aainstitutetext: School of Physical Sciences, National Institute of Science
Education and Research,
An OCC of Homi Bhabha National Institute, Jatni 752050, Odisha,
Indiabbinstitutetext: Department of Physics, Indian Institute of Technology
Kanpur, Kalyanpur, Kanpur 208016, India

# The zeroth law of black hole thermodynamics in arbitrary higher derivative
theories of gravity

Sayantani Bhattacharyya b , Parthajit Biswas a , Anirban Dinda b , Nilay Kundu
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We consider diffeomorphism invariant theories of gravity with arbitrary higher
derivative terms in the Lagrangian as corrections to the leading two
derivative theory of Einstein’s general relativity. We construct a proof of
the zeroth law of black hole thermodynamics in such theories. We assume that a
stationary black hole solution in an arbitrary higher derivative theory can be
obtained by starting with the corresponding stationary solution in general
relativity and correcting it order by order in a perturbative expansion in the
coupling constants of the higher derivative Lagrangian. We prove that surface
gravity remains constant on its horizon when computed for such stationary
black holes, which is the zeroth law. We argue that the constancy of surface
gravity on the horizon is related to specific components of the equations of
motion in such theories. We further use a specific boost symmetry of the near
horizon space-time of the stationary black hole to constrain the off-shell
structure of the equations of motion. Our proof for the zeroth law is valid up
to arbitrary order in the expansion in the higher derivative couplings.

###### Keywords:

Black hole entropy, Zeroth law of black hole thermodynamics, Diffeomorphism
invariant theories of gravity, Higher derivative theories of gravity.

##  1 Introduction

Black holes are fascinating gravitational objects with many properties that
are very surprising and peculiar. However, black holes share one common
property with ordinary matter: they also behave as thermodynamic objects. In
Einstein’s theory of two derivative classical gravity, also known as general
relativity, they are solutions to the equations of motion, and one can
associate geometric notions with corresponding thermodynamic properties, such
as the area of the event horizon as its entropy, the surface gravity as the
temperature, etc. It has long been understood that the laws of black hole
mechanics can be viewed as laws of thermodynamics Hawking:1971tu ;
Bardeen:1973gs ; Bekenstein:1973ur . We also know from Hawking:1974sw  that
this similarity is not an analogy. However, indeed one can derive the
temperature of a black hole related to its surface gravity in a rigorous way.

On the other hand, we know that general relativity is not a complete theory of
gravity. We must think of it as an effective theory valid at low energies or
large length scales. In a complete theory of quantum gravity, one can take its
low energy limit and would get general relativity as the leading theory.
Following this procedure, one would also generate corrections to general
relativity. Without detailed knowledge of the UV complete theory and the
process of taking a low energy limit, we cannot be sure what corrections to be
added to the leading two derivative theory. Nevertheless, on general grounds,
we expect that one would get various higher derivative terms in the Lagrangian
in addition to the Einstein-Hilbert piece. Different higher derivative
corrections will come with different dimensionful parameters as coefficients
in the Lagrangian, and this will signify the length scale, say $l_{\text{HD}}$
at which the higher derivative terms would be as important as the leading
Einstein gravity piece. We denote the higher derivative couplings collectively
as the dimensionless parameter $\alpha$ 111The higher derivative coupling will
have dimensions in general, however, with the use of appropriate powers of
$l_{\text{HD}}$ we can define a dimensionless coupling $\alpha$ and also
choose units by putting $l_{\text{HD}}=1$..

Once we extend the scope of gravity theories by including arbitrary higher
derivative corrections in addition to the leading two derivative theory, the
black holes still remain to be solutions of these new theories, and they
should also retain their thermodynamic properties. Therefore although the laws
of black hole mechanics were first understood in general relativity, one can
not ignore the importance of understanding the validity of a similar set of
laws for black hole thermodynamics in such higher derivative theories of
gravity. In PhysRevD.48.R3427 ; Iyer:1994ys , it was shown that a version of
the first law of black hole thermodynamics could indeed be argued for an
arbitrary diffeomorphism invariant theory of gravity. This construction also
suggested a geometric object defined on the horizon of the black hole as the
generalized definition of black hole entropy. This definition of black hole
entropy is known as the Wald entropy in the literature. It says that the
Noether charge associated with the Killing symmetry generator of the null
horizon should be identified as the entropy of black holes in such arbitrary
diffeomorphism invariant theory of gravity. Of course, this definition of
entropy reduces to the area of the horizon as one considers black holes in
general relativity. However, once out-of-equilibrium dynamic processes
involving black holes are considered, the Wald entropy suffers from possible
ambiguities known as the JKM ambiguities Jacobson:1993xs ; Jacobson:1993vj ;
Jacobson:1995uq . Additionally, there is no general proof that the Wald
entropy satisfies the second law of thermodynamics. There have been various
attempts at designing a proof for the second law that will be valid for
arbitrary higher derivative theories of gravity Iyer:1994ys ; Jacobson:1993xs
; Jacobson:1993vj ; Jacobson:1995uq ; Sarkar:2013swa ; Bhattacharjee:2015yaa ;
Wall:2015raa ; Bhattacharjee:2015qaa ; Wall:2011hj 222See the recent reviews
Wall:2018ydq ; Sarkar:2019xfd  and the references therein for a detailed
discussion on black hole thermodynamics for higher curvature theory of
gravity.. Also, recently the construction of an entropy current in such
theories was studied Bhattacharya:2019qal ; Bhattacharyya:2021jhr ;
Bhattacharyya:2022njk , following the work of Wall:2015raa .

However, in this paper, our aim is to focus on the zeroth law of black hole
thermodynamics in higher derivative theories of gravity. As in an ordinary
thermodynamic system, the zeroth law for black hole mechanics is a
characteristic signature of stationary or equilibrium black hole
configurations. Stationary black holes have a space-time metric that admits a
null hypersurface known as the Killing horizon, where a Killing vector becomes
null. Using the fact that event horizons for stationary black holes are
Killing horizons (due to the rigidity theorems), and also that the temperature
of the black hole is given by the surface gravity for such stationary black
hole metrics, one can make a precise statement of zeroth law as follows: _the
surface gravity of a stationary black hole is constant over the entire event
horizon_. This statement has been proven for two derivative theories of
gravity, with an additional assumption of the dominant energy condition for
the matter stress tensor Bardeen:1973gs , by analyzing the equations of motion
in general relativity. Alternative proofs have been constructed Zerothwald ;
Xie:2021bur , without any use of the equations of motion of the theory but
assuming extra symmetries of the space-time. If we do not use any additional
symmetry of the black hole space-time as mentioned above, it is an interesting
question to ask if one can extend the proof of the zeroth law to theories of
gravity beyond general relativity. Recently in Ghosh:2020dkk  such proof was
given for stationary black hole solutions in Gauss-Bonnet and Lovelock
theories of gravity by modifying and improving upon a previously reported
negative result in such theories Sarkar:2012wy ; also see Sang:2021rla ;
Dey:2021rke  for a similar result. As these results were worked out for
particular models of higher derivative theories of gravity, to the best of our
knowledge, a similar result is not yet known for arbitrary diffeomorphism
invariant theories of gravity. In this paper, we address this particular
question and find that the answer to this is in the affirmative: _we have been
able to construct a proof for the zeroth law in arbitrary diffeomorphism
invariant theories of gravity where the higher derivative terms in the
Lagrangian are added as a correction to the leading two derivative theory of
general relativity_.

An important assumption in our construction is the fact that it applies to
theories where all higher derivative terms, appearing in the Lagrangian
associated with a coupling parameter $\alpha$, are treated as corrections to a
leading two derivative theory of gravity, namely Einstein’s general
relativity. In operational terms, this means that for theories that we
consider, a smooth limit of taking the higher derivative coupling
$\alpha\rightarrow 0$ exists, and in that limit, we recover the general
relativity as the leading candidate theory. This, in particular, enables us to
obtain stationary black hole metrics as solutions in arbitrary higher
derivative theories of gravity as they can be constructed perturbatively
around some known stationary black hole solutions in two derivative general
relativity when $\alpha=0$. It is, however, important to highlight that our
proof only requires the existence of this perturbative higher derivative
coupling $\alpha$. However, it is valid for all orders in this
$\alpha$-expansion and is also valid for any number of higher derivative
coupling.

Let us now mention some of the salient features of the technical tools that we
have used in constructing our proof. We will be very brief here, and all of
these issues will be discussed in great detail in the subsequent sections.
Firstly, we will work with a particular choice for the metric of stationary
black holes. This does not lose any generality as one can always make these
gauge choices for any stationary black holes. For our analysis, we will focus
on Killing horizons where a Killing vector becomes null on the co-dimension
one null hypersurface 333Sometimes, we also call it the event horizon of the
black hole. However, one needs to account for various global issues in the
form of rigidity theorems to ensure that the local definition of a Killing
horizon can be associated with the global concept of an event horizon. It is
known to be true for general relativity, but it is still an open question to
prove rigidity theorems beyond general relativity. Therefore, to be precise,
we will actually be working with the Killing horizon in this paper. Further,
we will associate the constancy of surface gravity on the horizon with
specific components of the equations of motion. In other words, the off-shell
structure of some specific components of the equations of motion, when
evaluated at the horizon, will be related to (actually, be proportional to)
the derivative of the surface gravity with respect to the coordinates on the
horizon. Once we can establish this, the zeroth law would follow automatically
by equating these components of equations of motion to zero. However, we must
point out that we do not explicitly use equations of motion in our analysis
apart from this last step. Also, the main point here is to establish the
following fact - _it is always possible to express the off-shell structure of
a particular component of the equations of motion in arbitrary higher
derivative theories of gravity (with the assumption of them augmenting the
leading two derivative theory) in a form such that they get related to the
derivative of surface gravity with respect to coordinates tangent to the
horizon_ \- this is the main result of our analysis in this paper.

As mentioned before, we organize our calculations in a perturbative expansion
in the higher derivative coupling. Within such a perturbative framework, we
will use the method of induction to prove that a particular component of the
equation of motion has the desired off-shell structure at arbitrary order. We
first argue that at the leading order, i.e., when $\alpha=0$, the equations of
motion are indeed of the form expected, as this is just reviewing the proof of
zeroth law known in the literature. Next, we assume that the proof works at an
arbitrary order in the $\alpha$-expansion, say at $\mathcal{O}(\alpha^{m})$.
Then we show that the proof will also work at the next order
$\mathcal{O}(\alpha^{m+1})$. Therefore, following the method of induction, we
can conclude that the proof will work up to any arbitrary higher-order in the
$\alpha$-expansion.

In establishing our result, a crucial input used a residual gauge invariance
for our choice of the metric, named the boost symmetry. This boost symmetry is
the consequence of a Killing isometry for stationary black holes, and the
Killing horizon is mapped to itself under the flow generated by this boost
transformation. Any covariant tensor, e.g., the equations of motion, will
transform in a particular way under this boost transformation. This symmetry
was an essential input for several recent works in the context of black hole
thermodynamics. For example, in Wall:2015raa , a proof of linearized second
law for arbitrary higher derivative theories of gravity was developed using
this symmetry. Also, in Bhattacharya:2019qal  and Bhattacharyya:2021jhr , it
was crucial to determine the structure of the equations of motion to construct
an entropy current with non-negative divergence. In our present paper,
assuming that the zeroth law is being satisfied at the order,
$\mathcal{O}(\alpha^{m})$ of the $\alpha$-expansion, this boost-symmetry
enables us to constrain the off-shell structure of the equations of motion at
the next order $\mathcal{O}(\alpha^{m+1})$ as the desired one.

Finally, we end this section with an overview of how the paper is structured.
We begin with a description of the basic setup and an operational statement of
the problem at hand in §2. Here we discuss the particular choice of horizon
adapted coordinates that we will use throughout this paper and present a
schematic sketch of how various quantities can be organized in the
perturbative expansion in the higher derivative coupling $\alpha$. In the
following section §3, we present a detailed description of the boost-symmetry
and the basic rules following as a consequence of this, in connection to a
stationary black hole and the zeroth law. In the next section §4 we briefly
discuss and summarise the basic strategy of our proof without getting involved
in the technical details of it. This is followed by a technically rigorous
presentation of the main proof in §5. We divide this into several sub-
sections, each corresponding to various steps in the analysis following a
method of induction. We conclude this paper with some discussions in §6.
Important supplementary material with various technical results is presented
in the Appendices §A-§D.

##  2 Basic set-up and statement of the problem

In this section, we start by describing the basic set-up of our analysis, and
we will make a precise statement of the problem using that.

