aainstitutetext: Department of Physics, Indian Institute of Technology (Indian
School of Mines) Dhanbad, Jharkhand 826004, Indiabbinstitutetext: Department
of Physics, Indian Institute of Technology Kanpur, Kalyanpur, Kanpur 208016,
Indiaccinstitutetext: School of Physical Sciences, National Institute of
Science Education and Research,
An OCC of Homi Bhabha National Institute, Jatni-752050, India

# Iyer-Wald ambiguities and gauge covariance of Entropy current in Higher
derivative theories of gravity

Alokananda Kar b Prateksh Dhivakar c Shuvayu Roy a Binata Panda a Anowar
Shaikh  21dr0013@phy.iitism.ac.in,prateksh<EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In Bhattacharyya:2021jhr ; Biswas:2022grc  [arXiv:2105.06455,
arXiv:2206.04538], the authors have been able to argue for an ultra-local
version of the second law of black hole mechanics, for arbitrary
diffeomorphism invariant theories of gravity non-minimally coupled to matter
fields, by constructing an entropy current on the dynamical horizon with
manifestly positive divergence. This has been achieved by working in the
horizon-adapted coordinate system. In this work, we show that the local
entropy production through the divergence of the entropy current is covariant
under affine reparametrizations that leave the gauge of horizon-adapted
coordinates invariant. We explicitly derive a formula for how the entropy
current transforms under such coordinate transformations. This extends the
analysis of Bhattacharyya:2022njk  [arXiv:2204.08447] for arbitrary
diffeomorphism invariant theories of gravity non-minimally coupled to matter
fields. We also study the Iyer-Wald ambiguities of the covariant phase
formalism that generically plague the components of the entropy current.

##  1 Introduction

In modern theoretical physics, black holes serve as the best probe for
understanding the mysteries of Quantum gravity. Black holes are an interesting
class of solutions to Einstein’s General Relativity (GR) to which one can
associate thermodynamic notions like energy, temperature and, crucially,
entropy given by the area of the horizon PhysRevLett.26.1344 ; Bardeen:1973gs
; PhysRevD.7.2333 ; Hawking:1975vcx . Einstein’s GR is not UV complete.
Effective Field Theory tells us that any UV complete theory of Quantum gravity
in the low energy limit reduces to Einstein’s GR with corrections coming from
higher derivative interactions and couplings Donoghue:1994dn . Since we expect
to find a consistent statistical picture of thermodynamics for black holes, it
is of interest to study whether the laws of black hole thermodynamics continue
to hold for such higher derivative theories beyond GR PhysRevD.48.R3427 ;
Iyer:1994ys ; Jacobson:1993xs ; Jacobson:1993vj ; Jacobson:1995uq ;
Sarkar:2013swa ; Bhattacharjee:2015yaa 111For a review, see Wall:2018ydq ;
Sarkar:2019xfd ..

Iyer and Wald constructed an entropy functional that satisfied the first law
of black hole thermodynamics for arbitrary diffeomorphism invariant theories
of gravity PhysRevD.48.R3427 ; Iyer:1994ys . Using covariant phase space
methods 10.1063/1.528801 , they worked out the modifications to the area law
in the form of Iyer-Wald entropy that now satisfied the first law by
construction. Since the first law is a statement about equilibrium, the Iyer-
Wald entropy functional was unambiguously defined only for stationary Killing
horizons. It suffered from JKM ambiguities arising from the covariant phase
space formalism for dynamical situations Jacobson:1993vj . This is expected
because the second law is a statement about dynamics, and these higher
derivative terms modify Einstein’s Equations of Motion (EoM), which is crucial
for the proof of a second law.

Wall Wall:2015raa  222Wall’s algorithm has been extended to the non-linear
order in Hollands:2022fkn ; Davies:2023qaa ; Davies:2024fut . fixed the JKM
ambiguities in such a way that his entropy functional satisfied a second law
up to linear order in dynamics. The proof of the second law was constructed in
a dynamical situation where the black hole “is settling down to equilibrium”.
Thus, one is perturbatively close to the equilibrium configuration (with a
Killing horizon) at any point of evolution. Within this setup, the proof
amounted to establishing that the entropy functional increases in time up to
linear order in dynamics.

Following the lead of Wall:2015raa , recently in Bhattacharyya:2021jhr ;
Biswas:2022grc , the authors construct an ultra-local proof of the second law
for arbitrary diffeomorphism invariant theories of gravity up to linear order
in the dynamics 333As argued in Adami:2021kvx , it is reasonable to expect a
local formulation of the laws of thermodynamics arising from diffeomorphism
invariance.. In its strongest form, the second law could be expressed through
an entropy current with positive divergence. This entropy current on the black
hole horizon has not only a time component444 Here, the “time” component is
the component along the null generators of the horizon. Thus, the time is
within quotes because it is a null coordinate on the horizon. Though we
together call the time and spatial components a current, this is a non-
covariant current. These non-covariances arise from the fact that the metric
structure degenerates on the horizon. We will precisely quantify these non-
covariances in this paper. that is just Wall’s modifications to Iyer-Wald
entropy, but also crucially spatial components that capture the entropy re-
distribution along the spatial slices of the horizon.

Since dynamical situations are complicated even within the purview of GR, we
often resort to a choice of gauge that simplifies our analysis.
Bhattacharya:2019qal ; Bhattacharyya:2021jhr ; Biswas:2022grc  worked with a
horizon-adapted choice of coordinates detailed in Bhattacharyya:2016xfs  to
establish a local entropy current on the horizon. This essentially amounts to
choosing a specific slicing for the horizon. Let us now consider the following
situation. Some matter falls into the horizon of an initially stationary black
hole. The black hole finally settles down to a stationary solution. In this
process 555The Physical process version of the first law is very closely to
this setup of second law Gao:2001ut ; Amsel:2007mh ; Chatterjee:2011wj ;
Bhattacharjee:2014eea ; Mishra:2017sqs ., certain regions of the horizon might
be closer to equilibrium than other regions Bhattacharya:2019qal . This
physics is captured by a flow of entropy current between the different sub-
regions of the horizon. One would expect that the local entropy production
should not be affected by the choice of slicing of the horizon. If the entropy
production depends on the choice, then we could, in principle, choose a
particular slicing with a “sink” where entropy decreases locally. This would
not be desirable for an ultra-local version of the second
lawBhattacharya:2019qal ; Bhattacharyya:2021jhr ; Biswas:2022grc .

We aim to address this question of coordinate independence in this paper. To
linearized order in dynamics of amplitude, entropy production is directly
given by the divergence of the entropy current 666Recently in Davies:2023qaa ,
the authors have explored a non-perturbative formulation of the second law in
effective field theories.. By construction, this entropy current is
intricately tied to the horizon-adapted choice of coordinates
Bhattacharya:2019qal ; Bhattacharyya:2021jhr . Following up on some initial
observations in Bhattacharyya:2022njk , we will show that the entropy
production in black holes of arbitrary diffeomorphism invariant theories
admits a symmetry under coordinate transformations that preserve the form of
the horizon-adapted gauge. This fact was explicitly verified for the
particular case of Gauss-Bonnet theory by brute force computation in
Bhattacharyya:2022njk . The non-trivial part of our analysis is the fact that
the components of the codimension one entropy current “vector” are defined on
a null manifold 777As mentioned in footnote 4, though we loosely call it the
codimension one current, it should be understood that this is a “vector”
defined on a degenerate surface.. Thus, these components do not transform
covariantly under coordinate transformations. Despite these complications, we
w ill show that local entropy production remains unaffected under a change of
slicing of the horizon.

The symmetries that preserve the horizon-adapted gauge are affine
reparametrizations of the null generators of the horizon Bhattacharyya:2022njk
. The transformation of the components of the entropy current “vector” will
involve parts that are covariant, which are expected from the covariant
transformation of a vector, and additional parts that are non-covariant. The
non-covariant pieces can be interpreted as arising due to ambiguities in the
definition of the entropy current. Due to the dynamical situation, we are
interested in, the ambiguities of the covariant phase space formalism become
non-trivial Jacobson:1993vj . In this work, we study the Iyer-Wald ambiguities
of the covariant phase space formalism Jacobson:1993vj  that generically
plague the components of the entropy current constructed in
Bhattacharyya:2021jhr .

This paper is organized as follows. In Sec. 2, we give details of the
statement of the problem and summarize the main results of our paper. These
are: a formula for the transformation of the components of the entropy under
affine reparametrizations that relate the different classes of horizon-adapted
coordinates and the precise origin of ambiguities in the components of the
entropy current. In Sec. 3, we give a formal proof of the gauge covariance of
entropy production up to linear order in perturbations under affine
reparametrizations of the null generators. In Sec. 4, we relate the non-
covariances in the transformation of the components of the entropy current
with the Iyer-Wald ambiguities of the covariant phase space formalism. We will
also analyze the ambiguities in the physical process version of the first law
between arbitrary time slices of the horizon. In Sec. 5, we explicitly verify
the transformation formula of the components of the entropy current for
explicit diffeomorphism invariant theories of gravity. In Sec. 6, we look at
Gauss-Bonnet theory and study the gauge covariance to non-linear order in
perturbations. We finally conclude with some discussions of our results in
Sec. 7. The details of the computation are made explicit in various
appendices.

##  2 A symmetry of local entropy production

In this section, we lay out the results of the main calculation by stating the
problem. We will also briefly introduce the elements that will be crucial for
the following sections. The basic setup and the construction of a local
entropy current on the horizon are given in Appendix A. For more details, one
must consult section 2 of Bhattacharyya:2021jhr .

###  2.1 Statement of the problem and summary of results

As mentioned in the introduction, we would be interested in the physical
situation where the black hole is “settling down to equilibrium”. Thus, in the
far future, the dynamical horizon becomes a Killing horizon that satisfies the
Zeroth law of black hole mechanics Bhattacharyya:2022nqa ; Hollands:2022ajj ;
Davies:2024fut . This dynamical situation is effectively captured by the
horizon-adapted gauge of Bhattacharyya:2016xfs ; Bhattacharya:2019qal ;
Bhattacharyya:2021jhr . In this gauge, the spacetime of any $D$ dimensional
dynamical black solution is spanned by coordinates $y^{\mu}=\\{v,r,x^{i}\\}$
with the horizon located at $r=0$. The coordinate $v$ is chosen to be an
affine parameter along the null generators of the horizon $\partial_{v}$.
Along the constant $v$ slices on the horizon, we have the spatial coordinates
$x^{i}$ spanned by the spatial tangents $\partial_{i}$ of the horizon. $r$ is
an affine parameter along the null geodesics generated by $\partial_{r}$,
which make specific angles with $\partial_{v}$ and $\partial_{i}$. Thus, the
metric is given by

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

This metric is schematically given by the figure 1. The explicit examples of
Schwarzschild and Kerr black holes expressed in this form can be found in the
Appendix of Dhivakar:2023zqn . The dynamics are encoded in the metric
functions $X,\omega_{i},h_{ij}$ in the form of fluctuation atop an equilibrium
solution given by $g_{\mu\nu}=g^{eq}_{\mu\nu}+\epsilon\,\delta g_{\mu\nu}$.
Here $g^{eq}_{\mu\nu}$ is the equilibrium solution, and $\epsilon\ll 1$ is a
small parameter characterizing the deviations from an equilibrium solution.

Figure 1: Schematic representation of the horizon-adapted coordinates

Consider the most general diffeomorphism invariant theory of gravity given by
the following Lagrangian

$\mathcal{L}=\mathcal{L}\left(g_{\mu\nu},R_{\mu\nu\rho\sigma},D_{\alpha_{1}}R_{\mu\nu\rho\sigma},\dots,\phi,D_{\alpha_{1}}\phi,\dots,F_{\mu\nu},D_{\alpha_{1}}F_{\mu\nu},\dots\right)\,.$
(2)

Here $D_{\mu}$ denotes the covariant derivative with respect to the full
metric, $\phi$ denotes a scalar field, and $F_{\mu\nu}$ denotes the $U(1)$
gauge field strength tensor. Using constraints from diffeomorphism invariance
and a particular symmetry of the Killing horizon of $g^{eq}_{\mu\nu}$, the
authors of Bhattacharya:2019qal ; Bhattacharyya:2021jhr ; Biswas:2022grc  have
shown that the equations of motion projected on the horizon along the null
generators take a universal form at linear order, given by

$E_{vv}|_{r=0}=\partial_{v}\left(\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}\,\mathcal{J}^{v}\right)+\dfrac{1}{\sqrt{h}}\partial_{i}\left(\sqrt{h}\,\mathcal{J}^{i}\right)\right)+\mathcal{O}(\epsilon^{2})\,.$
(3)

($h=\text{det}(h_{ij})$) This special form of the $E_{vv}$ will lead to a
local second law at the linear order

$\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}\,\mathcal{J}^{v}\right)+\dfrac{1}{\sqrt{h}}\partial_{i}\left(\sqrt{h}\,\mathcal{J}^{i}\right)\geq
0~{}~{}\text{upto}~{}\mathcal{O}(\epsilon^{2})\,,$ (4)

as explained above eq.(146).

The structure of $E_{vv}$ crucially depends on the choice of slicing in eq.(1)
used. We ask the question as to what happens to the structure under a set of
transformations that change the slicing of the horizon but keep the gauge
intact. In other words, we are interested in understanding the covariance
under gauge preserving transformations. This covariance is important to
establish the robustness of the local entropy production at all regions of the
horizon. Affine reparametrizations of the form

$v\to\tau=v\,e^{-\zeta(\vec{x})}\,,~{}~{}~{}~{}x^{a}\to y^{a}=x^{a}\,,$ (5)

employ a change of coordinates $(v,r,x^{a})$ to $(\tau,\rho,y^{a})$ that leave
the form of the metric eq.(1) invariant 888As we will detail in section 2.2,
one also has to do a transformation of $r$ to ensure that the form eq.(1) is
intact away from the horizon $r=0$. Bhattacharyya:2022njk . Thus, it is
crucial to study the transformation of entropy currents under eq.(5) to see if
local entropy production is indeed preserved.

The components of the entropy current expressed through the structure of
$E_{vv}$ eq.(3) are also ambiguous in the following sense. Suppose we have the
quantities $\mathcal{J}^{v}_{\text{amb}}$ and $\mathcal{J}^{i}_{\text{amb}}$
satisfying

$\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}\,\mathcal{J}^{v}_{\text{amb}}\right)+\dfrac{1}{\sqrt{h}}\partial_{i}\left(\sqrt{h}\,\mathcal{J}^{i}_{\text{amb}}\right)=\mathcal{O}(\epsilon^{2})\,,$
(6)

then one can always add these currents to eq.(3) without changing $E_{vv}$
(i.e.)

$E_{vv}|_{r=0}=\partial_{v}\left(\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}\,\left(\mathcal{J}^{v}+\mathcal{J}^{v}_{\text{amb}}\right)\right)+\dfrac{1}{\sqrt{h}}\partial_{i}\left(\sqrt{h}\,\left(\mathcal{J}^{i}+\mathcal{J}^{i}_{\text{amb}}\right)\right)\right)+\mathcal{O}(\epsilon^{2})\,.$
(7)

Since we are relating the divergence of the current to local entropy
production as in eq.(4), the components of the current contain information of
entropy redistribution. Thus, it is important to study the origin of these
ambiguities and quantify them appropriately.

In this paper, we would like to address these questions. Given these details,
the main results of our paper are as follows:

  1. 1.

The components of entropy current $\mathcal{J}^{v}$ and $\mathcal{J}^{i}$
transform under eq.(5) in the following form on the horizon $r=0$:

$\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}\mathcal{J}^{v}\right)+\nabla_{i}\mathcal{J}^{i}=e^{-\zeta}\left[\dfrac{1}{\sqrt{\widetilde{h}}}\partial_{\tau}\left(\sqrt{\widetilde{h}}\widetilde{\mathcal{J}}^{\tau}\right)+\widetilde{\nabla}_{i}\widetilde{\mathcal{J}}^{i}\right]+\mathcal{O}(\epsilon^{2})\,.$
(8)

Here $\widetilde{\mathcal{J}}^{\tau}$ and $\widetilde{\mathcal{J}}^{a}$ 999We
will use tilde to denote quantities in the $(\rho,\tau,y^{i})$ coordinates.
are the components of the entropy current expressed in the $(\rho,\tau,y^{i})$
coordinates and are explicitly given by

$\begin{split}\mathcal{J}^{v}&=\widetilde{\mathcal{J}}^{\tau}+\tau\xi_{i}\widetilde{\mathcal{J}}^{i}-2\tau^{2}\xi_{i}\xi_{j}\partial_{\tau}\widetilde{L}^{ij}_{(1)}+2\tau\widetilde{\nabla}_{i}\left(\xi_{j}\widetilde{L}^{ij}_{(1)}\right)-2\tau\xi_{i}\xi_{j}\widetilde{L}^{ij}_{(1)}\,,\\\
\mathcal{J}^{i}&=e^{-\zeta}\widetilde{J}^{i}-2e^{-\zeta}(1+\tau\partial_{\tau})(\xi_{j}\widetilde{L}^{ij}_{(1)})\,,\end{split}$
(9)

where $\xi_{i}=\partial_{i}\zeta$ and $\widetilde{L}_{ij}$ is determined from
the Lagrangian $\mathcal{L}$ of eq.(2) as

$\frac{\partial\mathcal{L}}{\partial\tilde{R}_{\rho i\rho
j}}\equiv\tilde{E}_{R}^{\rho i\rho
j}\equiv\partial_{\tau}\widetilde{L}_{(1)}^{ij}+O(\epsilon^{2})$

From eq.(8), we can see that the local entropy production of eq.(4) is
unaffected. We show this in section 3.

  2. 2.

The extraction of $\mathcal{J}^{v}$ and $\mathcal{J}^{i}$ following eq. (3) is
ambiguous up to terms of the form

$\begin{split}\mathcal{J}^{v}&\rightarrow\mathcal{J}^{v}+\nabla_{i}C^{vi}\,,\\\
\mathcal{J}^{i}&\rightarrow\mathcal{J}^{i}-\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}\,C^{vi}\right)+\nabla_{j}C^{ij}\,,\end{split}$
(10)

where $C^{vi}$ and $C^{ij}$ are explicitly determined in terms of the
ambiguities in the symplectic potential $\Theta^{\mu}$ and Iyer-Wald Noether
Charge $Q^{\mu\nu}$ derived from the Lagrangian $\mathcal{L}$ of eq.(2). Iyer-
Wald ambiguities can occur in $\mathcal{L},\Theta^{\mu}$ and $Q^{\mu\nu}$ as
$\mathcal{L}\rightarrow\mathcal{L}+D_{\rho}M^{\rho},\Theta^{\mu}\rightarrow\Theta\mu+D_{\nu}Y^{\mu\nu}$and
$Q^{\mu\nu}\rightarrow Q^{\mu\nu}+D_{\rho}Z^{\mu\nu\rho}$. Of these, only the
$Y$-type and $Z$-type ambiguities contribute to $C^{vi}$ and $C^{ij}$. This is
derived in section 4.

These form the main results of our paper. At this point, we mention that a
special case of eq.(9) and eq.(10) has been considered in Hollands:2022fkn .
The existence of $\widetilde{L}^{ij}_{(1)}$ has not been discussed before.
This hints at a richer structure, and its existence is a consequence of
working on a null horizon with a degenerate structure. We will comment on the
comparison to Hollands:2022fkn  towards the end of section 3 and 4.

The basic setup of the construction of an entropy current on the horizon is
reviewed in Appendix A. We will now study how these affine reparametrizations
of eq.(5) relate to the various choices of horizon-adapted coordinates. This
will be crucial to answer the question of whether entropy production is
covariant under such transformations.

