}$ and evaluate both
sides of eq.(50) on the horizon $r=0$,

$\begin{split}\left(2\xi^{\mu}\xi^{\nu}E_{\mu\nu}\right)|_{r=0}=\,&2v^{2}E_{vv}|_{r=0}\\\
\left(\xi^{2}L-\Theta\cdot\xi+\xi_{\mu}\,D_{\nu}Q^{\mu\nu}\right)|_{r=0}=\,&v\left(-\Theta^{r}+D_{\nu}Q^{r\nu}\right)|_{r=0}\,.\end{split}$
(51)

Finally, cancelling one factor of $v$ from both sides, we arrive at

$\begin{split}2\,v\,E_{vv}\big{|}_{r=0}=~{}&\big{(}-\Theta^{r}+D_{\mu}Q^{r\mu}\big{)}\big{|}_{r=0}\,.\end{split}$
(52)

Equation (52) constitutes the main relation that we will use in §3 to prove
the existence of an entropy current in a diffeomorphism invariant theory of
gravity.

##  3 A proof of the existence of an entropy current

Following the fundamental concepts developed in §2, in this section, we will
carry out an extensive analysis to derive an expression of the entropy current
for dynamical black holes in any diffeomorphism invariant theory of gravity up
to linearized order in the amplitude of the out-of-equilibrium fluctuations.
In practice, we aim to establish that a specific component of the equations of
motion, namely $E_{vv}$, in such a theory can always be recast in a form
similar to the RHS of eq.(7), and thereby obtaining the form of ${\cal J}^{v}$
and ${\cal J}^{i}$. As we have already discussed in §2.4, showing eq.(7) is
equivalent to argue for a linearized version of the second law involving black
hole dynamics.

The starting point of our calculation would be to consider the important
relation, eq.(52), that was derived in §2.5. Here we re-write that equation
again for the sake of convenience,

$\begin{split}2\,v\,E_{vv}\big{|}_{r=0}=~{}&\big{(}-\Theta^{r}+D_{\mu}Q^{r\mu}\big{)}\big{|}_{r=0}\,.\end{split}$
(53)

With this at hand, our main job now is to establish that the expression for
$E_{vv}$ on the RHS of eq.(53) can always be rearranged as the following

$\begin{split}E_{vv}\big{|}_{r=0}=&\,\partial_{v}\bigg{[}{1\over\sqrt{h}}\partial_{v}\left({\sqrt{h}}\,{\cal
J}^{v}\right)+\nabla_{i}{\cal
J}^{i}\bigg{]}+\mathcal{O}(\epsilon^{2})\,,\end{split}$ (54)

which is the same as eq.(7).

For the sake of convenience of the reader, in the following, we will separate
the material to be presented into two sub-sections. To start with, in §3.1 we
will provide a brief schematic overview of the proof. The primary purpose of
this would be to convey the essential ingredients of the analysis postponing
the details of technical computations to the following sub-section §3.2.

###  3.1 A brief and schematic sketch of the proof highlighting the basic
strategy

In this sub-section, our primary goal is to provide an illustrative sketch of
the arguments justifying the construction of a local entropy current in an
arbitrary higher derivative theory of gravity. Before we dive into the
technical computations presented in the following sub-section, it will be
beneficial to develop an overall and intuitive understanding of the line of
logical sequence.

##### Basic strategy:

We start with a brief explanation of the basic strategy as follows. Since a
particular coordinate system, as in eq.(11), has been chosen for our analysis,
it is worth noting the explicit appearance of the coordinate $v$ on the LHS of
eq.(53). The off-shell structure of $E_{vv}$ cannot involve any factor of $v$
except through the metric functions $(X,\,\omega_{i},\,h_{ij})$ or their
derivatives. Therefore, our primary task going ahead would be to figure out
the explicit $v$-dependence in each of the two terms appearing on the RHS of
eq.(53) so that we can compare the terms with equal powers of $v$ on both
sides of it. This, in turn, would produce the desired structure of $E_{vv}$
that we are looking for.

We notice that on the RHS of eq.(53) two quantities determine the structure of
$E_{vv}$ when evaluated on the horizon. One of them is the $r$-component of
$\theta^{\mu}$, which is generated as a total derivative term in the variation
of the Lagrangian under a diffeomorphism, eq.(42). The second term involves
$Q^{\mu\nu}$, which is defined in eq.(49). In order to extract out the
explicit $v$-dependence in both of these terms, we will use their
corresponding definitions outlined in the construction of the Noether charge
for diffeomorphism invariant gravity Lagrangians, Iyer:1994ys . These
definitions are given in eq.(68) and eq.(69) for $\Theta^{\mu}$, and in
eq.(111) for $Q^{\mu\nu}$.

Furthermore, we will also need to consider the Killing symmetry of the
stationary background geometry, eq.(17), generated by the null generator of
the horizon $\xi^{\mu}$ eq.(15), and its breaking by a non-stationary
perturbation in the linearised amplitude approximation (see eq.(21)). Since
$\xi^{\mu}$ also generates the boost transformation eq.(14), we will have to
figure out how each of the two terms on the RHS of eq.(53) transforms under
it, i.e., the boost weights for each of them.

One crucial element for our computations would be determining the general
structure of an arbitrary covariant tensor quantity with a given boost weight.
At this point, it should be emphasized that every tensorial quantity appearing
in our analysis should be considered as if they are built out of the basic
building blocks as given in Table-1. In §2.3 we have already learned about how
to assign boost weights for the basic building blocks as well as for any
generic tensor built out of them by counting the number of $r$ and $v$
components of that particular tensor, see eq.(31).

Once we know how to assign boost weight for a generic covariant tensor, we can
easily determine its generic structure. For that, we need to analyze the
distribution of $\partial_{v}$-derivatives carefully. The fact that we will be
working in the linearised amplitude approximation will play a crucial role in
executing this. We should also remember another related and equally important
aspect, that is, eq.(53) has to be evaluated on the horizon located at $r=0$.
The guiding principles which are important in this context are summarised in
Table-2 and Table-3. Following these guidelines, we can compute the generic
structure of any covariant tensor with a positive boost weight, and the result
is given in eq.(66). The detailed arguments for deriving this relation would
be discussed in §3.2.1. We will call this relation as “Result: $1$” and it
will be repeatedly used in our calculations.

Next, we would directly implement the strategy outlined above for both the
terms on the RHS of eq.(53) one at a time. Using the definitions of
$\Theta^{\mu}$ and $Q^{\mu\nu}$ as given in eq.(68) and eq.(111) respectively,
we will analyze every term that contributes to each of them. Finally,
substituting the resulting expressions in eq.(52), we will derive the desired
structure for $E_{vv}$ as mentioned in eq.(7).

##### Schematic illustration of the proof:

As written above in eq.(54), the principal quantity that we are going to
examine is $E_{vv}$, the $vv$ component of the equation of motion. Following
the criteria set in eq.(31) and Table-3, $E_{vv}$ is a tensor component with
boost weight $+2$. Following the arguements presented in Wall:2015raa  and
also reviewed in section-(2.2.2) of Bhattacharya:2019qal , we have seen that
$E_{vv}$ being a term with boost weight $+2$ could be expressed, up to linear
order in the amplitude of the dynamical fluctuations, as

$E_{vv}\sim\partial_{v}^{2}C_{(0)}+A_{(0)}\,\partial_{v}^{2}B_{(0)}+\mathcal{O}(\epsilon^{2})\,.$
(55)

Here we have used the convention that the subscripts in $A_{(0)}$, $B_{(0)}$
and $C_{(0)}$ denote the boost weights of these quantities, and this will be
followed throughout in this paper. It is also essential to keep in mind that
the purpose of writing eq.(55) is to highlight the structural form on the RHS,
ignoring specific numerical factors, possible index contraction, and factors
of $\sqrt{h}$ to avoid the clutter.