We are considering any arbitrary higher curvature theory of gravity without
any matter couplings444For a proof of the zeroth law in Einstein gravity,
matter couplings can be introduced with the assumption of dominant energy
condition. We expect it would be straightforward to include matter couplings
(even non-minimal ones) in our set-up barring some subtleties with the
definition of temperature as discussed in Hajian:2020dcq . However, we do not
discuss them here. in $d$ space-time dimensions with coordinates denoted by
$x^{\mu}$. Following Iyer:1994ys , the requirement of diffemorphism invariance
restricts the Lagrangian for such theories to be of the following form

$\mathcal{L}=\mathcal{L}(g_{\mu\nu},\,R_{\mu\nu\alpha\beta},\,D_{\sigma}R_{\mu\nu\alpha\beta},\cdots)$
(1)

However, for our analysis in this paper, we will work with theories such that
the gravity action has the following form

$I=\frac{1}{4\pi}\int
d^{d}x\sqrt{-g}\left(R+\sum_{m=1}^{\infty}\alpha^{m}\,{\cal L}_{2m+2}\right)$
(2)

where the higher derivative couplings in the theory are denoted by the
parameter $\alpha$. The other parameter present in the Lagrangian (i.e. $m$)
counts the order of derivatives on the metric tensor (i.e. $g_{\mu\nu}$), the
field variable in our theory. Therefore, it should be clear that ${\cal
L}_{2m+2}$ is the $(2m+2)$-th order higher derivative term in the Lagrangian
involving $(2m+2)$-derivatives acting on $g_{\mu\nu}$. The leading term, i.e.
$m=0$, gives us the standard Einstein-Hilbert Lagrangian for general
relativity. It is important to mention that, apart from having $(2m+2)$ number
of derivatives on $g_{\mu\nu}$, $\mathcal{L}_{2m+2}$ has no other restrictions
and is, therefore, completely arbitrary.

Ideally, all such higher derivative terms can, in principle, appear in the
Lagrangian with different numerical coefficients. Hence, one should allow for
different coupling constants for each of them in different order of the
parameter $m$. Even within one same order of $m$-th derivative coupling,
different possible terms can appear with different coupling coefficients but
with the same dimensionality 555For example, let us consider the two terms at
$\mathcal{O}(\alpha^{2})$ in the Lagrangian:
$R_{\mu}^{\nu}R_{\nu}^{\rho}R_{\rho}^{\mu}$ and
$D_{\mu}R_{\nu\rho}D^{\mu}R^{\nu\rho}$. Both of them have six derivatives and
hence can appear in the Lagrangian in the following way
$\alpha^{2}\mathcal{L}_{6}\sim\alpha^{2}(c_{1}\,R_{\mu}^{\nu}R_{\nu}^{\rho}R_{\rho}^{\mu}+c_{2}\,D_{\mu}R_{\nu\rho}D^{\mu}R^{\nu\rho})\,,$
where $c_{1}$ and $c_{2}$ are two different but $\mathcal{O}(1)$
coefficients.. However, as we will see in the later parts of our analysis, the
only important thing for us is to have the Einstein-Hilbert term as the
leading contribution in a limiting sense when the higher derivative couplings
are taken to be small. In other words, all we need is to have theories with
arbitrary higher curvature terms in the Lagrangian, but any higher derivative
couplings can be taken to zero in a smooth limit, leaving us with two
derivative classical general relativity as the most significant one.
Therefore, without any loss of generality, we collectively denote every
possible higher derivative coupling by $\alpha^{m}$ for $m=1,\,2,\,\cdots$,
with a specific number of derivatives on $g_{\mu\nu}$ determined by the
corresponding value of $m$. We will treat $\alpha$ as a small parameter
allowing ourselves to perform a perturbative expansion in it. However, our
analysis will be valid for arbitrary higher-order in that $\alpha$ expansion,
as we have already mentioned before.

As we have described, we will be working in a perturbative expansion in the
parameter $\alpha$; it is obvious that the equations of motion (EoM) will have
the following structure,

$E_{\mu\nu}=E^{(0)}_{\mu\nu}+\alpha\,E^{(1)}_{\mu\nu}+\alpha^{2}\,E^{(2)}_{\mu\nu}+\,\cdots\,,$
(3)

where, $E^{(0)}_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R$, is the EoM coming
from Einstein’s general relativity.

Next, we would like to comment on another essential ingredient in setting up
our analysis related to obtaining stationary black hole solutions in arbitrary
higher derivative theories of gravity. For purposes of the arguments presented
in this paper, we do not need to know the exact form of the stationary black
hole metric as a solution to the equations of motion. However, we assume that
such solutions must exist in the higher derivative theory of gravity that we
are considering. One should, quite naturally, be able to construct such
solutions Cai_2002 ; Ma_2021  within our setup of perturbative expansion in
$\alpha$, the coupling of the higher derivative terms in our theory.

Let us suppose we start with a given stationary black hole solution, denoted
by $g^{(bh)}_{\mu\nu}$, in the leading order theory in $\alpha$ expansion,
which is Einstein’s general relativity. It is obvious that the stationary
$g_{\mu\nu}=g^{(bh)}_{\mu\nu}$ solves the equation of motion
$E^{(0)}_{\mu\nu}=0$. As a consequence of this, $g^{(bh)}_{\mu\nu}$ will have
a Killing horizon - a null hypersurface generated by a global Killing vector
field which we will denote by $\partial_{\tau}$. By definition,
$\partial_{\tau}$ will be a null geodesic on the horizon, and all the metric
components in $g^{(bh)}_{\mu\nu}$ will be independent of the $\tau$
coordinate. In the following paragraphs, we will make this more precise.

We will be working with a particular horizon adapted set of the space-time
coordinates along with a particular gauge choice for the metric
$g^{(bh)}_{\mu\nu}$. In a $d$-dimensional space-time we can always choose a
coordinate system $x^{\mu}=\\{\tau,\rho,x^{i}\\}$, where $i=1,...,d-2$, so
that the stationary metric $g^{(bh)}_{\mu\nu}$ takes the following form

$\begin{split}ds^{2}=g^{(bh)}_{\mu\nu}\,dx^{\mu}dx^{\nu}=2\,d\tau\,d\rho-\rho\,X(\rho,x^{i})\,d\tau^{2}+2\,\rho\,\omega_{i}(\rho,x^{i})\,d\tau\,dx^{i}+h_{ij}(\rho,x^{i})\,dx^{i}\,dx^{j}\,.\end{split}$
(4)

Let us briefly justify the gauge choice for the metric in eq.(4) (see
section-(2.1) and Appendix-A of Bhattacharyya:2016xfs  for the details). The
coordinates $\\{\tau,\,x^{i}\\}$ span the co-dimension one horizon which lies
on $\rho=0$. We should also note that $\rho=0$ is a null hypersurface, for
which the null generators are taken to be the vector $\xi=\partial_{\tau}$. By
construction, this is normal to itself and the other spatial generators
($\partial_{i}$) of the horizon. At constant values of the coordinates
$x^{i}$, the parameter $\tau$ runs along one null generator, whereas, for a
constant value of $\tau$, the coordinates $x^{i}$ parametrizes different null
generators on the horizon. The coordinate $\tau$ is not necessarily affinely
parametrized. To describe the geometry in the vicinity of a null hypersurface,
we need two null normals to it. Hence, apart from $\xi$, we have considered
the auxiliary vector $\chi=\partial_{\rho}$, which is also null. This gives us
the coordinate $\rho$, which parametrizes the distance away from the null
horizon. The coordinate $\rho$ has been chosen to be affinely parametrized,
and the inner products: $(\partial_{\tau},\partial_{\rho})|_{\rho=0}=1$ and
$(\partial_{i},\partial_{\rho})|_{\rho=0}=0$, define the angles with which the
null-vector $\partial_{\rho}$ pierces through the horizon at $\rho=0$.

Next, the additional requirement of stationarity should explain why the metric
coefficients (the functions $X$, $\omega_{i}$, and $h_{ij}$) are independent
of the coordinate $\tau$. To this we note that $\xi$ is a Killing vector for
the metric eq.(4), satisfying the Killing equation
$D_{\mu}\xi_{\nu}+D_{\nu}\xi_{\mu}=0$, where $D_{\mu}$ is the covariant
derivative with respect to the full black hole metric, $g^{(bh)}_{\mu\nu}$.
The norm of this Killing vector vanishes on the surface $\rho=0$. Thus, in our
choice of coordinates, $\rho=0$ hypersurface is a Killing horizon.

The vector field $\xi^{\mu}$ also satisfies the geodesic equation

$\xi^{\nu}\,D_{\nu}\xi^{\mu}=\kappa\,\xi^{\mu}\,.$ (5)

Note that, the RHS of the above equation is not zero since $\tau$ is not
necessarily an affine parameter. This equation could be considered as the
definition of the quantity $\kappa$, which is in general a function of the
coordinates $(\tau,\,x^{i})$ and is called the surface gravity. It can be
straightforwardly shown that the surface gravity for the black hole space-time
described by the metric given in eq.(4) can be written as

$\kappa=\left.\sqrt{-\frac{1}{2}\,(D_{\mu}\xi_{\nu})\,(D^{\mu}\xi^{\nu})}\,\right|_{\text{horizon}}\,.$
(6)

The surface gravity is related to the temperature of a stationary black hole,
and thus to prove the zeroth law, we must show that $\kappa$ is constant over
the horizon. It means that the surface gravity is constant not only for
evolutions along one null generator but also does not change across different
null generators of the null horizon. In other words, we would aim to prove
that, when evaluated on the horizon,

$\partial_{\tau}\kappa=0,\quad\text{and}\quad\partial_{i}\kappa=0\,.$ (7)

Following the definition in eq.(6), we can evaluate the surface gravity for
our choice of metric eq.(4) for $\xi=\partial_{\tau}$, to get the following
expression (see Appendix-A for details of the calculation)

$\kappa=\left.{1\over 2}X(\rho,x^{i})\,\right|_{\rho=0}.$ (8)

It is obvious from eq.(8) that $\kappa$ is independent of the coordinate
$\tau$. Basically, since $\xi$ is a Killing vector, we trivially obtain the
$\tau$ independence of $X(\rho,x^{i})$, and hence $\partial_{\tau}\kappa=0$.
Therefore, to prove the zeroth law we have to show the following on the
horizon

$\partial_{i}X(\rho,x^{i})\,|_{\rho=0}=0\,.$ (9)

##  3 Boost symmetry in the context of the zeroth law and stationarity

As we have laid down the statement of the problem in operational terms, in
this section, we would like to highlight one crucial significance of the
zeroth law or, equivalently, the constancy of surface gravity over the
horizon. Let us remind ourselves that the zeroth law is, in a sense, one
particular manifestation of stationarity for black hole solutions in our
theory. It is noteworthy that for our choice of the stationary black hole
metric in eq.(4) the coordinate $\tau$ runs along the null generators of the
horizon but is not affinely parametrized. However, a slightly different but
very useful choice of coordinate system as written below

$ds^{2}=\tilde{g}^{(bh)}_{\mu\nu}\,dx^{\mu}dx^{\nu}=2\,dv\,dr-r^{2}\,X(rv,\,x^{i})\,dv^{2}+2\,r\,\omega_{i}(rv,\,x^{i})\,dv\,dx^{i}+h_{ij}(rv,\,x^{i})\,dx^{i}\,dx^{j}\,,$
(10)

also describes metric of stationary black holes with the horizon being set at
$r=0$, see Wall:2015raa , Bhattacharya:2019qal , Bhattacharyya:2021jhr . The
crucial difference between this choice of metric in eq.(10), written in terms
of the new coordinates $(r,\,v,\,x^{i})$, compared to the one in eq.(4), is
the fact that the $v$ coordinate here is affinely parametrized along the null
generators $\partial_{v}$ of the horizon. It should also be noted that,
although, for the choice of metric in eq.(4) the metric coefficients are
independent of the parameter $\tau$, in eq.(10) the metric coefficients are
functions of the coordinate $v$. However, the dependence on $v$ is not
arbitrary but restricted to the product $r\,v$. The reason for this is the
following, for stationary metrics, the Killing generator $\partial_{\tau}$ and
the affinely parametrized null generators are not the same but proportional to
each other, see Appendix-A of Bhattacharya:2019qal  for a detailed discussion
on this.

Let us now highlight the usefulness of writing the stationary black hole
metric in the form of eq.(10) with $v$ being an affine parameter. This
particular choice does not fix the gauge completely and one still has some
residual freedom of performing further coordinate transformation.
Particularly, one can do the following scaling of the coordinates $(r,\,v)$

$r\rightarrow\lambda\,r\,,\quad\text{accompanied with}\quad
v\rightarrow{v\over\lambda}\,,$ (11)

where $\lambda$ is a constant parameter 666Actually, one can do a more general
residual coordinate transformation $v\rightarrow f_{1}(x^{i})v+f_{2}(x^{i}),$
along with appropriate redefinition of $r$, but here we have restricted
ourselves to a subclass of it.. It should be convincing that this
transformation should leave the metric invariant, since the metric functions
depend on the coordinates $(r,\,v)$ only through their product. This is called
the boost transformation and due to this the stationary black hole
configurations are said to enjoy a boost symmetry, see Bhattacharya:2019qal ,
Bhattacharyya:2021jhr  for details.