###  2.2 The various choices of horizon-adapted coordinates

To study the gauge covariance of local entropy production as explicated in
eq.(8), we must first study the precise relation between the different
coordinate systems related by an affine reparametrization of the form eq.(5).
We follow the analysis of section 3.1 of Bhattacharyya:2022njk . This affine
reparametrization leaves the form of the metric eq.(1) invariant. Eq.(5) is,
in fact, a residual symmetry of the horizon-adapted coordinates as reviewed in
eq.(132). The transformed metric is thus written as

$ds^{2}=2d\tau\,d\rho-\rho^{2}\,\widetilde{X}(\rho,\tau,y^{j})\,d\tau^{2}+2\rho\,\widetilde{\omega}(\rho,\tau,y^{j})\,d\tau\,dy^{i}+\widetilde{h}_{ij}(\rho,\tau,y^{k})\,dy^{i}\,dy^{j}\,.$
(11)

The transformation in eq.(5) is only valid on the horizon. To obtain the
transformation, if one moves away from the horizon, the $r$ coordinate must be
changed so that the angle between the tangents along the constant
$\\{\tau,y^{a}\\}$ and the coordinate vector is kept constant. This leads to
the redefinition of $r$ and the correction of coordinate transformation as one
moves away from the horizon. The full coordinates transformations are as
follows:

$\begin{split}r&=\rho
e^{-\zeta}\left(1+\sum_{n=1}(\rho\tau)^{n}R_{n}(\rho\tau,y)\right)\,,\\\
v&=\tau
e^{\zeta}\left(1+\sum_{n=1}(\rho\tau)^{n}V_{n}(\rho\tau,y)\right)\,,\\\
x^{a}&=y^{a}+\sum_{n=1}(\rho\tau)^{n}Z_{n}^{a}(\rho\tau,y)\,.\end{split}$ (12)

Since we will be dealing with structures on the horizon, it suffices to have
these expansion coefficients $(R_{n},V_{n},Z^{a}_{n})$ up to the $n=1$ term.
By imposing $D$ gauge conditions $\\{g_{\rho\rho}=0\,g_{\rho\tau}=0\,g_{\rho
a}=0\\}$ their expressions can be derived as follows Bhattacharyya:2022njk :

$\displaystyle Z_{1}^{i}$ $\displaystyle=-\widetilde{h}^{ij}\xi_{j}\,,$
$\displaystyle V_{1}$
$\displaystyle=-\frac{1}{2}\xi_{i}\xi_{j}\widetilde{h}^{ij}\,,$ $\displaystyle
R_{1}$
$\displaystyle=\frac{1}{2}\left(h^{ij}\omega_{i}\xi_{j}+\tau\xi_{i}\xi_{j}\widetilde{K}^{ij}\right)\,.$
(13)

Here, $\xi_{i}=\partial_{i}\zeta(x^{a})$. Using these expressions of the
expansion coefficients, the change of the metric coefficients
$(X,\omega_{i},h_{ij})$ up to the required orders of $\rho$ can be derived to
be:

$\begin{split}\widetilde{\omega}_{i}&={\omega}_{i}-2\xi_{i}+2\tau\xi_{k}\widetilde{h}^{jk}\widetilde{K}_{ij}+\mathcal{O}(\rho)\,,\\\
\widetilde{X}&=X+\omega_{i}\xi_{j}\widetilde{h}^{ij}-\xi_{i}\xi_{j}\widetilde{h}^{ij}-\tau\left(\xi_{i}\widetilde{h}^{ij}\partial_{\tau}\omega_{j}-2\omega_{i}\xi_{j}\widetilde{K}^{ij}+2\xi_{i}\xi_{j}\widetilde{K}^{ij}\right)\\\
&-\tau^{2}\left(\xi_{i}\xi_{l}\widetilde{h}^{kl}\widetilde{h}^{ij}\partial_{\tau}\widetilde{K}_{jk}\right)+\mathcal{O}(\rho)\,,\\\
\widetilde{h}_{ij}&={h}_{ij}-\rho\tau\left(\xi_{i}\omega_{j}+\omega_{i}\xi_{j}-2\xi_{i}\xi_{j}-2\xi_{ij}+\xi_{b}\widetilde{h}^{ab}\widetilde{\partial}_{i}\widetilde{h}_{ja}+\xi_{b}\widetilde{h}^{ab}\widetilde{\partial}_{j}\widetilde{h}_{ia}\right)+O(\rho^{2})\,.\\\
\end{split}$ (14)

The basis vectors in the $(r,v,x^{a})$ coordinates can be expressed in terms
of the $(\rho,\tau,y^{a})$ coordinates and the fundamental structures
$(\widetilde{X},\widetilde{\omega}_{i},\widetilde{h}_{ij})$ as

$\begin{split}\partial_{r}&=e^{\zeta}\left(\partial_{\rho}-\frac{1}{2}\tau^{2}\xi_{a}\xi_{b}\widetilde{h}^{ab}\partial_{\tau}+\tau\xi_{a}\widetilde{h}^{ab}\widetilde{\partial_{b}}\right)+O(\rho)\,,\\\
\partial_{v}&=e^{-\zeta}\partial_{\tau}+O(\rho)\,,\\\
\partial_{x^{i}}&=\widetilde{\partial}_{y^{i}}-\tau~{}\xi_{i}~{}\partial_{\tau}+\mathcal{O}(\rho)\,.\end{split}$
(15)

The details of how various quantities transform under a reparametrization of
the form eq.(5) are given in Appendix C. These structures will be important
when we study how the components of the entropy current transform in the
subsequent section 3.

##  3 Proof of gauge covariance of the entropy current

In this section, we will set out to prove the gauge covariance of local
entropy production for arbitrary diffeomorphism invariant theories of gravity
as predicted by eq.(8). This was checked for a particular case of Gauss-Bonnet
theory in Bhattacharyya:2022njk . We will generalize their result for
arbitrary diffeomorphism invariant theories of gravity non-minimally coupled
to matter fields of the form in eq.(2). The currents $\mathcal{J}^{v}$ and
$\mathcal{J}^{i}$ are quantities defined on a null surface; consequently, they
are non-covariant structures. Thus, they must transform highly non-trivially
for eq.(8) to be satisfied. We will explicitly derive the first result of our
paper given in eq.(9). The details of the calculations are presented in
Appendix B.

We will start with the key equation eq.(152) that goes into the proof of the
existence of the entropy current for arbitrary diffeomorphism invariant
theories of gravity Bhattacharyya:2021jhr :

$2v\,E_{vv}|_{r=0}=\left(-\Theta^{r}+D_{\mu}Q^{r\mu}\right)|_{r=0}\,.$ (16)

$\Theta^{r}$ and $Q^{r\mu}$ evaluated on the horizon $r=0$ take the form of
eq.(153):

$\begin{split}\Theta^{r}|_{r=0}&=(1+v\partial_{v})\mathcal{A}_{(1)}+v\partial^{2}_{v}\mathcal{B}_{(0)}+\mathcal{O}(\epsilon^{2})\,,\\\
Q^{r\mu}&=\mathcal{Q}^{r\mu}+v\,W^{r\mu}_{v}\,.\end{split}$ (17)

This results in eq.(157) and thus, we have proved eq.(3) with the components
of the current given by

$\mathcal{J}^{v}=-\dfrac{1}{2}\left(\mathcal{Q}^{rv}+\mathcal{B}_{(0)}\right)\,,~{}~{}~{}~{}~{}~{}\mathcal{J}^{i}=-\dfrac{1}{2}\left(\mathcal{Q}^{ri}-J^{i}_{(1)}\right)\,.$
(18)

The subscripts in the above expressions denote the boost weight (eq.(138)) of
various quantities. Under an affine reparametrization of the form eq.(5),

$v\to\tau=v\,e^{-\zeta(\vec{x})}\,,~{}~{}~{}~{}x^{a}\to y^{a}=x^{a}\,,$ (19)
$E_{vv}=e^{-2\zeta}E_{\tau\tau}\,,$ (20)

as EoM of any diffeomorphism invariant Lagrangian is covariant. The result
eq.(8) given by

$\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}\mathcal{J}^{v}\right)+\nabla_{i}\mathcal{J}^{i}=e^{-\zeta}\left[\dfrac{1}{\sqrt{\widetilde{h}}}\partial_{\tau}\left(\sqrt{\widetilde{h}}\widetilde{\mathcal{J}}^{\tau}\right)+\widetilde{\nabla}_{i}\widetilde{\mathcal{J}}^{i}\right]+\mathcal{O}(\epsilon^{2})\,,$
(21)

which we are trying to prove is certainly consistent with eq.(20). But to
argue that eq.(20) implies eq.(21) is non-trivial. This is because
$\mathcal{J}^{v}$ and $\mathcal{J}^{i}$ are non-covariant structures according
to eq.(18). The main element of our proof is that the non-covariant terms are
extracted from covariant quantities like $\Theta^{r}$ and $Q^{r\mu}$ through
eq.(17). We know how covariant quantities transform under eq.(5), and this, in
turn, will fix how the non-covariant quantities transform under eq.(5). The
main steps of the proof are summarized schematically in the figure 2.

Figure 2: Different slicings of the horizon-adapted coordinates

#### Structures in $Q^{r\mu}$:

We will first work out the transformation properties of structures extracted
from $Q^{r\mu}$. The Noether charge $Q^{r\mu}$ is covariant. The construction
of the Noether charge in Proposition 4.1 of Iyer:1994ys  has shown that

$Q^{\mu\nu}=W^{\mu\nu\rho}\xi_{\rho}-2E^{\mu\nu\alpha\beta}_{R}D_{[\alpha}\xi_{\beta]}+Y^{\mu\nu}+D_{\alpha}Z^{\mu\nu\alpha},$
(22)

where $Y^{\mu\nu}$ and $Z^{\mu\nu\alpha}$ are the possible ambiguities in the
definition of Noether Charge. The subtleties of the ambiguities will be
carefully analysed in section 4. Neglecting the ambiguities for the purposes
of this section, we have

$Q^{\mu\nu}=W^{\mu\nu\rho}\xi_{\rho}-2E^{\mu\nu\rho\sigma}_{R}D_{[\rho}\xi_{\sigma]}\,,$
(23)

where $E^{\mu\nu\rho\sigma}_{R}$ is defined from the Lagrangian in eq.(2) as
101010“$(\mu\nu)$” denote the symmetrization in $\mu$ and $\nu$ indices and
“$[\mu\nu]$” denote anti symmetrization. The convention used for
symmetrization/antisymmetrization of indices is as follows:
$A^{(\alpha}B^{\beta)}=({1}/{2})(A^{\alpha}B^{\beta}+A^{\beta}B^{\alpha}$).

$E^{\mu\nu\rho\sigma}_{R}=\dfrac{\partial\,\mathcal{L}}{\partial
R_{\mu\nu\rho\sigma}}-D_{\alpha_{1}}\left(\dfrac{\partial\,\mathcal{L}}{\partial
D_{\alpha_{1}}R_{\mu\nu\rho\sigma}}\right)+D_{(\alpha_{1}}D_{\alpha_{2})}\left(\dfrac{\partial\,\mathcal{L}}{\partial
D_{(\alpha_{1}}D_{\alpha_{2})}R_{\mu\nu\rho\sigma}}\right)+\dots\,,$ (24)

and $\xi^{\mu}$ is the Killing vector eq.(136). Since eq.(23) is linear in
$\xi^{\mu}$, the structure of $Q^{r\mu}$ on the horizon $r=0$ is given by
eq.(153)

$Q^{r\mu}|_{r=0}=\mathcal{Q}^{r\mu}+v\,W^{r\mu}_{v}\,,$ (25)

where the explicit expressions are

$\begin{split}\mathcal{Q}^{rv}&=4E^{rvrv}_{R}\,,~{}~{}~{}~{}~{}~{}\mathcal{Q}^{ri}=4E^{rirv}_{R}\,,\\\
W^{rv}_{v}&=W^{rvr}-2E^{rirv}_{R}\omega_{i}\,,\\\
W^{ri}_{v}&=W^{rir}-2E^{rirj}_{R}\omega_{j}=\partial_{v}J^{i}_{(1)}+\mathcal{O}(\epsilon^{2})\,.\end{split}$
(26)

In the expression of $W^{ri}_{v}$, we have chosen to express a boost weight
$+2$ quantity as $\partial_{v}J^{i}_{(1)}$ up to terms that are
$\mathcal{O}(\epsilon^{2})$ according to eq.(141). This structure of
$Q^{r\mu}$ in (26) makes it easier to study the transformation property of
$\mathcal{Q}^{r\mu}$ and $W^{r\mu}_{v}$. We will see that these transformation
properties will fix the transformation of $\mathcal{A}_{(1)}$ and subsequently
$\mathcal{B}_{(0)}$.

Since $\mathcal{L}$ of eq.(2) is a diffeomorphism invariant Lagrangian,
$E^{\mu\nu\rho\sigma}_{R}$ given by eq.(24) is a covariant tensor. Thus, under
a reparametrization of eq.(5), using the metric transformation eq.(14), we can
straightforwardly work out how the various quantities in eq.(26) transform.
For instance, the transformation of $\mathcal{Q}^{ri}$ and $W^{ri}_{v}$ is
given by

$\begin{split}\mathcal{Q}^{ri}&=e^{-\zeta}\widetilde{\mathcal{Q}}^{\rho
i}+4e^{-\zeta}\tau\xi_{b}\widetilde{E}^{\rho i\rho b}_{R}\,,\\\
W^{ri}_{v}&=e^{-2\zeta}\widetilde{W}^{\rho
i}_{\tau}-4e^{-2\zeta}\xi_{b}\widetilde{E}^{\rho i\rho b}_{R}\,.\end{split}$
(27)

This is checked in eq.(158) and eq.(159). As a sanity check, one can see that
the eq.(27) ensures that $Q^{ri}$ transforms covariantly as expected (i.e.) it
transforms as

$Q^{ri}=\dfrac{\partial r}{\partial\rho}\dfrac{\partial x^{i}}{\partial
y^{a}}\widetilde{Q}^{\rho a}=e^{-\zeta}\widetilde{Q}^{\rho i}\,.$ (28)

From the transformation of $W^{ri}_{v}$ of eq.(27), we can derive the
transformation of $J^{i}_{(1)}$. For this, we first define a boost weight $+1$
quantity $\widetilde{L}^{ij}_{(1)}$ through the following equation:

$\widetilde{E}^{\rho i\rho
j}_{R}=\partial_{\tau}\widetilde{L}^{ij}_{(1)}+\mathcal{O}(\epsilon^{2})\,.$
(29)

This is because $E^{\rho i\rho j}_{R}$ is a boost weight $+2$ tensor, and we
have used eq.(141)111111It has been recently proved Hollands:2022fkn  that the
notion of boost weight remains invariant under an affine reparametrization of
the form eq.(5).. Thus from eq.(27), we get

$J^{i}_{(1)}=e^{-\zeta}\widetilde{J}^{i}_{(1)}-4e^{-\zeta}\xi_{j}\widetilde{L}^{ij}_{(1)}\,.$
(30)

This is checked in eq.(160). Finally, from the transformation of
$\mathcal{Q}^{ri}$ of eq.(27) and $J^{i}_{(1)}$ of eq.(30), we get the
transformation for $\mathcal{J}^{i}$ of eq.(18) as

$\mathcal{J}^{i}=e^{-\zeta}\widetilde{\mathcal{J}}^{i}-2e^{-\zeta}(1+\tau\partial_{\tau})(\xi_{j}\widetilde{L}^{ij}_{(1)})\,.$
(31)

This is checked in eq.(161).

To look at how $\mathcal{J}^{v}$ of eq.(18) transforms, we need the
transformation of $\mathcal{Q}^{rv}$ and $\mathcal{B}_{(0)}$.
$\mathcal{Q}^{rv}$ of eq.(26) transforms as a tensor because of eq.(24):

$\mathcal{Q}^{rv}=\widetilde{\mathcal{Q}}^{\rho\tau}+2\tau\xi_{i}\widetilde{\mathcal{Q}}^{\rho
i}+4\tau^{2}\xi_{i}\xi_{j}\partial_{\tau}\widetilde{L}^{ij}_{(1)}\,.$ (32)

This is checked in eq.(162). Similarly, one can find the transformation of
$W^{rv}_{v}$ of eq.(26) as follows

$W^{rv}_{v}=e^{-\zeta}\widetilde{W}^{\rho\tau}_{\tau}+e^{-\zeta}\tau\xi_{i}\widetilde{W}^{\rho
i}_{\tau}-4e^{-\zeta}\xi_{i}\widetilde{E}^{\rho i\rho\tau}_{R}-4\tau
e^{-\zeta}\xi_{i}\xi_{j}\partial_{\tau}\widetilde{L}^{ij}_{(1)}\,.$ (33)

This is checked in eq.(163). Again, as a sanity check, using eq.(32) and
eq.(33), one can straightforwardly see that $Q^{rv}$ transforms as expected
(as a tensor):

$Q^{rv}=\widetilde{Q}^{\rho\tau}+\tau\xi_{i}\widetilde{Q}^{\rho i}\,.$ (34)

This is checked in eq.(164). This completes our analysis of the structures in
$Q^{r\mu}$.

#### Structures in $\Theta^{r}$:

We now focus on the structures in $\Theta^{r}$. This is the non-trivial part
of our exercise. The transformation of $\mathcal{Q}^{rv}$ of eq.(32),
$\mathcal{Q}^{ri}$ of eq.(27), and the transformation of $W^{rv}_{v}$ of
eq.(33) completely fix the transformation of $\mathcal{A}_{(1)}$ in the
structure of $\Theta^{r}$ given by eq.(17). This is crucially because of the
identity given by eq.(156) (eq.(3.69) of Bhattacharyya:2021jhr ):

$\mathcal{A}_{(1)}=\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}\,\mathcal{Q}^{rv}\right)+\dfrac{1}{\sqrt{h}}\partial_{i}\left(\sqrt{h}\,\mathcal{Q}^{ri}\right)+W^{rv}_{v}\,.$
(35)

From the transformation of the quantities in the RHS of eq.(35), one can
determine the transformation of $\mathcal{A}_{(1)}$ as

$\mathcal{A}_{(1)}=e^{-\zeta}\widetilde{\mathcal{A}}_{(1)}+e^{-\zeta}\tau\xi_{i}\partial_{\tau}\widetilde{\mathcal{Q}}^{\rho
i}+e^{-\zeta}\tau\widetilde{\nabla_{i}}\left(4\xi_{j}\partial_{\tau}\widetilde{L}^{ij}_{(1)}\right)+e^{-\zeta}\tau\xi_{i}\partial_{\tau}\widetilde{J}^{i}_{(1)}-4e^{-\zeta}\tau\xi_{i}\xi_{j}\partial_{\tau}\widetilde{L}^{ij}_{(1)}\,.$
(36)

This is derived in eq.(165). We now use the most important point that is
central to the proof. $\Theta^{r}$ is covariant, and thus, it transforms
covariantly under the reparametrization of eq.(5). But $\mathcal{A}_{(1)}$ and
$\mathcal{B}_{(0)}$ are not covariant. Thus, $\mathcal{A}_{(1)}$ and
$\mathcal{B}_{(0)}$ must transform non-trivially to
$\widetilde{\mathcal{A}}_{(1)}$ and $\widetilde{\mathcal{B}}_{(0)}$ such that
the following equation holds:

$\Theta^{r}|_{r=0}=\dfrac{\partial
r}{\partial\rho}\widetilde{\Theta}^{\rho}|_{\rho=0}=e^{-\zeta}\widetilde{\Theta}^{\rho}|_{\rho=0}\,.$
(37)

The transformation of $\mathcal{A}_{(1)}$ fixes the transformation of
$\mathcal{B}_{(0)}$ because we have to ensure that $\Theta^{r}$ transforms
like a tensor according to eq.(37). Thus, we have:

$\mathcal{B}_{(0)}=\widetilde{\mathcal{B}}_{(0)}-\tau\xi_{i}\widetilde{\mathcal{Q}}^{\rho
i}-\tau\xi_{i}\widetilde{J}^{i}_{(1)}-\tau\widetilde{\nabla}_{i}\left(4\xi_{j}\widetilde{L}^{ij}_{(1)}\right)+4\tau\xi_{i}\xi_{j}\widetilde{L}^{ij}_{(1)}\,.$
(38)

This is checked in eq.(167). From the transformation of $\mathcal{Q}^{rv}$ of
eq.(32), and that of $\mathcal{B}_{(0)}$ given in eq.(38), one can
straightforwardly work out how $\mathcal{J}^{v}$ of eq.(18) transforms.
$\mathcal{J}^{v}$ transforms as

$\mathcal{J}^{v}=\widetilde{\mathcal{J}}^{\tau}+\tau\xi_{i}\widetilde{\mathcal{J}}^{i}-2\tau^{2}\xi_{i}\xi_{j}\partial_{\tau}\widetilde{L}^{ij}_{(1)}+2\tau\widetilde{\nabla}_{i}\left(\xi_{j}\widetilde{L}^{ij}_{(1)}\right)-2\tau\xi_{i}\xi_{j}\widetilde{L}^{ij}_{(1)}\,.$
(39)

This is derived in eq.(168).

