we will see later in this paper, this particular term will be crucial to give
us the spatial components of an entropy current. Having said this, we should
also mention that one might come up with a particular theory of gravity where
such a term is not generated. Einstein’s two derivative gravity or $R^{2}$
theory of higher derivative gravity are such examples where $J^{i}_{(1)}$
turns out to be zero.

From eq.(116) we can then derive

$\nabla_{i}W^{ri}_{v}=\partial_{v}\left(\nabla_{i}J^{i}_{(1)}\right)+{\cal
O}(\epsilon^{2})\,.$ (117)

Finally, substituting eq.(115) and eq.(117) in equation eq.(114) we arrive at
the following expression for $D_{\mu}Q^{r\mu}$

$\begin{split}D_{\mu}Q^{r\mu}={1\over\sqrt{h}}\partial_{v}\left(\sqrt{h}\,\widetilde{Q}^{rv}\right)+\nabla_{i}\widetilde{Q}^{ri}+v\,\partial_{v}\left(\nabla_{i}J^{i}_{(1)}\right)+(1+v\,\partial_{v})W^{rv}_{v}+{\cal
O}(\epsilon^{2})\,.\end{split}$ (118)

For our purpose, we do not need to process this term any further.

####  3.2.4 Final analysis for the structure of $E_{vv}$

In the previous sub-sections, we have obtained the expressions for both
$\Theta^{r}$ and $D_{\mu}Q^{r\mu}$. For convenience, let us rewrite the final
structures of them here again,

$\begin{split}\Theta^{r}=&\,(1+v~{}\partial_{v}){\cal
A}_{(1)}+v~{}\partial_{v}^{2}{\cal B}_{(0)}+{\cal
O}(\epsilon^{2})\,,~{}\text{and}\,,~{}\\\
D_{\mu}Q^{r\mu}=&{1\over\sqrt{h}}\partial_{v}\left(\sqrt{h}\,\widetilde{Q}^{rv}\right)+\nabla_{i}\widetilde{Q}^{ri}+v~{}\partial_{v}\left(\nabla_{i}J^{i}_{(1)}\right)+(1+v~{}\partial_{v})W^{rv}_{v}+{\cal
O}(\epsilon^{2})\,,\end{split}$ (119)

where ${\cal A}_{(1)}$ and ${\cal B}_{(0)}$ are given in eq.(108). Next we
collect them together and substitute on the RHS of eq.(53) to obtain

$\begin{split}2\,v\,E_{vv}\big{|}_{r=0}=&\left(-\Theta^{r}+D_{\mu}Q^{r\mu}\right)\big{|}_{r=0}\\\
=&-{\cal
A}_{(1)}+{1\over\sqrt{h}}\partial_{v}\left(\sqrt{h}\,\widetilde{Q}^{rv}\right)+\nabla_{i}\widetilde{Q}^{ri}+W^{rv}_{v}\\\
&+v\,\partial_{v}\bigg{[}-{\cal
A}_{(1)}+W^{rv}_{v}+\nabla_{i}J^{i}_{(1)}-\partial_{v}{\cal
B}_{(0)}\bigg{]}+{\cal O}(\epsilon^{2})\,.\end{split}$ (120)

Since the dependence on $v$ is explicit in the equation above, we can compare
the coefficients of $v^{0}$ and $v$ on both sides of equation eq.(120) to find
the following two relations

$\begin{split}{\cal
A}_{(1)}&={1\over\sqrt{h}}\partial_{v}\left(\sqrt{h}\,\widetilde{Q}^{rv}\right)+\nabla_{i}\widetilde{Q}^{ri}+W^{rv}_{v}\,,\\\
2E_{vv}&=\partial_{v}\bigg{[}-{\cal
A}_{(1)}+W^{rv}_{v}+\nabla_{i}J^{i}_{(1)}-\partial_{v}{\cal
B}_{(0)}\bigg{]}\,.\end{split}$ (121)

Substituting ${\cal A}_{(1)}$ from the first relation into the second one of
eq.(121), we find

$\begin{split}2E_{vv}=\partial_{v}\bigg{[}-\left({1\over\sqrt{h}}\partial_{v}\left(\sqrt{h}\,\widetilde{Q}^{rv}\right)+\nabla_{i}\widetilde{Q}^{ri}\right)+\nabla_{i}J^{i}_{(1)}-\partial_{v}{\cal
B}_{(0)}\bigg{]}+{\cal O}(\epsilon^{2})\,.\end{split}$ (122)

As it was explained after eq.(108), we know that ${\cal B}_{(0)}$ is of boost
weight zero and it is always of order ${\cal O}(\epsilon)$, see eq.(109).
Therefore, we learn that

$\partial_{v}^{2}{\cal
B}_{(0)}={1\over\sqrt{h}}\partial_{v}\left[\sqrt{h}~{}\partial_{v}{\cal
B}_{(0)}\right]+{\cal O}\left(\epsilon^{2}\right)\,.$

Substituting this in the expression of $E_{vv}$ in eq.(122), we find

$2E_{vv}\big{|}_{r=0}=-\partial_{v}\bigg{(}{1\over\sqrt{h}}\partial_{v}\left[\sqrt{h}\,\left(\widetilde{Q}^{rv}+{\cal
B}_{(0)}\right)\right]+\nabla_{i}\left[\widetilde{Q}^{ri}-J^{i}_{(1)}\right]\bigg{)}+{\cal
O}\left(\epsilon^{2}\right)\,.$ (123)