Alternatively, we can also explain the boost symmetry, that we described
above, in the following way. In the coordinate system $\\{r,\,v,\,x\\}$, a
stationary black hole solution, as written in eq.(10), has a Killing vector

$\xi=\xi^{\mu}\partial_{\mu}=(v\partial_{v}-r\partial_{r})\,.$ (12)

In other words, the metric eq.(10) satisfies the following

$\mathcal{L}_{\xi}g_{\mu\nu}^{bh}=0\,,$ (13)

where $\mathcal{L}_{\xi}$ denotes the Lie derivative with respect to the
vector $\xi$. It can be easily checked that $\xi$ is also the generator of the
infinitesimal version of the boost transformation eq.(11).

As a consequence of this Killing symmetry, we can also confirm that the Lie
derivative of any arbitrary covariant tensor constructed out of the metric
should also vanish. The boost-symmetry is extremely useful in determining how
any general tensor quantity built out of the metric coefficients or various
derivatives of them, would transform under the aforementioned boost-
transformation. In particular, any covariant tensor, say $\mathcal{B}$, with
all components lowered, would transform in the following way

$\mathcal{B}\rightarrow\widetilde{\mathcal{B}}=\lambda^{w}\,\mathcal{B},\quad\text{under}\quad\left(r\rightarrow\tilde{r}=\lambda\,r\,,\,v\rightarrow\tilde{v}={v\over\lambda}\right)\,$
(14)

so that we define the boost-weight of $\mathcal{B}$ to be given by $w$.
Alternatively, we can also show that the boost-weight of any covariant tensor
would be given by the number of excess lower $v$-indices over the lower
$r$-indices, see Bhattacharyya:2021jhr  and Appendix-B for a justification in
favor of this.

Let us mention one important result that follows from the set up of boost-
symmetry discussed above, any quantity with positive boost-weight will always
vanish when computed using metric corresponding to a stationary configurations
and evaluated on the Killing horizon. In order to explain this statement, let
us first note that, from the definition of boost weight given in eq.(14) it
can be argued that the metric functions,
$X(rv,\,x^{i}),\,\omega_{i}(rv,\,x^{i}),\,h_{ij}(rv,\,x^{i})$ appearing in
eq.(10), are all boost invariant objects. Additionally the derivatives
$\partial_{v}$ and $\partial_{r}$ have boost weights given by $+1$ and $-1$
respectively,

$\partial_{v}\rightarrow\lambda\,\partial_{v}\,,\quad\text{and}\quad\partial_{r}\rightarrow\lambda^{-1}\,\partial_{r}\,.$
(15)

Therefore, any covariant tensor, say $\mathcal{B}(rv,\,x^{i})$, with positive
boost weight can generically be written as

$\mathcal{B}(rv,\,x^{i})\sim(\partial_{r})^{m_{r}}(\partial_{v})^{m_{v}}\,\widetilde{\mathcal{B}}(rv,\,x^{i})\,,\quad\text{with}\quad
m_{v}>m_{r}\,,$ (16)

where $\widetilde{\mathcal{B}}(rv,\,x^{i})$ can include derivatives with
respect to the spatial coordinates, but not any $\partial_{v}$ or
$\partial_{r}$. The functional dependence of $\mathcal{B}(rv,\,x^{i})$ or
$\widetilde{\mathcal{B}}(rv,\,x^{i})$ on the product of $rv$ signifies that
they are evaluated on stationary configurations. Because of $m_{v}>m_{r}$,
$\mathcal{B}$ has positive boost weight equal to $(m_{v}-m_{r})$. Now, it is
easy to convince ourselves that whenever one operates
$(\partial_{r})^{m_{r}}(\partial_{v})^{m_{v}}$ on
$\widetilde{\mathcal{B}}(rv,\,x^{i})$, or in that case any function of the
product $rv$, $(m_{v}-m_{r})$ factors of $r$ will be obtained, and hence it
will vanish when we further evaluate this on the horizon $r=0$. This will also
be very crucially used in our present paper.

In Bhattacharya:2019qal  and Bhattacharyya:2021jhr , this particular boost
symmetry was used to construct a local entropy current with non-negative
divergence on the horizon of a dynamically perturbed stationary black hole in
an arbitrary diffeomorphism invariant theory of gravity. In order to study
non-stationary dynamical processes, this boost-symmetry is broken slightly by
some matter source hitting the stationary black hole space-time. One can
organize the dynamics in a perturbative expansion around the initial
stationary configuration in the small amplitude of the external matter
disturbance. Up to linearized order in the expansion in this amplitude
expansion, the $vv$-component of the equations of motion (EoM) in any
diffeomorphism invariant theory of gravity attains a universal structure as
given below

$E_{vv}\sim\partial_{v}\left(\partial_{v}J^{v}+\nabla_{i}J^{i}\right)+\text{quadratic
fluctuations}\,,$ (17)

where the quantity $J^{v}$ represents local entropy density, reproducing the
Wald entropy expression upon taking the stationary limit. On the other hand,
the spatial components $J^{i}$ signify the spatial flow of entropy on constant
$v$-slices of the horizon. Using this result obtained in general gravity
theories, one further needs to use the null energy condition for the stress-
energy tensor coming from the matter sector to construct a proof for the local
version of a second law.

It was, therefore, indeed essential for the analysis in Bhattacharyya:2021jhr
to have the stationary metric written in the form given in eq.(10). In this
section, we will argue that if the zeroth law is satisfied one can perform a
coordinate transformation that changes the space-time metric from eq.(4) to
eq.(10). Although this was implicit in the calculations in Appendix-A of
Bhattacharyya:2021jhr , here we would like to make it very explicit.

Once zeroth law is satisfied, we get the surface gravity constant over the
horizon. Therefore, we should be able to solve eq.(9) and obtain the following
general solution for the metric coefficient function $X(\rho,\,x^{i})$

$X(\rho,\,x^{i})=c_{1}+\rho\,f(\rho,\,x^{i})\,,$ (18)

where $c_{1}$ is an integration constant and $f(\rho,\,x^{i})$ is some
arbitrary function of $(\rho,\,x^{i})$. Also, note that in order to satisfy
eq.(8), the constant $c_{1}$ gets fixed as $c_{1}=2\,\kappa$. We can
substitute this in eq.(4) to obtain

$\begin{split}ds^{2}=2\,d\tau\,d\rho-\rho\,\left(c_{1}+\rho\,f(\rho,\,x^{i})\right)\,d\tau^{2}+2\,\rho\,\omega_{i}(\rho,x^{i})\,d\tau\,dx^{i}+h_{ij}(\rho,x^{i})\,dx^{i}\,dx^{j}\,.\end{split}$
(19)

Next we perform the following coordinate transformation from the coordinates
$\\{\rho,\,\tau,\,x^{i}\\}$ to $\\{r,\,v,\,x^{i}\\}$ given by

$\tau\rightarrow
v=\frac{2}{c_{1}}\exp\left(\frac{c_{1}}{2}\tau\right),\quad\text{and},\quad\rho\rightarrow
r=\rho\exp\left(-\frac{c_{1}}{2}\tau\right)$ (20)

to arrive at

$ds^{2}=2\,dv\,dr-r^{2}\,f(c_{1}\,r\,v/2,\,x^{i})\,dv^{2}+r\,\omega_{i}(c_{1}\,r\,v/2,\,x^{i})\,dv\,dx^{i}+h_{ij}(c_{1}\,r\,v/2,\,x^{i})\,dx^{i}dx^{j}\,,$
(21)

which is of the form eq.(10). Note that the horizon stays at $r=0$ in the new
coordinates, and, also, the fact that $c_{1}=2\kappa$ is a constant was
crucially used while performing this coordinate transformation. Once written
in this coordinate system, we can straightforwardly use the consequences of
boost-symmetry that the metric in this form enjoys.

Finally, before we end this section, let us make one comment on how these
results that one derives using boost invariance of a stationary black hole
expressed in the coordinates as in eq.(21), would be helpful in the later
sections of this paper as we aim to prove zeroth law. This may seem puzzling
since, to derive these results, we have already used the zeroth law itself.
However, as we will explain later, we will follow a methodology for the proof
of zeroth law by organizing our calculations as a perturbative correction in
the higher derivative coupling $\alpha$ correcting the leading order two
derivative theory of general relativity. In that perturbative set-up, we will
construct the proof by using a method of induction. More precisely, with the
assumption that at $n$-th order in the $\alpha$-expansion, our construction
validates the zeroth law, we will aim to extend the proof to $n+1$-th order.
Therefore, while working at $n+1$-th order, the truncated and corrected metric
till the previous $n$-th order could be brought to the form as in eq.(21) and
thus would satisfy boost-invariance under the transformation given in eq.(11).
Consequently, when evaluated on the metric corrected and truncated up to
$n$-th order in $\alpha$-expansion, any covariant tensor would transform with
a particular boost-weight entirely determined by its index structure alone.
For our case, using these concepts, we will see that the $(vi)$-component of
EoM, i.e., $E_{vi}$, would have a boost weight equal to $+1$ and would thus
vanish for stationary black hole configurations.

##  4 Brief outline of the strategy

In this section, our goal is to present a broad outline of the strategy of the
proof without getting into operational details. Following this, in the next
section, we will construct a technically rigorous proof.

The crucial ingredient in our strategy to prove the zeroth law will be to
argue that eq.(9) follows from the vanishing of a particular component of the
equations of motion (EoM). More precisely, we will explicitly show that the
LHS of eq.(9) must be expressed in terms of the $\\{\tau\,i\\}$-component of
the EoM

$E_{\tau i}\,|_{\rho=0}\sim\partial_{i}X(\rho,x^{i})\,|_{\rho=0}\,,$ (22)

upto numerical factors, where, following eq.(3), $E_{\tau i}$ must include
contributions from all higher derivative terms present in the Lagrangian of
the theory in addition to the leading Einstein-Hilbert term. Next, we should
use $E_{\tau i}=0$, as the stationary black hole space-times must solve the
full EoM’s. It is clear that eq.(9) follows immediately.

With the explanations so far, let us summarise the main goal that we will
pursue in the rest of this paper: _In order to prove the zeroth law (or
equivalently, for proof of eq.( 9)), our primary goal would be to justify that
the off-shell structure of the entire $E_{\tau i}$ (including corrections due
to higher derivative coupling $\alpha$ as in eq.(3)) reproduces eq.(22). To
achieve this, we must evaluate $E_{\tau i}$ for a stationary black hole
solution obtained by treating the higher derivative coupling $\alpha$
perturbatively in an expansion around a stationary black hole solution of the
leading two-derivative theory._

Before we proceed further with describing our strategy, let us take a small
detour to highlight the significance of this particular component of EoM’s,
$E_{\tau i}$, in justifying the zeroth law. On general grounds, looking at
eq.(9), we should expect that $\partial_{i}X(\rho,x^{i})\,|_{\rho=0}$ should
get related to some components of $E_{\mu\nu}$ in order to be vanishing when
evaluated on on-shell configurations. The index structure then suggests that
one of the two indices in $E_{\mu\nu}$ must be the spatial indices (say,
$\mu=i$), leaving the other one to be either $\nu=\tau$, or, $\nu=\rho$. If we
now focus on Einstein’s gravity and compute the $(\tau,\,i)$ component of the
Einstein tensor, which is the EoM, we can immediately check that eq.(22) is
reproduced. Hence, the zeroth law is proved for the two derivative theory. In
our set-up, we treat arbitrary theories of gravity as perturbative corrections
in the higher derivative coupling $\alpha$ to a leading two derivative theory.
Hence, it is expected that even in such general theories, we must look into
the off-shell structure of $E_{\tau i}$ to justify eq.(9).