Furthermore, for a more ellaborate derivation of eq.(55), one should compare
it with eq.(2.22) in Bhattacharya:2019qal . This would also emphasise an
important difference in the structures of the two terms on the RHS of eq.(55).
In general, for a given theory of higher derivative gravity the quantities
$A_{(0)}$, $B_{(0)}$ and $C_{(0)}$ should be thought of as they are built out
of the metric functions $(X,\,\omega_{i},\,h_{ij})$ or their derivatives with
possible index contractions such that the resulting boost weights are zero for
all of them. However, $C_{(0)}$ is always a product of two terms with non-zero
boost weights equal in magnitude but opposite in signs as shown below

$C_{(0)}\sim\sum_{k}\widetilde{T}_{(-k)}~{}\partial_{v}^{(k)}T_{(0)}\,.$ (56)

It is obvious from this equation that $C_{(0)}$ has boost weight zero, but it
is product of two quantities $\widetilde{T}_{(-k)}$ and
$\partial_{v}^{(k)}T_{(0)}$ which have individual boost weights as $-k$ and
$+k$ respectively. However, the other term on the RHS of eq.(55) involves
$A_{(0)}$ and $B_{(0)}$ which could not be written like this. We name the
terms of the type $C_{(0)}$ as ‘JKM-type’ terms and the terms of the form
$A_{(0)}\partial_{v}^{2}B_{(0)}$ as the ‘zero-boost’ terms303030The fact that
the second term on the RHS of eq.(55) has been given the name ‘zero-boost’
terms should not be confused with the fact that this term
$A_{(0)}\,\partial_{v}^{2}B_{(0)}$ actually has a non-zero boost weight. To
see the origin of this nomenclature we refer the reader to section-(2.2.2) and
section-(2.2.3) in Bhattacharya:2019qal . Both the terms $A_{(0)}$ and
$B_{(0)}$ do not have any $v$ or $r$ derivatives in them which is not the case
for $C_{(0)}$.. Therefore, it follows from eq.(55) that any tensor component
with boost weight $+2$, such as $E_{vv}$, must have a set of such ‘zero-boost’
terms plus $\partial_{v}^{2}$ acting on ‘JKM type’ terms. The statements
mentioned above actually follows more generally from the important result that
determines the general structure of any covariant tensor quantity with
positive boost-weight, as given in eq.(66) named as “Result: $1$”.

Our goal in this paper is to show that the ‘zero-boost’ terms in $E_{vv}$
could also be written as313131This is very schematic, and there have to be
relevant factors of $\sqrt{h}$ to make everything consistent, but in this sub-
section, since we are aiming at a heuristic argument we shall be ignoring all
these subtleties.

$A_{(0)}\partial_{v}^{2}B_{(0)}~{}\sim~{}\partial_{v}^{2}{\cal
Q}_{(0)}+\partial_{v}\left(\nabla_{i}{\cal Q}_{(1)}^{i}\right)+{\cal
O}\left(\epsilon^{2}\right)\,,$ (57)

where ${\cal Q}_{(0)}$ is a quantity with vanishing boost weight and ${\cal
Q}_{(1)}^{i}$ is a quantity with boost weight $+1$ with a spatial index
$i=1,2,\cdots,D-2$. Also, $\nabla_{i}$ is the covariant derivative associated
with the induced metric $h_{ij}$ on the co-dimension two spatial slices of the
horizon. With this eq.(55) becomes

$E_{vv}\sim\partial_{v}^{2}\big{(}C_{(0)}+{\cal
Q}_{(0)}\big{)}+\partial_{v}\left(\nabla_{i}{\cal
Q}_{(1)}^{i}\right)+\mathcal{O}(\epsilon^{2})\,,$ (58)

In the following sub-section, we will carry out a detailed analysis to arrive
at eq.(123) which is actually a more precise version of eq.(58). It should
also be noted that the presence of the first term on the RHS of eq.(58) was
already explained in Wall:2015raa . The actual new result in this current
paper is to argue that in addition to the first term on the RHS, the second
term on the RHS can also, in general, be there in $E_{vv}$. This new term was
identified with an entropy current in Bhattacharya:2019qal .

In the following we will present a rough sketch for the arguments to argue
that eq.(58) is indeed true. For this we will use equation eq.(53) and analyze
both the terms on the RHS of this equation individually.

  * •

From the construction of Noether’s charge for diffeomorphism as pointed out in
Iyer:1994ys , we know that both $\Theta^{r}$ and $Q^{r\mu}$ are linear in
$\xi^{\mu}\partial_{\mu}=v\partial_{v}-r\partial_{r}$. The corresponding
expressions justifying this can be found in eq.(76) for $\Theta^{r}$ and
eq.(111) for $Q^{r\mu}$.

  * •

Since $\xi$ is the only source of explicit dependence on $v$, both
$\Theta^{r}$ and $Q^{r\mu}$, being linear in $\xi^{\mu}$, will have the
following structure

$\begin{split}\Theta^{r}|_{r=0}&=\Theta_{(1)}+v\,\Theta_{(2)}\,,~{}~{}~{}Q^{rv}|_{r=0}=Q_{(0)}+v\,Q_{(1)}\,,~{}~{}~{}Q^{ri}|_{r=0}=J^{i}_{(1)}+v\,\partial_{v}\tilde{J}^{i}_{(1)}\,.\end{split}$
(59)

Here also, the subscripts in the RHS denote the boost weights of the
corresponding quantities. In the last equation, we have also used the fact
that any spatial current with boost weight $+2$ could always be expressed as
$\partial_{v}$ acting on a current term with boost weight $+1$, up to
corrections non-linear in the amplitude of the dynamics. We have suppressed
some index structures to avoid cluttering in the presentation.

  * •

Using the fact that $\Theta^{\mu}$ could be expressed in terms of
$\mathcal{L}_{\xi}$ (and a single derivative of it)323232Here
$\mathcal{L}_{\xi}$ denotes Lie derivative along the vector $\xi^{\mu}$.
acting on covariant tensors (see the arguments leading to eq.(76) in §3.2.2),
one could give further structure to $\Theta^{r}$. Actually, as discussed in
Appendix-(D), one can show that

$\Theta_{(2)}=\partial_{v}\Theta_{(1)}+\partial^{2}_{v}{\cal M}_{(0)}\,,$ (60)

where ${\cal M}_{(0)}$ is a JKM-type term, i.e. a boost invariant term and
also a product of two factors each of which has opposite boost weights.

  * •

Now, substituting eq.(59) and eq.(60) in eq.(53) we find

$\begin{split}2v\,E_{vv}\big{|}_{r=0}=&\left[-\Theta_{(1)}+Q_{(1)}+\partial_{v}Q_{(0)}+\nabla_{i}J^{i}_{(1)}\right]\\\
&+v\,\partial_{v}\left[-\Theta_{(1)}-\partial_{v}{\cal
M}_{(0)}+Q_{(1)}+\nabla_{i}\tilde{J}^{i}_{(1)}\right]+\mathcal{O}(\epsilon^{2})\,.\end{split}$
(61)

Since the commutator of $\partial_{v}$ and $\nabla_{i}$ is of boost weight
$+1$, it has been neglected in the last line where it is acting on
$\tilde{J}^{i}_{(1)}$.

  * •

It is important to note that the $v$-dependence in eq.(61) is manifest
explicitly. In other words, it is an identity with no explicit factors of $v$
in $E_{vv}$, $\Theta$, $Q$ and $\tilde{J}^{i}$. Therefore, it follows that the
coefficients of different powers of $v$ must vanish independently

$\Theta_{(1)}-Q_{(1)}-\partial_{v}Q_{(0)}-\nabla_{i}J^{i}_{(1)}=0\\\ $ (62)
$\partial_{v}\left[-\Theta_{(1)}-\partial_{v}{\cal
M}_{(0)}+Q_{(1)}+\nabla_{i}\tilde{J}^{i}_{(1)}\right]=2E_{vv}\big{|}_{r=0}$
(63)

  * •

Finally, substituting eq.(62) in eq.(63) we get the desired result,

$2E_{vv}\big{|}_{r=0}=-\partial_{v}\left[\partial_{v}\left(Q_{(0)}+{\cal
M}_{(0)}\right)+\nabla_{i}\left(J^{i}_{(1)}-\tilde{J}^{i}_{(1)}\right)\right]+\mathcal{O}(\epsilon^{2})\,.$
(64)

By comparison, we find this is to be of the same form as on the RHS of
eq.(54). In the following sub-section, we will derive this relation with much
more technical rigour and arrive at eq.(123) with the expressions of ${\cal
J}^{v}$ and ${\cal J}^{i}$ in eq.(124).

###  3.2 An elaborate analysis of the proof with technical details

In this sub-section, our main goal is to methodically discuss in more detail
how eq.(53) can lead us to the construction of an entropy current in any
higher derivative theory of gravity. Simultaneously one also obtains the
components ${\cal J}^{v}$ and ${\cal J}^{i}$ as given in eq.(54) such that the
linearized second law is satisfied. For a better organization of the technical
computations to be presented in the following, we will separate them into
several sub-parts. In §3.2.1 we will start with obtaining an important result
regarding the generic structure of a general covariant tensor with any
positive boost weight. Using this result, in §3.2.2 and §3.2.3, the two terms
on the RHS of eq.(53) will be analyzed one at a time. Finally, we will collect
them all together in §3.2.4, to arrive at our final result for the structural
form of $E_{vv}$. In Fig:1, we provide a flow-diagram summarising the main
results from each of the following sub-sections.