This constitutes the main result of our paper. We have established that the
general structure of $E_{vv}$ as given above in eq.(123) exactly matches with
required expression mentioned in eq.(7). By comparing them we obtain the
following components of the entropy current

$\begin{split}{\cal J}^{v}=&-\dfrac{1}{2}\left(\widetilde{Q}^{rv}+{\cal
B}_{(0)}\right)\,,~{}~{}\text{and}~{}~{}{\cal
J}^{i}=-\dfrac{1}{2}\left(\widetilde{Q}^{ri}-J^{i}_{(1)}\right)\,.\end{split}$
(124)

In other words, we have also constructed an entropy current on the horizon,
with components given by ${\cal J}^{v}$ and ${\cal J}^{i}$ for an arbitrary
higher derivative theory of gravity.

We also note that $\widetilde{Q}_{rv}$, as worked out in eq.(113), is the Wald
entropy density. Additionally, as argued immediately after eq.(108), ${\cal
B}_{(0)}$ will contribute to the JKM ambiguities that are associated with Wald
entropy, and it will vanish on stationary horizons. Therefore, eq.(123) also
highlights how these ambiguous terms are fixed in order to argue for a
linearized second law, which was already pointed out in Wall:2015raa .

##  4 Verification of the proof for specific examples

In section §3 we have constructed a proof of the existence of an entropy
current in arbitrary higher derivative theories of gravity. In the following,
we would like to verify that the expression of the entropy current suggested
by the abstract proof, as given in eq.(123), indeed matches with the explicit
calculation of the same in some specific model of higher derivative gravity,
namely the four derivative theories. The entropy currents for four derivative
theories of gravity have already been worked out in (Bhattacharya:2019qal, ).
By comparing them with our result reported in this paper, we would, therefore,
have a conclusive check of the entropy currents derived via general Noetherian
principles and the boost symmetry of the horizon. For the reader’s
convenience, we summarize the main equations that we would like to explicitly
check. The Key equation that we work with is the following:

$2v\,E_{vv}|_{r=0}=[-\Theta^{r}+D_{\mu}Q^{r\mu}]|_{r=0}\,,$ (125)

where $\Theta^{r}$ has been argued to have the following form, see eq.(108),

$\Theta^{r}=(1+v~{}\partial_{v})\mathcal{A}_{(0)}+v~{}\partial^{2}_{v}\mathcal{B}_{(0)}\,.$
(126)

The Noether charge has the following form eq.(112):

$Q^{r\mu}=\widetilde{Q}^{r\mu}+v~{}W^{r\mu}_{v}$ (127)

Finally we have the expression for the entropy currents from $E_{vv}$ eq.(123)

$2E_{vv}=-\partial_{v}\bigg{(}{1\over\sqrt{h}}\partial_{v}\left[\sqrt{h}\,(\widetilde{Q}^{rv}+{\cal
B}_{(0)})\right]+\nabla_{i}\left[\widetilde{Q}^{ri}-J^{i}_{(1)})\right]\bigg{)}+{\cal
O}\left(\epsilon^{2}\right)\,.$ (128)

From Proposition 4.1 of Iyer:1994ys , we have the following general from for
the Noether charge 363636We have chosen not to consider the ambiguities of the
Noether charge here.

$Q^{\mu\nu}=W^{\mu\nu\alpha}\xi_{\alpha}-2E^{\mu\nu\alpha\beta}_{R}D_{[\alpha}\xi_{\beta]}$
(129)

If we evaluate the above expression in our gauge eq.(11), which we note below
for convenience

$\begin{split}ds^{2}=2\,dv\,dr-r^{2}\,X(r,v,x)~{}dv^{2}+2r\,\omega_{i}(r,v,x)\,dv\,dx^{i}+h_{ij}(r,v,x)\,dx^{i}\,dx^{j}\,.\end{split}$
(130)

and using the inverse metric components

$\begin{split}g^{rr}=r^{2}X(r,v,x^{i})+r^{2}\omega^{i}(r,v,x^{i})\omega_{i}(r,v,x^{i})=r^{2}X+r^{2}\omega^{2}\\\
g^{ri}=-r\omega^{i}\hskip 22.76228ptg^{rv}=1\hskip
22.76228ptg^{ij}=h^{ij}\hskip 22.76228ptg^{vv}=g^{vi}=0\end{split}$ (131)

we find the following quantities

$\begin{split}\widetilde{Q}^{rv}=4E^{rvrv}_{R}\,,\hskip
19.91684pt\widetilde{Q}^{ri}=4E^{rirv}_{R}\,,\\\
W^{ri}_{v}=-2E^{rirj}_{R}\omega_{j}+W^{rir}=\partial_{v}J^{i}_{(1)}\,.\end{split}$
(132)

We also define the following quantities related to the extrinsic curvature of
the horizon, as the final results have these structures:

$\begin{split}K_{ij}&=\dfrac{1}{2}\partial_{v}h_{ij}\hskip
22.76228ptK^{ij}=-\dfrac{1}{2}\partial_{v}h^{ij}\hskip
22.76228pt\bar{K}_{ij}=\dfrac{1}{2}\partial_{r}h_{ij}\hskip
22.76228pt\bar{K}^{ij}=-\dfrac{1}{2}\partial_{r}h^{ij}\\\
K&=\dfrac{1}{2}h^{ij}\partial_{v}h_{ij}=\dfrac{1}{\sqrt{h}}\partial_{v}\sqrt{h}\hskip
51.21504pt\bar{K}=\dfrac{1}{2}h^{ij}\partial_{r}h_{ij}=\dfrac{1}{\sqrt{h}}\partial_{r}\sqrt{h}\end{split}$
(133)

The results of the explicit calculations of the Noether charge $Q^{\mu\nu}$
and $\Theta^{\mu}$ for examples considered in (Bhattacharya:2019qal, ) are
given in Tables 4,5.