The arguments presented in the previous paragraph may appear to be heuristic.
However, a more rigorous justification can be devised to support the following
statement: for a proof of the zeroth law, one should investigate the off-shell
structure of $(\tau\,i)$-component of EoM. This has already been noted in the
literature. It is possible to show that (see Bardeen:1973gs  for proof)

$\begin{split}e^{\mu}_{i}D_{\mu}\,\kappa&=-R_{\mu\nu}\,\xi^{\mu}\,e^{\nu}_{i}\,,\end{split}$
(23)

where $e^{\mu}_{i}$ are the space-like tangent vectors to the horizon at
$\rho=0$. Most significantly, we should note that to derive eq.(23) one does
not need to use any EoM, and hence this is valid universally in any theory of
gravity. For two derivative Einstein gravity, once we use EoM, the RHS in
eq.(23) gets related to components of the stress-energy tensor, $T_{\mu\nu}$,
coming from the matter sector coupled to gravity, if any. One can further use
the dominant energy condition for the stress-energy tensor, and consequently,
the RHS in eq.(23) vanishes, proving the zeroth law. However, once we focus on
higher derivative theories of gravity, in this process of substituting
$R_{\mu\nu}$ in terms of $T_{\mu\nu}$ we get extra contributions in the EoM
due to the higher derivative terms in the Lagrangian

$E^{(0)}_{\mu\nu}+\alpha\,E^{\text{HD}}_{\mu\nu}=T_{\mu\nu}\,,$ (24)

where $E^{(0)}_{\mu\nu}=R_{\mu\nu}-(1/2)\,g_{\mu\nu}\,R$ is the Einstein
tensor, and $E^{\text{HD}}_{\mu\nu}$ is the EoM coming from the higher
derivative terms in the theory along with the coupling $\alpha$. With this, we
are convinced that in order to prove that the RHS of eq.(23) vanishes, we must
need to investigate the off-shell structure of
$E^{\text{HD}}_{\mu\nu}\,\xi^{\mu}\,e^{\nu}_{i}$, which, in our chosen
coordinate system, is precisely the $E^{\text{HD}}_{\tau i}$.

With the set-up that we have discussed so far, we are now in a position to
give a schematic overview of the operational strategy that will be followed to
argue that $E_{\tau i}$ is indeed of the form given in eq.(22). The main idea
will be to organize the analysis in a perturbative expansion in higher
derivative coupling $\alpha$ around the leading two derivative theory. The EoM
has already been written in eq.(3) in order by order expansion in $\alpha$. In
order to investigate the off-shell structure of $E_{\mu\nu}$ constructed out
of the space-time metric $g_{\mu\nu}$ and derivatives acting on it, we would
also need to take a similar ansatz for $g_{\mu\nu}$ expanded in powers of
$\alpha$

$g^{(bh)}_{\mu\nu}=g^{(0)}_{\mu\nu}+\alpha\,g^{(1)}_{\mu\nu}+\alpha^{2}\,g^{(2)}_{\mu\nu}+\cdots\,,$
(25)

where the superscript in $g^{(n)}_{\mu\nu}$ signifies that it corresponds to
the $n$-th order in the expansion of $\alpha$. We would demand that this
ansatz for $g^{(bh)}_{\mu\nu}$ solves the EoM given in eq.(3) order by order
in the expansion of $\alpha$. This, in turn, would allow us to justify that
eq.(22) is indeed true up to all orders in the perturbative expansion in
$\alpha$.

We will follow the method of induction to establish the desired off-shell
structure of $E_{\tau i}$ up to arbitrary order in the $\alpha$-expansion.
First, we would show that it is indeed the case at the leading order with
$\alpha=0$ for two-derivative Einstein gravity. Then we will extend this to
arbitrary order of $\mathcal{O}(\alpha^{m+1})$ in the $\alpha$-perturbative
expansion, assuming that things do work out till the previous order of
$\mathcal{O}(\alpha^{m})$. In this process, we will need to know specifics
about the generic structure of $E_{\mu\nu}$ in an arbitrary order of the
perturbation. To evaluate $E_{\mu\nu}$, we need to substitute for
$g^{(bh)}_{\mu\nu}$, given in eq.(25), in eq.(3) and isolate the terms
contributing at $\mathcal{O}(\alpha^{m+1})$. We will see that at this order
(i.e. at $\mathcal{O}(\alpha^{m+1})$), $E_{\mu\nu}$ can be partitioned into
two types of terms. The first type being the zeroth order EoM
$E^{(0)}_{\mu\nu}$ evaluated on
$g^{(0)}_{\mu\nu}+\alpha^{m+1}g^{(m+1)}_{\mu\nu}$, where we will treat
$\alpha^{m+1}g^{(m+1)}_{\mu\nu}$ as linearized perturbation around
$g^{(0)}_{\mu\nu}$. The second type of terms involve the coefficient of
$\alpha^{m+1}$ in the full EoM evaluated on the metric corrected till the
previous order, i.e. till $g^{(m)}_{\mu\nu}$. Schematically, this looks as the
following

$\text{At}\,\mathcal{O}(\alpha^{m+1}):\quad
E^{(0)}_{\mu\nu}[g^{(0)}_{\mu\nu}+\alpha^{m+1}\,g^{(m+1)}_{\mu\nu}]+E_{\mu\nu}[g^{(0)}_{\mu\nu}+\alpha\,g^{(1)}_{\mu\nu}+\alpha^{2}\,g^{(2)}_{\mu\nu}+\cdots+\alpha^{m}\,g^{(m)}_{\mu\nu}]=\mathcal{O}(\alpha^{m+2})\,,$
(26)

where for the second term on the LHS we should truncate it to
$\mathcal{O}(\alpha^{m+1})$.

The first term on the LHS of eq.(26) has an universal structure as it is
basically the Einstein’s tensor linearized around $g^{(0)}_{\mu\nu}$ for a
small perturbation given by $g^{(m+1)}_{\mu\nu}$.

To treat the second term on the LHS in eq.(26), however, we have to be more
careful. Since this term has no universal structure like the first one, a
further non-trivial argument must be invoked. We should take note of the fact
that this second term is an arbitrary covariant tensor of rank two, but most
importantly, built out of metric coefficients truncated at
$\mathcal{O}(\alpha^{m})$. When we are looking at the order ${\cal
O}(\alpha^{m+1})$, we will assume that eq.(22) and eq.(9) have been satisfied
till the order of $\mathcal{O}(\alpha^{m})$. This, in turn, enables us to
ascertain that the surface gravity, $\kappa$, computed with the corrected
metric till $\mathcal{O}(\alpha^{m})$, will be constant over the horizon. As a
consequence of this, we know that $\partial_{i}X(\rho,\,x^{i})|_{\rho=0}=0$ up
to $\mathcal{O}(\alpha^{m})$, and hence we can use the coordinate
transformation eq.(50) to write the metric in terms of coordinates
$(r,\,v,\,x^{i})$, as in eq.(51). As we have discussed before, once we have
succeeded in writing the space-time metric in this new coordinate system, we
can use the boost-symmetry. Consequently, we would now be able to assign boost
weights to various covariant tensors just by counting the difference in lower
$v$ and $r$ components. By standard rule of how tensors should transform under
coordinate transformation one can see that $E_{\tau i}$ in the original
$(\rho,\,\tau,\,i)$-coordinates will be proportional to $\tilde{E}_{vi}$ \-
the $(vi)$-component of the new EoM in the new coordinates. Finally, we note
that $\tilde{E}_{vi}$ has boost-weight$=+1$, and hence should vanish when
computed for stationary configurations. Therefore we conclude that the second
term on the LHS of eq.(26) will not contribute at $\mathcal{O}(\alpha^{m+1})$.
With this, we will be able to show that $E_{\tau i}$ has indeed the form
mentioned in eq.(22) and, thereby, we will be able to complete the proof of
the zeroth law.

##  5 Constructing the proof for the zeroth law

This section will construct the proof for the zeroth law with all details. The
discussions in this section will be divided into several sub-sections.
Following the outline of our strategy presented in §4, we would start with
understanding the general structure of the equations of motion (EoM) order by
order in an expansion in the higher derivative coupling $\alpha$. This would
be followed by explicitly working out the leading order term in this
expansion, which is actually the Einstein tensor coming from the Einstein-
Hilbert Lagrangian. We would review how zeroth law is satisfied at this
leading order. Next, we would extend this procedure to an arbitrary higher-
order in the $\alpha$-expansion adopting a method of induction. Finally, we
will also explicitly see how the application of boost symmetry for stationary
metric corrected up to a particular order of $\alpha$-expansion helps us
determine the $(\tau i)$-components of the EoM to the following order.

###  5.1 General structure for the equations of motion in $\alpha$-expansion

As we have already mentioned before in §2, the key assumption in our working
principle is that we could solve the EoM perturbatively in an expansion in the
higher derivative coupling $\alpha$. Thereby, the EoM has a structure given in
eq.(3), which we rewrite here for convenience

$E_{\mu\nu}=E^{(0)}_{\mu\nu}+\alpha\,E^{(1)}_{\mu\nu}+\alpha^{2}\,E^{(2)}_{\mu\nu}+\,\cdots\,,$
(27)

where $E^{(0)}_{\mu\nu}$ is the Einstein tensor - the EoM in the two
derivative theory of gravity. Also, $\alpha^{k}E^{(k)}_{\mu\nu}$, for $k\geq
1$, are all higher derivative corrections to the EoM. They depend on the
details of the theory. We have also noticed in §4, that as a consequence, the
metric will also admit a similar expansion, given in eq.(25). Here we present
that as well

$g^{(bh)}_{\mu\nu}=g^{(0)}_{\mu\nu}+\alpha\,g^{(1)}_{\mu\nu}+\alpha^{2}\,g^{(2)}_{\mu\nu}+\cdots\,.$
(28)

In our choice of coordinates this will lead to an expansion of the metric
components $X$, $\omega_{i}$ and $h_{ij}$ as given below

$\begin{split}X(\rho,x^{i})&=X^{(0)}(\rho,x^{i})+\alpha\,X^{(1)}(\rho,x^{i})+\alpha^{2}\,X^{(2)}(\rho,x^{i})+\cdots\,,\\\
\omega_{i}(\rho,x^{i})&=\omega_{i}^{(0)}(\rho,x^{i})+\alpha\,\omega_{i}^{(1)}(\rho,x^{i})+\alpha^{2}\,\omega_{i}^{(2)}(\rho,x^{i})+\cdots\,,\\\
h_{ij}(\rho,x^{i})&=h^{(0)}_{ij}(\rho,x^{i})+\alpha\,h^{(1)}_{ij}(\rho,x^{i})+\alpha^{2}\,h^{(2)}_{ij}(\rho,x^{i})+\cdots\,.\end{split}$
(29)

We are viewing the EoM in eq.(27) to be evaluated on the metric
$g^{(bh)}_{\mu\nu}$ in eq.(28), and the corresponding structures should be
analysed order by order in the $\alpha$-expansion. Let us now see what we can
learn at the very leading order, i.e. at $\mathcal{O}(\alpha^{0})$. At this
order very leading order the metric functions
$X^{(0)}(\rho,x^{i}),\,\omega_{i}^{(0)}(\rho,x^{i})$ and
$h^{(0)}_{ij}(\rho,x^{i})$ should be exact solutions of the zeroth order
equation (i.e. the Einstein’s equation for two derivative theory of gravity)

$E^{(0)}_{\mu\nu}[g^{(0)}_{\mu\nu}]=0$ (30)

Therefore, we should be viewing this as a differential equation for
$g^{(0)}_{\mu\nu}$, which is the unknown variable at leading order, and by
solving this, we would be able to fix $g^{(0)}_{\mu\nu}$.

Now, let us suppose that we want to solve the EoM $E_{\mu\nu}=0$ upto the
first sub-leading order (i.e., upto order ${\cal O}(\alpha^{1})$). We have
already determined $g^{(0)}_{\mu\nu}$ while working at the previous order
${\cal O}(\alpha^{0})$. At this order of ${\mathcal{O}}(\alpha^{1})$, we
realise that $g^{(1)}_{\mu\nu}$ (or the metric functions
$X^{(1)}(\rho,x^{i}),\,\omega_{i}^{(1)}(\rho,x^{i})$ and
$h^{(1)}_{ij}(\rho,x^{i})$) are the unknowns. To find out the relevant part of
EoM from eq.(27) at this order, we will basically have to evaluate the tensor
$E_{\mu\nu}$ on the metric $g_{\mu\nu}^{(bh)}$, neglecting all terms
proportional to quadratic or higher powers of $\alpha$. In other words, it is
obvious that $E^{(n)}_{\mu\nu}$ and $g^{(n)}_{\mu\nu}$ for every $n\geq 2$ are
negligible at this order of ${\cal O}(\alpha^{1})$. As a result, the
differential equation for the unknowns will have the following structure

$E_{\mu\nu}^{(0)}\left[g^{(0)}_{\alpha\beta}+\alpha~{}g^{(1)}_{\alpha\beta}\right]+\alpha~{}E_{\mu\nu}^{(1)}\left[g^{(0)}_{\alpha\beta}\right]=\mathcal{O}(\alpha^{2})\,,$
(31)

Here, in eq.(31), the first term on the LHS is basically the Einstein tensor,
linearised around $g^{(0)}_{\mu\nu}$ where $g_{\mu\nu}^{(1)}$ plays the role
of the small fluctuation metric. The second term is actually not of any
universal structure like Einstein tensor since the explicit form would depend
on the type of higher derivative theory that we are focussing on at linear
order in $\alpha$. However, for our purpose, that is not at all a problem
since we just need to know that $E_{\mu\nu}^{(1)}$ is a covariant tensor of
rank two (constructed out of appropriate contractions of Riemann tensors and
its covariant derivatives) evaluated on the exact stationary black hole
solution, i.e., $g^{(0)}_{\mu\nu}$, of the two-derivative theory of gravity.
From eq.(31), it is clear that this term will act as a source term in the
inhomogeneous PDE for $g^{(1)}_{\mu\nu}$. The first term, on the other hand,
is homogeneous in $g^{(1)}_{\mu\nu}$ and has a known and universal structure.