Figure 1: A flow diagram summarising the main technical result in each sub-
section.

####  3.2.1 The generic structure of a covariant tensor with positive boost
weight

We shall first write down an important result involving the boost weight of a
covariant tensor and possible distributions of partial differential operator
$\partial_{v}$, evaluated on the horizon. The typical structure of any
covariant tensor component with positive boost weight $w=a+1>0$, restricted to
the horizon and up to linear order in the amplitude of the non-stationary
perturbation, will be of the form

$t^{(k)}_{(a+1)}\big{|}_{r=0}=\widetilde{T}_{(-k)}\,\partial_{v}^{(k+a+1)}T_{(0)}\big{|}_{r=0}+{\cal
O}(\epsilon^{2})\,.$ (65)

Here the subscripts denote the boost weight of the corresponding quantity and
for convenience we have suppressed the index structure for
$t^{(k)}_{(a+1)},\,\widetilde{T}_{(-k)}$ and $T_{(0)}$. In writing eq.(65) we
have followed the guidelines summarised in both Table-2 and Table-3. More
specifically, we made use of the fact that one $\partial_{v}$-derivative
operator carries a boost weight of $(+1)$. The quantities
$\widetilde{T}_{(-k)}$ and $T_{(0)}$, appearing on the RHS of eq.(65) are
arbitrary tensor quantities with boost weight ‘$-k$’ and zero respectively.
The superscript ‘$k$’ on $t^{(k)}_{(a+1)}$ signifies the fact that any
covariant tensor with boost weight $w=a+1>0$, can be represented by a member
in an one parameter family of terms denoted by $k=\,0,\,1,\,2,\cdots$.

Once a typical covariant tensor with positive boost weight has been written as
in eq.(65), one can rearrange the $\partial_{v}$-derivatives and recast it in
the following form

$\left.\begin{aligned}
\hfil\displaystyle\begin{split}t^{(k)}_{(a+1)}\big{|}_{r=0}=\,&\widetilde{T}_{(-k)}\,\partial_{v}^{(k+a+1)}T_{(0)}+{\cal
O}(\epsilon^{2})\\\
=\,&\partial_{v}^{(a+1)}\left[\sum_{m=0}^{k-1}(-1)^{m}\,\left[{{}^{m+a}C}_{m}\right]\,\widetilde{T}_{(-k+m)}~{}\partial_{v}^{(k-m)}T_{(0)}\right]~{}~{}~{}~{}\\\
\,&+(-1)^{k}\left[{{}^{k+a}C}_{a}\right]\widetilde{T}_{(0)}~{}\partial_{v}^{a+1}T_{(0)}+{\cal
O}(\epsilon^{2})\,,\end{split}\end{aligned}\right\\}\text{\, \, Result:\,$1$}$
(66)

where $\widetilde{T}_{(-k+m)}=\partial_{v}^{m}\widetilde{T}_{(-k)}$ and the
numerical coefficient $\left[{{}^{k+a}C}_{a}\right]={(k+a)!\over k!\,a!}$. The
detailed arguments justifying that such a rearrangement is always possible
allowing us to write $t^{(k)}_{(a+1)}$ as the RHS of eq.(66), is presented in
Appendix-(E).

This is a significant result which we will use quite extensively in the rest
of this section. For that reason, we have also identified eq.(66) as “Result:
$1$” and will refer to it accordingly whenever we use it.

Before we proceed, let us mention a notable feature of the final expression in
eq.(66). In the first term on the RHS, the quantity within the square bracket
is boost invariant, however, it is product of two terms which are individually
not boost invariant: one of them has positive boost weight and the other one
has equal boost weight but of negative sign,

$\text{The first term on the RHS of
eq.\eqref{sq01}}~{}\sim~{}\partial_{v}^{(a+1)}\left[\widetilde{T}_{(-k+m)}~{}\partial_{v}^{(k-m)}T_{(0)}\right]\,$
(67)

where $\widetilde{T}_{(-k+m)}$ and $\partial_{v}^{(k-m)}T_{(0)}$ have boost
weights equal to $(-k+m)$ and $(k-m)$ respectively. Following our set of
guidelines presented in Table:2, we can therefore conclude that boost
invariant terms of this kind are non-zero only due to dynamics and will vanish
when evaluated on stationary configurations.

####  3.2.2 Analyzing the structure of the first term (i.e. $\Theta^{r}$) in
$E_{vv}$

In this subsection, our focus will be on studying the first term’s structure
on the RHS of eq.(53), i.e., $\Theta^{r}$.

For any diffeomorphism invariant theory of gravity, the functional dependence
of the Lagrangian is given in eq.(2). Under an arbitrary change of the space-
time metric denoted by $g_{\mu\nu}\rightarrow g_{\mu\nu}+\delta g_{\mu\nu}$,
the variation in the Lagrangian, written in eq.(42), contains a total
derivative term, i.e. $D_{\mu}\theta^{\mu}$. Furthermore, following the Lemma
$3.1$ of Iyer:1994ys , (see also eq.(166) and eq.(168) in Appendix-(C)), the
general structure of $\theta^{\mu}$ can be obtained as the following

$\Theta^{\mu}[g_{\beta_{1}\beta_{2}},\delta
g_{\beta_{1}\beta_{2}},\delta\phi]=2E^{\mu\nu\alpha\beta}_{R}D_{\beta}\delta
g_{\nu\alpha}+\Theta^{\prime\mu}\,,$ (68)

where

$\begin{split}\Theta^{\prime\mu}=S^{\mu\alpha\beta}(g_{\beta_{1}\beta_{2}},\phi)\,\delta
g_{\alpha\beta}&+\sum_{i=0}^{m-1}T^{\mu\alpha\beta\rho\sigma\alpha_{1}\dots\alpha_{i}}_{i}(g_{\beta_{1}\beta_{2}},\phi)\,\delta
D_{(\alpha_{1}}\dots D_{\alpha_{i})}R_{\alpha\beta\rho\sigma}\\\
&+\sum_{i=0}^{l-1}U^{\mu\alpha_{1}\dots\alpha_{i}}_{i}(g_{\beta_{1}\beta_{2}},\phi)\,\delta
D_{(\alpha_{1}}\dots D_{\alpha_{i})}\phi\,,\end{split}$ (69)

where $E^{\mu\nu\alpha\beta}_{R}$ is given by

$E^{\mu\nu\alpha\beta}_{R}=\dfrac{\partial\,L}{\partial
R_{\mu\nu\alpha\beta}}-D_{\rho_{1}}\dfrac{\partial\,L}{\partial
D_{\rho_{1}}R_{\mu\nu\alpha\beta}}+\dots+(-1)^{m}D_{(\rho_{1}}\dots
D_{\rho_{m})}\dfrac{\partial\,L}{\partial D_{(\rho_{1}}\dots
D_{\rho_{m})}R_{\mu\nu\alpha\beta}}\,,$ (70)

which is a covariant tensor with the same symmetry properties of
$R_{\mu\nu\rho\sigma}$ and $\phi$ is the matter field that are also present in
the Lagrangian.

It should be noted that in the expression of $\theta^{\mu}$ on the RHS of
eq.(68), the variation that is denoted by $\delta$, can be put to the left of
the covariant derivatives everywhere in $\Theta^{\prime\mu}$ except for the
term $E^{\mu\nu\alpha\beta}_{R}D_{\beta}\delta g_{\nu\alpha}$. This
distinction between the two terms on the RHS of eq.(68) will be crucial in the
following.

Let us rewrite eq.(68) schematically as the following

$\Theta^{\mu}=2E^{\mu\nu\alpha\beta}_{R}D_{\beta}\delta
g_{\nu\alpha}+\sum_{k}{\cal
T}^{\mu\alpha_{1}\alpha_{2}\cdots\alpha_{k}}\,\delta{\cal
S}_{\alpha_{1}\alpha_{2}\cdots\alpha_{k}}\,,$ (71)

where on the RHS, the second term collectively includes all the contributions
that are in $\theta^{\prime\mu}$, eq.(69). Therefore, the quantity ${\cal
S}_{\alpha_{1}\alpha_{2}\cdots\alpha_{k}}$ is actually a representative member
of the family of covariant tensors as indicated below

$\begin{split}{\cal
S}_{\alpha_{1}\alpha_{2}\cdots\alpha_{k}}:\big{\\{}&\,g_{\alpha\beta},\,R_{\alpha\beta\rho\sigma},\,D_{\alpha_{1}}R_{\alpha\beta\rho\sigma},\cdots,D_{(\alpha_{1}}\dots
D_{\alpha_{m})}R_{\mu\nu\alpha\beta},\cdots,\\\
&\,\phi,\,D_{\alpha_{1}}\phi,\cdots,D_{(\alpha_{1}}\dots
D_{\alpha_{i})}\phi,\cdots\big{\\}}\,.\end{split}$ (72)

It should be noted that in our discussion so far, there has been neither any
reference to the diffeomorphism transformation nor have we used any invariance
of the theory under diffeomorphism. Hence, eq.(68) is true for arbitrary
variations $\delta g_{\alpha\beta}$.