1 |  Einstein’s theory: $L=R$ |  $\Theta^{\mu}=g^{\mu\alpha}D^{\sigma}\delta g_{\alpha\sigma}-g^{\rho\sigma}D^{\mu}\delta g_{\rho\sigma}$
---|---|---
2 |  Ricci Scalar Squared Theory: $L=R^{2}$ |  $\Theta^{\mu}=2\left[R(g^{\mu\alpha}D^{\nu}\delta g_{\nu\alpha}-g^{\nu\alpha}D^{\mu}\delta g_{\nu\alpha})\right.$ $\quad\quad\left.-g^{\mu\alpha}(D^{\beta}R)\delta g_{\alpha\beta}+(D^{\mu}R)g^{\alpha\beta}\delta g_{\alpha\beta}\right]$
3 |  Ricci Tensor Squared Theory: $L=R_{\mu\nu}R^{\mu\nu}$ |  $\Theta^{\mu}=2g^{\mu\alpha}R^{\sigma\nu}D_{\sigma}\delta g_{\nu\alpha}-2(D^{\alpha}R^{\mu\nu})\delta g_{\nu\alpha}$ $-R^{\alpha\beta}D^{\mu}\delta g_{\alpha\beta}+(D^{\mu}R^{\alpha\beta})\delta g_{\alpha\beta}$ $\quad\quad-g^{\rho\sigma}R^{\mu\nu}D_{\nu}\delta g_{\rho\sigma}+\dfrac{1}{2}(D^{\mu}R)g^{\alpha\beta}\delta g_{\alpha\beta}$
4 |  Riemann Tensor Squared Theory: $L=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$ |  $\Theta^{\mu}=4R^{\mu\rho\nu\sigma}D_{\sigma}\delta g_{\nu\rho}-4(D_{\sigma}R^{\mu\nu\rho\sigma})\delta g_{\nu\rho}$
Table 4: Symplectic Potential for Einstein gravity and other four derivative theories of gravity 1 |  Einstein’s theory: $L=R$ |  $Q^{\mu\nu}=(D^{\nu}\xi^{\mu}-D^{\mu}\xi^{\nu})$
---|---|---
2 |  Ricci Scalar Squared Theory: $L=R^{2}$ |  $Q^{\mu\nu}=[4(D^{\mu}R)g^{\nu\alpha}-4(D^{\nu}R)g^{\mu\alpha}]\xi_{\alpha}$ $+2R(D^{\nu}\xi^{\mu}-D^{\mu}\xi^{\nu})$
3 |  Ricci Tensor Squared Theory: $L=R_{\mu\nu}R^{\mu\nu}$ |  $Q^{\mu\nu}=\big{(}2D^{\mu}R^{\nu\alpha}-2D^{\nu}R^{\mu\alpha}+(D^{\mu}R)g^{\nu\alpha}$ $\quad\quad-(D^{\nu}R)g^{\mu\alpha}\big{)}\xi_{\alpha}+2R^{\nu\beta}D_{\beta}\xi^{\mu}-2R^{\mu\beta}D_{\beta}\xi^{\nu}$
4 |  Riemann Tensor Squared Theory: $L=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$ |  $Q^{\mu\nu}=4R^{\mu\nu\alpha\sigma}D_{\sigma}\xi_{\alpha}-8(D_{\sigma}R^{\mu\nu\alpha\sigma})\xi_{\alpha}$
Table 5: Noether Charges for Einstein gravity and other four derivative
theories of gravity

Using the above expressions of Tables 4,5, it is straightforward to evaluate
$\Theta^{r}$ and the various quantities of eq.(132) which are given in Table
6.