Consequently, at the next sub-leading order at ${\mathcal{O}}(\alpha^{2})$,
the unknowns are $g^{(2)}_{\mu\nu}$ (or, as usual,
$X^{(2)}(\rho,x^{i}),\,\omega_{i}^{(2)}(\rho,x^{i})$ and
$h^{(2)}_{ij}(\rho,x^{i})$), and the PDE for them should look like

$E_{\mu\nu}^{(0)}\left[g^{(0)}_{\alpha\beta}+\alpha^{2}\,g^{(2)}_{\alpha\beta}\right]+E_{\mu\nu}\left[g^{(0)}_{\alpha\beta}+\alpha\,g^{(1)}_{\alpha\beta}\right]={\mathcal{O}}(\alpha^{3})\,,$
(32)

where the first term on LHS is the homogeneous piece and the other term is the
source term for $g^{(2)}_{\mu\nu}$. The source term, again, is evaluated on
metric functions $g^{(1)}_{\mu\nu}$ and $g^{(0)}_{\mu\nu}$, which are already
solved in the previous iteration at ${\mathcal{O}}(\alpha^{2})$.

Once we have studied the EoM to the second sub-leading order in the
$\alpha$-expansion, we should be able to extend this analysis to any arbitrary
order in $\alpha$. Let us assume that we are currently focussing on the
$(m+1)$-th order term in the expansion. We have learned that if we would like
to determine the solution correctly up to order ${\mathcal{O}}(\alpha^{m+1})$,
we have to evaluate $E_{\mu\nu}$ on $g_{\mu\nu}^{(bh)}$ neglecting all terms
of order ${\mathcal{O}}(\alpha^{m+2})$ and higher. At this order, i.e. at
${\cal O}(\alpha^{m+1})$, the unknowns would be the components of
$g^{(m+1)}_{\mu\nu}$ or in our choice of gauge, the metric functions
$X^{(m+1)}(\rho,x^{i}),\,\omega_{i}^{(m+1)}(\rho,x^{i})$ and
$h^{(m+1)}_{ij}(\rho,x^{i})$. Now, we would like to know what would be the
structure of the EoM at this order. As it is true for any perturbative
solution technique, the homogeneous part of the equation at every order has an
universal structure. In this case it is the Einstein equation
$E^{(0)}_{\mu\nu}$ linearized around the zeroth order black hole metric
$g^{(0)}_{\mu\nu}$, but now the role of the fluctuation metric will be played
by $g^{(m+1)}_{\mu\nu}$ 777The reason for this universality is as follows. The
correction to the solution i.e., $g^{(m+1)}_{\mu\nu}$ already carries a factor
of $\alpha^{m+1}$. Since we are interested in evaluating the equation at order
${\cal O}(\alpha^{m+1})$ and also if we want to collect only those terms that
involves $g^{(m+1)}_{\mu\nu}$, everything else in the equation must be of
zeroth order in $\alpha$. It follows that at order
${\mathcal{O}}(\alpha^{m+1})$ none of the $E_{\mu\nu}^{(m)}$, for $m>0$ can
contribute to terms that has $g^{(m+1)}_{\mu\nu}$ and the same is true for
product terms of the form $g^{(m)}\,g^{(n)}$, for $m>0$. Therefore at order
${\mathcal{O}}(\alpha^{m+1})$, terms that contain $g^{(m+1)}_{\mu\nu}$ can
only come from the EoM at zeroth order linearized around the zeroth order
solution..

But the source terms (analogous to the second term in eq.(31)) will not have
this universal form. At order ${\cal O}(\alpha^{m+1})$, where the metric upto
order ${\cal O}(\alpha^{m})$ is already fixed by solving the equations at
previous orders, the source terms will be the coefficient of $\alpha^{m+1}$ in
$E_{\mu\nu}$ once evaluated on $g^{(bh)}_{\mu\nu}$ as in eq.(28) but corrected
upto ${\mathcal{O}}(\alpha^{m})$, that is

$g^{(bh)}_{\mu\nu}\,|_{\text{corrected
upto}~{}{\mathcal{O}}(\alpha^{m})}\,=\,g^{(0)}_{\mu\nu}+\alpha~{}g^{(1)}_{\mu\nu}+\cdots+\alpha^{m}~{}g^{(m)}_{\mu\nu}\,.$

Therefore, at ${\mathcal{O}}(\alpha^{m+1})$, the equation looks like the
following

$E^{(0)}_{\mu\nu}\left[g_{\mu\nu}^{(0)}+\alpha^{m+1}\,g^{(m+1)}_{\mu\nu}\right]+E_{\mu\nu}\left[g^{(0)}_{\mu\nu}+\alpha\,g^{(1)}_{\mu\nu}+\alpha^{2}\,g^{(2)}_{\mu\nu}+\cdots+\alpha^{m}\,g^{(m)}_{\mu\nu}\right]=\mathcal{O}(\alpha^{m+2})\,,$
(33)

However, all the terms contributing to the source term in eq.(33) are obtained
from metric functions, which are all solved until the previous order in this
iterative construction. Interestingly, for our proof, we do not need the
details of the source term, except for the fact that at any given order, it is
a covariant tensor evaluated on a metric that solves the EoM up to the
previous order 888 We must emphasize that for this perturbative technique to
work at a given order (say ${\cal O}(\alpha^{m+1})$), EoM must be solved till
the previous order. This will ensure that the source term i.e., $E_{\mu\nu}$,
evaluated on
$(g^{(0)}_{\mu\nu}+\alpha\,g^{(1)}_{\mu\nu}+\cdots+\alpha^{m}\,g^{(m)}_{\mu\nu})$
will be non-zero only at order ${\cal O}(\alpha^{m+1})$..

###  5.2 Zeroth law for two derivative theories of gravity, at leading order
in $\alpha$-expansion

In the previous subsection, we have described the general structure of the EoM
at any given order in $\alpha$-expansion. We have seen that the starting point
must be Einstein’s two derivative gravity, and $g^{(0)}_{\mu\nu}$ must be an
exact stationary black hole solution of the Einstein equations
$E^{(0)}_{\mu\nu}$.

It is well known that in the two derivative theory of gravity, the temperature
of a stationary black hole is constant over the horizon. In this sub-section,
we will review, following the strategy outlined in §4, how this can be proved
in our choice of coordinate system eq.(4). As we have already mentioned, to
achieve this, we must look into the off-shell structure of the $(\tau i)$
component of the zeroth-order EoM $E^{(0)}_{\mu\nu}$. It will turn out that
$E^{(0)}_{\tau i}$ is indeed of the form eq.(22). When EoMs are satisfied by
stationary black hole configurations, we will readily obtain eq.(9). This is,
therefore, enough to prove the desired result in two-derivative theories of
gravity. In the following, we will argue that eq.(22) is indeed true.

Let us consider two derivative theories of gravity without any matter field.
To be more explicit, let us reiterate that the equation of motion eq.(3) is

$E_{\mu\nu}=E^{(0)}_{\mu\nu}\,.$ (34)

The metric eq.(4) upto order ${\cal O}(\alpha^{0})$ in the horizon adapted
coordinate system is

$\begin{split}ds^{2}&=2d\tau~{}d\rho-\rho
X^{(0)}(\rho,x^{i})d\tau^{2}+2\rho~{}\omega_{i}^{(0)}(\rho,x^{i})d\tau
dx^{i}+h^{(0)}_{ij}(\rho,x^{i})dx^{i}dx^{j}\,.\end{split}$ (35)

We would calculate $\tau i$ component of the EoM on the horizon.

$\begin{split}E_{\tau i}&=R_{\tau i}-\frac{1}{2}Rg_{\tau
i}\quad\Rightarrow\quad E_{\tau i}|_{\rho=0}=R_{\tau
i}|_{\rho=0}\,.\end{split}$ (36)

We must now compute $R_{\tau i}$ for our choice of metric eq.(35). Using the
expression of $R_{\tau i}$ (see appendix C), we get

$E_{\tau i}|_{\rho=0}=-\frac{1}{2}(\partial_{i}X^{(0)})|_{\rho=0}$ (37)

Using EoM, we can straightforwardly conclude

$\partial_{i}X^{(0)}(\rho,\,x^{i})|_{\rho=0}=0\,,$ (38)

which is basically eq.(9) upto ${\cal O}(\alpha^{0})$, and therefore, implies
zeroth law at the same order.

###  5.3 Zeroth law for higher curvature theories of gravity at arbitrary
order in $\alpha$-expansion

After establishing the zeroth law at the leading order in $\alpha$-expansion
for two derivative theories, in this section, we aim to extend this to
arbitrary higher-order in the perturbative $\alpha$ expansion. We will
construct our proof using a method of induction. It will be shown that if the
temperature is constant over the horizon till order
${\mathcal{O}}(\alpha^{n})$, then it will remain constant at order
${\mathcal{O}}(\alpha^{n+1})$. Following our strategy described in §4, and
just like what we did at the zeroth order, we will again use the off-shell
structure of the $(\tau i)$ component of the EoM to show this.

For convenience, let us first re-write the metric and its $\alpha$-expansion,
eq.(28) and eq.(40),

$\begin{split}ds^{2}=g^{(bh)}_{\mu\nu}dx^{\mu}dx^{\nu}=2\,d\tau\,d\rho-\rho\,X(\rho,x^{i})d\tau^{2}+2\rho\,\omega_{i}(\rho,x^{i})d\tau
dx^{i}+h_{ij}(\rho,x^{i})dx^{i}dx^{j}\,,\end{split}$ (39)

where,

$\begin{split}X(\rho,x^{i})&=X^{(0)}(\rho,x^{i})+\alpha\,X^{(1)}(\rho,x^{i})+\alpha^{2}\,X^{(2)}(\rho,x^{i})+\cdots\\\
\omega_{i}(\rho,x^{i})&=\omega_{i}^{(0)}(\rho,x^{i})+\alpha\,\omega_{i}^{(1)}(\rho,x^{i})+\alpha^{2}\,\omega_{i}^{(2)}(\rho,x^{i})+\cdots\\\
h_{ij}(\rho,x^{i})&=h^{(0)}_{ij}(\rho,x^{i})+\alpha\,h^{(1)}_{ij}(\rho,x^{i})+\alpha^{2}\,h^{(2)}_{ij}(\rho,x^{i})+\cdots\,.\end{split}$
(40)

We start with the statement that we have solved the EoM accurately upto order
${\cal O}(\alpha^{m})$. Also, following the same logic, we are assuming that
the temperature is constant on the horizon upto order ${\cal O}(\alpha^{m})$.
In terms of equation it implies

$\partial_{i}\left(X^{(0)}(\rho,\,x^{i})+\alpha~{}X^{(1)}(\rho,\,x^{i})+\cdots+\alpha^{m}~{}X^{(m)}(\rho,\,x^{i})\right)\Big{|}_{\rho=0}=0,$
(41)

Given this, now, we would like to solve the EoM at order
${\mathcal{O}}(\alpha^{m+1})$. As we have discussed in the previous sub-
section §5.1, in the context of the general structure of the EoM at an
arbitrary order of the $\alpha$-expansion, working at
${\mathcal{O}}(\alpha^{m+1})$, we will get a linear partial differential
equation for the unknown $g^{(m+1)}_{\mu\nu}$. This will be a linear PDE with
two types of terms; one is the homogeneous term along with another source
term. In the following, we will analyze these two terms one by one.

From eq.(33) we have learned that the homogeneous part of the equation could
be universally evaluated as linearisation of the Einstein tensor
$\left(\text{i.e., }E^{(0)}_{\mu\nu}\right)$ around $g^{(0)}_{\mu\nu}$,

$\text{Homogeneous part of the PDE
at}\,{\mathcal{O}}\left(\alpha^{m+1}\right)\,=\,E^{(0)}_{\mu\nu}\left[g^{(0)}_{\mu\nu}+\alpha^{m+1}g^{(m+1)}_{\mu\nu}\right]+{\cal
O}(\alpha^{m+2})\,.$ (42)

Note in the above equation the RHS will have the leading contribution at order
${\cal O}(\alpha^{m+1})$ since by construction
$E^{(0)}_{\mu\nu}\left[g^{(0)}_{\mu\nu}\right]=0$. For our purpose, we just
need to look at the $(\tau i)$ component of the EoM. By explicit evaluation in
our choice of coordinate system we could show (see appendix D)

$\begin{split}&E^{(0)}_{\tau
i}\left[g^{(0)}_{\mu\nu}+\alpha^{m+1}g^{(m+1)}_{\mu\nu}\right]_{\rho=0}=-\left.\frac{1}{2}\,\alpha^{m+1}\,\left(\partial_{i}X^{(m+1)}\right)\right|_{\rho=0}+{\cal
O}(\alpha^{m+2})\end{split}$ (43)

It should be noted that in deriving eq.(43), we have used the result obtained
in eq.(38) for the leading order two derivative theory.