Thus putting together eq.(30) and eq.(39), we have the final transformation of
$\mathcal{J}^{v}$ and $\mathcal{J}^{i}$ as follows:

$\begin{split}\mathcal{J}^{v}&=\widetilde{\mathcal{J}}^{\tau}+\tau\xi_{i}\widetilde{\mathcal{J}}^{i}-2\tau^{2}\xi_{i}\xi_{j}\partial_{\tau}\widetilde{L}^{ij}_{(1)}+2\tau\widetilde{\nabla}_{i}\left(\xi_{j}\widetilde{L}^{ij}_{(1)}\right)-2\tau\xi_{i}\xi_{j}\widetilde{L}^{ij}_{(1)}\,,\\\
\mathcal{J}^{i}&=e^{-\zeta}\widetilde{J}^{i}-2e^{-\zeta}(1+\tau\partial_{\tau})(\xi_{j}\widetilde{L}^{ij}_{(1)})\,.\end{split}$
(40)

Here $\widetilde{L}^{ij}_{(1)}$ is defined by eq.(29). This proves the first
result of our paper given in eq.(9). We will verify the formula of eq.(40) for
various Lagrangians in section 5. To understand the significance of
$\widetilde{L}^{ij}_{(1)}$, notice that if you set
$\widetilde{L}^{ij}_{(1)}=0$, then $\mathcal{J}^{v}$ and $\mathcal{J}^{i}$
would transform covariantly. $\mathcal{J}^{v}$ would transform like a
covariant vector with an upper index, and $\mathcal{J}^{i}$ would transform
like a covariant boost weight $+1$ quantity. Thus, $\widetilde{L}^{ij}_{(1)}$
signifies a non-covariance associated with the transformation of the
components of entropy current defined on the dynamical horizon. The non-
trivial consequence of this analysis is that despite these non-covariances,
the local entropy production remains covariant (eq.(21)) under affine
reparametrizations of the form eq.(5).

At this point, it is useful to comment on the relation between eq.(40) and the
result of eq(131) of Hollands:2022fkn  which, when translated to our notation,
becomes:

$\begin{split}\widetilde{\mathcal{J}}^{\tau}&=\mathcal{J}^{v}-\nabla_{i}B^{vi}\,,\\\
e^{-\zeta}\widetilde{\mathcal{J}}^{i}&=\mathcal{J}^{i}+\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}\,B^{vi}\right)+\nabla_{j}B^{ij}\,.\end{split}$
(41)

Our result eq.(40) expressed in the $(r,v,x^{i})$ coordinates is given by

$\begin{split}\widetilde{\mathcal{J}}^{\tau}&=\mathcal{J}^{v}-v\xi_{i}\left[\mathcal{J}^{i}+\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}\,2v\,\xi_{j}L^{ij}_{(1)}\right)\right]+\nabla_{i}\left(-2v\,\xi_{j}L^{ij}_{(1)}\right)\,,\\\
e^{-\zeta}\widetilde{\mathcal{J}}^{i}&=\mathcal{J}^{i}+\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}\,2v\,\xi_{j}L^{ij}_{(1)}\right)\,.\end{split}$
(42)

However, to compare eq.(42) with eq.(41), one must set $v=\tau=0$ because
eq.(41) has been derived for $v=\tau=0$ slice of the black hole horizon
Hollands:2022fkn . Setting $v=0$ in eq.(42), we get

$\widetilde{\mathcal{J}}^{\tau}=\mathcal{J}^{v}\,,~{}~{}~{}~{}~{}~{}e^{-\zeta}\widetilde{\mathcal{J}}^{i}=\mathcal{J}^{i}+2\xi_{j}L^{ij}_{(1)}\,.$
(43)

Comparing eq.(43) to eq.(41), we see that to have a non-trivial $B^{vi}$, one
must not restrict to the $v=\tau=0$ slice. Thus, in order to see the richer
structure in eq.(41), one must not restrict to the $v=\tau=0$ slice as we have
done in eq.(40).

We emphasize the difference between the two approaches: Hollands:2022fkn  is
concerned with the gauge invariance of the definition of dynamical entropy
$\mathcal{J}^{v}$, which in the linear order is the statement of eq.(43). Our
approach looks at the gauge covariance of entropy production through eq.(21).
This is slightly different because in our approach, both $\mathcal{J}^{v}$ and
$\mathcal{J}^{i}$ transform non-trivially to ensure that the covariant
relation of eq.(20) is respected. This non-triviality is due to the presence
of $\widetilde{L}^{ij}_{(1)}$ in eq.(40). In the subsequent section, we will
study the Iyer-Wald ambiguities in the context of entropy current, and we will
show that they have a structure similar to the non-covariant terms in eq.(42).

##  4 Iyer-Wald Ambiguities of the entropy current

In this section, our aim will be to establish the second result of our paper
given in eq.(10). Consider the structure of $E_{vv}$ at the linear order given
by eq.(3)

$E_{vv}=\partial_{v}\left[\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}\mathcal{J}^{v}\right)+\nabla_{i}\mathcal{J}^{i}\right]+\mathcal{O}(\epsilon^{2})\,.$
(44)

One can straightforwardly see that

$\begin{split}\mathcal{J}^{v}&\rightarrow\mathcal{J}^{v}+\nabla_{i}C^{vi}\,,\\\
\mathcal{J}^{i}&\rightarrow\mathcal{J}^{i}-\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}\,C^{vi}\right)+\nabla_{j}C^{ij}\,,\end{split}$
(45)

with $C^{ij}=-C^{ji}$ will ensure that $E_{vv}$ remains the same though the
components of entropy current change. Thus the additional terms in eq.(45)
given by

$\mathcal{J}^{v}_{\text{amb}}=\nabla_{i}C^{vi}\,,~{}~{}~{}~{}~{}\mathcal{J}^{i}_{\text{amb}}=-\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}\,C^{vi}\right)+\nabla_{j}C^{ij}\,,$
(46)

constitute an ambiguity in the definition of the entropy currents
$\mathcal{J}^{v}$ and $\mathcal{J}^{i}$. This is because eq.(46) satisfies
eq.(6). Given the generic existence of ambiguities via eq.(45), it is
important to understand the origin of these ambiguities from the fundamental
elements. We will show that the ambiguities of the form eq.(45) arise from the
ambiguities in $\Theta^{r}$ and $Q^{r\mu}$ of the defining equation eq.(16).

###  4.1 The different types of ambiguities

We want to arrive at eq.(45) from the first principle arguments. We will see
that it arises from the Iyer-Wald ambiguities Iyer:1994ys  that are part of
the definition of the Lagrangian $\mathcal{L}$, $\Theta^{\mu}$ and
$Q^{\mu\nu}$. From the variation of the Lagrangian of the form eq.(2), we have

$\delta(\sqrt{-g}\mathcal{L})=\sqrt{-g}E^{\mu\nu}\delta
g_{\mu\nu}+\sqrt{-g}G^{\mu}\delta A_{\mu}+\sqrt{-g}D_{\mu}\Theta^{\mu}[\delta
g_{\alpha\beta},\delta A_{\gamma}]\,.$ (47)

Here $E^{\mu\nu}$ are the gravitational EoM and $G^{\mu}$ is the gauge field
EoM. We see that there is a $Y-$type ambiguity in the definition of
$\Theta^{\mu}$ itself, which is given by

$\Theta^{\mu}\rightarrow\Theta^{\mu}+D_{\nu}Y^{\mu\nu}\,,$ (48)

where $Y^{\mu\nu}=-Y^{\nu\mu}$. Thus, the definition of $\Theta^{\mu}$ is
ambiguous up to this $Y^{\mu\nu}$. The definition of the Lagrangian
$\mathcal{L}$ itself is ambiguous up to a total derivative term:

$\mathcal{L}\rightarrow\mathcal{L}+D_{\mu}M^{\mu}\,.$ (49)

eq.(49) doesn’t affect the equations of motion but leaves an ambiguity in
$\Theta^{\mu}$ given by

$\Theta^{\mu}\rightarrow\Theta^{\mu}+\delta
M^{\mu}+\dfrac{1}{2}M^{\mu}g^{\alpha\beta}\delta g_{\alpha\beta}\,.$ (50)

Combining eq.(48) and eq.(50), we have

$\Theta^{\mu}[\delta g_{\alpha\beta},\delta
A_{\gamma}]\rightarrow\Theta^{\mu}[\delta g_{\alpha\beta},\delta
A_{\gamma}]+D_{\nu}Y^{\mu\nu}[\delta g_{\alpha\beta},\delta A_{\gamma}]+\delta
M^{\mu}+\dfrac{1}{2}M^{\mu}g^{\alpha\beta}\delta g_{\alpha\beta}\,.$ (51)

One can obtain the Iyer-Wald Noether charge $Q^{\mu\nu}$ from eq.(47) if one
considers the variation of the Lagrangian under a diffeomorphism given by
eq.(149). The definition of $Q^{\mu\nu}$ is through eq.(151) 121212$\Lambda$
is the $U(1)$ gauge transformation. Thus one has $\delta
A_{\mu}=\mathcal{L}_{\xi}A_{\mu}+D_{\mu}\Lambda$. Substituting this in
$\Theta^{\mu}$ brings out a factor of $A_{\nu}\xi^{\nu}+\Lambda$. This is
worked out in Section 6.1 of Biswas:2022grc  (see, for instance eq.(6.6) of
Biswas:2022grc ).:

$\Theta^{\mu}[\mathcal{L}_{\xi}g_{\mu\nu},\dots]-\xi^{\mu}\mathcal{L}=-2E^{\mu\nu}\xi_{\nu}-G^{\mu}(A_{\nu}\xi^{\nu}+\Lambda)+D_{\nu}Q^{\mu\nu}\,.$
(52)

This definition of $Q^{\mu\nu}$ will automatically induce an ambiguity of the
form

$Q^{\mu\nu}\rightarrow
Q^{\mu\nu}+D_{\rho}Z^{\mu\nu\rho}~{}~{}~{}~{}\text{where}~{}Z^{\mu\nu\rho}=Z^{[\mu\nu\rho]}\,.$
(53)

$Q^{\mu\nu}$ will also inherit the ambiguities of $\Theta^{\mu}$ of eq.(51).
To find out the contributions due to the $Y-$type ambiguity of eq.(48) and the
$M-$type ambiguity of eq.(50), we start with the $\Theta^{\mu}$ in the
definition of $Q^{\mu\nu}$ in eq.(52):

$\begin{split}\Theta^{\mu}[\mathcal{L}_{\xi}g_{\alpha\beta},\mathcal{L}_{\xi}A_{\gamma}+D_{\gamma}\Lambda]&\rightarrow\Theta^{\mu}[\mathcal{L}_{\xi}g_{\alpha\beta},\mathcal{L}_{\xi}A_{\gamma}+D_{\gamma}\Lambda]+D_{\nu}Y^{\mu\nu}[\mathcal{L}_{\xi}g_{\alpha\beta},\mathcal{L}_{\xi}A_{\gamma}+D_{\gamma}\Lambda]\\\
&~{}~{}~{}~{}+\mathcal{L}_{\xi}M^{\mu}+\dfrac{1}{2}M^{\mu}g^{\alpha\beta}\mathcal{L}_{\xi}g_{\alpha\beta}\\\
&\rightarrow\Theta^{\mu}[\mathcal{L}_{\xi}g_{\alpha\beta},\mathcal{L}_{\xi}A_{\gamma}+D_{\gamma}\Lambda]+D_{\nu}Y^{\mu\nu}[\mathcal{L}_{\xi}g_{\alpha\beta},\mathcal{L}_{\xi}A_{\gamma}+D_{\gamma}\Lambda]\\\
&~{}~{}~{}~{}+\xi^{\mu}D_{\rho}M^{\rho}+D_{\nu}\left[M^{\mu}\xi^{\nu}-M^{\nu}\xi^{\mu}\right]\,.\end{split}$
(54)

Note that crucially, we use $\delta M^{\mu}=\mathcal{L}_{\xi}M^{\mu}$ since we
consider total derivative terms that are diffeomorphism covariant and $U(1)$
gauge invariant. Thus, we will not have anything to say about Chern-Simons
theories, which are diffeomorphism invariant ($U(1)$ invariant) up to total
derivatives only. These theories have been recently analysed in Deo:2023vvb .
From eq.(52), we see that the contribution of eq.(51) to $Q^{\mu\nu}$ is given
by

$Q^{\mu\nu}\rightarrow
Q^{\mu\nu}+Y^{\mu\nu}+M^{\mu}\xi^{\nu}-M^{\nu}\xi^{\mu}\,.$ (55)

Hence the combined ambiguity in $Q^{\mu\nu}$ from eq.(53) and eq.(55) is given
by

$Q^{\mu\nu}\rightarrow
Q^{\mu\nu}+Y^{\mu\nu}+M^{\mu}\xi^{\nu}-M^{\nu}\xi^{\mu}+D_{\rho}Z^{\mu\nu\rho}\,.$
(56)

One can straightforwardly see that the ambiguities in $\Theta^{\mu}$ (eq.(51))
and the ambiguities of $Q^{\mu\nu}$ (eq.(56)) cancel out in the combination of
$E_{vv}$ given by

$2E^{\mu\nu}\xi_{\mu}\xi_{\nu}+\xi^{\mu}G_{\mu}(A_{\nu}\xi^{\nu}+\Lambda)=-\xi_{\mu}\Theta^{\mu}+\xi_{\mu}D_{\nu}Q^{\mu\nu}\,.$
(57)

This is expected because equations of motion for a theory are unambiguous, so
the ambiguities should be canceled out in this combination. The ambiguities,
however directly contribute to $\Theta^{r}$ and $Q^{r\nu}$ and thus, we expect
that they should contribute to the entropy current even though they do not
contribute to $E_{vv}$. We will see that these ambiguities will contribute in
the form of eq.(46). Thus, the cancellation of ambiguities is more
illuminating in our gauge eq.(1).

We will note down the following results from Appendix A (eq.(153) and
eq.(157)) for the sake of clarity:

$\begin{split}\Theta^{r}|_{r=0}&=(1+v\partial_{v})\mathcal{A}_{(1)}+v\partial^{2}_{v}\mathcal{B}_{(0)}\,,\\\
Q^{r\mu}&=\mathcal{Q}^{r\mu}+vW^{r\mu}_{v}\,.\end{split}$ (58)

Using eq.(16) (or eq.(152)) given by

$2v\,E_{vv}|_{r=0}=\left(-\Theta^{r}+D_{\mu}Q^{r\mu}\right)|_{r=0}\,,$ (59)

the components of the entropy current can be worked out to be

$\mathcal{J}^{v}=-\dfrac{1}{2}(\mathcal{Q}^{rv}+\mathcal{B}_{(0)})\,,~{}~{}~{}~{}~{}\mathcal{J}^{i}=-\dfrac{1}{2}(\mathcal{Q}^{ri}-J^{i}_{(1)})\,,$
(60)

where $W^{ri}_{v}=\partial_{v}J^{i}_{(1)}+\mathcal{O}(\epsilon^{2})$. We will
now analyse the $M-$type (eq.(49)), $Y-$type (eq.(48)) and $Z-$type (eq.(53))
ambiguities in the following. Such ambiguities have been analysed in
Bueno:2016gnv ; Speranza:2017gxd ; Haehl:2017sot . They also play a role in
the holographic entanglement entropy functionals Dong:2013qoa ; Camps:2013zua
; Miao:2014nxa . We will explicitly show that they are of the form given in
eq.(6).

### Analysis of the $M$-type ambiguity:

The $M-$type ambiguity in $\Theta^{\mu}$ is given by eq.(50):

$\Theta^{\mu}[\mathcal{L}_{\xi}g_{\alpha\beta},\mathcal{L}_{\xi}A_{\gamma}+D_{\gamma}\Lambda]\rightarrow\Theta^{\mu}+\xi^{\mu}D_{\rho}M^{\rho}+D_{\nu}\left[M^{\mu}\xi^{\nu}-M^{\nu}\xi^{\mu}\right]\,.$
(61)

This is valid only for a diffeomorphism covariant $M^{\mu}$. Thus, at $r=0$,
we have

$\begin{split}\Theta^{r}&\rightarrow\Theta^{r}+D_{\nu}(M^{r}\xi^{\nu})\\\
&=\Theta^{r}+\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}\,M^{r}v\right)=\Theta^{r}+(1+v\partial_{v})M^{r}+\mathcal{O}(\epsilon^{2})\,.\end{split}$
(62)

Thus, the entropy current of eq.(60) is unaffected by this ambiguity because
$\mathcal{B}_{(0)}$ is unchanged. Hence, $M-$type ambiguity does not
contribute to the entropy current.

### Analysis of the $Y$-type ambiguity:

We note that this $Y-$type ambiguity has been looked at in the context of
defining entropy before in Bueno:2016gnv ; Haehl:2017sot . Here
$Y^{\mu\nu}[\delta g_{\alpha\beta},\delta A_{\gamma}]$ of eq.(48) is assumed
to be diffeomorphism covariant and $U(1)$ gauge invariant. Consider
$Y^{ri}=Y^{ri}[\mathcal{L}_{\xi}g_{\alpha\beta},\mathcal{L}_{\xi}A_{\gamma}]$.
This has the same structure as $\Theta^{r}$ and thus, one can essentially
borrow the analysis of Bhattacharyya:2021jhr ; Biswas:2022grc  to arrive at
131313Notation: Here the subscripts are used to denote different terms. It
shouldn’t be confused with boost weights written within $()$ in subscripts.

$\begin{split}Y^{ri}&=(1+v\partial_{v})\mathcal{Y}^{\prime\,i}_{(1)}+v\partial^{2}_{v}\mathcal{Y}^{i}_{(0)}\,,\\\
\implies\nabla_{i}Y^{ri}&=(1+v\partial_{v})\nabla_{i}\mathcal{Y}^{\prime\,i}_{(1)}+v\partial^{2}_{v}\nabla_{i}\mathcal{Y}^{i}_{(0)}\,.\end{split}$
(63)

See section 3.2.2 of Bhattacharyya:2021jhr  and section 6.3 (also eq.(6.32))
of Biswas:2022grc  for details of this construction. The analysis of section
3.2.2 of Bhattacharyya:2021jhr  deals with $\delta g_{\alpha\beta}$ terms in
$\Theta^{r}$ and section 6.3 of Biswas:2022grc  deals with the $\delta
A_{\mu}$ terms.

Consider
$Y^{rv}=Y^{rv}[\mathcal{L}_{\xi}g_{\alpha\beta},\mathcal{L}_{\xi}A_{\gamma}+D_{\gamma}\Lambda]$.
$Y^{rv}$ is boost weight zero and it is linear in $\xi$. It also depends on
$\mathcal{L}_{\xi}A_{\gamma}+D_{\gamma}\Lambda$ through $\delta A_{\mu}$. From
the analysis of the $\delta A_{\mu}$ terms in eq.(6.26) of Biswas:2022grc
((i.e.) the $\delta A_{\mu}$ terms in boost weight zero quantities), the
structure of $Y^{rv}$ is of the following form

$Y^{rv}=\bar{Y}^{rv}_{1}+v\mathcal{Y}^{rv}_{v}+\bar{Y}^{v}_{2}\partial_{v}(vA_{v}+\Lambda)\,.$
(64)

There is an additional $vA_{v}+\Lambda$ piece compared to the structure of
$Q^{rv}$ in eq.(58). Here it should be noted that
$(vA_{v}+\Lambda)\sim\mathcal{O}(\epsilon)$ according to the analysis
presented in Appendix B of Biswas:2022grc . Thus, from eq.(64), we have

$\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}\,Y^{rv}\right)=\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}\,\bar{Y}^{rv}_{1}\right)+(1+v\partial_{v})\mathcal{Y}^{rv}_{v}+\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}\,\bar{Y}^{v}_{2}\partial_{v}(vA_{v}+\Lambda)\right)\,.$
(65)

We now have,

$\begin{split}D_{\nu}Y^{r\nu}&=\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}Y^{rv}\right)+\nabla_{i}Y^{ri}\\\
&=(1+v\partial_{v})\left[\mathcal{Y}^{rv}_{v}+\nabla_{i}\mathcal{Y}^{\prime\,i}_{(1)}+\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}\,\bar{Y}^{rv}_{1}\right)\right]\\\
&~{}~{}~{}+v\partial_{v}\left[\dfrac{1}{\sqrt{h}}\partial_{v}\left[\sqrt{h}(\nabla_{i}\mathcal{Y}^{i}_{(0)}-\bar{Y}^{rv}_{1})\right]\right]+\dfrac{1}{\sqrt{h}}\partial_{v}\left[\sqrt{h}\left(\bar{Y}^{v}_{2}(vA_{v}+\Lambda)\right)\right]+\mathcal{O}(\epsilon^{2})\,.\end{split}$
(66)

In the second step, we have used eq.(65).

eq.(66) represents the ambiguity in $\Theta^{r}$ of the form eq.(48). Thus,
the terms in eq.(66) are ambiguities in the definition of $\mathcal{A}_{(1)}$
and $\mathcal{B}_{(0)}$ of $\Theta^{r}$ of eq.(58). For instance,
$\Theta^{r}\rightarrow\Theta^{r}+D_{\nu}Y^{r\nu}$ from eq.(48) implies

$\begin{split}\mathcal{A}_{(1)}&\rightarrow\mathcal{A}_{(1)}+\mathcal{Y}^{rv}_{v}+\nabla_{i}\mathcal{Y}^{\prime\,i}_{(1)}+\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}\,\bar{Y}^{rv}_{1}\right)\,,\\\
\mathcal{B}_{(0)}&\rightarrow\mathcal{B}_{(0)}+\nabla_{i}\mathcal{Y}^{i}_{(0)}-\bar{Y}^{rv}_{1}\,.\end{split}$
(67)

Now since $Q^{r\nu}\rightarrow Q^{r\nu}+Y^{r\nu}$, the last term proportional
to $vA_{v}+\Lambda$ in eq.(66) cancels out in the combination of eq.(59). As
$Q^{rv}=\mathcal{Q}^{rv}+vW^{rv}_{v}$ from eq.(58), $Y^{rv}$ of eq.(64)
implies

$\mathcal{Q}^{rv}\rightarrow\mathcal{Q}^{rv}+\bar{Y}^{rv}_{1}\,.$ (68)

Also, $Q^{ri}=\mathcal{Q}^{ri}+vW^{ri}_{v}$ from eq.(58), $Y^{ri}$ of eq.(63)
implies

$\begin{split}\mathcal{Q}^{ri}&\rightarrow\mathcal{Q}^{ri}+\mathcal{Y}^{i}_{(0)}\,,\\\
W^{ri}_{v}&\rightarrow
W^{ri}_{v}+\partial_{v}\mathcal{Y}^{\prime\,i}_{(1)}+\partial^{2}_{v}\mathcal{Y}^{i}_{(0)}\,.\end{split}$
(69)

But $W^{ri}_{v}=\partial_{v}J^{i}_{(1)}$, and thus eq.(69) implies

$J^{i}_{(1)}\rightarrow
J^{i}_{(1)}+\mathcal{Y}^{\prime\,i}_{(1)}+\partial_{v}\mathcal{Y}^{i}_{(0)}\,.$
(70)

Thus collecting eq.(67), eq.(68), eq.(69) and eq.(70) in eq.(60), we get

$\begin{split}\mathcal{J}^{v}&\rightarrow\mathcal{J}^{v}-\dfrac{1}{2}\nabla_{i}\mathcal{Y}^{i}_{(0)}\,,\\\
\mathcal{J}^{i}&\rightarrow\mathcal{J}^{i}+\dfrac{1}{2}\partial_{v}\mathcal{Y}^{i}_{(0)}\,.\end{split}$
(71)

This comprises the effect of $Y-$type ambiguity. $\mathcal{Y}^{i}_{(0)}$ by
definition is $\mathcal{O}(\epsilon)$ and thus this analysis answers why JKM
ambiguties arise from $Y-$type ambiguities Bueno:2016gnv ; Haehl:2017sot . The
ambiguities in eq.(71) are clearly of the form eq.(6).