1 |  Einstein’s theory: $L=R$ |  $\Theta^{r}=(1+v\partial_{v})(-2K)$ $\widetilde{Q}^{rv}=-2\hskip 19.91684pt\widetilde{Q}^{ri}=0\hskip 19.91684ptJ^{i}_{(1)}=0$
---|---|---
2 |  Ricci Scalar Squared Theory: $L=R^{2}$ |  $\Theta^{r}=(1+v\partial_{v})(-4RK)$ $\widetilde{Q}^{rv}=-4R\hskip 19.91684pt\widetilde{Q}^{ri}=0\hskip 19.91684ptJ^{i}_{(1)}=0$
3 |  Ricci Tensor Squared Theory: $L=R_{\mu\nu}R^{\mu\nu}$ |  $\Theta^{r}=(1+v~{}\partial_{v})[-2R^{ij}K_{ij}-2R_{vr}K+2K\partial_{v}\bar{K}$ $-2\partial_{v}(K\bar{K})]+v~{}\partial^{2}_{v}[2K\bar{K}]$ $\widetilde{Q}^{rv}=-4R_{rv}$ $\widetilde{Q}^{ri}=h^{ij}\partial_{v}\omega_{j}-2\nabla_{n}K^{in}+2\nabla^{i}K+\omega^{i}K$ $J^{i}_{(1)}=\omega^{i}K-h^{ij}\partial_{v}\omega_{j}+2\nabla_{j}K^{ij}$
4 |  Riemann Tensor Squared Theory: $L=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$ |  $\Theta^{r}=(1+v~{}\partial_{v})[8R_{vijr}K^{ij}+8K_{ij}\partial_{v}\bar{K}^{ij}$ $-8\partial_{v}(\bar{K}^{ij}K_{ij})]+v~{}\partial^{2}_{v}(8\bar{K}^{ij}K_{ij})$ $\widetilde{Q}^{rv}=8R_{rvrv}$ $\widetilde{Q}^{ri}=4h^{ij}\partial_{v}\omega_{j}+4\omega_{k}K^{ik}$ $J^{i}_{(1)}=-4h^{ij}\partial_{v}\omega_{j}+4\omega_{j}K^{ij}+8\nabla_{j}K^{ij}$
Table 6: Summary of the various quantities needed to compute the Entropy
currents

From the above results, it is clear that in examples considered, $\Theta^{r}$
is really of the form given in eq.(108). Using the expressions of Table 6, it
is straightforward to evaluate the entropy currents
$\mathcal{J}^{v}$,$\mathcal{J}^{i}$ which are summarized in Table 7.

1 |  Einstein’s theory: $L=R$ |  $\mathcal{J}^{v}=1\hskip 19.91684pt\mathcal{J}^{i}=0$
---|---|---
2 |  Ricci Scalar Squared Theory: $L=R^{2}$ |  $\mathcal{J}^{v}=2R\hskip 19.91684pt\mathcal{J}^{i}=0$
3 |  Ricci Tensor Squared Theory: $L=R_{\mu\nu}R^{\mu\nu}$ |  $\mathcal{J}^{v}=2R_{rv}-K\bar{K}$ $\mathcal{J}^{i}=-h^{ij}\partial_{v}\omega_{j}+2\nabla_{j}K^{ij}-\nabla^{i}K$
4 |  Riemann Tensor Squared Theory: $L=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$ |  $\mathcal{J}^{v}=-4R_{rvrv}-4\bar{K}^{ij}K_{ij}$ $\mathcal{J}^{i}=-4h^{ij}\partial_{v}\omega_{j}+4\nabla_{j}K^{ij}$
Table 7: Entropy currents for Einstein gravity and other four derivative
theories of gravity

Finally, we provide the expressions for Gauss-Bonnet Theory:
$L=R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$.
$\Theta^{\mu}$ is given by

$\begin{split}\Theta^{\mu}=2R(g^{\mu\alpha}D^{\sigma}\delta
g_{\alpha\sigma}-g^{\rho\sigma}D^{\mu}\delta
g_{\rho\sigma})-2g^{\mu\nu}(D^{\alpha}R)\delta g_{\alpha\nu}\\\
-8g^{\mu\alpha}R^{\sigma\nu}D_{\sigma}\delta
g_{\nu\alpha}+4(D^{\alpha}R^{\mu\nu})\delta
g_{\nu\alpha}+4R^{\alpha\beta}D^{\mu}\delta g_{\alpha\beta}\\\
+4g^{\rho\sigma}R^{\mu\nu}D_{\nu}\delta
g_{\rho\sigma}+4R^{\mu\nu\rho\sigma}D_{\sigma}\delta g_{\nu\rho}\end{split}$
(134)

The Noether charge is given by

$Q^{\mu\nu}=2R(D^{\nu}\xi^{\mu}-D^{\mu}\xi^{\nu})+8(R^{\mu\beta}D_{\beta}\xi^{\nu}-R^{\nu\beta}D_{\beta}\xi^{\mu})+4R^{\mu\nu\alpha\sigma}D_{\sigma}\xi_{\alpha}$
(135)

Thus, $\Theta^{r}$ and the quantities in eq.(132) are straightforward to
compute:

$\begin{split}\Theta^{r}&=(1+v\partial_{v})[-4RK+8R^{ij}K_{ij}+8R_{rv}K-8K\partial_{v}\bar{K}\\\
&+8R_{vijr}K^{ij}+8K_{ij}\partial_{v}\bar{K}^{ij}+8\partial_{v}(K\bar{K}-8K_{ij}\bar{K}^{ij})]\\\
&+v\partial^{2}_{v}[-8K\bar{K}+8\bar{K}^{ij}K_{ij}]\end{split}$ (136)
$\begin{split}\widetilde{Q}^{rv}&=-4R+16R_{rv}+8R_{rvrv}\\\
\widetilde{Q}^{ri}&=8\nabla_{j}K^{ij}-8\nabla^{i}K-4\omega^{i}K+4\omega_{j}K^{ij}\\\
J^{i}_{(1)}&=-4\omega^{i}K+4\omega_{j}K^{ij}\end{split}$ (137)

The Entropy currents are therefore given by

$\begin{split}\mathcal{J}^{v}&=2R-8R_{rv}-4R_{rvrv}+4K\bar{K}-4\bar{K}^{ij}K_{ij}=2\mathcal{R}\\\
\mathcal{J}^{i}&=-4\nabla_{j}K^{ij}+4\nabla^{i}K\end{split}$ (138)

where $\mathcal{R}$ is the intrinsic Ricci scalar of the horizon.