Now we come to the source terms. These are the known terms at order ${\cal
O}(\alpha^{m+1})$. These could be computed by evaluating the EoM, keeping
terms up to order ${\cal O}(\alpha^{m+1})$, and ignoring all higher-order
terms on the metric corrected up to order ${\mathcal{O}}(\alpha^{m})$.

Before we proceed, let us introduce a new notation here for convenience. For
any function $Y$ that admits an $\alpha$ expansion, $Y^{(m)}$ denotes the
coefficient of $\alpha^{m}$ and $Y^{[m]}$ denotes the expansion of $Y$ correct
upto order ${\cal O}(\alpha^{m})$. In other words, if $Y$ could be written as
$Y=\sum_{m=0}^{\infty}\alpha^{m}\,Y^{(m)}$, then $Y^{[m]}$ denotes

$Y^{[m]}\equiv\sum_{k=0}^{m}\alpha^{k}\,Y^{(k)}\,.$ (44)

According to this notation, then, for the corrected and truncated metric and
EoM till order ${\mathcal{O}}(\alpha^{m})$, we get

$g^{[m]}_{\mu\nu}\equiv\sum_{i=0}^{m}\alpha^{m}\,g_{\mu\nu}^{(m)},\quad\text{and}\quad
E_{\mu\nu}^{[m]}\equiv\sum_{i=0}^{m}\alpha^{m}\,E_{\mu\nu}^{(m)}\,.$ (45)

Using this new notation, let us now write down the source term in the PDE for
$g^{(m)}_{\mu\nu}$, working at order ${\mathcal{O}}(\alpha^{m+1})$, as the
following

$\begin{split}\text{Source terms of the PDE
at}~{}{\mathcal{O}}(\alpha^{m+1})\,=\left.E^{[m+1]}_{\mu\nu}\,\right|_{\text{evaluated
on}\,g^{[m]}_{\mu\nu}}+{\mathcal{O}}(\alpha^{m+2})\end{split}$ (46)

Note that according to our assumptions, $g^{[m]}_{\mu\nu}$ solves the EoM up
to order ${\mathcal{O}}(\alpha^{m})$. It follows that the source terms as
written above will have the first non-zero contribution at order
${\mathcal{O}}(\alpha^{m+1})$. As we have mentioned before, the source terms
do not have any universal structure, unlike the homogeneous piece. However,
for the constancy of the temperature, we need to analyze only the $(\tau
i)$-component of the EoM and that too only at the horizon, i.e., $\rho=0$
hypersurface in our choice of coordinates. This will simplify our analysis.

####  5.3.1 Boost symmetry for the truncated metric and vanishing of the
source term

In the previous sub-section, we have shown that the homogeneous part of the
$(\tau i)$ component of the EoM at this order is simply proportional to
$\partial_{i}X^{(m+1)}$, see eq.(43). Therefore, what is left to be checked is
that the source term in the PDE for $g^{(m+1)}_{\mu\nu}$ vanishes at this
order ${\mathcal{O}}(\alpha^{m+1})$. In the following, we will argue that this
is exactly what will turn out to be true.

As we have described before, the source term at order ${\cal
O}(\alpha^{(m+1)})$, given in eq.(46), is simply the leading piece (in terms
of $\alpha$-expansion) in $E^{[m]}_{\mu\nu}$ evaluated on $g^{[m]}_{\mu\nu}$.
We should remember that, the corrected space-time metric $g^{[m]}_{\mu\nu}$ is
truncated at ${\mathcal{O}}(\alpha^{m})$. Also, it is corrected, because, it
solves the EoM till the same order. The truncated black hole metric
$g_{\mu\nu}^{[m]}$, leads to the following line element

$\begin{split}ds_{[m]}^{2}&=2\,d\tau\,d\rho-\rho\,X^{[m]}(\rho,x^{i})\,d\tau^{2}+2\,\rho\,\omega_{i}^{[m]}(\rho,x^{i})\,d\tau\,dx^{i}+h^{[m]}_{ij}(\rho,x^{i})\,dx^{i}dx^{j}\end{split}$
(47)

As a consequence of our assumption in eq.(41), the zeroth law can be assumed
to be satisfied for the metric $g^{[m]}_{\mu\nu}$ till
${\mathcal{O}}(\alpha^{m})$. So, we are allowed to use the fact that
$X^{[m]}(\rho=0,\,x^{i})$ is constant over the horizon ($\rho=0$
hypersurface),

$\left.\partial_{i}\left(\sum_{m\leq
n}\left(X^{(0)}(\rho,x^{i})+\alpha^{m}X^{(m)}(\rho,x^{i})\right)\right)\right|_{\rho=0}=\,\mathcal{O}(\alpha^{m+1})\,.$
(48)

This in turn enables us to ascertain that the surface gravity, $\kappa$,
computed with the corrected metric till $\mathcal{O}(\alpha^{m})$, will be
constant over the horizon. In other words $X^{[m]}$ could be written, by
solving eq.(48), as

$X^{[m]}(\rho,x^{i})=C^{[m]}+\rho\,F^{[m]}(\rho,x^{i})\,,$ (49)

where $C^{[m]}$ is a constant, and $F^{[m]}(\rho,x^{i})$ is an arbitrary
function of $(\rho,x^{i})$, and both of them are corrected upto
$\mathcal{O}(\alpha^{m})$.

Following our discussion in §3 (see eq.(20)) we would like to perform the
following coordinate transformation from $x^{\mu}=\\{\tau,\rho,x^{i}\\}$ to
$\widetilde{x}^{\mu}=\\{v,r,x^{i}\\}$ where, $v$ is the affine parameter along
the null generator of the horizon,

$v=\frac{2}{C^{[m]}}\exp\left(\frac{C^{[m]}}{2}\tau\right),\quad
r=\rho\,\exp\left(-\frac{C^{[m]}}{2}\tau\right)\,.$ (50)

The truncated metric in the new coordinates takes the form

$ds_{[m]}^{2}=2dv\,dr-r^{2}\,F^{[m]}(C^{[m]}rv/2,x^{i})dv^{2}+2r\,\omega^{[m]}_{i}(C^{[m]}rv/2,x^{i})dv\,dx^{i}+h^{[m]}_{ij}(C^{[m]}rv/2,x^{i})dx^{i}dx^{j}$
(51)

Now, $E^{[m]}_{\mu\nu}$ is just a covariant tensor of rank two, constructed
out of appropriate contractions of the product of Riemann tensors and/or their
covariant derivatives. So, without knowing any details about it, we could tell
how its component would transform under the above-mentioned coordinate
transformation. By which we mean that $\widetilde{E}^{[m]}_{\mu\nu}$ in the
new coordinates will be related to $E^{[m]}_{\mu\nu}$ in the old coordinates,
as follows

$E^{[m]}_{\mu\nu}(\rho,\tau,x^{i})=\frac{\partial\widetilde{x}^{\alpha}}{\partial
x^{\mu}}\,\frac{\partial\widetilde{x}^{\beta}}{\partial
x^{\nu}}\,\widetilde{E}^{[m]}_{\alpha\beta}(r,,v,x^{i})\,.$ (52)

For our purpose we just need to study the $\tau i$-component of
$E^{[m]}_{\mu\nu}(\rho,\tau,x^{i})$. Also, we can readily obtain the relevant
components of $\frac{\partial\widetilde{x}^{\alpha}}{\partial x^{\mu}}$ from
eq.(50),

${\partial
v\over\partial\tau}=\exp\left(\frac{C^{[m]}}{2}\tau\right),\quad\text{and}\quad{\partial
r\over\partial\tau}=-\rho\frac{C^{[m]}}{2}\exp\left(-\frac{C^{[m]}}{2}\tau\right)\,.$
(53)

Using them we obtain

$\begin{split}\left.E^{[m]}_{\tau
i}(\rho,\tau,x^{i})\right|_{\rho=0}=\left.\exp\left(\frac{C^{[m]}}{2}\tau\right)\widetilde{E}^{[m]}_{vi}(r,,v,x^{i})\right|_{r=0}\end{split}$
(54)

It is important to note that we have obtained $E^{[m]}_{\tau i}$ is
proportional to $\widetilde{E}^{[m]}_{vi}$ when evaluated on the horizon. To
decide about $\widetilde{E}^{[m]}_{vi}$ in the new coordinate system
$(r,,v,x^{i})$ we can directly use the boost-invariance of the metric eq.(47).
As we can see that $\widetilde{E}^{[m]}_{vi}$ contains one extra lower
$v$-index compared to $r$-index. According to the arguments due to this boost-
symmetry the boost-weight assigned to $\widetilde{E}^{[m]}_{vi}$ comes out to
be $+1$. Therefore, if we compute $\widetilde{E}^{[m]}_{vi}$ on the stationary
metric eq.(21) at $r=0$ it will simply vanish. This in turn shows that, by
using eq.(54), in our old $(\rho,\tau,x^{i})$ coordinates $E^{[m]}_{\tau i}$
also vanishes.

Therefore, we have now established the fact that at
$\mathcal{O}(\alpha^{m+1})$ the source term contribution to the PDE for
$g^{(m+1)}_{\mu\nu}$ vanishes. The homogeneous piece is the only contribution
and that too is of the form argued in eq.(43). With this we have also
successfully demonstrated that

$\partial_{i}X^{(k)}(\rho,\,x^{i})|_{\rho=0}=0,\quad\text{for}\quad
k=(m+1)\,,$ (55)

once we start with the assumption of
$\partial_{i}X^{(k)}(\rho,\,x^{i})|_{\rho=0}=0$ for $k\leq m$. Finally, by
method of induction, we, therefore also prove that, starting with a positive
result in the leading two derivative gravity, the zeroth law is true upto all
orders in the perturbative expansion in the higher derivative coupling
$\alpha$.

##  6 Discussions

In this paper, we have worked out a proof for the zeroth law of black hole
thermodynamics in diffeomorphism invariant theories of gravity. Our analysis
crucially depends on the fact that we consider only such theories of gravity
where arbitrary higher derivative theories of gravity augment the leading two
derivative theory of general relativity. We assumed that the higher derivative
coupling (denoted by $\alpha$ in this paper) could be taken to zero in a
smooth limit leaving us with the leading two derivative theory. This, in turn,
allows us to organize our analysis in a perturbative expansion in the higher
derivative coupling $\alpha$. For example, suppose we have an exact solution
in the form of a black hole metric of the equations of motion coming from the
two-derivative Einstein’s equation. We can correct this solution in that
perturbation scheme and expect to obtain the corresponding black hole solution
in the higher derivative theory of gravity. The zeroth law is a statement
about stationary configurations. We used a particular coordinate system, like
choosing a particular gauge, to write down the space-time metric of a
stationary black hole. Since the temperature of a black hole is identified
with the geometric quantity called surface gravity, working within our choice
of metric gauge, our main aim was to prove that the surface gravity is
constant over the horizon for stationary black holes. We want to stress here
that for our construction of the proof we did not need to use any extra
symmetry, we have only used the boost-symmetry which follows from
stationarity.

The crucial ingredient in our construction for the proof was to use specific
components of the equations of motion (EoM). We expanded the EoM order by
order in a perturbation series in the higher derivative coupling $\alpha$,
with the leading term (with $\alpha$ = 0) being Einstein’s equation. The
metric was also expanded in a similar perturbative expansion in $\alpha$, with
the leading order term being the stationary black hole solution in Einstein
gravity. We followed a method of induction for the proof. First, we showed
that the components of EoM have the desired off-shell structure needed for the
proof to go through at the leading order in Einstein’s gravity. Then we
assumed that this is true at the $n$-th order, and we argued that it should be
satisfied at the following order in $\alpha$-expansion.

Working at $\mathcal{O}(\alpha^{m+1})$, once we assumed that the zeroth law is
satisfied at the previous order, i.e., till $\mathcal{O}(\alpha^{m})$, we made
use of a specific residual gauge symmetry to perform a coordinate
transformation. In this new coordinate system, the coordinate along the null
generators of the horizon happens to be affinely parametrized, and the new
metric enjoys a symmetry called the boost-symmetry of the stationary black
holes. Using this symmetry, we could predict the structure of the components
of the EoM without knowing its explicit form. In other words, we viewed the
EoM at $\mathcal{O}(\alpha^{m+1})$ as a covariant tensor built out of the
metric components corrected till $\mathcal{O}(\alpha^{m})$ to satisfy the
zeroth law. Then, by knowing how the EoM for any arbitrary higher derivative
theory should transform under the coordinate transformation (boost
transformation), we could prove that it indeed has the required structure to
satisfy the zeroth law at $\mathcal{O}(\alpha^{m+1})$.