Now, if $\delta g_{\alpha\beta}$ is generated due to a diffeomorphism given by
$x^{\mu}\rightarrow x^{\mu}+\zeta^{\mu}(x)$ where $\zeta^{\mu}$ is some vector
field, we can write

$\delta g_{\alpha\beta}={\cal
L}_{\zeta}g_{\alpha\beta}=D_{\alpha}\zeta_{\beta}+D_{\beta}\zeta_{\alpha}\,,$
(73)

where ${\cal L}_{\zeta}$ is the Lie derivative along $\zeta^{\mu}$. If we
substitute for $\delta g_{\alpha\beta}$ from the equation above in eq.(68),
the corresponding $\Theta^{\mu}$ for diffemorphism will be obtained, which we
denote by $\Theta^{\mu}[g,\,{\cal L}_{\zeta}g]$. Implementing this in eq.(71)
leads us to the following

$\Theta^{\mu}=2E^{\mu\nu\alpha\beta}_{R}D_{\beta}({\cal
L}_{\zeta}g_{\nu\alpha})+\sum_{k}{\cal
T}^{\mu\alpha_{1}\alpha_{2}\cdots\alpha_{k}}\,\delta{\cal
S}_{\alpha_{1}\alpha_{2}\cdots\alpha_{k}}[\delta
g_{\alpha\beta}\rightarrow{\cal L}_{\zeta}g_{\alpha\beta}]\,.$ (74)

In the second term on the RHS of eq.(74), the meaning of the substitution
$\delta g_{\alpha\beta}\rightarrow{\cal L}_{\zeta}g_{\alpha\beta}$ in the
argument of $\delta{\cal S}_{\alpha_{1}\alpha_{2}\cdots\alpha_{k}}$ is obvious
in the following sense: for each possible terms in $\delta{\cal
S}_{\alpha_{1}\alpha_{2}\cdots\alpha_{k}}$ listed in eq.(72), one should first
reduce it to a form involving $\delta g_{\alpha\beta}$ and arbitrary number of
derivatives on it, and then substitute for $\delta
g_{\alpha\beta}\rightarrow{\cal L}_{\zeta}g_{\alpha\beta}$.

Next, we adopt a specific choice for the vector field that generates
diffeomorphism as $\xi^{\mu}$ in place of $\zeta^{\mu}$. In our coordinate
system $\\{r,v,x^{i}\\}$ in eq.(11), $\xi^{\mu}$ is given by
$\xi^{\mu}\partial_{\mu}=v\partial_{v}-r\partial_{r}$. From our discussions
both in §2.2 and §2.3, we immediately identify this $\xi^{\mu}$ as the
generator of the boost transformation given in eq.(14) as well as the Killing
vector, eq.(26), for the most general stationary black holes in eq.(17).

Furthermore, with the input of diffeomorphism invariance of our theory, we can
show that

$\delta{\cal S}_{\alpha_{1}\alpha_{2}\cdots\alpha_{k}}[\delta
g_{\alpha\beta}\rightarrow{\cal L}_{\xi}g_{\alpha\beta}]={\cal L}_{\xi}{\cal
S}_{\alpha_{1}\alpha_{2}\cdots\alpha_{k}}\,,$ (75)

and thus obtain the following expression for $\theta^{\mu}$

$\Theta^{\mu}=2E^{\mu\nu\alpha\beta}_{R}D_{\beta}({\cal
L}_{\xi}g_{\nu\alpha})+\sum_{k}{\cal
T}^{\mu\alpha_{1}\alpha_{2}\cdots\alpha_{k}}\,{\cal L}_{\xi}{\cal
S}_{\alpha_{1}\alpha_{2}\cdots\alpha_{k}}\,.$ (76)

We refer the readers to Appendix-(F) for a detailed clarification regarding
eq.(75).

In what follows, we will consider each of the two terms on the RHS of eq.(76)
one at a time and try to obtain the general structure of them. As explained in
§3.1, we would like to know the explicit $v$-dependence of each of the two
terms that determines $\Theta^{\mu}$ in eq.(76). In both of these quantities,
the source of $v$-dependence is solely from the factor of $\xi^{\mu}$. We know
that $\xi$ dependence in $\Theta$ comes from ${\cal L}_{\xi}$, which is linear
in $\xi$ and therefore linear in $v$. From here, we learn that, when evaluated
on the horizon, the $v$-dependence in $\Theta^{\mu}$ can at most be linear in
$v$, and schematically we can write

$\Theta^{\mu}\big{|}_{r=0}=\Theta_{1}^{\mu}+v\,\Theta_{2}^{\mu}\,,$ (77)

where $\Theta_{1}^{\mu}$ and $\Theta_{2}^{\mu}$ are, as of now, components of
covariant tensors without any explicit $v$ dependence. Focussing on
$\Theta^{r}$, which is our primary object, and following our rules of boost
weight counting, we learn that the corresponding weights of $\Theta_{(1)}^{r}$
and $\Theta_{(2)}^{r}$ are $(+1)$ and $(+2)$ respectively, as also denoted by
their subscripts. With this knowledge of boost weights for $\Theta_{(1)}^{r}$
and $\Theta_{(2)}^{r}$, we can use Result- $1$ from eq.(66) to predict the
following schematic structures for each of them (ignoring the explicit
constant coefficients for now)

$\begin{split}\Theta_{(1)}^{r}\big{|}_{r=0}\sim~{}&\partial_{v}{\cal
X}+\widetilde{\cal X}_{(0)}\partial_{v}{\cal X}_{(0)}\\\
\Theta_{(2)}^{r}\big{|}_{r=0}\sim~{}&\partial_{v}^{2}{\cal Y}+\widetilde{\cal
Y}_{(0)}~{}\partial_{v}^{2}{\cal Y}_{(0)}\,,\end{split}$ (78)

where ${\cal X},\,{\cal X}_{(0)},\,\widetilde{\cal X}_{(0)},\,{\cal Y},\,{\cal
Y}_{(0)},\,\widetilde{\cal Y}_{(0)}$ are all boost invariant quantities.
Recall that according to Result- $1$ (in eq.(66)), although ${\cal Y}$ and
${\cal X}$ are both boost invariant terms, each of them could have pieces that
are always product of two terms, one with strictly positive and the other with
with strictly negative boost weight and therefore vanish on stationary
horizons (see the discussions around eq.(67)).

We now readily obtain the following form for $\Theta^{r}$

$\begin{split}\Theta^{r}\big{|}_{r=0}\sim\,\partial_{v}{\cal
X}+\widetilde{\cal X}_{(0)}\,\partial_{v}{\cal
X}_{(0)}+v\,\left[\partial_{v}^{2}{\cal Y}+\widetilde{\cal
Y}_{(0)}\,\partial_{v}^{2}{\cal Y}_{(0)}\right]\,.\end{split}$ (79)

Following our discussion so far, it might appear that $\Theta_{(1)}^{r}$ and
$\Theta_{(2)}^{r}$ are two independent quantities and therefore their
structures as written in eq.(78) are not related to each other. However, this
is not true and due to the fact that both of them are actually generated by an
action of the operator ${\cal L}_{\xi}$ in eq.(76), we can constrain the
structure of $\Theta^{r}$ further. In particular we will explicitly show that
one can relate the quantities appearing in eq.(78) up to non-linear order
corrections in the amplitude of the fluctuations,

$\widetilde{\cal Y}_{(0)}\,\partial_{v}^{2}{\cal Y}_{(0)}+{\cal
O}(\epsilon^{2})=\widetilde{\cal X}_{(0)}\,\partial_{v}^{2}{\cal
X}_{(0)}+{\cal O}(\epsilon^{2})=\partial_{v}\left(\widetilde{\cal
X}_{(0)}\,\partial_{v}{\cal X}_{(0)}\right)+{\cal O}(\epsilon^{2})\,.$ (80)

Substituting this relation in equation eq.(79) we find the final schematic
structure for $\Theta^{r}$

$\begin{split}\Theta^{r}\big{|}_{r=0}=&\,(1+v~{}\partial_{v}){\cal
A}_{(1)}+v~{}\partial_{v}^{2}{\cal B}_{(0)}+{\cal O}(\epsilon^{2})\\\
\text{such that,}~{}~{}{\cal A}_{(1)}=&\,\tilde{\cal
X}_{(0)}~{}\partial_{v}{\cal X}_{(0)}+\partial_{v}{\cal
X}\,,~{}~{}\text{and}~{}~{}{\cal B}_{(0)}=\,{\cal Y}-{\cal X}\,.\end{split}$
(81)

Therefore, in the following, our job is to justify the critical relation
eq.(80), for both the terms in the expression of $\Theta^{r}$ as given in
eq.(76).