### Analysis of $Z$-type ambiguity:

We now analyze the $Z-$type ambiguity of eq.(53). We have

$Q^{rv}\rightarrow Q^{rv}+D_{\rho}Z^{rv\rho}=Q^{rv}+\nabla_{i}Z^{rvi}\,.$ (72)

Thus eq.(58) implies

$\mathcal{Q}^{rv}\rightarrow\mathcal{Q}^{rv}+\nabla_{i}Z^{rvi}\,.$ (73)

Similarly

$Q^{ri}\rightarrow
Q^{ri}+D_{\rho}Z^{ri\rho}=Q^{ri}+\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}\,Z^{riv}\right)+\nabla_{j}Z^{rij}\,.$
(74)

Notice how $Z^{\mu\nu\rho}=Z^{[\mu\nu\rho]}$ played a crucial role in arriving
at this structure. We thus have

$\mathcal{Q}^{ri}\rightarrow\mathcal{Q}^{ri}+\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}\,Z^{riv}\right)+\nabla_{j}Z^{rij}\,.$
(75)

Collecting eq.(73) and eq.(75) in eq.(60), we have

$\begin{split}\mathcal{J}^{v}&\rightarrow\mathcal{J}^{v}-\dfrac{1}{2}\nabla_{i}Z^{rvi}\,,\\\
\mathcal{J}^{i}&\rightarrow\mathcal{J}^{i}+\dfrac{1}{2}\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}\,Z^{rvi}\right)-\dfrac{1}{2}\nabla_{j}Z^{rij}\,.\end{split}$
(76)

### Summary:

To summarise, there are three types of Iyer-Wald ambiguities:

$\begin{split}&\text{(i)}~{}~{}\mathcal{L}\rightarrow\mathcal{L}+D_{\rho}M^{\rho}\,,\\\
&\text{(ii)}~{}~{}\Theta^{\mu}\rightarrow\Theta^{\mu}+D_{\nu}Y^{\mu\nu}\,,\\\
&\text{(iii)}~{}~{}Q^{\mu\nu}\rightarrow
Q^{\mu\nu}+D_{\rho}Z^{\mu\nu\rho}\,.\end{split}$ (77)

The corresponding contributions to the entropy current are given by eq.(71)
and eq.(76)

$\begin{split}&\text{(i)}~{}~{}\text{No contribution to $\mathcal{J}^{v}$ and
$\mathcal{J}^{i}$.}\\\
&\text{(ii)}~{}~{}\mathcal{J}^{v}\rightarrow\mathcal{J}^{v}-\dfrac{1}{2}\nabla_{i}\mathcal{Y}^{i}_{(0)}\,,~{}~{}\mathcal{J}^{i}\rightarrow\mathcal{J}^{i}+\dfrac{1}{2}\partial_{v}\mathcal{Y}^{i}_{(0)}\,,~{}~{}\mathcal{Y}^{i}_{(0)}\sim\mathcal{O}(\epsilon)\,.\\\
&\text{(iii)}~{}~{}\mathcal{J}^{v}\rightarrow\mathcal{J}^{v}-\dfrac{1}{2}\nabla_{i}Z^{rvi}\,,~{}~{}\mathcal{J}^{i}\rightarrow\mathcal{J}^{i}+\dfrac{1}{2}\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}\,Z^{rvi}\right)-\dfrac{1}{2}\nabla_{j}Z^{rij}\,.\end{split}$
(78)

Thus, all the entropy current constructions of the previous works
Bhattacharya:2019qal ; Bhattacharyya:2021jhr ; Biswas:2022grc ; Deo:2023vvb
inherently suffer from ambiguities that arise from the covariant phase
formalism used 141414It is useful to note that the ambiguities (ii) and (iii)
do not affect the total entropy which is given by the integral of
$\mathcal{J}^{v}$ over a co-dimension 2 slice of the horizon. This is because
the ambiguous terms are total derivatives. This was also recently noted and
carefully studied in Wall:2024lbd .. We have proved the second result of our
paper in eq.(10). The ambiguities in eq.(7) are of the form eq.(10), and they
arise from the ambiguities in $\Theta^{r}$ and $Q^{r\mu}$.

###  4.2 Ambiguities in the physical process version of the first law

In this section, we will analyse the ambiguities in the setup of the physical
process version of the first law (PPFL) discussed in Bhattacharyya:2021jhr .
The setup of the second law proved through the null projected EoM structure of
eq.(3) can be straightforwardly used to argue for a physical process version
of the first law Gao:2001ut ; Amsel:2007mh ; Chatterjee:2011wj ;
Bhattacharjee:2014eea ; Mishra:2017sqs . PPFL is a relation between the change
of entropy of a Black hole $\delta S$ and the infalling matter generated by a
small matter stress tensor $\delta T_{\mu\nu}$, which is given by eq.5.1 of
Bhattacharyya:2021jhr :

$\delta
S=2\pi\int_{\mathcal{H}}dv\,d^{d-2}x\,\sqrt{h}\chi^{\mu}\xi^{\nu}\delta
T_{\mu\nu}\,.$ (79)

Section 2.2.3 of Bhattacharya:2019qal  discusses this equation in detail, and
the derivation is given in section 2 of Jacobson:1995uq . Here $\mathcal{H}$
is the codimension 1 horizon $r=0$ of eq.(135),
$\chi^{\mu}=(\partial_{v})^{\mu}$ is the affinely parameterized null generator
of the horizon of eq.(135) and $\xi^{\mu}$ is the Killing vector eq.(136). We
will consider the PPFL between arbitrary time slices of the horizon which
naturally leads to an expression for a dynamical entropy for an arbitrary non-
stationary slice of the horizon. We will then focus on the effect of
ambiguities in this version of PPFL.

#### PPFL between arbitrary time slices of the horizon

In writing eq.(79), we made the following assumptions: the horizon is stable
under perturbations; perturbations do not shift the location of the horizon
151515For example, if in some coordinate system, the dynamical horizon is
located at $R=R_{H}(v)$, then to transform to the coordinates as in (1), the
corresponding transformation would be $r=R-R_{H}(v)$ Bhattacharyya:2022nqa .
The first assumption is a notion of the Weak cosmic censorship hypothesis,
which is a desirable property. The second assumption follows because we are
working with the Gaussian null coordinate system of eq.(1). The perturbations
are of the form eq.(137), and they clearly do not shift the horizon at $r=0$
as the effect of these perturbations is already taken into account in the
definition of the $r$-coordinate. An important consequence of this assumption
is that the event horizon of the dynamical spacetime coincides with the event
horizon of the equilibrium spacetime. This is a well-motivated assumption
because we look at situations when a black hole is “settling down to
equilibrium”. This allows us to do the integral over the affine coordinate $v$
on a dynamical horizon because $v$ remains null throughout the entire
evolution. As is usually done for the first law, we assume the statement of
the zeroth law IstvánRácz_1996 ; Bhattacharyya:2022nqa ; Davies:2024fut  to be
valid.161616On the horizon, the surface gravity $\kappa$ arises from the
relation $\xi^{\mu}=\kappa v\chi^{\mu}$.

The RHS of eq.(79) represents the mass $\delta M$ falling into the horizon
$\mathcal{H}$ and thus the proof of the PPFL will be complete if we can
extract a quantity $S$ that satisfies $T\delta S=\delta M$. We will precisely
find an $S$ that satisfies the required condition. For this, we first use the
on-shell EoM eq.(142) given by $E_{\mu\nu}=-\delta T_{\mu\nu}$ in eq.(79).
After substituting for $\chi^{\mu}$ and $\xi^{\mu}$, we get

$\delta S=-2\pi\int_{\mathcal{H}}dv\,d^{d-2}x\,\sqrt{h}\,vE_{vv}\,.$ (80)

Using eq.(3) in eq.(80), section 5 of Bhattacharyya:2021jhr  extracted an $S$
from eq.(80) thereby proving the PPFL. In order to work this out, we must
integrate over $v$ from some past time slice to a future time slice of
codimension 1 horizon. Thus, we have

$\delta
S=-2\pi\int^{v=v_{1}}_{v=v_{2}}dv\,\int_{\mathcal{H}_{v}}d^{d-2}x\,\sqrt{h}\,vE_{vv}\,.$
(81)

Here $\mathcal{H}_{v}$ denotes the constant $v$ spatial slices. In
Bhattacharyya:2021jhr , the authors considered the case of entropy increase
from an initial equilibrium configuration in the far past to a final
equilibrium configuration in the far future. Thus, the entropy increase is
given by the difference of the equilibrium Wald entropy evaluated in both
slices. This is a geometric quantity that is clearly gauge invariant.
Bhattacharyya:2022njk  argued for eq.(8) as a consequence of gauge invariance
of this difference of Wald entropy. Thus, we naturally have a notion of gauge
invariance of entropy production.

However, there is nothing special about the choice of constant-$v$ slices made
in Bhattacharyya:2021jhr . The setup of section 5 of Bhattacharyya:2021jhr
can be used to argue for a PPFL for arbitrary $v_{1}$ and $v_{2}$. Such an
analysis was already carried out for Lovelock theories in Mishra:2017sqs . As
a byproduct of this analysis, we can obtain an expression for the entropy of a
dynamical black hole in an arbitrary diffeomorphism invariant theory of
gravity for arbitrary time slices on the horizon. Such an expression would be
consistent with the PPFL. To derive that expression, we first substitute
eq.(3) in eq.(81) to obtain

$\delta
S=-\int^{v=v_{1}}_{v=v_{2}}dv\,\int_{\mathcal{H}_{v}}d^{d-2}x\,\sqrt{h}\,v\partial_{v}\Theta\,,$
(82)

where $\Theta$ is the generalized expansion given by

$\Theta=\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}\,\mathcal{J}^{v}\right)+\nabla_{i}\mathcal{J}^{i}\,.$
(83)

In writing eq.(82), we have normalized the Lagrangian to get rid of the $2\pi$
factor. We can now basically follow the steps of section 5 of
Bhattacharyya:2021jhr . One can do an integration by parts on eq.(82) to
obtain

$\begin{split}\delta
S&=-\left[\int_{\mathcal{H}_{v}}d^{d-2}x\,\sqrt{h}\,v\,\Theta\right]^{v=v_{1}}_{v=v_{2}}+\int^{v=v_{1}}_{v=v_{2}}dv\int_{\mathcal{H}_{v}}d^{d-2}x\,\partial_{v}(v\sqrt{h})\,\Theta\\\
&=-\left[\int_{\mathcal{H}_{v}}d^{d-2}x\,\sqrt{h}\,v\,\Theta\right]^{v=v_{1}}_{v=v_{2}}+\int^{v=v_{1}}_{v=v_{2}}dv\int_{\mathcal{H}_{v}}d^{d-2}x\,\sqrt{h}\,\Theta+\mathcal{O}(\epsilon^{2})\,,\end{split}$
(84)

where the second step arises from the fact that
$\Theta\sim\mathcal{O}(\epsilon)$ and
$\partial_{v}(v\sqrt{h})=\sqrt{h}+\mathcal{O}(\epsilon)$. Now,

$\begin{split}\int^{v=v_{1}}_{v=v_{2}}dv\int_{\mathcal{H}_{v}}d^{d-2}x\sqrt{h}\,\Theta&=\int^{v=v_{1}}_{v=v_{2}}dv\int_{\mathcal{H}_{v}}d^{d-2}x\sqrt{h}\,\left[\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}\,\mathcal{J}^{v}\right)+\nabla_{i}\mathcal{J}^{i}\right]\\\
&=\int^{v=v_{1}}_{v=v_{2}}dv\int_{\mathcal{H}_{v}}d^{d-2}x\partial_{v}\left(\sqrt{h}\mathcal{J}^{v}\right)\\\
&=\left[\int
d^{d-2}x\,\sqrt{h}\,\mathcal{J}^{v}\right]^{v=v_{1}}_{v=v_{2}}\end{split}$
(85)

In the second step, we have assumed that the constant $v$ slices
$\mathcal{H}_{v}$ are compact to integrate out $\nabla_{i}\mathcal{J}^{i}$.
Substituting the result of eq.(85) in eq.(84), we get

$\delta
S=-\left[\int_{\mathcal{H}_{v}}d^{d-2}x\,\sqrt{h}\,v\,\Theta\right]^{v=v_{1}}_{v=v_{2}}+\left[\int_{\mathcal{H}_{v}}d^{d-2}x\,\sqrt{h}\,\mathcal{J}^{v}\right]^{v=v_{1}}_{v=v_{2}}\,.$
(86)

eq.(86) gives an expression for the entropy of a dynamical black hole for an
arbitrary time slice of the horizon in a general diffeomorphism invariant
theory of gravity. This is given by

$S|_{\mathcal{H}_{v}}=\int_{\mathcal{H}_{v}}d^{d-2}x\,\sqrt{h}\,\mathcal{J}^{v}-\int_{\mathcal{H}_{v}}d^{d-2}x\,\sqrt{h}\,v\,\Theta$
(87)

At this point, it is useful to mention that the presence of an additional
boundary term proportional to $v\,\Theta$ in eq.(86) for arbitrary time slices
of the horizon has already been emphasized in Mishra:2017sqs . By changing the
null parameterization to a non-affine parameterization, the authors of
Mishra:2017sqs  have interpreted this extra term as a change in energy $\delta
E$ of the horizon membrane obtained through the membrane paradigm picture.

If we consider, Einstein gravity given by $L=R$, we have
$\mathcal{J}^{v}=1,\,\mathcal{J}^{i}=0$. Thus, eq.(87) becomes

$\begin{split}S[L=R]|_{\mathcal{H}_{v}}&=\int_{\mathcal{H}_{v}}d^{d-2}x\,\sqrt{h}-\int_{\mathcal{H}_{v}}d^{d-2}x\,\sqrt{h}\,v\,\theta\\\
&=A[\mathcal{H}_{v}]-\int_{\mathcal{H}_{v}}d^{d-2}x\,\sqrt{h}\,v\,\theta\,,\end{split}$
(88)

where $A[\mathcal{H}_{v}]$ 171717In our choice of normalization, we dropped
the $2\pi$ factor in eq.(82) and thus we don’t get the standard $\frac{1}{4}$
factor. If the Lagrangian is of the form $\frac{1}{16\pi}R$, then we do get
the usual $\frac{1}{4}$ factor from the Wald entropy. denotes the area of the
constant $v$ slice $\mathcal{H}_{v}$ and
$\theta=\dfrac{1}{\sqrt{h}}\partial_{v}\sqrt{h}$ is the expansion of the null
generators $\chi^{\mu}=(\partial_{v})^{\mu}$ 181818$S$ of eq.(88) was recently
obtained in Hollands:2024vbe  using covariant phase methods of
PhysRevD.61.084027 . This $S$ was interpreted to be the Area of the apparent
horizon.. For arbitrary higher derivative theories of gravity of the form
$\mathcal{L}$ of eq.(2), the dynamical entropy is given by eq.(87). The
$\Theta$ clearly contains the information of the entropy current
$\mathcal{J}^{i}$ through eq.(83). We could alternately choose to integrate
out $\mathcal{J}^{i}$ by assuming $\mathcal{H}_{v}$ is compact to obtain
191919eq.(89) was obtained in Hollands:2024vbe . The entropy expression also
satisfies the PPFL between arbitrary time slices.

$S|_{\mathcal{H}_{v}}=\int_{\mathcal{H}_{v}}d^{d-2}x\,\sqrt{h}\,\mathcal{J}^{v}-v\partial_{v}\left[\int_{\mathcal{H}_{v}}d^{d-2}x\,\sqrt{h}\mathcal{J}^{v}\right]\,.$
(89)

It is useful to note that the expressions in eq.(87) and eq.(89) are valid for
arbitrary non-minimally coupled theories of the form eq.(2). This is because
these quantities are directly derived from the structure of $E_{vv}$ of the
eq.(3), which in turn follows from the fact that we have a Killing horizon at
$r=0$ for an equilibrium configuration. The structure of $E_{vv}$ for eq.(2)
has already been established in Biswas:2022grc  and it takes the form of
eq.(3). The currents $\mathcal{J}^{v}$ and $\mathcal{J}^{i}$ would receive
contributions from non-minimal matter couplings. A linearized PPFL for such
couplings has been proved in section 7 of Biswas:2022grc . The subtlety of
carefully incorporating both the matter and the gravitational contributions
from non-minimal couplings to the PPFL has been cleanly worked out in section
7 of Biswas:2022grc .

We now study the effect of the Iyer-Wald ambiguities summarized in eq.(78) on
the general expression for the dynamical entropy given by eq.(87). It is clear
from eq.(83) that $\Theta$ remains invariant under the ambiguities of eq.(78).
We should only analyse the effect of ambiguities on by $\mathcal{J}^{v}$,
which is given by $\mathcal{J}^{v}\to\mathcal{J}^{v}+\nabla_{i}A^{i}$. This is
a total derivative term that drops out when you integrate over a compact
$\mathcal{H}_{v}$. Thus, the Iyer-Wald ambiguities do not affect the PPFL up
to linear order in perturbations. This has also been emphasized recently in
Wall:2024lbd .

##  5 Verification of the proof

In this section, we will consider explicit examples of Lagrangians considered
in Bhattacharyya:2021jhr  and Biswas:2022grc  to illustrate the general
formulae for the transformation of the entropy currents derived in eq.(40). In
what follows, we will do a brute force computation of how the components of
entropy current transform under a reparametrization eq.(5). We will
extensively use the transformation properties of various structures collected
in Appendix C to analyse how the components of the entropy current in explicit
examples transform. In all cases, the brute force results match the prediction
eq.(40) of the abstract proof.