These results exactly match with the explicit results of
(Bhattacharya:2019qal, ). To see this, we must compare the expressions written
above for ${\cal J}^{v}$ and ${\cal J}^{i}$ as listed in Table-7 with
section-$3.1$ in Bhattacharya:2019qal 373737It must be noted that there is an
overall negative sign in the equations of motion used in the calculations
above ((i.e.) $E_{vv}$ ) when compared to the expressions used in
(Bhattacharya:2019qal, ). Due to an additional overall negative sign in
definitions of the entropy currents, they exactly match with the results
provided here.. Thus, the results in this section are validation of the proof
provided above.

##  5 A proof of the physical process version of the first law

So far in this paper, our focus has been to construct an entropy current with
the motivation to produce a proof of the linearized version of the second law
of thermodynamics for dynamical black holes. However, a similar framework
could be used to construct an alternative proof of a particular version of the
first law, namely, the physical process version of it, Jacobson:1995uq ;
Gao:2001ut ; Amsel:2007mh ; Bhattacharjee:2014eea ; Chakraborty:2017kob ;
Chatterjee:2011wj ; Kolekar:2012tq . Interestingly, one should note that in
Wall:2015raa  the validity of this version of the first law was actually in a
subtle way needed to prove the linearized version of the second law (see
section-$(2.2.3)$ in Bhattacharya:2019qal  for a detailed discussion on this
point).

The purpose of this section is to reorganize the setup that we have already
developed in this paper in order to devise an alternative proof of the
physical process version of the first law applicable to the most general
higher derivative theories of gravity. The main ingredient that we will use to
achieve this, is essentially the fact that $\chi^{\mu}\xi^{\nu}E_{\mu\nu}$,
383838Recall $\chi=\chi^{a}\partial_{a}=\partial_{v}$ is the affinely
parametrized null generator of the horizon and
$\xi=\xi^{\mu}\partial_{\mu}=v\partial_{v}-r\partial_{r}$ is the Killing
vector field. could always be expressed in the form of eq.(7).

Here we are restricting ourselves to a situation where the dynamical
fluctuation around a stationary horizon is of small amplitude. It is initiated
by a small in-falling matter perturbing a black hole configuration at
equilibrium. By small matter, what we mean is more precisely the introduction
of a perturbation in the system of equations by a small matter stress tensor
denoted by $\delta T_{\mu\nu}$. The perturbed horizon eventually settles down
to a new stationary black hole configuration with slightly shifted asymptotic
parameters (identified with the total energy, angular momentum, etc.) and
slightly altered geometry around the horizon (identified with the total
entropy). Since the same stress tensor component generates all such
infinitesimal shifts, they are related, and this relation is nothing but the
statement of the first law.

Let us denote the change in entropy due to the evolution from the initial to
the final equilibrium configuration as $\delta S$. The relevant component of
the stress tensor which gets related to $\delta S$ can be identified with
$\chi^{\mu}\xi^{\nu}\,\delta T_{\mu\nu}$ and that helps us writing down the
first law more concretely as follows393939A detailed derivation of this can be
found in section $2$ of Jacobson:1995uq . Also see section-(2.2.3) of
Bhattacharya:2019qal  for more discussions regarding the same.

$\begin{split}\delta S=&\,2\pi\int_{\cal
H}dv\,d^{d-2}x\,\sqrt{h}\,\chi^{\mu}\xi^{\nu}\delta
T_{\mu\nu}=\,-2\pi\int_{-\infty}^{+\infty}dv\int_{{\cal
H}_{v}}d^{d-2}x\,\sqrt{h}\,v\,E_{vv}\,\end{split}$ (139)

where $\cal H$ is the co-dimension one horizon at $r=0$, ${\cal H}_{v}$ is the
constant $v$-slice of the horizon. Also for the last equality above we have
used the fact that the equation of motion is given by $E_{\mu\nu}=-\delta
T_{\mu\nu}$ (see eq.(36)), along with $\chi^{v}=1$ and $\xi^{v}=v$ are the
only non-zero components of $\chi^{\mu},\,\xi^{\nu}$ in our set up.

Next, let us recollect that the main result of §3 was to obtain $E_{vv}$ in
the form given in eq.(123), which we write here again for convenience,

$\begin{split}E_{vv}=&\,\partial_{v}\left({1\over\sqrt{h}}\partial_{v}\left(\sqrt{h}\,{\cal
J}^{v}\right)+\nabla_{i}{\cal J}^{i}\right)\,+{\cal O}(\epsilon^{2}),\\\
\text{with}~{}~{}{\cal J}^{v}=&\,-\dfrac{1}{2}\left(\widetilde{Q}^{rv}+{\cal
B}_{(0)}\right)\,,~{}~{}{\cal
J}^{i}=\,-\dfrac{1}{2}\left(\widetilde{Q}^{ri}-J^{i}_{(1)}\right)\,.\end{split}$
(140)

In what follows, we will argue that the form of $E_{vv}$ as given in eq.(140)
is indeed consistent with eq.(139). Therefore, to show that eq.(140) is true
in arbitrary higher derivative theories of gravity is equivalent to verify the
physical process version of the first law in such theories.