#### Analysis of the second term in $\Theta^{r}$ from eq.(76):

For reasons that will be clear later, to start with, we focus first on the
second term on the RHS of(76). As we are considering the $r$-component of
$\Theta^{\mu}$, it follows from our rule of assigning boost weights that
$\Theta^{r}$ must have a weight of $(+1)$ and consequently both the terms on
the RHS of eq.(76) will have boost weights equal to $(+1)$. With this
knowledge let us now write down the schematic form of the second term in
$\Theta^{r}$ for our convenience

$\sum_{j}{\cal T}^{r\alpha_{1}\alpha_{2}\cdots\alpha_{j}}\,{\cal L}_{\xi}{\cal
S}_{\alpha_{1}\alpha_{2}\cdots\alpha_{j}}=~{}\sum_{k\geq
0}\widetilde{T}_{(-k)}{\cal L}_{\xi}T_{(k+1)}+\sum_{k\geq 0}S_{(k+1)}{\cal
L}_{\xi}\widetilde{S}_{(-k)}\,,$ (82)

where we have suppressed the details of the index structures of the covariant
tensor components $T$, $\widetilde{T}$, $S$ and $\widetilde{S}$, and the
subscripts denote the boost weights for each of them. One can see that the two
terms appearing on the RHS of eq.(82), are two different cases where the Lie
derivative ${\cal L}_{\xi}$ is acting on quantities with positive and negative
boost weight respectively. It should be noted that here we are just doing an
operation which is in principle possible without stating anything specific
about the structures of the quantities on the RHS above. Thus all the
quantities $T$, $\widetilde{T}$, $S$ and $\widetilde{S}$ on the RHS of the
equation above are covariant tensors which are otherwise arbitrary apart from
having the specific boost weights assigned to them. We can then perform
further manipulations as shown below to bring it to a form as given in the
following

$\begin{split}\sum_{j}&{\cal T}^{r\alpha_{1}\alpha_{2}\cdots\alpha_{j}}\,{\cal
L}_{\xi}{\cal S}_{\alpha_{1}\alpha_{2}\cdots\alpha_{j}}\\\ =~{}&\sum_{k\geq
0}\widetilde{T}_{(-k)}{\cal L}_{\xi}T_{(k+1)}+\sum_{k\geq 0}S_{(k+1)}{\cal
L}_{\xi}\widetilde{S}_{(-k)}\\\ =~{}&\sum_{k\geq 0}\widetilde{T}_{(-k)}{\cal
L}_{\xi}T_{(k+1)}+{\cal L}_{\xi}\bigg{(}\sum_{k\geq
0}\left[S_{(k+1)}\widetilde{S}_{(-k)}\right]\bigg{)}-\sum_{k\geq
0}\widetilde{S}_{(-k)}{\cal L}_{\xi}S_{(k+1)}\\\
=~{}&\left(1+v\partial_{v}\right)\bigg{(}\sum_{k\geq
0}\left[S_{(k+1)}\widetilde{S}_{(-k)}\right]\bigg{)}+\sum_{k\geq
0}\widetilde{T}_{(-k)}{\cal L}_{\xi}T_{(k+1)}-\sum_{k\geq
0}\widetilde{S}_{(-k)}{\cal L}_{\xi}S_{(k+1)}\\\
=~{}&\left(1+v\partial_{v}\right){\mathbb{S}}_{(1)}+\sum_{k\geq
0}\widetilde{\mathbb{T}}_{(-k)}~{}{\cal L}_{\xi}{\mathbb{T}}_{(k+1)}\\\
=~{}&\left(1+v\partial_{v}\right){\mathbb{S}}_{(1)}+\sum_{k\geq
0}\widetilde{\mathbb{T}}_{(-k)}~{}\left(k+1+v\partial_{v}\right){\mathbb{T}}_{(k+1)}\,.\end{split}$
(83)

In the second last line we have defined new quantities for brevity

${\mathbb{S}}_{(1)}\equiv\sum_{k\geq
0}\left(S_{(k+1)}\widetilde{S}_{(-k)}\right),~{}~{}\text{and}~{}~{}\widetilde{\mathbb{T}}_{(-k)}{\cal
L}_{\xi}{\mathbb{T}}_{(k+1)}\equiv\widetilde{T}_{(-k)}{\cal
L}_{\xi}T_{(k+1)}-\widetilde{S}_{(-k)}{\cal L}_{\xi}S_{(k+1)}\,.$ (84)

Comparing with eq.(81), it is clear from the final expression on the RHS of
eq.(83) that the first term there is already of the desired form.

In the following we will process the second term in eq.(83). Schematically,
this has two different structures in it, namely
$\widetilde{\mathbb{T}}_{(-k)}\,{\mathbb{T}}_{(k+1)}$ and
$\widetilde{\mathbb{T}}_{(-k)}\,\partial_{v}{\mathbb{T}}_{(k+1)}$. Our
strategy would be to use Result: $1$ from eq.(66) repeatedly for each of them.
Firstly, we see that, according to Result: $1$, ${\mathbb{T}}_{(k+1)}$ will
have the following schematic structure (ignoring the coefficients)

${\mathbb{T}}_{(k+1)}=\partial_{v}^{k+1}C_{(0)}+A_{(0)}~{}\partial_{v}^{(k+1)}B_{(0)}\,.$
(85)

Using this we define the quantities $t^{(k)}_{(1)}$ and $t^{(k)}_{(2)}$ as
given by

$\begin{split}t^{(k)}_{(1)}\,\equiv\,&\widetilde{\mathbb{T}}_{(-k)}{\mathbb{T}}_{(k+1)}=~{}\widetilde{\mathbb{T}}_{(-k)}\,\partial_{v}^{k+1}C_{(0)}+A_{(0)}\,\widetilde{\mathbb{T}}_{(-k)}\,\partial_{v}^{(k+1)}B_{(0)}\,,\\\
t^{(k)}_{(2)}\,\equiv\,&\widetilde{\mathbb{T}}_{(-k)}\,\partial_{v}{\mathbb{T}}_{(k+1)}=~{}\widetilde{\mathbb{T}}_{(-k)}\,\partial_{v}^{k+2}C_{(0)}+A_{(0)}\,\widetilde{\mathbb{T}}_{(-k)}\,\partial_{v}^{(k+2)}B_{(0)}\,,\end{split}$
(86)

which then leads us to the following form of the second term in eq.(83)

$\begin{split}\sum_{k\geq
0}\widetilde{\mathbb{T}}_{(-k)}\,\mathcal{L}_{\xi}{\mathbb{T}}_{(k+1)}=\sum_{k\geq
0}\widetilde{\mathbb{T}}_{(-k)}\left(k+1+v\partial_{v}\right){\mathbb{T}}_{(k+1)}=\sum_{k\geq
0}\left((k+1)~{}t^{(k)}_{(1)}+v~{}t^{(k)}_{(2)}\right)\,.\end{split}$ (87)

Furthermore, from eq.(86) we also learn that, in this case, both the terms
that are present in $t^{(k)}_{(1)}$ has the general structure as

$t^{(k)}_{(1)}\sim X_{(-k)}\,\partial_{v}^{k+1}Y_{(0)}\,,$ (88)

with the following possibilities

$\begin{split}\text{either}~{}~{}X_{(-k)}\sim\,\widetilde{\mathbb{T}}_{(-k)}\,,&~{}~{}Y_{(0)}\sim\,C_{(0)}\,;~{}~{}\text{or}~{}~{}X_{(-k)}\sim\,A_{(0)}\,\widetilde{\mathbb{T}}_{(-k)},~{}~{}Y_{(0)}\sim\,B_{(0)}\,.\end{split}$
(89)

However, for both the cases mentioned above $t^{(k)}_{(2)}$ must also have the
form given by

$t^{(k)}_{(2)}\sim X_{(-k)}\,\partial_{v}^{k+2}Y_{(0)}\,.$ (90)