###  5.1 Non-minimally coupled scalar field

We first consider a case of Non-minimally coupled scalar field with a
Lagrangian given by

$\mathcal{L}{(\phi,g_{\mu\nu})}=R_{\mu\nu}(D^{\mu}\phi)(D^{\nu}\phi)\,,$ (90)

for which the components of the entropy current were worked out to be
Biswas:2022grc  (eq.(5.26) of Biswas:2022grc )

$\mathcal{J}^{v}=(\partial_{v}\phi)(\partial_{r}\phi)\,,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mathcal{J}^{i}=(\nabla^{i}\phi)(\partial_{v}\phi)\,.$
(91)

Under a reparametrization of the form eq.(5), the components of the entropy
current transform as

$\begin{split}\mathcal{J}^{v}&=(\partial_{r}\phi)(\partial_{v}\phi)\\\
&=(\partial_{\rho}\phi)(\partial_{\tau}\phi)+\tau\xi_{a}(\partial^{a}\phi)(\partial_{\tau}\phi)\\\
&=\widetilde{\mathcal{J}}^{\tau}+\tau\xi_{a}\widetilde{\mathcal{J}}^{a}\,,\\\
\mathcal{J}^{i}&=(\nabla^{i}\phi)(\partial_{v}\phi)\\\
&=e^{-\zeta}\widetilde{\mathcal{J}}^{i}\,.\end{split}$ (92)

These expressions agree with the general result eq.(40) once we see that

$\widetilde{E}^{\rho i\rho
j}_{R}=\mathcal{O}(\epsilon^{2})\implies\widetilde{L}^{ij}_{(1)}=\mathcal{O}(\epsilon^{2})\,.$
(93)

One important point to notice here is that
$\mathcal{B}_{(0)}=\mathcal{O}(\epsilon^{2})$ remains
$\mathcal{O}(\epsilon^{2})$ under the transformation given by eq.(38). This
can be seen from the following transformations as

$\begin{split}\mathcal{Q}^{rv}&=-2(\partial_{v}\phi)(\partial_{r}\phi)=\widetilde{\mathcal{Q}}^{\rho\tau}+2\tau\xi_{a}\widetilde{\mathcal{Q}}^{\rho
a}\,,\\\
\mathcal{Q}^{ri}&=-(\nabla^{i}\phi)(\partial_{v}\phi)=e^{-\zeta}\mathcal{Q}^{\rho
i}\,,\\\
J^{i}_{(1)}&=(\nabla^{i}\phi)(\partial_{v}\phi)=e^{-\zeta}\widetilde{J}^{i}_{(1)}\,.\end{split}$
(94)

The answers for these structures have been worked out in eq.(5.25) of
Biswas:2022grc . Therefore, under the reparametrization, $\mathcal{B}_{(0)}$
transforms to

$\mathcal{B}_{(0)}=\widetilde{\mathcal{B}}_{(0)}-\tau\xi_{i}\widetilde{\mathcal{Q}}^{\rho
i}-\tau\xi_{i}\widetilde{J}^{i}_{(1)}-\tau\widetilde{\nabla}_{i}\left(4\xi_{j}\widetilde{L}^{ij}_{(1)}\right)+4\tau\xi_{i}\xi_{j}\widetilde{L}^{ij}_{(1)}=\mathcal{O}(\epsilon^{2})\,.$
(95)

This is an important validation of the general result in eq.(40) because a
simple reparametrization shouldn’t generate any non-trivial expression from
trivial expressions.

###  5.2 Non-minimally coupled gauge field

Consider a non-minimally coupled gauge field with a Lagrangian of the form

$\mathcal{L}{(A_{\mu},g_{\mu\nu})}=R_{\mu\nu\rho\sigma}F^{\mu\nu}F^{\rho\sigma}\,.$
(96)

The components of the entropy current were worked out to be Biswas:2022grc
(eq.(6.44) of Biswas:2022grc )

$\mathcal{J}^{v}=-2F^{rv}F^{rv}\,,~{}~{}~{}~{}~{}~{}\mathcal{J}^{i}=-4F^{rv}F^{ri}\,.$
(97)

Since $F^{\mu\nu}$ is a covariant tensor, the transformation of
$\mathcal{J}^{v}$ and $\mathcal{J}^{i}$ under eq.(5) can be worked out to be
(upto linear order in $\epsilon$)

$\begin{split}\mathcal{J}^{v}&=-2F^{rv}F^{rv}=-2\frac{\partial{r}}{\partial{\rho}}\frac{\partial{v}}{\partial{\tau}}F^{\rho\tau}\frac{\partial{r}}{\partial{\rho}}\frac{\partial{v}}{\partial{\tau}}F^{\rho\tau}\\\
&=\widetilde{\mathcal{J}}^{\tau}-4\tau\xi_{a}F^{\rho a}F^{\rho\tau}\\\
&=\widetilde{\mathcal{J}}^{\tau}+\tau\xi_{a}\widetilde{\mathcal{J}}^{a}\,,\\\
\mathcal{J}^{i}&=-4F^{rv}F^{ri}=e^{-\zeta}\widetilde{\mathcal{J}}^{i}\,.\end{split}$
(98)

These expressions agree with eq.(40) once we use the fact that

$\widetilde{E}^{\rho i\rho
j}_{R}=\mathcal{O}(\epsilon^{2})\implies\widetilde{L}^{ij}_{(1)}=\mathcal{O}(\epsilon^{2})\,.$
(99)

Similar to the non-minimally coupled scalar case, here also we have
$\mathcal{B}_{(0)}=\mathcal{O}(\epsilon^{2})$. To see that it remains
$\mathcal{O}(\epsilon^{2})$ under reparametrization, we use the expressions
for $\mathcal{Q}^{rv}$,$\mathcal{Q}^{ri}$ and $J^{i}_{(1)}$ worked out in
eq.(6.47) of Biswas:2022grc :

$\begin{split}\mathcal{Q}^{rv}&=4F^{rv}F^{rv}\\\
\mathcal{Q}^{ri}&=4F^{ri}F^{rv}\\\ J^{i}_{(1)}&=-4F^{ri}F^{rv}\end{split}$
(100)

Thus, we straightforwardly have

$\mathcal{B}_{(0)}=\widetilde{\mathcal{B}}_{(0)}-\tau\xi_{i}\widetilde{\mathcal{Q}}^{\rho
i}-\tau\xi_{i}\widetilde{J}^{i}_{(1)}-\tau\widetilde{\nabla}_{i}\left(4\xi_{j}\widetilde{L}^{ij}_{(1)}\right)+4\tau\xi_{i}\xi_{j}\widetilde{L}^{ij}_{(1)}=\mathcal{O}(\epsilon^{2})\,.$
(101)

where, again, we see that the reparametrization of the geodesics doesn’t
affect these trivial terms.

###  5.3 Ricci scalar squared theory

In the following subsections, we will consider the effects of the
reparametrization on some higher-curvature gravity theories. Starting with the
Ricci scalar squared theory, the Lagrangian is given in terms of the Ricci
scalar for the full metric ($R$) as

$\mathcal{L}=R^{2}\,.$ (102)

The expressions for $\mathcal{J}^{v}$ and $\mathcal{J}^{i}$ corresponding to
this theory were worked out in Table 1 of Bhattacharya:2019qal  (also see
Table 7 above eq.(4.10) of Bhattacharyya:2021jhr ):

$\begin{split}\mathcal{J}^{v}=2R\,,~{}~{}~{}~{}~{}~{}\mathcal{J}^{i}=0\,.\end{split}$
(103)

Since $R$ is the Ricci tensor for the full metric, it is invariant under a
coordinate transformation. One can also see this if we use the complicated
metric transformations of eq.(14) in eq.(174). This serves as a consistency
check of the formulae derived in eq.(14). We thus have,

$\begin{split}\widetilde{\mathcal{J}}^{\tau}=2\widetilde{R}\,,~{}~{}~{}~{}~{}~{}\widetilde{\mathcal{J}}^{i}=0\,.\end{split}$
(104)

These expressions exactly match with the general expression given in eq.(40)
if one uses

$\widetilde{E}^{\rho i\rho j}_{R}=0\implies\widetilde{L}^{ij}_{(1)}=0\,.$
(105)

###  5.4 Ricci tensor squared theory

Next, we consider the Ricci tensor squared theory with the Lagrangian of the
form

$\mathcal{L}=R_{\mu\nu}R^{\mu\nu}\,.$ (106)

The corresponding $\mathcal{J}^{v}$ and $\mathcal{J}^{i}$ were worked out in
Table 1 of Bhattacharya:2019qal  to be:

$\begin{split}\mathcal{J}^{v}&=2R_{rv}-K\bar{K}\,,\\\
\mathcal{J}^{i}&=-h^{ij}\partial_{v}\omega_{j}+2\nabla_{j}K^{ij}-\nabla^{i}K\,.\\\
\end{split}$ (107)

In $(\rho,\tau,y^{a})$ coordinates, they can be written as

$\begin{split}\widetilde{\mathcal{J}}^{\tau}&=2\widetilde{R}_{\rho\tau}-\widetilde{K}\widetilde{\bar{K}}\,,\\\
\widetilde{\mathcal{J}}^{i}&=-\widetilde{h}^{ij}\partial_{\tau}\widetilde{\omega}_{j}+2\widetilde{\nabla}_{j}\widetilde{K}^{ij}-\widetilde{\nabla}^{i}\widetilde{K}\,.\\\
\end{split}$ (108)

Using the change of structures under reparametrization eq.(14) and eq.(15),
the transformation of the components of the entropy current can be obtained
as:

$\begin{split}\mathcal{J}^{v}-\widetilde{\mathcal{J}}^{\tau}&=\tau\left(2\xi_{i}\widetilde{\nabla}_{j}\widetilde{K}^{ij}-2\xi\cdot\widetilde{\nabla}\widetilde{K}-\widetilde{K}\widetilde{\nabla}\cdot\xi-\xi\cdot\partial_{\tau}\widetilde{\omega}+\xi^{2}\widetilde{K}\right)+\tau^{2}\xi^{2}\partial_{\tau}\widetilde{K}\,,\\\
e^{\zeta}\mathcal{J}^{i}-\widetilde{\mathcal{J}}^{i}&=\xi_{j}\widetilde{h}^{ij}\left(1+\tau\partial_{\tau}\right)\widetilde{K}\,.\end{split}$
(109)

These expressions match in the linear order with the general expression given
in eq.(40) if we use

$\widetilde{E}^{\rho i\rho
j}_{R}=\partial_{\tau}\left(-\dfrac{1}{2}\widetilde{h}^{ij}\widetilde{K}\right)\implies\widetilde{L}^{ij}_{(1)}=-\dfrac{1}{2}\widetilde{h}^{ij}\widetilde{K}\,.$
(110)

###  5.5 Riemann tensor squared theory

The Lagrangian for the Riemann Tensor Squared Theory is

$\mathcal{L}=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\,.$ (111)

The entropy current and density were worked out in Table 1 of
Bhattacharya:2019qal  to be:

$\begin{split}\mathcal{J}^{v}&=-4\left(K^{ij}\bar{K}_{ij}+R_{rvrv}\right)=-4\left(K^{ij}\bar{K}_{ij}+X+\frac{\omega^{2}}{4}\right)\,,\\\
\mathcal{J}^{i}&=4\left(\nabla_{j}K^{ij}-h^{ij}\partial_{v}\omega_{j}\right)\,.\\\
\end{split}$ (112)

In $(\rho,\tau,y^{a})$ coordinates, they can be written as

$\begin{split}\widetilde{\mathcal{J}}^{\tau}&=-4\left(\widetilde{K}^{ij}\widetilde{\bar{K}}_{ij}+\widetilde{R}_{\rho\tau\rho\tau}\right)=-4\left(\widetilde{K}^{ij}\widetilde{\bar{K}}_{ij}+\widetilde{X}+\frac{\widetilde{\omega}^{2}}{4}\right)\,,\\\
\widetilde{\mathcal{J}}^{i}&=4\left(\widetilde{\nabla}_{j}\widetilde{K}^{ij}-\widetilde{h}^{ij}\partial_{\tau}\widetilde{\omega}_{j}\right)\,.\\\
\end{split}$ (113)

Again, using the change of structures under reparametrization eq.(14) and
eq.(15), we get the transformation of the components of the entropy current to
be:

$\begin{split}\mathcal{J}^{v}-\widetilde{\mathcal{J}}^{\tau}&=4\tau\left(\xi_{a}\xi_{b}\widetilde{K}^{ab}-\widetilde{K}^{ab}\widetilde{\nabla}_{a}\xi_{b}-\xi_{a}\widetilde{h}^{ab}\partial_{\tau}\widetilde{\omega}_{b}\right)+4\tau^{2}\left(\xi_{a}\xi_{b}\partial_{\tau}\widetilde{K}^{ab}\right)\,,\\\
\mathcal{J}^{i}-e^{-\zeta}\widetilde{\mathcal{J}}^{i}&=4e^{-\zeta}\xi_{j}\left(1+\tau\partial_{\tau}\right)\widetilde{K}^{ij}\,.\end{split}$
(114)

These expressions exactly match with the general expression given in eq.(40)
if we use

$\widetilde{E}^{\rho i\rho
j}_{R}=-2\partial_{\tau}\widetilde{K}^{ij}\implies\widetilde{L}^{ij}_{(1)}=-2\widetilde{K}^{ij}\,.$
(115)

###  5.6 Gauss Bonnet theory

Combining the Lagrangians of the last three subsections, the Gauss-Bonnet
Lagrangian can be obtained as

$\mathcal{L}=R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\,.$
(116)

The transformations of the entropy currents were explicitly worked out in
section 4 of Bhattacharyya:2022njk . The transformation of the components of
the entropy current under reparametrization can be calculated from the
previous results to be

$\begin{split}\mathcal{J}^{v}-\widetilde{\mathcal{J}}^{\tau}&=2\mathcal{R}-2\widetilde{\mathcal{R}}\\\
&=2\left[-2\tau\left(-\xi\cdot\widetilde{K}\cdot\xi+4\widetilde{K}^{ij}\widetilde{\nabla}_{i}\xi_{j}+4\xi_{i}\widetilde{\nabla}_{j}\widetilde{K}^{ij}\right.\right.\\\
&\left.\left.-2\xi\cdot\widetilde{\nabla}\widetilde{K}+\xi^{2}\widetilde{K}-2\xi_{i}\widetilde{\nabla_{j}}\widetilde{K}^{ij}-3\widetilde{K}^{ij}\widetilde{\nabla_{j}}\xi_{i}-\widetilde{K}\widetilde{\nabla}\cdot\xi\right)\right.\\\
&\left.-2\tau^{2}\left(\xi^{2}\partial_{\tau}\widetilde{K}-\xi\cdot\partial_{\tau}\widetilde{K}\cdot\xi\right)\right]\,,\\\
\mathcal{J}^{i}-e^{-\zeta}\widetilde{\mathcal{J}}^{i}&=4e^{-\zeta}\left(\xi_{j}\widetilde{K}^{ij}+\tau\xi_{j}\partial_{\tau}\widetilde{K}^{ij}-\xi_{j}\widetilde{h}^{ij}\widetilde{K}-\tau\xi_{j}\widetilde{h}^{ij}\partial_{\tau}\widetilde{K}\right)\,,\end{split}$
(117)

which match exactly with the ones obtained in Bhattacharyya:2022njk . We see
that we can obtain the expressions already derived there from the general
result of eq.(40) with

$\widetilde{E}^{\rho i\rho
j}_{R}=\partial_{\tau}\left(2\widetilde{h}^{ij}\widetilde{K}-2\widetilde{K}^{ij}\right)\implies\widetilde{L}^{ij}_{(1)}=2\widetilde{h}^{ij}\widetilde{K}-2\widetilde{K}^{ij}\,.$
(118)

This completes our check of the validation of eq.(40) for explicit example
theories.

##  6 Gauge covariance to the non-linear order - a case study

In this section, we make some preliminary observations of the consequences of
a reparametrization of the form eq.(5) in the components of the entropy
current to the non-linear order in the amplitude of dynamics. The analysis in
section 3 and 4 was restricted to the linear order in dynamics. We want to see
if eq.(8) continues to hold after including the non-linear terms are added.
For this, we require the non-linear terms in eq.(3). Treating higher
derivative terms in a Lagrangian like eq.(147) in an Effective field theory
(EFT) sense, one can show that Hollands:2022fkn ; Davies:2022xdq ;
Davies:2023qaa  the structure of $E_{vv}$ is given by

$E_{vv}|_{r=0}=\underbrace{\partial_{v}\left(\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}\,\mathcal{J}^{v}\right)+\nabla_{i}\mathcal{J}^{i}\right)}_{E}\underbrace{-(K_{ij}+X_{ij})(K^{ij}+X^{ij})+D_{i}Y^{i}}_{F}+\mathcal{O}(l^{N})$
(119)

This structure was first argued for Lovelock class of theories in
Bhattacharyya:2016xfs  202020This structure was also recently argued for non-
minimally coupled Lagrangians of the form eq.(2) Davies:2024fut  treated in an
EFT sense.. The Lagrangian in eq.(147) will be treated like a formal sum of
terms starting with the Einstein two derivative gravity with higher
derivatives suppressed by the UV length scale $l$. In the above eq.(119), we
assume that the EFT Lagrangian is explicitly known up to $N$ or fewer
derivatives only.

In eq.(119), we have chosen to indicate the entropy current structure as ‘$E$’
and the non-linear structures as ‘$F$’. We want to see if $E$ and $F$
individually transform like eq.(20). This is the case for the $(2+1)$
dimensional pure gravitational Chern-Simons term worked out in section 4.1 of
Biswas:2022grc . We want to take the simplest example of the Gauss-Bonnet
theory to check whether $E$ and $F$ transform like eq.(20). We will see that
there is an additional structure to the transformation, and $E$ and $F$ won’t
individually transform as eq.(20).

#### Gauss Bonnet theory

We will start with an effective field approach with Gauss-Bonnet theory as the
sub-leading order term in the Lagrangian:

$\mathcal{L}_{\text{gauss}}=\frac{1}{2}\int{d^{d}x\sqrt{-g}\bigg{[}R+l^{2}(R^{2}-4R_{\mu\nu}R^{\mu\nu}+4R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma})\bigg{]}}\,.$
(120)

The papers Bhattacharyya:2016xfs ; Hollands:2022fkn  have investigated
$E_{vv}$ at the non-linear level. Following Hollands:2022fkn  and eq.(119),
$F$ is given by

$F=E_{vv}|_{r=0}-\partial_{v}\left(\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}\,\mathcal{J}^{v}\right)+\nabla_{i}\mathcal{J}^{i}\right)\,,$
(121)

where $F$ contains quadratic or higher order quantities of positive boost
weight up to $\mathcal{O}(\epsilon^{2})$ terms as indicated in eq.(119). The
second term on the RHS is $E$, the local entropy production term which
contains both linear and non-linear pieces as denoted below

$\partial_{v}\left(\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}\,\mathcal{J}^{v}\right)+\nabla_{i}\mathcal{J}^{i}\right)=A(\epsilon)+B(\epsilon^{2})\,.$
(122)

We have shown that local entropy production is gauge covariant under a
reparametrization as in eq.(8). This split of the linear and non-linear terms
will show the additional structure in the transformation properties, as will
be clear soon.