It is essential to highlight that this particular boost symmetry can be used
only when the metric can be cast in the new coordinate system we mentioned
above. This was crucially used in constructing the entropy current, using
which the local version of the second law was argued for arbitrary
diffeomorphism invariant theory of gravity. We also understood that one could
write down the stationary black hole metric in these new coordinates if the
zeroth law is satisfied. This was an important assumption in constructing the
entropy current in Bhattacharya:2019qal  and Bhattacharyya:2021jhr .
Therefore, our proof of zeroth law in this paper justifies this important
assumption that was made in those works aimed at proving the second law.

Another critical point in constructing our proof in this paper is the
assumption that it only applies to such theories where, in an appropriate
$\alpha$ expansion, the leading order piece has to be Einstein’s two
derivative theory. It was one significant input in our proof. However, we have
not argued that there cannot be any other proof that will not require this
assumption of starting the perturbation series from Einstein’s gravity. From
the perspective of a UV complete theory of quantum gravity, it is pretty
natural to expect that the low energy effective theories following from any
quantum theory of gravity would organize themselves in such a perturbative
framework starting with two derivative general relativity. However, it is
exciting to note that without having access to the details of how UV
completion is achieved and staying entirely within a low energy perspective,
principles like the laws of black hole thermodynamics also hint toward general
relativity as the leading theory in a perturbative framework.

Although, our proof is perturbative in higher derivative coupling constant
$\alpha$, we want to stress that it works up to arbitrary order in the
perturbative expansion. With this statement, we might hope that finding a
proof of zeroth law for theories non-perturbatively connected to general
relativity will be worth exploring. We leave that for future work.

###### Acknowledgements.

We want to thank Prateksh Dhivakar for collaboration at the initial stage. We
would also like to thank Diptarka Das, Suchetan Das, Prateksh Dhivakar, Suman
Kundu, Alok Laddha, Mangesh Mandlik, Milan Patra, Shuvayu Roy, Sudipta Sarkar
for useful discussions. PB would like to acknowledge the support provided by
the grants: SPO/SERB/PHY/2017504 and SB/SJF/2019-20/08. We would finally like
to thank the people of India for their steady support for research in basic
sciences.

##  Appendix A Computing the Christoffel symbols and the surface gravity for
the metric eq.(4)

###  A.1 Computing the Christoffel symbols

In this appendix, we will calculate the Christoffel symbols for the metric
eq.(4) upto order ${\cal O}(\alpha)$.

$\begin{split}ds^{2}&=2d\tau~{}d\rho-\rho\left[X^{(0)}(\rho,x^{i})+\alpha
X^{(1)}(\rho,x^{i})\right]d\tau^{2}+2\rho\left[\omega_{i}^{(0)}(\rho,x^{i})+\alpha\omega_{i}^{(1)}(\rho,x^{i})\right]d\tau
dx^{i}\\\ &+\left[h^{(0)}_{ij}(\rho,x^{i})+\alpha
h^{(1)}_{ij}(\rho,x^{i})\right]dx^{i}dx^{j}\end{split}$ (56)

Different components of the metric are

$\begin{split}&g_{\tau\tau}=-\rho\left[X^{(0)}(\rho,x^{i})+\alpha
X^{(1)}(\rho,x^{i})\right],~{}~{}g_{\tau\rho}=1,~{}~{}g_{\tau
i}=\rho\left[\omega_{i}^{(0)}(\rho,x^{i})+\alpha\omega_{i}^{(1)}(\rho,x^{i})\right],\\\
&g_{\rho\rho}=0,~{}~{}g_{\rho
i}=0,~{}~{}g_{ij}=\left[h^{(0)}_{ij}(\rho,x^{i})+\alpha
h^{(1)}_{ij}(\rho,x^{i})\right]\end{split}$ (57)

Different components of the inverse metric up to order ${\cal O}(\alpha)$ are

$\begin{split}&g^{\tau\tau}=0,~{}~{}g^{\tau\rho}=1,~{}~{}g^{\tau i}=0,\\\
&g^{\rho\rho}=\rho\left[X^{(0)}+\alpha
X^{(1)}\right]+\rho^{2}h_{(0)}^{ij}\omega^{(0)}_{i}\omega^{(0)}_{j}+\alpha\rho^{2}\left[2h^{ij}_{(0)}\omega^{(0)}_{i}\omega^{(1)}_{j}-h_{(1)}^{ij}\omega^{(0)}_{i}\omega^{(0)}_{j}\right]\\\
&g^{\rho
i}=-\rho\left[h^{ij}_{(0)}\omega_{j}^{(0)}+\alpha\left(h_{(0)}^{ij}\omega^{(1)}_{j}-h_{(1)}^{ij}\omega^{(0)}_{j}\right)\right],~{}~{}g^{ij}=h^{ij}_{(0)}-\alpha
h^{ij}_{(1)}\end{split}$ (58)

where, $h_{(0)}^{ij}$ is defined as $h_{(0)}^{ik}h^{(0)}_{kj}=\delta^{i}_{j}$
and $h_{(1)}^{ij}$ is defined as
$h_{(1)}^{ij}=h_{(0)}^{im}h_{(0)}^{jn}h^{(1)}_{mn}$.
Now we will compute different components of Christoffel symbols. We would
require the expression of one component of the Christoffel symbol
$\Gamma^{\rho}_{i\tau}$ off the horizon. The expression of
$\Gamma^{\rho}_{i\tau}$ up to order ${\cal O}(\rho)$ is

$\Gamma^{\rho}_{i\tau}=-\frac{\rho}{2}\partial_{i}\left(X^{(0)}+\alpha
X^{(1)}\right)-\frac{1}{2}\rho\left(X^{(0)}\omega_{i}^{(0)}+\alpha
X^{(1)}\omega^{(0)}_{i}+\alpha X^{(0)}\omega^{(1)}_{i}\right)$ (59)

The rest of the components are on the horizon

$\begin{split}&\Gamma^{\rho}_{\rho\tau}=-\frac{1}{2}\left[X^{(0)}+\alpha
X^{(1)}\right],~{}~{}\Gamma^{\tau}_{i\tau}=-\frac{1}{2}\left[\omega_{i}^{(0)}+\alpha\omega^{(1)}_{i}\right],~{}~{}\Gamma^{\rho}_{\rho
j}=\frac{1}{2}\left[\omega_{i}^{(0)}+\alpha~{}\omega^{(1)}_{i}\right],\\\
&\Gamma^{j}_{i\tau}=0,~{}~{}\Gamma^{j}_{\rho\tau}=\frac{1}{2}\left[\omega^{j}_{(0)}+\alpha~{}\omega_{(1)}^{j}-\alpha
h_{(1)}^{jk}\omega^{(0)}_{k}\right],~{}~{}\Gamma^{\rho}_{ij}=0,~{}~{}\Gamma^{\tau}_{\tau\tau}=\frac{1}{2}\left[X^{(0)}+\alpha
X^{(1)}\right],\\\
&\Gamma^{\tau}_{\tau\rho}=0,~{}~{}\Gamma^{\tau}_{i\rho}=0,~{}~{}\Gamma^{j}_{\tau\tau}=0,~{}~{}\Gamma^{\tau}_{ij}=-\frac{1}{2}\left(\partial_{\rho}h^{(0)}_{ij}+\alpha\partial_{\rho}h^{(1)}_{ij}\right),~{}~{}\Gamma^{\rho}_{\tau\tau}=0,~{}~{}\Gamma^{i}_{\tau\tau}=0,\end{split}$
(60)

Where, $\omega^{i}_{(0)}$ and $\omega^{i}_{(1)}$ are defined as
$\omega^{i}_{(0)}=h^{ij}_{(0)}\omega^{(0)}_{j}$ and
$\omega^{i}_{(1)}=h^{ij}_{(0)}\omega^{(1)}_{j}$

###  A.2 Computing the surface gravity

The metric of the space-time is given in eq.(4) and we write it here again for
convenience

$\begin{split}ds^{2}=2d\tau\,d\rho-\rho
X(\rho,x^{i})d\tau^{2}+2\rho~{}\omega_{i}(\rho,x^{i})d\tau
dx^{i}+h_{ij}(\rho,x^{i})dx^{i}dx^{j}\end{split}$ (61)

This metric admits a Killing vector $\xi=\partial_{\tau}$ with the horizon
being chosen to be at $\rho=0$. The definition of surface gravity is given by

$\kappa=\left.\sqrt{-\frac{1}{2}(\nabla_{\mu}\xi_{\nu})(\nabla^{\mu}\xi^{\nu})}\,\right|_{\rho=0}\,.$
(62)

We use the inverse metric expressions written in eq.(58) to obtain the
following components of $\xi_{\mu}$,

$\xi_{\rho}=\,1,\,\xi_{\tau}=\,-\rho\,X(\rho,x^{i}),\,\xi_{i}=\,\rho\,\omega_{i}(\rho,x^{i})\,.$
(63)

Next we compute the components of $\nabla_{\mu}\xi_{\nu}$ evaluated on
$\rho=0$ and the non-vanishing components are as follows

$\begin{split}\nabla_{\tau}\xi_{\rho}|_{\rho=0}&=-\,\nabla_{\rho}\xi_{\tau}|_{\rho=0}=\,-{1\over
2}\,X(\rho=0,x^{i})\,,\\\
\nabla_{\rho}\xi_{i}|_{\rho=0}&=-\,\nabla_{i}\xi_{\rho}|_{\rho=0}=\,{1\over
2}\,\omega_{i}(\rho=0,x^{i})\,,\end{split}$ (64)

Using these, we obtain

$(\nabla_{\mu}\xi_{\nu})(\nabla^{\mu}\xi^{\nu})\,|_{\rho=0}\,=2g^{\tau\rho}g^{\tau\rho}\,(\nabla_{\tau}\xi_{\rho})\,(\nabla_{\rho}\xi_{\tau})\,|_{\rho=0}=-{1\over
2}\,X^{2}(\rho=0,x^{i})$ (65)

Finally, we obtain the surface gravity as the following

$\kappa=\left.{1\over 2}X(\rho,x^{i})\,\right|_{\rho=0}\,.$ (66)

##  Appendix B Few details regarding the boost weight of covariant tensors

In this appendix we aim to provide some more detail regarding the boost
invariance of the stationary metric written in eq.(10). We write the metric
here again for convenience,

$ds^{2}=\tilde{g}^{(bh)}_{\mu\nu}\,dx^{\mu}dx^{\nu}=2\,dv\,dr-r^{2}\,X(rv,\,x^{i})\,dv^{2}+2\,r\,\omega_{i}(rv,\,x^{i})\,dv\,dx^{i}+h_{ij}(rv,\,x^{i})\,dx^{i}\,dx^{j}\,.$
(67)

The vector $\xi$ defined in eq.(12) ,generates Killing symmetry of the
stationary background with the metric in eq.(67). Due to this, as we have
already mentioned before, when we operate Lie derivative with respect to $\xi$
on any covariant tensor constructed out of the stationary metric eq.(67), will
vanish. To be more precise, acting with the Lie derivative with respect to
$\xi$, on a covariant tensor, say $\mathcal{B}_{\mu_{1}\mu_{2}\cdots\mu_{k}}$
with all lowered indices, will produce the following,

$\begin{split}{\cal
L}_{\xi}\mathcal{B}_{\mu_{1}\mu_{2}\cdots\mu_{k}}=\,&\xi^{\beta}\partial_{\beta}\mathcal{B}_{\mu_{1}\mu_{2}\cdots\mu_{k}}+\left(\partial_{\mu_{1}}\xi^{\beta}\right)\mathcal{B}_{\beta\mu_{2}\cdots\mu_{k}}+\left(\partial_{\mu_{2}}\xi^{\beta}\right)\mathcal{B}_{\mu_{1}\beta\cdots\mu_{k}}+\cdots\\\
&+\left(\partial_{\mu_{k}}\xi^{\beta}\right)\mathcal{B}_{\mu_{1}\mu_{2}\cdots\beta}\,.\end{split}$
(68)

Furthermore, when we evaluate this for the metric eq.(67), and with $xi$ given
in eq.(12), we will get

$\begin{split}{\cal
L}_{\xi}\mathcal{B}_{\mu_{1}\mu_{2}\cdots\mu_{k}}=\,\left[w+(v\partial_{v}-r\partial_{r})\right]\mathcal{B}_{\mu_{1}\mu_{2}\cdots\mu_{k}}\,,\end{split}$
(69)

where $w$ is the boost weight of $\mathcal{B}_{\mu_{1}\mu_{2}\cdots\mu_{k}}$
and from eq.(69) we can also confirm that $w$ counts the excess number of
lower $v$ indices compared to lower $r$ indices in
$\mathcal{B}_{\mu_{1}\mu_{2}\cdots\mu_{k}}$. Following this argument, it is
also obvious that the $vi$-component of EoM, $E_{vi}$ will have boost weight
equal to $+1$, and hence, will vanish for stationary configurations when
evaluated on the horizon. This is the main ingredient that we have used in
§5.3.1.