We should remember that eq.(80) is the key equation that we need to show to
recast $\Theta^{r}$ into the desired structure given by eq.(81). In this case,
for the choices of $t^{(k)}_{(1)}$ and $t^{(k)}_{(2)}$ given by eq.(88) and
eq.(90) respectively in terms of $X_{(-k)}$ and $Y_{(0)}$, eq.(80) results in
the following statement:
If we apply Result: $1$ to $t^{(k)}_{(1)}$ in eq.(88), it will generate a term
of the form $X_{(0)}\,\partial_{v}Y_{(0)}$ and similarly $t^{(k)}_{(2)}$ in
eq.(90), will generate a term of the form $X_{(0)}\,\partial_{v}^{2}Y_{(0)}$.
The constant coefficient multiplying $X_{(0)}\partial_{v}Y_{(0)}$ in
$t^{(k)}_{(1)}$ is $(k+1)$ times the constant coefficient multiplying
$X_{(0)}\partial_{v}^{2}Y_{(0)}$ in the expansion of $t^{(k)}_{(2)}$, see
eq.(87) .
The italicized statement written above follows trivially, as we show
explicitly below, once we apply Result: $1$ to $t^{(k)}_{(1)}$ and
$t^{(k)}_{(2)}$ and keep track of the coefficients of these two particular
terms. This also recasts the term $\widetilde{\mathbb{T}}_{(-k)}\,{\cal
L}_{\xi}{\mathbb{T}}_{(k+1)}$ in the following form,

$\begin{split}\widetilde{\mathbb{T}}_{(-k)}\,&{\cal
L}_{\xi}{\mathbb{T}}_{(k+1)}\\\
=~{}&(k+1)\left(X_{(-k)}\partial_{v}^{k+1}Y_{(0)}\right)+v~{}\left(X_{(-k)}\partial_{v}^{k+2}Y_{(0)}\right)\\\
=~{}&(k+1)\bigg{[}\partial_{v}\left(\sum_{m=0}^{k-1}(-1)^{m}X_{(-k+m)}\partial_{v}^{k-m}Y_{(0)}\right)+(-1)^{k}X_{(0)}\partial_{v}Y_{(0)}\bigg{]}\\\
&+v~{}\partial_{v}^{2}\left(\sum_{m=0}^{k-1}(-1)^{m}(m+1)X_{(-k+m)}\partial_{v}^{k-m}Y_{(0)}\right)\\\
&+v~{}(-1)^{k}(k+1)\left(X_{(0)}\partial_{v}^{2}Y_{(0)}\right)\\\
=~{}&(k+1)\bigg{[}\partial_{v}\left(\sum_{m=0}^{k-1}(-1)^{m}X_{(-k+m)}\partial_{v}^{k-m}Y_{(0)}\right)+(-1)^{k}X_{(0)}\partial_{v}Y_{(0)}\bigg{]}\\\
&+v~{}\partial_{v}^{2}\left(\sum_{m=0}^{k-1}(-1)^{m}\left[(k+1)-(k-m)\right]X_{(-k+m)}\partial_{v}^{k-m}Y_{(0)}\right)\\\
&+v~{}(-1)^{k}(k+1)\left(X_{(0)}\partial_{v}^{2}Y_{(0)}\right)\\\
=~{}&(1+v\partial_{v})\bigg{[}(k+1)\left(\sum_{m=0}^{k-1}(-1)^{m}\partial_{v}\left[X_{(-k+m)}\partial_{v}^{k-m}Y_{(0)}\right]+(-1)^{k}X_{(0)}\partial_{v}Y_{(0)}\right)\bigg{]}\\\
&-v~{}\partial_{v}^{2}\left(\sum_{m=0}^{k-1}(-1)^{m}(k-m)X_{(-k+m)}\partial_{v}^{k-m}Y_{(0)}\right)+{\cal
O}\left(\epsilon^{2}\right)\,.\end{split}$ (91)

In the last line we have simply rearranged the terms and used the fact that

$X_{(0)}\,\partial_{v}^{2}Y_{(0)}=\partial_{v}\left(X_{(0)}\,\partial_{v}Y_{(0)}\right)+{\cal
O}\left(\epsilon^{2}\right)\,.$

Next, we substitute eq.(91) in eq.(83) and obtain the following form for the
second term in $\Theta^{r}$ eq.(76),

$\begin{split}\sum_{j}{\cal T}^{r\alpha_{1}\alpha_{2}\cdots\alpha_{j}}\,{\cal
L}_{\xi}{\cal
S}_{\alpha_{1}\alpha_{2}\cdots\alpha_{j}}=~{}&\left(1+v\partial_{v}\right){\mathbb{S}}_{(1)}+\sum_{k\geq
0}\widetilde{\mathbb{T}}_{(-k)}~{}{\cal L}_{\xi}{\mathbb{T}}_{(k+1)}+{\cal
O}\left(\epsilon^{2}\right)\\\ =~{}&(1+v~{}\partial_{v}){\cal
M}_{(1)}+v~{}\partial_{v}^{2}{\cal N}_{(0)}+{\cal
O}\left(\epsilon^{2}\right)\,,\end{split}$ (92)

such that

$\begin{split}{\cal M}_{(1)}=&\,{\cal S}_{(1)}+\sum_{k\geq
0}(k+1)\left(\sum_{m=0}^{k-1}(-1)^{m}\partial_{v}\left[X_{(-k+m)}\partial_{v}^{k-m}Y_{(0)}\right]+(-1)^{k}X_{(0)}\partial_{v}Y_{(0)}\right)\,,\\\
\text{and}~{}~{}{\cal N}_{(0)}=&\sum_{k\geq
0}\sum_{m=0}^{k-1}(-1)^{m+1}\,(k-m)\,X_{(-k+m)}\,\partial_{v}^{k-m}Y_{(0)}\,.\end{split}$
(93)

It should be noted that the form in eq.(92) is consistent with eq.(81).

Finally, we conclude this sub-section with the following comments regarding
the structure of ${\cal N}_{(0)}$ that we have obtained in eq.(93). It is
evident that ${\cal N}_{(0)}$ is a boost invariant quantity of the type
mentioned in eq.(67). From its expression, we note that it is a product of two
terms that are not individually boost invariant, i.e.

${\cal N}_{(0)}\,\sim\,X_{(-k+m)}\,\partial_{v}^{k-m}Y_{(0)}\,,$ (94)

where $X_{(-k+m)}$ has boost weight of $(-k+m)$ but
$\partial_{v}^{k-m}Y_{(0)}$ has a boost weight of $(k-m)$. Following our
guidelines listed in Table-2 (see Rule: $2$) we can conclude that ${\cal
N}_{(0)}$ is evaluated to be linear in the amplitude of the dynamical
fluctuations, i.e. ${\cal N}_{(0)}\,\sim\,\mathcal{O}(\epsilon)$. Furthermore,
using Rule: $1$ from the same Table-2 we also learn that ${\cal N}_{(0)}$ will
vanish when the stationarity limit is taken by turning the dynamics off. From
these properties it is evident that ${\cal N}_{(0)}$ is a possible candidate
to generate the JKM ambiguities associated with the Wald entropy, which will
be used in our subsequent analysis.

#### Analysis of the first term in $\Theta^{r}$ from eq.(76):

Next, we consider, for the $r$ component of $\Theta^{\mu}$, the first term on
the RHS of eq.(76)

$\begin{split}&2\,E^{r\nu\rho\sigma}_{R}\,D_{\sigma}\left[{\cal
L}_{\xi}g_{\nu\rho}\right]=2\,D_{\sigma}\left(E^{r\nu\rho\sigma}_{R}\,{\cal
L}_{\xi}g_{\nu\rho}\right)-2\,D_{\sigma}E^{r\nu\rho\sigma}_{R}\,{\cal
L}_{\xi}g_{\nu\rho}\,.\end{split}$ (95)

Now the second term on the RHS of the above equation has the same structure as
that of the second term in eq.(76). Therefore one can do a similar analysis
for that term following the steps mentioned in §3.2.

###### For the first term in eq.(95):

In the following, we will only process the first term in eq.(95).