We first substitute for $E_{vv}$ from eq.(140) in eq.(139) followed by certain
manipulations involving integrations by parts to move the overall
$\partial_{v}$ on the RHS of eq.(140) in $E_{vv}$ to obtain

$\begin{split}\delta S&=-\int_{-\infty}^{\infty}dv\int_{{\cal
H}_{v}}d^{d-2}x~{}v\,\sqrt{h}~{}\partial_{v}\bigg{(}{1\over\sqrt{h}}\partial_{v}\left(\sqrt{h}~{}{\cal
J}^{v}\right)+\nabla_{i}{\cal J}^{i}\bigg{)}\\\ &=-\bigg{[}\int_{{\cal
H}_{v}}d^{d-2}x~{}v\,\bigg{(}\partial_{v}\left(\sqrt{h}~{}{\cal
J}^{v}\right)+\partial_{i}\left(\sqrt{h}\,{\cal
J}^{i}\right)\bigg{)}\bigg{]}_{v=-\infty}^{v=+\infty}\\\
&~{}~{}+\int_{-\infty}^{\infty}dv\int_{{\cal
H}_{v}}d^{d-2}x~{}{\partial_{v}\left(v\sqrt{h}\right)\over\sqrt{h}}\bigg{(}\partial_{v}\left(\sqrt{h}\,{\cal
J}^{v}\right)+\partial_{i}\left(\sqrt{h}\,{\cal
J}^{i}\right)\bigg{)}\,.\end{split}$ (141)

Note that the combination “$\partial_{v}\left(\sqrt{h}~{}{\cal
J}^{v}\right)+\partial_{i}\left(\sqrt{h}\,{\cal J}^{i}\right)$” is of boost
weight $(+1)$ and therefore must vanish when evaluated on any stationary
configuration. Due to this and the fact that at $v=\pm\infty$ we have two
stationary configurations at the beginning and the end of the dynamical
process, the first term on the RHS of eq.(141) (appearing on the second line)
simply vanishes.

Now, for the second term on the RHS of eq.(141)(appearing on the third line),
we note that it is a product of two factors, one of which is the same quantity
with boost weight $(+1)$ that we mentioned above and therefore this is already
linear in the amplitude of the dynamics at ${\cal O}(\epsilon)$. So, from the
second factor, we have to consider only that piece which is boost-invariant,
neglecting the other ${\cal O}(\epsilon)$ piece, i.e.

$\partial_{v}(v\sqrt{h})=\sqrt{h}+\underbrace{v\,\partial_{v}\sqrt{h}}_{\sim{\cal
O}(\epsilon)}\,,$

ensuring that the RHS of eq.(141) is calculated up to ${\cal O}(\epsilon)$.
Therefore we obtain

$\begin{split}\delta S&\approx\int_{-\infty}^{\infty}dv\int_{{\cal
H}_{v}}d^{d-2}x\,\bigg{(}\partial_{v}\left(\sqrt{h}\,{\cal
J}^{v}\right)+\,\partial_{i}\left(\sqrt{h}\,{\cal
J}^{i}\right)\bigg{)}\,+{\cal O}(\epsilon^{2})\,.\end{split}$ (142)

In eq.(142) the second term on the RHS being a total spatial derivative,
integrates to boundary terms on ${\cal H}_{v}$, the constant $v$ slices of the
horizon. Therefore, it vanishes if we assume that the horizon is compact.
Neglecting this term, we finally arrive at

$\begin{split}\delta
S&\approx\int_{-\infty}^{\infty}dv~{}\partial_{v}\left[\int_{{\cal
H}_{v}}d^{d-2}x\,\sqrt{h}\,{\cal J}^{v}\right]\,+{\cal
O}(\epsilon^{2})=\left[\int_{{\cal H}_{v}}d^{d-2}x\,\sqrt{h}\,{\cal
J}^{v}\right]_{v=-\infty}^{v=+\infty}\,+{\cal O}(\epsilon^{2})\end{split}$
(143)

From eq.(143) it is clear that the change in entropy during this process is
given by the change in the quantity ${\cal J}^{v}$ evaluated at the two
stationary configurations at $v=-\infty$ and $v=+\infty$ respectively. We also
obtain an expression that has to be identified with the total entropy on every
constant $v$ slices,

$S\approx\int_{{\cal H}_{v}}d^{d-2}x\,\sqrt{h}\,{\cal
J}^{v}=-\dfrac{1}{2}\int_{{\cal
H}_{v}}d^{d-2}x\,\sqrt{h}\,\left(\widetilde{Q}^{rv}+{\cal B}_{(0)}\right)\,,$
(144)

where we have substituted for the expression of ${\cal J}^{v}$ from eq.(140).

From eq.(144) we immediately recognize $\widetilde{Q}^{rv}$ as the Wald
entropy density for equilibrium black holes, which we have already argued in
the previous sections. The other term on the RHS of eq.(144), i.e., ${\cal
B}_{(0)}$, has also previously been identified with the JKM ambiguities
present in the out-of-equilibrium extensions of Wald entropy; see the
discussion after eq.(108). Since the JKM ambiguities vanish on stationary
configurations, the term ${\cal B}_{(0)}$ will not contribute to the RHS of
eq.(143), justifying that the physical process version of the first law is not
affected by these terms. However, as we have already seen, they still play a
crucial role in obtaining the linearized version of the second law.