For the Einstein-Gauss-Bonnet theory of eq.(120), the entropy density
$(J^{v})$ and entropy current $(J^{i})$ on the horizon are given by Table 7
and eq.(4.14) of Bhattacharyya:2021jhr :

$\begin{split}J^{v}&=1+2l^{2}\mathcal{R}\,,\\\
J^{i}&=l^{2}\bigl{[}-4\nabla_{j}K^{ij}+4\nabla^{i}K\bigr{]}\,.\end{split}$
(123)

Therefore, the local entropy production term can be expanded up to
$\mathcal{O}(\epsilon^{2})$ in terms of metric components as

${}\begin{split}\partial_{v}\left(\frac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}J^{v}\right)+\nabla_{i}J^{i}\right)&=l^{2}[2\mathcal{R}\partial_{v}K+2K\partial_{v}\mathcal{R}+2\partial_{v}{}^{2}\mathcal{R}+4\partial_{v}\nabla_{i}\nabla^{i}K\\\
&-4\partial_{v}\nabla_{i}\nabla_{j}K^{ij}]+\mathcal{O}(\epsilon^{3})\,.\end{split}$
(124)

Thus, we get the explicit form of $A(\epsilon)$ and $B(\epsilon^{2})$ as
defined in eq.(122)

$\begin{split}A(\epsilon)&=l^{2}\big{[}2\mathcal{R}\partial_{v}K+2\partial_{v}{}^{2}\mathcal{R}+4\partial_{v}\nabla_{i}\nabla^{i}K-4\partial_{v}\nabla_{i}\nabla_{j}K^{ij}\big{]}\,,\\\
B(\epsilon^{2})&=l^{2}\big{[}2K\partial_{v}\mathcal{R}\big{]}\,.\end{split}$
(125)

We now look into the structure of $E_{vv}$ in eq.(121) for eq.(120), which is
given in eq.(3.57) of Bhattacharya:2019qal

$E_{vv}=R_{vv}+l^{2}(2RR_{vv}-4R^{\alpha\beta}R_{v\alpha
v\beta}-4R^{~{}\alpha}_{v}R_{v\alpha}+2R^{~{}\alpha\beta\sigma}_{v}R_{v\alpha\beta\sigma})\,.$
(126)

Eq.(126) can be written explicitly in terms of metric components as

${}\begin{split}E_{vv}&=(-\partial_{v}K-K_{ij}K^{ij})+2l^{2}\bigl{[}2\mathcal{R}_{ij}\partial_{v}K^{ij}+6\mathcal{R}_{ij}K_{k}^{i}K^{jk}-\mathcal{R}\partial_{v}K-\mathcal{R}K_{ij}K^{ij}\\\
&-2\omega^{k}K_{jk}\nabla_{i}K^{ij}+2\omega_{i}K^{ij}\nabla_{j}K+K\omega_{i}\omega_{j}K^{ij}-2\nabla_{i}K^{ij}\nabla_{k}K_{j}^{k}+4\nabla_{i}K\nabla_{j}K^{ij}\\\
&+2K\omega_{i}\nabla_{j}K^{ij}-2\nabla_{i}K\nabla^{i}K-2K\omega_{i}\nabla^{i}K-\frac{1}{2}K^{2}\omega^{2}+\frac{1}{2}\omega^{2}K_{ij}K^{ij}-\omega_{i}\omega^{k}K^{ij}K_{jk}\\\
&+2\nabla^{c}K^{ab}\left(\nabla_{c}K_{ab}-\nabla_{b}K_{ac}\right)+2\omega^{c}K^{ab}\nabla_{c}K_{ab}-2\omega^{c}K^{ab}\nabla_{b}K_{ac}-2\partial_{v}KK^{ij}\bar{K}_{ij}\\\
&+2\bar{K}K\partial_{v}K-2\bar{K}K^{ij}\partial_{v}K_{ij}-2K\bar{K}_{ij}\partial_{v}K^{ij}+4\partial_{v}K^{ij}\bar{K}_{jk}K_{i}^{k}\bigr{]}+\mathcal{O}(\epsilon^{3})\,.\end{split}$
(127)

The $E_{vv}$ can be split into linear and non-linear pieces
($E_{vv}(\epsilon),E_{vv}(\epsilon^{2})$). Hence, we may rewrite the
definition of $F$ of eq.(121) as

$F=E_{vv}(\epsilon)+E_{vv}(\epsilon^{2})-A(\epsilon)-B(\epsilon^{2})\,.$ (128)

However, from eq.(3) we know that for linearized approximation in $\epsilon$,

${}E_{vv}(\epsilon)=A(\epsilon)\,,$ (129)

where the linear terms of $E_{vv}$ from eq.(127) are given by

$E_{vv}(\epsilon)=2l^{2}\left(2\mathcal{R}_{ij}\partial_{v}K^{ij}-\mathcal{R}\partial_{v}K\right)\,.$
(130)

We have already proved in section 3 that $A(\epsilon)$ transforms
homogeneously up to linear order as per eq.(21). Hence, $E_{vv}(\epsilon)$
piece will also transform homogeneously up to linear order. However, one can
check that the non-linear piece, i.e., $B(\epsilon^{2})$ does not transform
homogeneously:

$\begin{split}B-e^{-2\zeta}\tilde{B}&=4e^{-2\zeta}\tilde{K}\left\\{\left(\xi_{i}\xi_{j}\tilde{K}^{ij}-2\xi_{i}\tilde{\nabla}_{j}\tilde{K}^{ij}-\tilde{K}^{ij}\tilde{\nabla}_{i}\xi_{j}+2\xi^{i}\tilde{\nabla}_{i}\tilde{K}-\xi^{2}\tilde{K}+\tilde{K}\tilde{\nabla}_{i}\xi^{i}\right)\right.\\\
&+\tau\left(3\xi_{i}\xi_{j}\partial_{\tau}\tilde{K}^{ij}-2\xi_{j}\partial_{\tau}\tilde{\nabla}_{i}\tilde{K}^{ij}-\tilde{\nabla}_{i}\xi_{j}\partial_{\tau}\tilde{K}^{ij}+2\xi^{i}\partial_{\tau}\tilde{\nabla}_{i}\tilde{K}-3\xi^{2}\partial_{\tau}\tilde{K}+\tilde{\nabla}_{i}\xi^{i}\partial_{\tau}\tilde{K}\right)\\\
&\left.+\tau^{2}\left(\xi_{i}\xi_{j}\partial_{\tau}{}^{2}\tilde{K}^{ij}-\xi^{2}\partial_{\tau}{}^{2}\tilde{K}\right)\right\\}+\mathcal{O}(\epsilon^{3})\,.\end{split}$
(131)

This explicitly shows that for Einstein-Gauss-Bonnet theory, the ‘$E$’ term of
eq.(122) does not transform homogeneously to non-linear order. From this, one
can also conclude that $F$ of eq.(121) (and hence eq.(119)) does not transform
homogeneously. The additional structure is that $E_{vv}(\epsilon^{2})$ of
eq.(128) which consists of both $B$ and $F$ transforms homogeneously as
eq.(20). Thus, there is a hierarchical structure to the $E_{vv}$ of eq.(119):
$\mathcal{O}(\epsilon)$ terms transform homogeneously up to
$\mathcal{O}(\epsilon^{2})$ terms; $\mathcal{O}(\epsilon^{2})$ terms transform
homogeneously up to $\mathcal{O}(\epsilon^{3})$ terms and so on.

##  7 Discussions

Let us now briefly summarize what we have been able to achieve in our paper.
We wanted to prove the covariance of entropy production explicated in eq.(8)
under a reparametrization of the form eq.(5). This result is explicitly
checked in section 3. Consequently, we have derived eq.(9) in eq.(40). eq.(9)
is an explicit formula that tells us how the components of the entropy current
transform under a reparametrization of the form eq.(5). In particular, the
formula had non-homogeneous terms in addition to the homogeneous
transformation pieces that represented the transformation of a covariant co-
dimension one vector on the horizon. These non-homogeneous pieces can be
interpreted as arising due to the intrinsic ambiguities of the form eq.(7).
The effect of these ambiguities was detailed in section 4. They originated in
the ambiguities arising from the covariant phase space quantities (namely
$\Theta^{\mu}$ and $Q^{\mu\nu}$), and the final effect on the components of
the entropy current are summarized in eq.(78). Following this in section 4.2,
we carefully analysed the physical process version of the first law for
arbitrary time slices of the horizon. As a byproduct of this analysis, we
naturally obtained an expression for the dynamical entropy of a black hole in
a theory of the form eq.(2), which was derived to be eq.(87). Then, in section
5, we explicitly verified the formulae eq.(9), thereby validating the abstract
proof of section 3. Finally, in section 6, we performed a preliminary analysis
of the reparametrization covariance of entropy production to non-linear order
in the dynamics. We observed that the entropy current structure $E$ of
eq.(119) doesn’t transform homogeneously by itself. However, the non-covariant
structures $E$ and $F$ of eq.(119) together transform homogeneously as
expected in eq.(20).

We now make some remarks on the possible future directions that could follow
from our analysis. Although we were mainly motivated by extending the initial
analysis of Bhattacharyya:2022njk  to arbitrary diffeomorphism invariant
theories of gravity non-minimally coupled to matter fields, we also wanted to
analyse the approach of Hollands:2022fkn  in proving the gauge invariance of
the definition of a dynamical entropy for black holes in such theories. We
emphasize that Hollands:2022fkn  was concerned with the question of the gauge
invariance of the definition of entropy. In contrast, we are concerned with
the gauge invariance of entropy production as in eq.(8). The definition of
entropy is given by the integral of $\mathcal{J}^{v}$ over a compact
codimension two slice of the dynamical horizon. The approach of
Hollands:2022fkn  argues that this definition is gauge invariant up to non-
linear order in perturbations. However, in Davies:2022xdq , it was shown that
the gauge invariance will be lost if one considers Lagrangian theories with
more than eight derivatives. Thus, it is useful to check whether our formalism
of working with covariant structures like $E_{vv}$ to argue for the gauge
invariance might help resolve this issue. Since the structure of eq.(119) was
argued in Hollands:2022fkn  using multiple components of EoM, a possible
future direction is to consider different linear combinations of EoM and see
if one can argue for gauge covariance (as EoM components transform
homogeneously as in eq.(20)).

One possible extension of this work would be to see if one can utilize the
Iyer-Wald ambiguities to cancel out the non-homogeneous pieces in the
transformation. It would be interesting to develop a systematic algorithm for
such cancellations in arbitrary higher-derivative theories.

Another future direction is to extend our analysis to generic Chern-Simons
theories of gravity considered in Deo:2023vvb . The non-trivial point is that
$\Theta^{r}$ and $Q^{r\mu}$ are no longer covariant and hence they do not
transform like eq.(37) and eq.(34) respectively under a reparametrization
eq.(5). Thus, we expect to have additional non-covariant terms in the final
result of eq.(40).

The different coordinate systems of section 2.2 are related by affine
transformations. It is interesting to study the consequences of exploring non-
affine transformations that will take one away from the gauge of eq.(1).
Certain aspects of these non-affine coordinate transformations have been
studied in the context of fluid gravity correspondence in
Chandranathan:2022pfx . These transformations were the map relating the bulk
horizon coordinates to the boundary fluid coordinates. It turns out that under
this map, the boundary entropy current for Gauss-Bonnet theory is not solely
dependent upon the fluid variables (like velocity, temperature and their
derivatives) but also on the mapping functions between the affine coordinates
on the horizon and the non-affine coordinates on the boundary. It would be
interesting to see whether these ambiguities can be used to redefine the
entropy currents in the affine coordinates so that they transform in a well-
behaved manner to entropy currents of the boundary fluid, which depend only on
fluid variables.

Finally, an interesting open direction would be the connection to the
intrinsic BMS symmetries of the null horizon Donnay:2015abr ; Blau:2015nee ;
Donnay:2016ejv ; Grumiller:2019fmp ; Adami:2021nnf ; Adami:2023fbm . In this
language, eq.(5) should be thought of as a “superrotation”. Now, there is an
isomorphism between BMS symmetries in $d+1$ dimensions and conformal
Carrollian symmetries in $d$ dimensions Duval:2014uva . The transformation
eq.(9) is very reminiscent of how a Carrollian vector field transforms
Ciambelli:2018ojf . This is expected because, on the horizon, the metric
degenerates. The precise connection between the non-covariant terms in eq.(9)
and the intrinsic Carrollian symmetries is worth exploring. It is also
interesting to ask if this entropy current arises from some Carrollian
description akin to Donnay:2019jiz . We hope to address these questions in
future works.

## Acknowledgements

We are greatly indebted to Sayantani Bhattacharyya and Nilay Kundu for
valuable insights and collaboration on related ideas. We also thank Parthajit
Biswas, Iain Davies, Krishna Jalan, Alok Laddha, Milan Patra, Harvey Reall,
Amitabh Virmani, Aron Wall, Zihan Yan, and Mangesh Mandlik for useful
discussions.

We are grateful to the organizers and IIT Mandi for their hospitality at the
“Student Talks in Trending Topics in Theory (ST4)” workshop where some part of
the work was carried out. We would like to thank the organizers of ISM 2023
for giving us an opportunity to present this work as a poster. PD acknowledges
the warm hospitality of University of Cambridge, University of Southampton,
Durham University, King’s College London, University of Edinburgh, IIT Bombay,
Institute of Mathematical Sciences Chennai, Chennai Mathematical Institute,
and BITS Goa during the course of this work. SR would like to acknowledge the
warm hospitality of Chennai Mathematical Institute during the final stages of
the work. PD would like to duly acknowledge the Council of Scientific and
Industrial Research (CSIR), New Delhi, for financial assistance through the
Senior Research Fellowship (SRF) scheme and the partial support from the Royal
Society of London international exchange grant with the University of
Edinburgh.

##  Appendix A The basic setup and review of entropy current construction

In this Appendix, we will briefly review the essential elements that go into
the construction of the entropy current Bhattacharyya:2021jhr . For more
details, the reader is referred to section 2 of Bhattacharyya:2021jhr .

#### Residual gauge transformations in the gauge:

The metric structure of the near horizon solution eq.(1) is schematically
described by figure 1. This, however, will not fix the coordinates completely
on constant $r$ and $v$ slices. We can do the following coordinate
transformations while preserving the form of the gauge:

  1. 1.

Given two arbitrary functions $g_{1}(x^{i})$ and $g_{2}(x^{i})$, the
transformation

$v\to\tilde{v}=g_{1}(x^{i})\,v+g_{2}(x^{i})\,,$ (132)

with a suitable re-definition of $r$ leaves the metric eq.(1) invariant. In
section 2.2, we will study this by explicitly constructing the functions
$g_{1}(x^{i})$ and $g_{2}(x^{i})$. We will see that if one considers the form
of the metric away from the horizon, these functions will, in principle,
depend on the affine coordinates $r$ and $v$.

  2. 2.

A residual diffeomorphism invariance of the co-dimension two slice of the
horizon

$x^{i}\to\tilde{x}^{i}=f^{i}(x^{j})\,.$ (133)

This allows us to define a covariant derivative $\nabla_{i}$ compatible with
the metric $h_{ij}$.

The affine reparametrization eq.(5) is a special case of these residual
coordinate transformations.

#### Dynamical structures in the gauge:

As mentioned below eq.(1), we are interested in a dynamical situation near
equilibrium. This is described by the perturbative approximation
$g_{\mu\nu}=g^{eq}_{\mu\nu}+\epsilon\,\delta g_{\mu\nu}$, where
$g^{eq}_{\mu\nu}$ is the equilibrium metric and $\epsilon\ll 1$ is the
parameter characterizing the deviations from equilibrium. Thus, we need to
construct an equilibrium solution with any available residual symmetry of the
metric. We choose the following rescaling of coordinates to be the symmetry of
the equilibrium metric:

$r\to\tilde{r}=b\,r\,,~{}~{}~{}v\to\tilde{v}=\dfrac{v}{b}\,,$ (134)

where $b$ is a constant. This transformation is called ‘Boost transformation’.
The equilibrium solution invariant under the boost transformation is given by

$ds^{2}=2dv\,dr-r^{2}\,X(rv,x^{i})\,dv^{2}+2r\,\omega_{i}(rv,x^{i})\,dv\,dx^{i}+h_{ij}(rv,x^{i})\,dx^{i}\,dx^{j}\,,$
(135)

where the $v$ dependence of metric components is fixed through the product
$rv$. The Killing vector that generates the horizon $r=0$ of the equilibrium
solution is

$\xi^{\alpha}\partial_{\alpha}=v\partial_{v}-r\partial_{r}\,.$ (136)

The perturbations about eq.(135) are of the form

$\begin{split}X(r,v,x^{i})&=X^{eq}(rv,x^{i})+\epsilon\,\delta X(r,v,x^{i})\\\
\omega_{i}(r,v,x^{i})&=\omega^{eq}_{i}(rv,x^{i})+\epsilon\,\delta\omega_{i}(r,v,x^{i})\\\
h_{ij}(r,v,x^{i})&=h^{eq}_{ij}(rv,x^{i})+\epsilon\,\delta
h_{ij}(r,v,x^{i})\end{split}$ (137)

Fixing the equilibrium solution allows us to classify the non-equilibrium
structures using the notion of “Boost weight”. To see this consider a quantity
$\mathcal{B}$ that transforms in the following way under the boost
transformation eq.(134):

$\mathcal{B}\to\widetilde{\mathcal{B}}=b^{w}\mathcal{B}\,,~{}~{}~{}\text{under
eq.\eqref{eq:boosttransf}}\,.$ (138)

$w$ is called the Boost weight of $\mathcal{B}$. If $w>0$, then $\mathcal{B}$
is a dynamical structure on the horizon $r=0$. This is because $\mathcal{B}$
will be of the form $(\partial_{r})^{m}(\partial_{v})^{m+w}\mathcal{C}$ and
$\mathcal{C}$ itself is given by

$\mathcal{C}(r,v,x^{i})=\mathcal{C}^{eq}(rv,x^{i})+\epsilon\,\delta\mathcal{C}(r,v,x^{i})\,.$
(139)

This is because $\mathcal{C}$ is only constructed out of
$X,\omega_{i},h_{ij},\nabla_{i}$. Thus, we have

$(\partial_{r})^{m}(\partial_{v})^{m+w}\mathcal{C}|_{r=0}=\epsilon(\partial_{r})^{m}(\partial_{v})^{m+w}\delta\mathcal{C}|_{r=0}\sim\mathcal{O}(\epsilon)\,.$
(140)

From the boost weight criterion eq.(138), it is clear that any covariant lower
indexed tensor that has more number of $v$ indices when compared to $r$
indices is a dynamical quantity. The notion of boost weight in eq.(138) also
gives additional structure to covariant quantities. For instance, according to
“Result: 1” of Bhattacharyya:2021jhr , a typical covariant tensor with boost
weight $a+1$ on the horizon $r=0$ can be written as

$\begin{split}t^{(k)}_{(a+1)}|_{r=0}&=M_{(-k)}\partial^{k+a+1}_{v}N_{(0)}+\mathcal{O}(\epsilon^{2})\\\
&=\partial^{(a+1)}_{v}\left[\sum^{k-1}_{m=0}(-1)^{m}\,[^{m+a}C_{m}]\,M_{(-k+m)}\partial^{k-m}_{v}N_{(0)}\right]\\\
&~{}~{}+(-1)^{k}\,[^{k+a}C_{a}]\,M_{(0)}\partial^{(a+1)}_{v}N_{(0)}+\mathcal{O}(\epsilon^{2})\,.\end{split}$
(141)

#### Local entropy production on the horizon:

We will see how the structure of $E_{vv}$ described by eq.(3) will lead to a
local second law as eq.(4). Given the Lagrangian $\mathcal{L}$ of eq.(2), we
add a null energy condition satisfying matter sector Lagrangian
$\mathcal{L}_{\text{mat}}$. Thus, the full equation of motion (EoM) is given
by

$\mathcal{E}^{full}_{\mu\nu}=E_{\mu\nu}+T_{\mu\nu}\,,$ (142)

where $E_{\mu\nu}$ is the contribution from eq.(2) and $T_{\mu\nu}$ is the
contribution from $\mathcal{L}_{\text{mat}}$. We now project the full EoM onto
the null generators $\partial_{v}$ and then go on-shell. We, thus, have

$\mathcal{E}^{full}_{vv}=\partial_{v}\left(\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}\,\mathcal{J}^{v}\right)+\dfrac{1}{\sqrt{h}}\partial_{i}\left(\sqrt{h}\,\mathcal{J}^{i}\right)\right)+T_{vv}+\mathcal{O}(\epsilon^{2})=0\,,$
(143)

where we have used eq.(3). The Null energy condition in our gauge eq.(1)
essentially amounts to $T_{vv}\geq 0$ and hence we have from eq.(143)

$\partial_{v}\left(\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}\,\mathcal{J}^{v}\right)+\dfrac{1}{\sqrt{h}}\partial_{i}\left(\sqrt{h}\,\mathcal{J}^{i}\right)\right)\leq
0~{}~{}~{}\text{upto}~{}\mathcal{O}(\epsilon^{2})\,.$ (144)

We now use the physical boundary condition that the black hole settles down to
some equilibrium solution in the far future as $v\to\infty$. Since
$\mathcal{J}^{v}$ has boost weight $0$ and $\mathcal{J}^{i}$ has boost weight
$+1$, we have

$\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}\,\mathcal{J}^{v}\right)+\dfrac{1}{\sqrt{h}}\partial_{i}\left(\sqrt{h}\,\mathcal{J}^{i}\right)\to
0~{}~{}~{}\text{as}~{}~{}~{}v\to\infty\,.$ (145)

Using eq.(144) and eq.(145), we straightforwardly have

$\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}\,\mathcal{J}^{v}\right)+\dfrac{1}{\sqrt{h}}\partial_{i}\left(\sqrt{h}\,\mathcal{J}^{i}\right)\geq
0~{}\text{upto}~{}\mathcal{O}(\epsilon^{2})~{}~{}~{}\text{for all finite
$v$}\,.$ (146)

This signifies ultra-local entropy production on the horizon as in eq.(4).