Let us define the following quantity

$H^{\mu\sigma}\equiv E^{\mu\nu\rho\sigma}_{R}\,{\cal L}_{\xi}g_{\nu\rho}\,,$
(96)

and using the symmetries of $E^{\mu\nu\rho\sigma}_{R}$ under exchanging its
indices, we note that $H^{\mu\sigma}$ is a symmetric tensor,

$\begin{split}H^{\sigma\mu}=E^{\sigma\nu\rho\mu}_{R}\,{\cal
L}_{\xi}g_{\nu\rho}=E^{\sigma\rho\nu\mu}_{R}\,{\cal
L}_{\xi}g_{\rho\nu}=E^{\nu\mu\sigma\rho}_{R}\,{\cal
L}_{\xi}g_{\nu\rho}=E^{\mu\nu\rho\sigma}_{R}\,{\cal
L}_{\xi}g_{\nu\rho}=H^{\mu\sigma}\,.\end{split}$ (97)

Next, we calculate $D_{\sigma}H^{r\sigma}$ and evaluate that at the horizon to
obtain the following expression333333We have used the following formula to
compute the divergence of a symmetric tensor
$D_{\sigma}H^{\sigma}_{\mu}={1\over\sqrt{g}}\partial_{\sigma}\left[\sqrt{g}H^{\sigma}_{\mu}\right]-{1\over
2}H^{\sigma\nu}\partial_{\mu}g_{\sigma\nu}$

$\begin{split}D_{\sigma}&H^{r\sigma}\big{|}_{r=0}=g^{r\mu}D_{\sigma}H^{\sigma}_{\mu}\big{|}_{r=0}=D_{\sigma}H^{\sigma}_{v}\big{|}_{r=0}\\\
&=\left[{1\over\sqrt{g}}\partial_{\sigma}\left[\sqrt{g}H^{\sigma}_{v}\right]-{1\over
2}H^{\mu\rho}\partial_{v}g_{\mu\rho}\right]_{r=0}\\\
&=\left[{1\over\sqrt{g}}\partial_{r}\left[\sqrt{g}H^{r}_{v}\right]+{1\over\sqrt{g}}\partial_{v}\left[\sqrt{g}H^{v}_{v}\right]+{1\over\sqrt{g}}\partial_{i}\left[\sqrt{g}H^{i}_{v}\right]-{1\over
2}H^{ij}\partial_{v}g_{ij}\right]_{r=0}\\\
&=\left[{1\over\sqrt{g}}\partial_{r}\left[\sqrt{g}H^{r\mu}g_{\mu
v}\right]+{1\over\sqrt{g}}\partial_{v}\left[\sqrt{g}H^{v\rho}g_{\rho
v}\right]+{1\over\sqrt{g}}\partial_{i}\left[\sqrt{g}H^{i\rho}g_{\rho
v}\right]-{1\over 2}H^{ij}\partial_{v}g_{ij}\right]_{r=0}\\\
&=\left[{1\over\sqrt{g}}\partial_{r}\left[\sqrt{g}H^{rr}\right]+\omega_{i}H^{ri}+{1\over\sqrt{g}}\partial_{v}\left[\sqrt{g}H^{vr}\right]+{1\over\sqrt{g}}\partial_{i}\left[\sqrt{g}H^{ir}\right]-{1\over
2}H^{ij}\partial_{v}g_{ij}\right]_{r=0}\,.\end{split}$ (98)

To proceed from here we need to compute different components of
$H^{\mu\sigma}$ for our metric gauge and evaluate that at the horizon. First
we define $\Delta_{\mu\nu}\equiv{\cal L}_{\xi}g_{\mu\nu}$, and compute its
non-zero components as given below

$\begin{split}\Delta_{vv}=r^{2}(v\partial_{v}-r\partial_{r})X,~{}~{}\Delta_{vi}=r(v\partial_{v}-r\partial_{r})\omega_{i},~{}~{}\Delta_{ij}=(v\partial_{v}-r\partial_{r})h_{ij}\,.\end{split}$
(99)

This enables us to obtain the following expressions for different components
of $H^{\mu\sigma}$

$\begin{split}H^{rr}&=E^{rvvr}_{R}\,\Delta_{vv}+E^{rvir}_{R}\,\Delta_{vi}+E^{rijr}_{R}\,\Delta_{ij}=r\,v\,E^{rvir}_{R}\,\partial_{v}\omega_{i}+E^{rijr}_{R}(v\partial_{v}-r\partial_{r})h_{ij}+{\cal
O}\left(r^{2}\right)\\\
H^{rv}&=H^{vr}=E^{rvjv}_{R}\,\Delta_{vj}+E^{rjkv}_{R}\,\Delta_{jk}=v\,E^{rjkv}_{R}\,\partial_{v}h_{jk}+{\cal
O}(r)\\\
H^{ri}&=H^{ir}=E^{rvvi}_{R}\Delta_{vv}+2E^{rvji}_{R}\,\Delta_{vj}+E^{rjki}_{R}\Delta_{jk}=v\,E^{rjki}_{R}\,\partial_{v}h_{jk}+{\cal
O}(r)\\\
H^{ij}&=E^{ivvj}_{R}\,\Delta_{vv}+2E^{ivkj}_{R}\,\Delta_{vk}+E^{ikk^{\prime}j}_{R}\,\Delta_{kk^{\prime}}=v\,E^{ikk^{\prime}j}_{R}\,\partial_{v}h_{kk^{\prime}}+{\cal
O}\left(r\right)\end{split}$ (100)

Remembering the rule of assigning boost weights, we note that
$H^{ri}|_{r=0}={\cal O}(\epsilon^{2})$ and $H^{ij}|_{r=0}={\cal
O}\left(\epsilon\right)$, and, consequently we conclude
$H^{ij}\partial_{v}g_{ij}=H^{ij}\partial_{v}h_{ij}={\cal O}(\epsilon^{2})$.
Implementing all of these results in eq.(98) we are left with

$D_{\sigma}H^{r\sigma}=\left[{1\over\sqrt{g}}\partial_{r}\left[\sqrt{g}H^{rr}\right]+{1\over\sqrt{g}}\partial_{v}\left[\sqrt{g}H^{vr}\right]\right]_{r=0}+{\cal
O}\left(\epsilon^{2}\right)\,,$ (101)

with $H^{rr}$, $H^{vr}$ and $H^{ir}$ being given in eq.(100). Next we evaluate
the terms on the RHS of eq.(101). The first term can be manipulated using
eq.(100) as

$\begin{split}{1\over\sqrt{g}}\partial_{r}\left[\sqrt{g}H^{rr}\right]\big{|}_{r=0}&=E^{rijr}_{R}\,(v\partial_{r}\partial_{v}-\partial_{r})h_{ij}+{\cal
O}\left(\epsilon^{2}\right)=E^{rijr}_{R}\,(-1+v\partial_{v})(\partial_{r}h_{ij})+{\cal
O}\left(\epsilon^{2}\right)\\\ &=\left[E^{rijr}_{R}\,{\cal
L}_{\xi}(\partial_{r}h_{ij})\right]_{r=0}+{\cal O}\left(\epsilon^{2}\right)\\\
\end{split}$ (102)

It should be noted that $\partial_{r}h_{ij}$ cannot be identified as a
component of any covariant tensor constructed out of metric. However, it has a
definite boost weight $(=-1)$ and therefore, the action of ${\cal L}_{\xi}$,
once it is expressed in terms of boost weight and $(\xi\cdot\partial)$ as in
eq.(30), is well-defined. We have used this in arriving at the last equality
above in eq.(102).

We can clearly see from the RHS of eq.(102) that it has the same schematic
structure as the second term in equation eq.(76), analyzed in previous
subsection §3.2. Though $(\partial_{r}h_{ij})$ is not a covariant tensor
component, the analysis of the previous subsection will go through here since
it uses only the boost weight defined through the eq.(30). Therefore following
the same analysis as in §3.2 we get

$\begin{split}{1\over\sqrt{g}}\partial_{r}\left(\sqrt{g}H^{rr}\right)\big{|}_{r=0}=~{}&E^{rijr}_{R}\,{\cal
L}_{\xi}(\partial_{r}h_{ij})\big{|}_{r=0}\\\ =~{}&(1+v\,\partial_{v}){\cal
C}_{(1)}+v\,\partial_{v}^{2}{\cal D}_{(0)}+{\cal
O}\left(\epsilon^{2}\right)\,,\end{split}$ (103)

where ${\cal C}_{(1)}$ is a tensor quantity with boost weight $(+1)$ and
${\cal D}_{(0)}$ is a boost invariant quantity.

For the second term in eq.(101), using the corresponding relation from
eq.(100), we obtain the following form

$\begin{split}{1\over\sqrt{g}}\partial_{v}\left(\sqrt{g}\,H^{rv}\right)&={1\over\sqrt{g}}\partial_{v}\left[\sqrt{g}\,v\,E^{rjkv}_{R}\,\partial_{v}h_{jk}\right]=E^{rjkv}_{R}\,\partial_{v}h_{jk}+v\left({1\over\sqrt{h}}\partial_{v}\left[\sqrt{h}\,E^{rjkv}_{R}\,\partial_{v}h_{jk}\right]\right)\\\
&=(1+v\,\partial_{v}){\cal E}_{(1)}+{\cal O}(\epsilon^{2})\,,\end{split}$
(104)

where

$\begin{split}{\cal E}_{(1)}=&E^{rjkv}_{R}\,\partial_{v}h_{jk}\,.\end{split}$
(105)

###### For the second term in eq.(95):

As we have mentioned before the second term on the RHS of eq.(95) is already
in the form of the second term in eq.(76) which has been analyzed in detail in
the previous sub-section §3.2 . Therefore following a similar method we can
obtain the following form for this term as shown below

$\begin{split}-2\,D_{\sigma}E^{r\nu\rho\sigma}_{R}\,{\cal
L}_{\xi}g_{\nu\rho}\,\big{|}_{r=0}=~{}(1+v\,\partial_{v}){\cal
F}_{(1)}+v~{}\partial_{v}^{2}{\cal G}_{(0)}+{\cal
O}\left(\epsilon^{2}\right)\,,\end{split}$ (106)

where again, ${\cal F}_{(1)}$ is a tensor quantity with boost weight $(+1)$
and ${\cal G}_{(0)}$ is a boost invariant quantity.