It is worth highlighting that the crucial input for this derivation was the
structure of $E_{vv}$ as predicted in eq.(7). In other words, our proof of the
second law within the approximation of linearized dynamics also allows us to
construct the proof of the first law using the physical process version in the
most general higher derivative theory of gravity.

Finally, we conclude this section with the following clarification regarding
the implications of our derivation of this version of the first law. As it
turns out, for this physical process version of the first law to be true in
its most general form for an arbitrary higher derivative theory of gravity,
the $vv$-component of the equations of motion, i.e. $E_{vv}$, has to satisfy
the following restriction imposed on its off-shell structure, (see eq.(25) in
Sarkar:2019xfd ),

$\int_{-\infty}^{\infty}dv\int_{{\cal
H}_{v}}d^{d-2}x\,\sqrt{h}\,v\left(\partial_{v}^{2}s_{w}-s_{w}\,R_{vv}+E^{\text{HD}}_{vv}\right)={\cal
O}(\epsilon^{2})\,,$ (145)

where $s_{w}=s^{\text{HD}}_{w}+s_{cor}$, such that $s^{\text{HD}}_{w}$ is the
contribution to the Wald entropy density from higher derivative part of the
Lagrangian defined in eq.(32), and $s_{cor}$ is the possible out-of-
equilibrium extension or correction to $s^{\text{HD}}_{w}$ defined in eq.(33).
Also $E^{\text{HD}}_{vv}$ is the component of equations of motion getting
contributions only from the higher derivative part of the Lagrangian; see
eq.(36) 404040The condition written in eq.(145) refers to the physical process
version of the first law in a manner that is only determined by the geometry
near the dynamical horizon and, most importantly, independent of the
information at the asymptotic boundary of space-time.. It has been shown that
eq.(145) is true for $f(R)$ theories Jacobson:1995uq , and for Gauss-Bonnet or
Lovelock theories Chatterjee:2011wj ; Kolekar:2012tq ; Sarkar:2013swa .
However, it is not yet known if this condition is true for arbitrary higher
derivative theories of gravity 414141For the arguments to derive eq.(145) and
a detailed discussion on this we refer the reader to section-$2$ of
Chakraborty:2017kob . This is also discussed in section-$3$ of the review
Sarkar:2019xfd ..

In this section, we have shown that the explicit form of $E_{vv}$ given in
eq.(140) indeed satisfies this condition up to linearised order in the
amplitude expansion. To see this, we should compute the $vv$-component of the
Ricci Tensor, i.e., $R_{vv}$ in our metric gauge, eq.(11), which yields

$R_{vv}=-\partial_{v}\left({1\over\sqrt{h}}\partial_{v}(\sqrt{h}\,)\right)+{\cal
O}(\epsilon^{2})\,.$ (146)

Using eq.(146) one can show that the terms inside the parenthesis in the
integrand on the LHS of eq.(145), can be expressed as the following

$\partial_{v}^{2}s_{w}-s_{w}\,R_{vv}+E^{\text{HD}}_{vv}=\partial_{v}\left({1\over\sqrt{h}}\partial_{v}(\sqrt{h}\,s_{w})\right)+E^{\text{HD}}_{vv}+{\cal
O}(\epsilon^{2})\,.$ (147)

With eq.(147) at hand, the restriction on $E_{vv}$ as written above in
eq.(145) takes the following form

$\int_{-\infty}^{\infty}dv\int_{{\cal
H}_{v}}d^{d-2}x\,\sqrt{h}\,v\left(\partial_{v}\left({1\over\sqrt{h}}\partial_{v}(\sqrt{h}\,s_{w})\right)+E^{\text{HD}}_{vv}\right)={\cal
O}(\epsilon^{2})\,,$ (148)

It is now straightforward to check that eq.(148) follows trivially from
eq.(39), which we have already encountered in §2.4. Hence, it is also obvious
that the form of $E_{vv}$ given in eq.(140) with the specific identifications
of ${\cal J}^{v}$ and ${\cal J}^{i}$, satisfies the condition in eq.(145).

One crucial outcome of our analysis in this paper is to justify that certain
total spatial derivatives in the form of $\nabla_{i}{\cal J}^{i}$ are
generically present in the structure of $E_{vv}$ as in eq.(140). However, we
now see that the presence of such total spatial divergences does not affect
the arguments leading to this version of the first law. As we know, in the
statement of the first law only the total entropies between two equilibrium
configurations are compared and the expressions of this total entropy involve
a spatial integration on the co-dimension two hyper-surface ${\cal H}_{v}$.
Consequently, for compact horizons, it is easy to check that the spatial total
derivatives integrate to zero,

$\int_{-\infty}^{\infty}dv\int_{{\cal
H}_{v}}d^{d-2}x\,\sqrt{h}\,v\,\partial_{v}\left(\nabla_{i}{\cal
J}^{i}\right)\sim{\cal O}(\epsilon^{2})\,.$ (149)

##  6 Discussions and future directions

In this paper, we have constructed a proof of the second law of black hole
thermodynamics in arbitrary higher derivative theories of gravity working with
a particular choice of coordinates. We have focussed on situations where the
dynamical fluctuations connecting the two stationary black holes are of small
amplitude such that a linearized amplitude expansion around the initial
stationary background is possible. The dynamics are initiated by, e.g.,
throwing a small matter into the equilibrium black hole, and we have assumed a
null energy condition for that matter stress tensor. In the course of
designing the proof, we have also constructed an entropy current on the
horizon. The divergence of this entropy current being non-negative signifies
entropy production locally at each point of the space-time. The ‘time’
component of this entropy current produces the correct expression for the Wald
entropy density and also fixes the associated JKM ambiguities as a result of
imposing the local second law. The spatial components of the entropy current
provide us a mechanism of entropy equilibration via the spatial flow of
entropy on the constant ‘time’-slice of the horizon. This, in turn, provides
us with a possible out-of-equilibrium extension of the equilibrium Wald
entropy function, maintaining the second law in its strongest possible form.
As a by-product of our analysis, we have also completed an alternative proof
of the first law using physical processes like in-falling matter initiating
black hole dynamics in the most general higher derivative theories of gravity.