#### Proof via an off-shell identity:

We will now briefly sketch the proof of the structure of $E_{vv}$ (eq.(3)) for
arbitrary higher derivative theories of gravity. Certain elements of the proof
will be crucial for the analysis in sections 3 and 4. We will use the
structures of the covariant phase space formalism to facilitate this. For the
sake of simplicity, we will only review the case of pure gravitational
Lagrangian of the form

$L=L(g_{\mu\nu},R_{\mu\nu\rho\sigma},D_{\alpha_{1}}R_{\mu\nu\rho\sigma},\dots)\,.$
(147)

The variation of the Lagrangian $L$ of eq.(147) with respect to the metric is
given by

$\delta[\sqrt{-g}L]=\sqrt{-g}E^{\mu\nu}_{g}\delta
g_{\mu\nu}+\sqrt{-g}D_{\mu}\Theta^{\mu}[\delta g]\,.$ (148)

The crucial off-shell identity comes about if we consider a variation that is
produced due to an infinitesimal diffeomorphism $x^{\mu}\to
x^{\mu}+\zeta^{\mu}$ (i.e.) $\delta
g_{\mu\nu}=D_{\mu}\zeta_{\nu}+D_{\nu}\zeta_{\mu}$ for which we have

$\delta\left[\sqrt{-g}L\right]=\sqrt{-g}D_{\mu}(\zeta^{\mu}L)\,.$ (149)

Substituting eq.(149) in eq.(148) and using the Generalized Bianchi identity
$D_{\mu}E^{\mu\nu}=0$ 10.1063/1.528801 , we have

$D_{\mu}\left(\zeta^{\mu}L-2E^{\mu\nu}\zeta_{\nu}-\Theta^{\mu}[\mathcal{L}_{\xi}g_{\mu\nu}]\right)=0\,.$
(150)

Since we have a conserved vector, we can express the above equation in terms
of the “Iyer-Wald” charge $Q^{\mu\nu}$

$\zeta^{\mu}L-2E^{\mu\nu}\zeta_{\nu}-\Theta^{\mu}[\mathcal{L}_{\xi}g_{\mu\nu}]=-D_{\nu}Q^{\mu\nu}\,,$
(151)

for an antisymmetric $Q^{\mu\nu}=-Q^{\nu\mu}$. The equation eq.(151) is true
for any $\zeta^{\mu}$ but we will choose $\zeta^{\mu}=\xi^{\mu}$ given by the
Killing vector of the equilibrium solution eq.(136). Contracting eq.(151) with
$\xi^{\mu}$ on $r=0$, we finally get

$2v\,E_{vv}|_{r=0}=\left(-\Theta^{r}+D_{\mu}Q^{r\mu}\right)|_{r=0}\,.$ (152)

This is an important equation in the analysis of Bhattacharyya:2021jhr ;
Biswas:2022grc  because while we have no general structures for the equations
of motion for arbitrary higher derivative theories of gravity, we have general
structures for $\Theta^{r}$ and $Q^{r\mu}$ Iyer:1994ys . Using the general
structures, one can show that in our gauge eq.(1) they take the following form
on the horizon $r=0$:

$\begin{split}\Theta^{r}|_{r=0}&=(1+v\partial_{v})\mathcal{A}_{(1)}+v\partial^{2}_{v}\mathcal{B}_{(0)}+\mathcal{O}(\epsilon^{2})\,,\\\
Q^{r\mu}&=\mathcal{Q}^{r\mu}+v\,W^{r\mu}_{v}\,.\end{split}$ (153)

Substituting eq.(153) in eq.(152), we have

$\begin{split}2v\,E_{vv}=&\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}\,\mathcal{Q}^{rv}\right)+\nabla_{i}\mathcal{Q}^{ri}+W^{rv}_{v}-\mathcal{A}_{(1)}\\\
&+v\partial_{v}\left[W^{rv}_{v}+\nabla_{i}J^{i}_{(1)}-\mathcal{A}_{(1)}-\partial_{v}\mathcal{B}_{(0)}\right]+\mathcal{O}(\epsilon^{2})\,.\end{split}$
(154)

In writing the second line of eq.(154), we have used

$W^{ri}_{v}=\partial_{v}J^{i}_{(1)}+\mathcal{O}(\epsilon^{2})\implies\nabla_{i}W^{ri}_{v}=\partial_{v}\left(\nabla_{i}J^{i}_{(1)}\right)+\mathcal{O}(\epsilon^{2})\,,$
(155)

which is a consequence of eq.(141). Since then we have explicated the $v$
dependence in eq.(154), we can explicitly compare the coefficients of $v^{0}$
and $v^{1}$ in eq.(154) to write

$\begin{split}\mathcal{A}_{(1)}&=\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}\,\mathcal{Q}^{rv}\right)+\nabla_{i}\mathcal{Q}^{ri}+W^{rv}_{v}\,,\\\
2E_{vv}&=\partial_{v}\left[W^{rv}_{v}+\nabla_{i}J^{i}_{(1)}-\mathcal{A}_{(1)}-\partial_{v}\mathcal{B}_{(0)}\right]\,.\end{split}$
(156)

eq.(154) is justified in Appendix A.1.1 of Deo:2023vvb  and this has also been
emphasized in Hollands:2022fkn . Substituting for $\mathcal{A}_{(1)}$ from the
first equation of eq.(156) into the second equation, we have

$2E_{vv}|_{r=0}=-\partial_{v}\left[\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}\left(\mathcal{Q}^{rv}+\mathcal{B}_{(0)}\right)\right)+\nabla_{i}\left[\mathcal{Q}^{ri}-J^{i}_{(1)}\right]\right]+\mathcal{O}(\epsilon^{2})\,,$
(157)

where $\nabla_{i}$ is the covariant derivative associated with $h_{ij}$. Thus,
we have proved eq.(3), concluding our brief review of the construction of
entropy current. The generalization to $U(1)$ gauge theories requires more
technical machinery described in Biswas:2022grc  to bring the structure of
$\Theta^{r}$ and $Q^{r\mu}$ to the form of eq.(153). We will be using eq.(153)
and eq.(157) to argue for the gauge covariance of local entropy production in
section 3.

##  Appendix B Intricate details of the computation

In this Appendix, we collect the details of the computations made in section
3. For the computations, we keep terms only up to $\mathcal{O}(\epsilon)\,$.

  * •

Transformation of $\mathcal{Q}^{ri}$ and $W^{ri}_{v}$

$\begin{split}\mathcal{Q}^{ri}&=4E^{rirv}_{R}\\\ &=4\frac{\partial
r}{\partial\rho}\frac{\partial x^{i}}{\partial y^{a}}\frac{\partial
r}{\partial\rho}\frac{\partial v}{\partial\tau}\widetilde{E}^{\rho
a\rho\tau}_{R}+4\frac{\partial r}{\partial\rho}\frac{\partial x^{i}}{\partial
y^{a}}\frac{\partial r}{\partial\rho}\frac{\partial v}{\partial
y^{b}}\widetilde{E}^{\rho a\rho b}_{R}\\\ &=4e^{-\zeta}\widetilde{E}^{\rho
i\rho\tau}_{R}+4e^{-\zeta}\tau\xi_{b}\widetilde{E}^{\rho i\rho b}_{R}\\\
&=e^{-\zeta}\widetilde{\mathcal{Q}}^{\rho
i}+4e^{-\zeta}\tau\xi_{b}\widetilde{E}^{\rho i\rho b}_{R}\end{split}$ (158)

$\begin{split}W^{ri}_{v}&=-2E^{rirj}_{R}\omega_{j}+W^{rir}\\\
&=-2\frac{\partial r}{\partial\rho}\frac{\partial x^{i}}{\partial
y^{a}}\frac{\partial r}{\partial\rho}\frac{\partial x^{j}}{\partial
y^{b}}\widetilde{E}^{\rho a\rho b}_{R}\omega_{b}+\frac{\partial
r}{\partial\rho}\frac{\partial x^{i}}{\partial y^{a}}\frac{\partial
r}{\partial\rho}\widetilde{W}^{\rho a\rho}\\\
&=-2e^{-2\zeta}\widetilde{E}^{\rho i\rho
b}_{R}(\widetilde{\omega}_{b}+2\xi_{b}-2\tau\xi_{k}\widetilde{h}^{jk}\widetilde{K}_{bj})+e^{-2\zeta}\widetilde{W}^{\rho
i\rho}\\\ &=-2e^{-2\zeta}\widetilde{E}^{\rho i\rho
b}_{R}\widetilde{\omega}_{b}-4e^{-2\zeta}\xi_{b}\widetilde{E}^{\rho i\rho
b}_{R}+e^{-2\zeta}\widetilde{W}^{\rho i\rho}\\\
&=e^{-2\zeta}\widetilde{W}^{\rho
i}_{\tau}-4e^{-2\zeta}\xi_{b}\widetilde{E}^{\rho i\rho b}_{R}.\end{split}$
(159)

These equations prove eq.(27).

  * •

Transformation of $J^{i}_{(1)}$

By definition of eq.(26)
$W^{ri}_{v}=\partial_{v}J^{i}_{(1)}+\mathcal{O}(\epsilon^{2})$. As obtained
above,

$\begin{split}&W^{ri}_{v}=e^{-2\zeta}\widetilde{W}^{\rho
i}_{\tau}-4e^{-2\zeta}\xi_{b}\widetilde{E}^{\rho i\rho b}_{R}\\\
&\implies\partial_{v}{J^{i}_{(1)}}=e^{-2\zeta}\partial_{\tau}{\widetilde{J}^{i}_{(1)}}-4e^{-2\zeta}\xi_{b}\widetilde{E}^{\rho
i\rho b}_{R}\\\ &\implies
e^{-\zeta}\partial_{\tau}{J^{i}_{(1)}}=e^{-2\zeta}\partial_{\tau}{\widetilde{J}^{i}_{(1)}}-4e^{-2\zeta}\xi_{b}\partial_{\tau}{\widetilde{L}^{ib}_{(1)}}\\\
&\implies{J^{i}_{(1)}}=e^{-\zeta}{\widetilde{J}^{i}_{(1)}}-4e^{-\zeta}\xi_{b}\widetilde{L}^{ib}_{(1)}\end{split}$
(160)

This proves eq.(30).

  * •

Transformation of $\mathcal{J}^{i}$

$\begin{split}\mathcal{J}^{i}&=-\dfrac{1}{2}\left(\mathcal{Q}^{ri}-J^{i}_{(1)}\right)\,.\\\
&=-\dfrac{1}{2}(e^{-\zeta}\widetilde{\mathcal{Q}}^{\rho
i}+4e^{-\zeta}\tau\xi_{b}\widetilde{E}^{\rho i\rho
b}_{R}-e^{-\zeta}{\widetilde{J}^{i}_{(1)}}+4e^{-\zeta}\xi_{b}\widetilde{L}^{ib}_{(1)})\\\
&=-\dfrac{1}{2}(e^{-\zeta}\widetilde{\mathcal{Q}}^{\rho
i}+4e^{-\zeta}\tau\xi_{b}\partial_{\tau}{\widetilde{L}^{ib}_{(1)}}-e^{-\zeta}{\widetilde{J}^{i}_{(1)}}+4e^{-\zeta}\xi_{b}\widetilde{L}^{ib}_{(1)})\\\
&=-\dfrac{1}{2}(e^{-\zeta}\widetilde{\mathcal{Q}}^{\rho
i}-e^{-\zeta}{\widetilde{J}^{i}_{(1)}})-2e^{-\zeta}(1+\tau\partial_{\tau})(\xi_{b}\widetilde{L}^{ib}_{(1)})\\\
&=e^{-\zeta}\widetilde{\mathcal{J}}^{i}-2e^{-\zeta}(1+\tau\partial_{\tau})(\xi_{j}\widetilde{L}^{ij}_{(1)})\\\
\end{split}$ (161)

This proves eq.(31).

  * •

Transformation of $\mathcal{Q}^{rv}$ and $W^{rv}_{v}$

$\begin{split}\mathcal{Q}^{rv}&=4E^{rvrv}_{R}\\\ &=4\frac{\partial
r}{\partial\rho}\frac{\partial v}{\partial\tau}\frac{\partial
r}{\partial\rho}\frac{\partial
v}{\partial\tau}\widetilde{E}^{\rho\tau\rho\tau}_{R}+4\frac{\partial
r}{\partial\rho}\frac{\partial v}{\partial y^{i}}\frac{\partial
r}{\partial\rho}\frac{\partial v}{\partial\tau}\widetilde{E}^{\rho
i\rho\tau}_{R}+4\frac{\partial r}{\partial\rho}\frac{\partial
v}{\partial\tau}\frac{\partial r}{\partial\rho}\frac{\partial v}{\partial
y^{i}}\widetilde{E}^{\rho\tau\rho i}_{R}\\\ &+4\frac{\partial
r}{\partial\rho}\frac{\partial v}{\partial y^{i}}\frac{\partial
r}{\partial\rho}\frac{\partial v}{\partial y^{j}}\widetilde{E}^{\rho i\rho
j}_{R}\\\
&=4\widetilde{E}^{\rho\tau\rho\tau}_{R}+8\tau\xi_{i}\widetilde{E}^{\rho
i\rho\tau}_{R}+4\tau^{2}\xi_{i}\xi_{j}\widetilde{E}^{\rho i\rho j}_{R}\\\
&=\widetilde{\mathcal{Q}}^{\rho\tau}+2\xi_{i}\tau\widetilde{\mathcal{Q}}^{\rho
i}+4\tau^{2}\xi_{i}\xi_{j}\widetilde{E}^{\rho i\rho j}\end{split}$ (162)

This proves eq.(32).

$\begin{split}W^{rv}_{v}&=-2E^{rirv}_{R}\omega_{i}+W^{rvr}\\\ &=\frac{\partial
r}{\partial\rho}\frac{\partial v}{\partial\tau}\frac{\partial
r}{\partial\rho}\widetilde{W}^{\rho\tau\rho}+\frac{\partial
r}{\partial\rho}\frac{\partial v}{\partial y^{i}}\frac{\partial
r}{\partial\rho}\widetilde{W}^{\rho i\rho}-2\frac{\partial
r}{\partial\rho}\frac{\partial x^{i}}{\partial y^{a}}\frac{\partial
r}{\partial\rho}\frac{\partial v}{\partial\tau}\widetilde{E}^{\rho
a\rho\tau}_{R}(\widetilde{\omega_{i}}+2\xi_{i})\\\ &-2\frac{\partial
r}{\partial\rho}\frac{\partial x^{i}}{\partial y^{a}}\frac{\partial
r}{\partial\rho}\frac{\partial v}{\partial y^{b}}\widetilde{E}^{\rho a\rho
b}_{R}(\widetilde{\omega_{i}}+2\xi_{i})+\mathcal{O}(\epsilon^{2})\,\\\
&=e^{-\zeta}\widetilde{W}^{\rho\tau\rho}+\tau
e^{-\zeta}\xi_{i}\widetilde{W}^{\rho i\rho}-2e^{-\zeta}\widetilde{E}^{\rho
i\rho\tau}_{R}(\widetilde{\omega_{i}}+2\xi_{i})-2e^{-\zeta}\tau\xi_{b}\widetilde{E}^{\rho
i\rho b}_{R}(\widetilde{\omega_{i}}+2\xi_{i})\\\
&=e^{-\zeta}(\widetilde{W}^{\rho\tau\rho}-2\widetilde{E}^{\rho
i\rho\tau}_{R}\widetilde{\omega_{i}})+\tau
e^{-\zeta}\xi_{i}(\widetilde{W}^{\rho i\rho}-2\widetilde{E}^{\rho
i\rho\tau}_{R}\widetilde{\omega_{b}})-4e^{-\zeta}\xi_{i}\widetilde{E}^{\rho
i\rho\tau}_{R}\\\ &-4\tau e^{-\zeta}\xi_{i}\xi_{j}\widetilde{E}^{\rho i\rho
j}_{R}\\\ &=e^{-\zeta}(\widetilde{W}^{\rho\tau\rho}-2\widetilde{E}^{\rho
i\rho\tau}_{R}\widetilde{\omega_{i}})+\tau
e^{-\zeta}\xi_{i}(\widetilde{W}^{\rho i\rho}-2\widetilde{E}^{\rho i\rho
b}_{R}\widetilde{\omega_{b}})-4e^{-\zeta}\xi_{i}\widetilde{E}^{\rho
i\rho\tau}_{R}\\\ &-4\tau e^{-\zeta}\xi_{i}\xi_{j}\widetilde{E}^{\rho i\rho
j}_{R}+\\\
&=e^{-\zeta}\left(\widetilde{W}^{\rho\tau}_{\tau}+\tau\xi_{i}\widetilde{W}^{\rho
i}_{\tau}-4\xi_{i}\widetilde{E}^{\rho
i\rho\tau}_{R}-4\tau\xi_{i}\xi_{j}\widetilde{E}^{\rho i\rho
j}_{R}\right),\end{split}$ (163)

This proves eq.(33).

  * •

Transformation of $Q^{rv}$

$\begin{split}Q^{rv}&=\mathcal{Q}^{rv}+vW^{rv}_{v}\\\
&=\widetilde{\mathcal{Q}}^{\rho\tau}+2\xi_{i}\tau\widetilde{\mathcal{Q}}^{\rho
i}+4\tau^{2}\xi_{i}\xi_{j}\widetilde{E}^{\rho i\rho j}\\\ &~{}+\tau
e^{\zeta}\left[e^{-\zeta}\left(\widetilde{W}^{\rho\tau}_{\tau}+\tau\xi_{i}\widetilde{W}^{\rho
i}_{\tau}-4\xi_{i}\widetilde{E}^{\rho
i\rho\tau}_{R}-4\tau\xi_{i}\xi_{j}\widetilde{E}^{\rho i\rho
j}_{R}\right)\right]\\\
&=\widetilde{\mathcal{Q}}^{\rho\tau}+\tau\widetilde{W}^{\rho\tau}_{\tau}+2\xi_{i}\tau\widetilde{\mathcal{Q}}^{\rho
i}-\tau 4\xi_{i}\widetilde{E}^{\rho i\rho\tau}_{R}\\\
&=\widetilde{Q}^{\rho\tau}+2\xi_{i}\tau\widetilde{\mathcal{Q}}^{\rho
i}-\tau\xi_{i}\widetilde{\mathcal{Q}}^{\rho i}\\\
&=\widetilde{Q}^{\rho\tau}+\tau\xi_{i}\widetilde{Q}^{\rho i}\end{split}$ (164)

This proves eq.(34).