Before we conclude this appendix let us summarise the useful points that we
should remember while using the boost weight analysis,

  1. 1.

We should think about any component of a covariant tensor to have a structure
with some number of $\partial_{r}$, $\partial_{v}$ and $\nabla_{i}$ operators
acting on the metric coefficients in eq.(67): ($X$, $\omega_{i}$, and
$h_{ij}$) or product of such structures.

  2. 2.

The boost weight of any covariant tensor can be obtained by looking at the
factor $w$ in eq.(69), when a Lie derivative ${\cal L}_{\xi}$, with respect to
$\xi\left(=v\partial_{v}-r\partial_{r}\right)$, acts on it.

  3. 3.

Any expression with positive boost weight will vanish when evaluated on the
horizon for a stationary metric.

For more details we refer the reader to section-(2.3) and Appendix-B of
Bhattacharyya:2021jhr .

##  Appendix C More detailed calculation for Einstein’s gravity

In this appendix, we will calculate $\tau i$ component of equation of motion
$E_{\tau i}^{(0)}$ for Einstein’s gravity.

$R_{\tau i}={R^{\tau}}_{\tau\tau i}+{R^{\rho}}_{\tau\rho i}+{R^{j}}_{\tau ji}$
(70)

Using the expressions of Christoffel symbols computed in Appendix-(A.1), we
can calculate different components of Riemann tensor upto order ${\cal
O}(\alpha^{0})$

$\begin{split}{R^{\tau}}_{\tau\tau
i}|_{\rho=0}&=\partial_{\tau}\Gamma^{\tau}_{i\tau}-\partial_{i}\Gamma^{\tau}_{\tau\tau}+\Gamma^{\tau}_{\tau
E}\Gamma^{E}_{i\tau}-\Gamma^{\tau}_{iE}\Gamma^{E}_{\tau\tau}\\\
&=-\frac{1}{2}\partial_{i}X^{(0)}\end{split}$ (71)
$\begin{split}{R^{\rho}}_{\tau\rho
i}|_{\rho=0}&=\partial_{\rho}\Gamma^{\rho}_{i\tau}-\partial_{i}\Gamma^{\rho}_{\rho\tau}+\Gamma^{\rho}_{\rho
E}\Gamma^{E}_{i\tau}-\Gamma^{\rho}_{iE}\Gamma^{E}_{\rho\tau}\\\
&=0\end{split}$ (72) $\begin{split}{R^{j}}_{\tau
ji}|_{\rho=0}&=\partial_{j}\Gamma^{j}_{i\tau}-\partial_{i}\Gamma^{j}_{j\tau}+\Gamma^{j}_{jE}\Gamma^{E}_{i\tau}-\Gamma^{j}_{iE}\Gamma^{E}_{j\tau}\\\
&=0\end{split}$ (73)

Finally we get

$E_{\tau i}|_{\rho=0}=R_{\tau
i}|_{\rho=0}=-\frac{1}{2}(\partial_{i}X^{(0)})|_{\rho=0}$ (74)

##  Appendix D Calculation of the homogeneous part

In this appendix, we will derive the expression of the homogeneous part
eq.(43). As has been discussed in sub-section §5.1, we have to linearize
$E^{(0)}_{\mu\nu}$ around $g^{(0)}_{\alpha\beta}$. We have to calculate
$E^{(0)}_{\mu\nu}\left[g^{(0)}_{\alpha\beta}+\delta g_{\alpha\beta}\right]$,
where, we will treat $\delta
g_{\alpha\beta}\equiv\alpha^{m+1}g_{\alpha\beta}^{(m+1)}$ as linearized
perturbations around $g^{(0)}_{\alpha\beta}$. $E^{(0)}_{\mu\nu}$ is the
Einstein’s tensor

$E^{(0)}_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}$ (75)

As, $g^{(0)}_{\mu\nu}$ is an exact solution of $E^{(0)}_{\mu\nu}$

$E^{(0)}_{\mu\nu}\left[g^{(0)}_{\alpha\beta}+\delta
g_{\alpha\beta}\right]\equiv\delta E^{(0)}_{\mu\nu}=\delta
R_{\mu\nu}-\frac{1}{2}g^{(0)}_{\mu\nu}\delta R-\frac{1}{2}R^{(0)}\delta
g_{\mu\nu}$ (76)

$R^{(0)}$ is the Ricci scalar evaluated on the metric $g^{(0)}_{\mu\nu}$. We
have to calculate $\tau i$ component of the above equation at $\rho=0$. We can
compute $\delta E^{(0)}_{\tau i}$ off the horizon, but for our purpose that is
not required.

$\delta E^{(0)}_{\tau i}|_{\rho=0}=\delta R_{\tau i}-\frac{1}{2}g^{(0)}_{\tau
i}\delta R-\frac{1}{2}R^{(0)}\alpha^{m+1}g^{(m+1)}_{\tau i}=\delta R_{\tau i}$
(77)

Since we have denoted the coordinates by $\\{\tau,\rho,x^{i}\\}$, for
notational convenience, instead of using $\mu,\nu$ we will be denoting the
spacetime coordinates by $\\{A,B,C...\\}$. We will be using this notation only
for this appendix. If we calculate the Christoffel symbols on
$g_{AB}^{(0)}+\alpha^{m+1}g^{(m+1)}_{AB}$ we can decompose it as follows

$\Gamma^{A}_{BC}=\bar{\Gamma}^{A}_{BC}+\delta\Gamma^{A}_{BC}$ (78)

where $\bar{\Gamma}^{A}_{BC}$ is the Christoffel symbols for
$g^{(0)}_{\mu\nu}$. Linearized Ricci tensor is

$\begin{split}\delta
R_{AB}&=\nabla_{D}\delta\Gamma^{D}_{AB}-\nabla_{B}\delta\Gamma^{D}_{AD}\\\
\end{split}$ (79)

We can very easily read-off the expressions of $\bar{\Gamma}^{A}_{BC}$ and
$\delta\Gamma^{A}_{BC}$ from eq.(60).

$\delta
R_{AB}=\partial_{D}\left(\delta\Gamma^{D}_{AB}\right)+\bar{\Gamma}^{D}_{DE}\delta\Gamma^{E}_{AB}-\bar{\Gamma}^{E}_{DA}\delta\Gamma^{D}_{EB}-\bar{\Gamma}^{E}_{DB}\delta\Gamma^{D}_{AE}-\partial_{B}\delta\Gamma^{D}_{AD}+\bar{\Gamma}^{E}_{BA}\delta\Gamma^{D}_{ED}$
(80) $\delta R_{\tau i}=\partial_{D}\left(\delta\Gamma^{D}_{\tau
i}\right)+\bar{\Gamma}^{D}_{DE}\delta\Gamma^{E}_{\tau
i}-\bar{\Gamma}^{E}_{D\tau}\delta\Gamma^{D}_{Ei}-\bar{\Gamma}^{E}_{Di}\delta\Gamma^{D}_{\tau
E}-\partial_{i}\delta\Gamma^{D}_{\tau
D}+\bar{\Gamma}^{E}_{i\tau}\delta\Gamma^{D}_{ED}$ (81)

Now, we will compute different terms of the above equation on $\rho=0$
separately

$\begin{split}\partial_{D}\delta\Gamma^{D}_{\tau
i}&=\partial_{\rho}\delta\Gamma^{\rho}_{\tau
i}+\partial_{\tau}\delta\Gamma^{\tau}_{\tau
i}+\partial_{j}\delta\Gamma^{j}_{\tau i}\\\
&=-\frac{1}{2}\alpha^{m+1}\partial_{i}\left(X^{(m+1)}\right)-\frac{1}{2}\alpha^{m+1}\left(X^{(m+1)}\omega_{i}^{(0)}+X^{(0)}\omega_{i}^{(m+1)}\right)\end{split}$
(82) $\begin{split}\bar{\Gamma}^{D}_{DE}\delta\Gamma^{E}_{\tau
i}&=\bar{\Gamma}^{D}_{D\tau}\delta\Gamma^{\tau}_{\tau i}\\\
&=\left(\bar{\Gamma}^{\tau}_{\tau\tau}+\bar{\Gamma}^{\rho}_{\rho\tau}\right)\delta\Gamma^{\tau}_{\tau
i}\\\
&=\left(\frac{1}{2}X^{(0)}-\frac{1}{2}X^{(0)}\right)\left(-\frac{1}{2}\alpha^{m+1}\omega_{i}^{(m+1)}\right)\\\
&=0\end{split}$ (83)
$\begin{split}\bar{\Gamma}^{E}_{D\tau}\delta\Gamma^{D}_{Ei}&=\bar{\Gamma}^{\tau}_{D\tau}\delta\Gamma^{D}_{\tau
i}+\bar{\Gamma}^{\rho}_{D\tau}\delta\Gamma^{D}_{\rho
i}+\bar{\Gamma}^{j}_{D\tau}\delta\Gamma^{D}_{ji}\\\
&=\bar{\Gamma}^{\tau}_{\tau\tau}\delta\Gamma^{\tau}_{\tau
i}+\bar{\Gamma}^{\rho}_{\rho\tau}\delta\Gamma^{\rho}_{\rho
i}+\bar{\Gamma}^{j}_{\rho\tau}\delta\Gamma^{\rho}_{ji}\\\
&=\frac{1}{2}X^{(0)}\left(-\frac{1}{2}\alpha^{m+1}\omega_{i}^{(m+1)}\right)-\frac{1}{2}X^{(0)}\frac{1}{2}\alpha^{m+1}\omega_{i}^{(m+1)}\\\
&=-\frac{1}{2}X^{(0)}\alpha^{m+1}\omega_{i}^{(m+1)}\end{split}$ (84)
$\begin{split}\bar{\Gamma}^{E}_{Di}\delta\Gamma^{D}_{\tau
E}&=\bar{\Gamma}^{\tau}_{Di}\delta\Gamma^{D}_{\tau\tau}+\bar{\Gamma}^{\rho}_{Di}\delta\Gamma^{D}_{\tau\rho}+\bar{\Gamma}^{j}_{Di}\delta\Gamma^{D}_{\tau
j}\\\ &=\bar{\Gamma}^{\tau}_{\tau
i}\delta\Gamma^{\tau}_{\tau\tau}+\left(\bar{\Gamma}^{\rho}_{\rho
i}\delta\Gamma^{\rho}_{\tau\rho}+\bar{\Gamma}^{\rho}_{ji}\delta\Gamma^{j}_{\tau\rho}\right)+\bar{\Gamma}^{j}_{\tau
i}\delta\Gamma^{\tau}_{\tau j}\\\
&=-\frac{1}{2}\omega_{i}^{(0)}\frac{1}{2}\alpha^{m+1}X^{(m+1)}+\frac{1}{2}\omega_{i}^{(0)}\left(-\frac{1}{2}\alpha^{m+1}X^{(m+1)}\right)\\\
&=-\frac{1}{2}\omega_{i}^{(0)}\alpha^{m+1}X^{(m+1)}\end{split}$ (85)
$\begin{split}\partial_{i}\delta\Gamma^{D}_{\tau
D}&=\partial_{i}\left(\delta\Gamma^{\rho}_{\tau\rho}+\delta\Gamma^{\tau}_{\tau\tau}\right)\\\
&=\partial_{i}\left(-\frac{1}{2}\alpha^{m+1}X^{(m+1)}+\frac{1}{2}\alpha^{m+1}X^{(m+1)}\right)\\\
&=0\end{split}$ (86)
$\begin{split}\bar{\Gamma}^{E}_{i\tau}\delta\Gamma^{D}_{ED}&=\bar{\Gamma}^{\tau}_{i\tau}\delta\Gamma^{D}_{\tau
D}\\\
&=\bar{\Gamma}^{\tau}_{i\tau}\left(\delta\Gamma^{\rho}_{\tau\rho}+\delta\Gamma^{\tau}_{\tau\tau}\right)\\\
&=-\frac{1}{2}\omega_{i}^{(0)}\left(-\frac{1}{2}\alpha^{m+1}X^{(m+1)}+\frac{1}{2}\alpha^{m+1}X^{(m+1)}\right)\\\
&=0\end{split}$ (87)

Substituting eq.(82) - eq.(87) in eq.(81) we get

$\delta R_{\tau i}=-\frac{1}{2}\alpha^{m+1}\partial_{i}\left(X^{(m+1)}\right)$
(88)

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