Finally, collecting all the results derived in parts so far from eq.(103),
eq.(104) and eq.(106), we obtain the expression for the first term in eq.(76)
as

$\begin{split}2\,E^{r\nu\rho\sigma}_{R}\,D_{\sigma}\left[{\cal
L}_{\xi}g_{\nu\rho}\right]\,\big{|}_{r=0}=~{}(1+v\,\partial_{v}){\cal
P}_{(1)}+v~{}\partial_{v}^{2}{\cal Q}_{(0)}+{\cal
O}\left(\epsilon^{2}\right)\,,\\\ \text{such that}~{}~{}{\cal P}_{(1)}=\,{\cal
C}_{(1)}+{\cal E}_{(1)}+{\cal F}_{(1)}\,,~{}~{}\text{and}~{}~{}{\cal
Q}_{(0)}=\,{\cal D}_{(0)}+{\cal G}_{(0)}\,.\end{split}$ (107)

which is, again, consistent with the expected form of eq.(81).

With all these results at our disposal, we are in a position to collect them
together in eq.(76) for $\Theta^{r}$ as predicted in eq.(81), to write down

$\begin{split}\Theta^{r}\,\big{|}_{r=0}=~{}(1+v\,\partial_{v}){\cal
A}_{(1)}+v~{}\partial_{v}^{2}{\cal B}_{(0)}+{\cal
O}\left(\epsilon^{2}\right)\,,\\\ \text{such that}~{}~{}{\cal A}_{(1)}=\,{\cal
M}_{(1)}+{\cal P}_{(1)}\,,~{}~{}\text{and}~{}~{}{\cal B}_{(0)}=\,{\cal
N}_{(0)}+{\cal Q}_{(0)}\,.\end{split}$ (108)

Before we conclude, let us make the following comment regarding ${\cal
B}_{(0)}$. The term ${\cal Q}_{(0)}$ in eq.(107) has the same origin as the
term ${\cal N}_{(0)}$ in eq.(93). To be more precise, both of them are
produced from the first term on the RHS of eq.(66), i.e. terms of the kind as
given in eq.(67). Therefore following the discussion around eq.(94), we can
argue that ${\cal B}_{(0)}$ is a boost invariant term, although it is product
of two terms who are not individually boost invariant. Such terms will vanish
in stationary configurations and are non-zero only because of dynamical
fluctuations. Using the summary listed in Table-2 we can also argue that

${\cal B}_{(0)}~{}\sim~{}\mathcal{O}(\epsilon)$ (109)

As we will see later ${\cal B}_{(0)}$ will, therefore, contribute to the JKM
ambiguities of the Wald entropy.

####  3.2.3 Analyzing the structure of the second term (i.e.
$D_{\mu}Q^{r\mu}$) in $E_{vv}$

Next we turn our focus to the second term on the RHS of eq.(53) in the
expression of $E_{vv}$. This term involves the quantity $Q^{\mu\nu}$, which
was defined in eq.(49). This $Q^{\mu\nu}$ is the Noether charge corresponding
to diffeomorphism invariance. The general form for this $Q^{\mu\nu}$ in a
diffeomorphism invariant theory of gravity was constructed in Iyer:1994ys .
Following the Proposition $4.1$ in Iyer:1994ys  (also see eq.(187) and
eq.(188) in Appendix-(C)), we can obtain this as given below

$Q^{\mu\nu}=W^{\mu\nu\rho}\,\xi_{\rho}-2E^{\mu\nu\alpha\beta}_{R}\,D_{[\alpha}\xi_{\beta]}+Y^{\mu\nu}+D_{\alpha}Z^{\mu\nu\alpha}\,,$
(110)

where we have suppressed the functional dependence of the various quantities,
e.g. $W^{\mu\nu\rho}(g_{\beta_{1}\beta_{2}},\phi)$, etc. and
$E^{\mu\nu\rho\sigma}_{R}$ is already defined in eq.(70). It should be noted
that $Y^{\mu\nu}$ and $Z^{\mu\nu\alpha}$ are the possible ambiguities in the
definition of the Noether charge. They are also present in the definition of
$\theta^{\mu}$, but, interestingly, get cancelled in the combination
$(-\Theta^{\mu}+D_{\nu}Q^{\nu\mu})$ which is important for our calculations,
see eq.(53). Therefore, in whatever follows we will neglect them, focussing
only on

$Q^{\mu\nu}=W^{\mu\nu\rho}\,\xi_{\rho}-2E^{\mu\nu\alpha\beta}_{R}\,D_{[\alpha}\xi_{\beta]}\,.$
(111)

It should be noted that the two terms present on the RHS above are linear
either in $\xi^{\mu}$ or in $D_{\alpha}\xi_{\beta}$. Therefore, following the
same argument as in the case of $\Theta^{r}$, we could decompose $Q^{r\mu}$,
by using $\xi^{\mu}$ from eq.(26) and explicitly working out the $v$
dependence as

$Q^{r\mu}=\widetilde{Q}^{r\mu}+v\,W^{r\mu}_{v}\,,$ (112)

where $\widetilde{Q}^{r\mu}$ and $W^{r\mu}_{v}$ have no explicit $v$
dependence343434Since $Q^{\nu\mu}$ is anti-symmetric in its two indices, for
$Q^{r\mu}$ we should note that possible entries for $\mu$ are
$\mu=v,i,j,k\cdots$, except $r$.. In the process of deriving eq.(112) from
eq.(111), we can also obtain the following expression for
$\widetilde{Q}^{r\mu}$

$\widetilde{Q}^{rv}=4\,E^{rvrv}_{R}\,,$ (113)

from which it can be concluded that $\widetilde{Q}^{r\mu}$ actually reproduces
the equilibrium Wald entropy density. This will be important later in our
analysis.

Following our rules for determining boost weight that we have already declared
in Table-3 using index counting, $Q^{rv}$ is boost invariant but $Q^{ri}$ has
boost weight $+1$. Next, substituting this decomposition for $Q^{r\mu}$ from
eq.(112) in the expression of $D_{\mu}Q^{r\mu}$ we find

$\begin{split}D_{\mu}Q^{r\mu}=~{}&D_{\mu}\widetilde{Q}^{r\mu}+W^{rv}_{v}+v~{}D_{\mu}W^{r\mu}_{v}\\\
=~{}&{1\over\sqrt{h}}\partial_{v}\left(\sqrt{h}~{}\widetilde{Q}^{rv}\right)+\nabla_{i}\widetilde{Q}^{ri}+W^{rv}_{v}+v\left[{1\over\sqrt{h}}\partial_{v}\left(\sqrt{h}~{}W^{rv}_{v}\right)+\nabla_{i}W^{ri}_{v}\right]\,.\end{split}$
(114)

Let us now process the last term on the RHS above. It should be noted that
$W^{rv}_{v}$ has boost weight $(+1)$ and therefore must be of order ${\cal
O}(\epsilon)$. It follows that

${1\over\sqrt{h}}\partial_{v}\left(\sqrt{h}~{}W^{rv}_{v}\right)=\partial_{v}W^{rv}_{v}+{\cal
O}(\epsilon^{2})\,.$ (115)

Following the same logic, we learn that $W^{ri}_{v}$ is a term of boost weight
$(+2)$. Therefore, applying Result: $1$ from eq.(66) it could be written
as353535We are not careful with the coefficients dictated by Result: $1$.

$W^{ri}_{v}=\partial_{v}J^{i}_{(1)}+{\cal O}(\epsilon^{2})\,$ (116)

such that $J^{i}_{(1)}$ is some spatial vector with boost weight $(+1)$.

It should be noted here that the form of $W^{ri}_{v}$ written above is a
simple representation of its possible structure. For an arbitrary covariant
tensor with boost weight equal to $+2$, we will generally obtain a much more
complicated structure. However, here our purpose is to justify the existence
of such a term in general. We should also point out that the first term on the
RHS of Result: $1$ (in eq.(66)) will always generate a term like
$\partial_{v}J^{i}_{(1)}$ and there is no obstruction to that in principle. As