  * •

Transformation of $\mathcal{A}_{(1)}$.
By the definition of eq.(156)

$\begin{split}\mathcal{A}_{(1)}&=\dfrac{1}{\sqrt{h}}\partial_{v}\left(\sqrt{h}\,\mathcal{Q}^{rv}\right)+\dfrac{1}{\sqrt{h}}\partial_{i}\left(\sqrt{h}\,\mathcal{Q}^{ri}\right)+W^{rv}_{v}\,\\\
&=\dfrac{1}{\sqrt{\widetilde{h}}}\partial_{v}\left(\sqrt{\widetilde{h}}\,(\widetilde{\mathcal{Q}}^{\rho\tau}+2\xi_{i}\tau\widetilde{\mathcal{Q}}^{\rho
i}+4\tau^{2}\xi_{i}\xi_{j}\widetilde{E}^{\rho i\rho
j}_{R})\right)+\dfrac{1}{\sqrt{\widetilde{h}}}(\widetilde{\partial_{i}}-\tau\xi_{i}\partial_{\tau})\\\
&\times\left(\sqrt{\widetilde{h}}(e^{-\zeta}\widetilde{\mathcal{Q}}^{\rho
i}+4e^{-\zeta}\tau\xi_{b}\widetilde{E}^{\rho i\rho
b}_{R})\right)+e^{-\zeta}\left(\widetilde{W}^{\rho\tau}_{\tau}+\tau\xi_{i}\widetilde{W}^{\rho
i}_{\tau}-4\xi_{i}\widetilde{E}^{\rho
i\rho\tau}_{R}-4\tau\xi_{i}\xi_{j}\widetilde{E}^{\rho i\rho j}_{R}\right)\\\
&=\dfrac{1}{\sqrt{\widetilde{h}}}e^{-\zeta}\partial_{\tau}(\sqrt{\widetilde{h}}\widetilde{\mathcal{Q}}^{\rho\tau})+2e^{-\zeta}\xi_{i}\widetilde{\mathcal{Q}}^{\rho
i}+2e^{-\zeta}\tau\xi_{i}\partial_{\tau}\widetilde{\mathcal{Q}}^{\rho
i}+8e^{-\zeta}\tau\xi_{i}\xi_{j}\widetilde{E}^{\rho i\rho j}_{R}\\\
&+4\tau^{2}e^{-\zeta}\xi_{i}\xi_{j}\partial_{\tau}\widetilde{E}^{\rho i\rho
j}_{R}+\dfrac{1}{\sqrt{\widetilde{h}}}e^{-\zeta}\widetilde{\partial}_{i}(\sqrt{\widetilde{h}}\widetilde{\mathcal{Q}}^{\rho
i})-e^{-\zeta}\xi_{i}\widetilde{\mathcal{Q}}^{\rho
i}-e^{-\zeta}\tau\xi_{i}\partial_{\tau}\widetilde{\mathcal{Q}}^{\rho i}\\\
&+\dfrac{1}{\sqrt{\widetilde{h}}}e^{-\zeta}\widetilde{\partial}_{i}(\sqrt{\widetilde{h}}4\tau\xi_{b}{\partial}_{\tau}\widetilde{L}^{ib}_{(1)})-4\tau
e^{-\zeta}\xi_{i}\xi_{b}\widetilde{E}^{\rho i\rho
b}_{R}+e^{-\zeta}\widetilde{W}^{\rho\tau}_{\tau}+e^{-\zeta}\tau\xi_{i}\widetilde{W}^{\rho
i}_{\tau}\\\ &-e^{-\zeta}\xi_{i}\widetilde{\mathcal{Q}}^{\rho i}-4\tau
e^{-\zeta}\xi_{i}\xi_{j}\partial_{\tau}\widetilde{E}^{\rho i\rho j}_{R}-4\tau
e^{-\zeta}\xi_{i}\xi_{j}\partial_{\tau}\widetilde{E}^{\rho i\rho
j}_{R}-4\tau^{2}e^{-\zeta}\xi_{i}\xi_{j}\partial_{\tau}\widetilde{E}^{\rho
i\rho j}_{R}\\\
&=e^{-\zeta}\left(\dfrac{1}{\sqrt{\widetilde{h}}}e^{-\zeta}\partial_{\tau}+\dfrac{1}{\sqrt{\widetilde{h}}}e^{-\zeta}\widetilde{\partial}_{i}(\sqrt{\widetilde{h}}\widetilde{\mathcal{Q}}^{\rho
i})+\widetilde{W}^{\rho\tau}_{\tau}\right)+e^{-\zeta}\tau\xi_{i}\partial_{\tau}\widetilde{\mathcal{Q}}^{\rho
i}\\\
&+e^{-\zeta}\tau\widetilde{\nabla}_{i}(4\xi_{j}\partial_{\tau}\widetilde{L}^{ij}_{(1)})+e^{-\zeta}\tau\xi_{i}\partial_{\tau}\widetilde{J}^{i}_{(1)}-4e^{-\zeta}\tau\xi_{i}\xi_{j}\partial_{\tau}\widetilde{L}^{ij}_{(1)}\\\
&=e^{-\zeta}\widetilde{\mathcal{A}}_{(1)}+e^{-\zeta}\tau\xi_{i}\partial_{\tau}\widetilde{\mathcal{Q}}^{\rho
i}+e^{-\zeta}\tau\widetilde{\nabla_{i}}\left(4\xi_{j}\partial_{\tau}\widetilde{L}^{ij}_{(1)}\right)+e^{-\zeta}\tau\xi_{i}\partial_{\tau}\widetilde{J}^{i}_{(1)}\\\
&-4e^{-\zeta}\tau\xi_{i}\xi_{j}\partial_{\tau}\widetilde{L}^{ij}_{(1)}.\end{split}$
(165)

This proves eq.(36).

  * •

Transformation of $\mathcal{B}_{(0)}$

We begin with the expression of $\Theta^{r}$ on the horizon $r=0$ i.e,

$\begin{split}\Theta^{r}|_{r=0}&=(1+v\partial_{v})\mathcal{A}_{(1)}+v\partial^{2}_{v}\mathcal{B}_{(0)}\\\
&=e^{-\zeta}(1+\tau\partial_{\tau})\widetilde{\mathcal{A}}_{(1)}+e^{-\zeta}\tau\partial^{2}_{\tau}{\mathcal{B}}_{(0)}+e^{-\zeta}(1+\tau\partial_{\tau})(\tau\xi_{i}\partial_{\tau}\widetilde{\mathcal{Q}}^{\rho
i})\\\
&+e^{-\zeta}(1+\tau\partial_{\tau})(\tau\xi_{i}\partial_{\tau}\widetilde{{J}}^{i}_{(1)})+e^{-\zeta}(1+\tau\partial_{\tau})\left(\tau\widetilde{\nabla_{i}}\left(4\xi_{j}\partial_{\tau}\widetilde{L}^{ij}_{(1)}\right)\right)\\\
&-e^{-\zeta}(1+\tau\partial_{\tau})\left(-4\tau\xi_{i}\xi_{j}\partial_{\tau}\widetilde{L}^{ij}_{(1)}\right)\\\
&=e^{-\zeta}(1+\tau\partial_{\tau})\widetilde{\mathcal{A}}_{(1)}+e^{-\zeta}\partial^{2}_{\tau}{\mathcal{B}}_{(0)}+e^{-\zeta}\tau(2+\tau\partial_{\tau})\left(\xi_{i}\partial_{\tau}\widetilde{\mathcal{Q}}^{\rho
i}+\xi_{i}\partial_{\tau}\widetilde{{J}}^{i}_{(1)}\right.\\\ &\hskip
85.35826pt\left.+\widetilde{\nabla_{i}}(4\xi_{j}\partial_{\tau}\widetilde{L}^{ij}_{(1)})-4\xi_{i}\xi_{j}\partial_{\tau}\widetilde{L}^{ij}_{(1)}\right)\\\
&\equiv
e^{-\zeta}(1+\tau\partial_{\tau})\widetilde{\mathcal{A}}_{(1)}+e^{-\zeta}\tau\partial^{2}_{\tau}\widetilde{{\mathcal{B}}}_{(0)}\end{split}$
(166)

where
$\partial^{2}_{\tau}\widetilde{{\mathcal{B}}}_{(0)}=\partial^{2}_{\tau}{\mathcal{B}}_{(0)}+(2+\tau\partial_{\tau})\left(\xi_{i}\partial_{\tau}\widetilde{\mathcal{Q}}^{\rho
i}+\xi_{i}\partial_{\tau}\widetilde{{J}}^{i}_{(1)}+\widetilde{\nabla_{i}}(4\xi_{j}\partial_{\tau}\widetilde{L}^{ij}_{(1)})-4\xi_{i}\xi_{j}\partial_{\tau}\widetilde{L}^{ij}_{(1)}\right)$

$\begin{split}\partial^{2}_{\tau}\widetilde{{\mathcal{B}}}_{(0)}&=\partial^{2}_{\tau}{\mathcal{B}}_{(0)}+(2+\tau\partial_{\tau})\left(\xi_{i}\partial_{\tau}\widetilde{\mathcal{Q}}^{\rho
i}+\xi_{i}\partial_{\tau}\widetilde{{J}}^{i}_{(1)}+\widetilde{\nabla_{i}}(4\xi_{j}\partial_{\tau}\widetilde{L}^{ij}_{(1)})-4\xi_{i}\xi_{j}\partial_{\tau}\widetilde{L}^{ij}_{(1)}\right)\\\
\partial^{2}_{v}{\mathcal{B}}_{(0)}&=e^{-2\zeta}\left(\partial^{2}_{\tau}\widetilde{{\mathcal{B}}}_{(0)}-(2+\tau\partial_{\tau})\left(\xi_{i}\partial_{\tau}\widetilde{\mathcal{Q}}^{\rho
i}+\xi_{i}\partial_{\tau}\widetilde{{J}}^{i}_{(1)}+\widetilde{\nabla_{i}}(4\xi_{j}\partial_{\tau}\widetilde{L}^{ij}_{(1)})-4\xi_{i}\xi_{j}\partial_{\tau}\widetilde{L}^{ij}_{(1)}\right)\right)\end{split}$

Considering,

$(2+\tau\partial_{\tau})(\xi_{i}\partial_{\tau}\widetilde{\mathcal{Q}}^{\rho
i})=\partial_{\tau}\left((1+\tau\partial_{\tau})(\xi_{i}\widetilde{\mathcal{Q}}^{\rho
i})\right)=\partial_{\tau}\left(\partial_{\tau}(\tau\xi_{i}\widetilde{\mathcal{Q}}^{\rho
i})\right)$

We have,

$\begin{split}\partial^{2}_{v}{\mathcal{B}}_{(0)}&=e^{-2\zeta}\left(\partial^{2}_{\tau}\widetilde{{\mathcal{B}}}_{(0)}-\partial^{2}_{\tau}\left(\tau\xi_{i}\widetilde{\mathcal{Q}}^{\rho
i}+\tau\xi_{i}\widetilde{{J}}^{i}_{(1)}+\tau\widetilde{\nabla_{i}}(4\xi_{j}\widetilde{L}^{ij}_{(1)})-4\tau\xi_{i}\xi_{j}\widetilde{L}^{ij}_{(1)}\right)\right)\\\
&=e^{-2\zeta}\partial^{2}_{\tau}\widetilde{\mathcal{B}}_{(0)}\\\
\implies{\mathcal{B}}_{(0)}&=\widetilde{\mathcal{B}}_{(0)}-\left(\tau\xi_{i}\widetilde{\mathcal{Q}}^{\rho
i}+\tau\xi_{i}\widetilde{{J}}^{i}_{(1)}+\tau\widetilde{\nabla_{i}}(4\xi_{j}\widetilde{L}^{ij}_{(1)})-4\tau\xi_{i}\xi_{j}\widetilde{L}^{ij}_{(1)}\right)\end{split}$
(167)

This proves eq.(38).

  * •

Transformation of $\mathcal{J}^{v}$

$\begin{split}&\mathcal{J}^{v}=-\dfrac{1}{2}\left(\mathcal{Q}^{rv}+\mathcal{B}_{(0)}\right)\\\
&=-\dfrac{1}{2}\bigg{[}\widetilde{\mathcal{Q}}^{\rho\tau}+2\xi_{i}\tau\widetilde{\mathcal{Q}}^{\rho
i}+4\tau^{2}\xi_{i}\xi_{j}\widetilde{E}^{\rho i\rho
j}+\widetilde{\mathcal{B}}_{(0)}-\bigg{(}\tau\xi_{i}\widetilde{\mathcal{Q}}^{\rho
i}+\tau\xi_{i}\widetilde{{J}}^{i}_{(1)}+\tau\widetilde{\nabla_{i}}(4\xi_{j}\widetilde{L}^{ij}_{(1)})\\\
&-4\tau\xi_{i}\xi_{j}\widetilde{L}^{ij}_{(1)}\bigg{)}\bigg{]}\\\
&=\widetilde{\mathcal{J}}^{\tau}-\dfrac{1}{2}\left(\xi_{i}\tau\widetilde{\mathcal{Q}}^{\rho
i}-\tau\xi_{i}\widetilde{{J}}^{i}_{(1)}+4\tau^{2}\xi_{i}\xi_{j}\partial_{\tau}\widetilde{L}^{ij}_{(1)}-\tau\widetilde{\nabla_{i}}(4\xi_{j}\widetilde{L}^{ij}_{(1)})+4\tau\xi_{i}\xi_{j}\widetilde{L}^{ij}_{(1)}\right)\\\
&=\widetilde{\mathcal{J}}^{\tau}+\tau\xi_{i}\widetilde{\mathcal{J}}^{i}-2\tau^{2}\xi_{i}\xi_{j}\partial_{\tau}\widetilde{L}^{ij}_{(1)}+2\tau\widetilde{\nabla}_{i}\left(\xi_{j}\widetilde{L}^{ij}_{(1)}\right)-2\tau\xi_{i}\xi_{j}\widetilde{L}^{ij}_{(1)}\,.\end{split}$
(168)

This proves eq.(39).

##  Appendix C Change of structures under reparametrization

In this Appendix, we will list the transformation of various structures under
a reparametrization. On the horizon, the Christoffel symbols corresponding to
the induced metrics $h_{ij}$ and $\widetilde{h}_{ij}$ are related by

$\Gamma_{a,bc}(h)=\widetilde{\Gamma}_{a,bc}(\widetilde{h})-\tau\left(\xi_{c}\widetilde{K}_{ab}+\xi_{b}\widetilde{K}_{ac}-\xi_{a}\widetilde{K}_{bc}\right)\,,$
(169)

where the extrinsic curvatures $(K,\bar{K})$ and
$(\widetilde{K},\widetilde{\bar{K}})$ are defined as

$\displaystyle K_{ij}$ $\displaystyle=\frac{1}{2}\partial_{v}h_{ij}$
$\displaystyle K$ $\displaystyle=h^{ij}K_{ij}$
$\displaystyle\widetilde{K}_{ij}$
$\displaystyle=\frac{1}{2}\partial_{\tau}\widetilde{h}_{ij}$
$\displaystyle\widetilde{K}$
$\displaystyle=\widetilde{h}^{ij}\widetilde{K}_{ij}$ (170)
$\displaystyle\bar{K}_{ij}$ $\displaystyle=\frac{1}{2}\partial_{r}h_{ij}$
$\displaystyle\bar{K}$ $\displaystyle=h^{ij}\bar{K}_{ij}$
$\displaystyle\widetilde{\bar{K}}_{ij}$
$\displaystyle=\frac{1}{2}\partial_{\rho}\widetilde{h}_{ij}$
$\displaystyle\widetilde{\bar{K}}$
$\displaystyle=\widetilde{h}^{ij}\widetilde{\bar{K}}_{ij}$ (171)

The extrinsic curvature tensors transform under the coordinate transformation
as

$\begin{split}K_{ij}&=e^{-\zeta}\widetilde{K}_{ij}+\mathcal{O}(\rho)~{}~{}~{}~{}~{}~{}K=e^{-\zeta}\widetilde{K}+\mathcal{O}(\rho)\\\
\bar{K}_{ij}&=e^{\zeta}\widetilde{\bar{K}}_{ij}-e^{\zeta}\bigg{[}\tau\left\\{\xi_{i}\xi_{j}-\widetilde{\nabla_{i}}\xi_{j}+\frac{1}{2}\left(\xi_{j}\widetilde{\omega}_{i}+\xi_{i}\widetilde{\omega}_{j}\right)\right\\}-\tau^{2}\bigg{(}-\frac{1}{2}\xi_{a}\xi_{b}\widetilde{h}^{ab}\widetilde{K}_{ij}+\xi_{j}\xi_{k}\widetilde{K}_{i}^{k}\\\
&+\xi_{i}\xi_{k}\widetilde{K}_{j}^{k}\bigg{)}\bigg{]}+\mathcal{O}(\rho)\\\
\bar{K}&=e^{\zeta}\widetilde{\bar{K}}-e^{\zeta}\left[\tau\widetilde{h}^{ij}\left\\{\xi_{i}\xi_{j}-\widetilde{\nabla_{i}}\xi_{j}+\xi_{j}\widetilde{\omega}_{i}\right\\}-\tau^{2}\left(2\xi_{i}\xi_{j}\widetilde{K}^{ij}-\frac{1}{2}\widetilde{K}\xi_{a}\xi_{b}\widetilde{h}^{ab}\right)\right]+\mathcal{O}(\rho)\end{split}$
(172)

where the $\widetilde{\nabla}_{i}$s are covariant derivatives with respect to
the induced metric $\widetilde{h}_{ij}$ in the $(\rho,\tau,y^{a})$ coordinate
system.

Some more structures that would be useful for calculating the change in
entropy density and entropy current structures are given below.

$\begin{split}\partial_{v}\bar{K}&=\partial_{\tau}\widetilde{\bar{K}}-\left(\xi^{2}-\widetilde{\nabla}\cdot\xi+\widetilde{\omega}\cdot\xi\right)\\\
&+\tau\left(6\xi\cdot\widetilde{K}\cdot\xi-2\xi_{i}\widetilde{\nabla}_{j}\widetilde{K}^{ij}-2\widetilde{K}^{ij}\widetilde{\nabla}_{i}\xi_{j}+\xi\cdot\widetilde{\nabla}\widetilde{K}-\xi\cdot\partial_{\tau}\widetilde{\omega}+2\widetilde{\omega}\cdot\widetilde{K}\cdot\xi-\xi^{2}\widetilde{K}\right)\\\
&+\tau^{2}\left(2\xi\cdot\partial_{\tau}\widetilde{K}\cdot\xi-\frac{1}{2}\xi^{2}\partial_{\tau}\widetilde{K}\right)+\mathcal{O}(\rho)\\\
K\bar{K}&=\widetilde{K}\widetilde{\bar{K}}+\tau\widetilde{K}\left(-\xi^{2}+\widetilde{\nabla}\cdot\xi-\widetilde{\omega}\cdot\xi\right)+\mathcal{O}(\rho)\\\
\nabla\cdot\omega&=\widetilde{\nabla}\cdot\widetilde{\omega}+2\widetilde{\nabla}\cdot\xi\\\
&+\tau\left(-2\xi_{i}\widetilde{\nabla}_{j}\widetilde{K}^{ij}-2\widetilde{K}^{ij}\widetilde{\nabla}_{j}\xi_{i}+2\xi\cdot\widetilde{K}\cdot\widetilde{\omega}-\widetilde{K}\widetilde{\omega}\cdot\xi+6\xi\cdot\widetilde{K}\cdot\xi-2\xi^{2}\widetilde{K}-\xi\cdot\partial_{\tau}\widetilde{\omega}\right)\\\
&+2\tau^{2}\xi_{i}\xi_{j}\partial_{\tau}\widetilde{K}^{ij}\\\
\partial_{\tau}(\widetilde{\nabla}\cdot\xi)&=-2\widetilde{K}^{ij}\widetilde{\nabla}_{i}\xi_{j}-2\xi_{i}\widetilde{\nabla}_{j}\widetilde{K}^{ij}+\xi\cdot\widetilde{\nabla}\widetilde{K}\\\
\omega^{2}&=\widetilde{\omega}^{2}+4\xi_{a}\left(\widetilde{h}^{ab}\left(\widetilde{\omega}_{b}+\xi_{b}\right)\right)-4\tau\xi_{a}\left(\widetilde{K}^{ab}\left(\widetilde{\omega}_{b}+2\xi_{b}\right)\right)\\\
K^{ij}\bar{K}_{ij}&=\widetilde{K}^{ij}\widetilde{\bar{K}}_{ij}-\tau\left(\xi_{a}\widetilde{\omega}_{b}\widetilde{K}^{ab}+\xi_{a}\xi_{b}\widetilde{K}^{ab}-\widetilde{K}^{ab}\widetilde{\nabla}_{a}\xi_{b}\right)\end{split}$
(173)

The required Ricci and Riemann tensor components of the full metrics are
expressed as

$\begin{split}R_{rv}&=-\chi-\frac{\omega^{2}}{2}-\partial_{v}\bar{K}-K^{ij}\bar{K}_{ij}+\frac{1}{2}\nabla\cdot\omega\\\
\widetilde{R}_{\rho\tau}&=-\widetilde{\chi}-\frac{\widetilde{\omega}^{2}}{2}-\partial_{\tau}\widetilde{\bar{K}}-\widetilde{K}^{ij}\widetilde{\bar{K}}_{ij}+\frac{1}{2}\widetilde{\nabla}\cdot\widetilde{\omega}\\\
R_{rvrv}&=\chi+\frac{1}{4}\omega^{2}\\\
\widetilde{R}_{\rho\tau\rho\tau}&=\widetilde{\chi}+\frac{1}{4}\widetilde{\omega}^{2}\\\
R&=\mathcal{R}-2\chi-\frac{3\omega^{2}}{2}-4\partial_{v}\bar{K}+2\nabla\cdot\omega-2K^{ij}\bar{K}_{ij}-2K\bar{K}\\\
\widetilde{R}&=\widetilde{\mathcal{R}}-2\widetilde{\chi}-\frac{3\widetilde{\omega}^{2}}{2}-4\partial_{\tau}\widetilde{\bar{K}}+2\widetilde{\nabla}\cdot\widetilde{\omega}-2\widetilde{K}^{ij}\widetilde{\bar{K}}_{ij}-2\widetilde{K}\widetilde{\bar{K}}\end{split}$
(174)

The Ricci scalar corresponding to the induced metric $\widetilde{h}_{ij}$
transforms on the horizon as

$\begin{split}\mathcal{R}-\widetilde{\mathcal{R}}=-2\tau\left(-\xi\cdot\widetilde{K}\cdot\xi+\xi^{2}\widetilde{K}+2\xi_{i}\widetilde{\nabla_{j}}\widetilde{K}^{ij}+\widetilde{K}^{ij}\widetilde{\nabla_{j}}\xi_{i}-\widetilde{K}\widetilde{\nabla}\cdot\xi-2\xi\cdot\widetilde{\nabla}\widetilde{K}\right)\\\
-2\tau^{2}\left(\xi^{2}\partial_{\tau}\widetilde{K}-\xi\cdot\partial_{\tau}\widetilde{K}\cdot\xi\right)\end{split}$
(175)

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