Let us now comment on some possible future directions of our work:

##### Extending beyond linear order in amplitude expansion:

From our analysis in this paper, it might appear that our results up to linear
order in amplitude expansion are enough to prove the second law completely in
higher derivative gravitational theory up to all orders. In other words, there
is apparently no need to worry about the non-linear terms as long as we are
treating the higher derivative terms as perturbative corrections to the
leading two derivative Einstein gravity. The reason for this is as follows.
When considering the terms that are non-linear in the amplitude of the
dynamics, we could see entropy production at two derivative order itself.
Furthermore, the higher derivative terms, being considered as a small
perturbative correction, should always be suppressed compared to the two
derivative Einstein-Hilbert terms in the gravity Lagrangian. Hence, they could
never change the sign of the rate of entropy production. Therefore, one might
conclude that the higher derivative corrections in the gravity Lagrangian have
to be analyzed carefully only when the two derivative contributions to the
entropy production vanish, namely at the linear order in the amplitude of the
dynamics.

However, as it was shown in Bhattacharyya:2016xfs , there are situations where
such an argument does not work, at least locally, and one needs to add further
corrections to the entropy density (and also presumably to the current) that
are manifestly non-linear in the amplitude. In view of this observation, the
immediate question is whether we could extend our proof to non-linear orders
in amplitude.

##### Interpretation of the entropy current via fluid-gravity duality:

Another interesting future direction in our opinion is to explore the
following question: what does this gravitational entropy density along with
the spatial current translate to in the non-gravitational dual systems,
whenever they exist, like fluid dynamics Bhattacharyya:2008xc  and membrane
dynamics Dandekar:2019hyc ; Saha:2020zho . We already have algorithms to
construct an entropy current on these dual systems that will maintain the
second law locally even at nonlinear order in the amplitude of the dynamics
Bhattacharyya:2013lha ; Bhattacharyya:2014bha ; Dandekar:2019hyc ;
Saha:2020zho . If we can cleanly work out the map between the coordinates of
this current paper and the different gauges or coordinates used to construct
these dual fluid or membrane systems, it would presumably tell us a lot about
the non-linear completion of the gravitational entropy current.

##### Exploring a “covariant version” of the entropy current:

In the dual systems that we have mentioned above, in any given coordinate
system, the space and time components of the entropy current are always
related by the constraints imposed by the freedom of coordinate
transformation. However, our proof relies heavily on the horizon adapted
coordinate system that we have chosen, and our effective ‘time’ is basically
an affine parameter along a null direction. So naive mixing of time and space
is not possible here. In other words, we have been able to justify the
existence of an entropy current following the construction of Iyer:1994ys ,
but only working in a particular choice of the space-time metric as in
eq.(11). Nevertheless, we believe that the notion of non-negative divergence
of an entropy current, if it exists, should be a robust feature in a covariant
sense as well. It would be nice if we could obtain a proof supporting this in
a more coordinate invariant fashion.

Additionally, the components of the entropy current that we have obtained in
our analysis also crucially depend on the particular slicing of the null
horizon. It would be interesting to understand how they transform under an
arbitrary redefinition of the constant $v$-slices of the horizon, e.g.,
eq.(12). Such an attempt might lead to a covariant formula also for the
spatial entropy current. One possible direction going ahead would be to
explore the problem with a viewpoint of covariant phase space analysis along
the lines of Harlow:2019yfa .

We hope to come back to these issues in our future works.

###### Acknowledgements.

We want to thank Jyotirmoy Bhattacharya for initial collaboration, useful
discussions, and important inputs. We are grateful to Aron Wall for his
efforts in carefully reading our manuscript and providing us with his valuable
suggestions and several significant comments. We would also like to thank
Shamik Banerjee, Parthajit Biswas, Arunabha Saha for helpful discussions. M.P.
would like to thank Shankha Singha Mahapatra for valuable discussions. P.D.
would like to duly acknowledge the Council of Scientific and Industrial
Research (CSIR), New Delhi for financial assistance through the Junior
Research Fellowship (JRF) scheme.

##  Appendix A Notations, conventions and useful definitions

  * •

We work in $d$-dimensional space-time with coordinates denoted by $x^{\mu}$.

  * •

Lower-case Greek indices:
$\\{\alpha,\beta,\cdots,\mu,\nu,\cdots,\alpha_{1},\alpha_{2},\cdots\\}$, are
used for space-time coordinates.

  * •

The space-time metric is denoted by $g_{\mu\nu}$ and the associated covariant
derivative is denoted by the operator $D_{\mu}$.

  * •

The horizon, a co-dimension one hyper-surface $\cal H$, is located at $r=0$.
Here $r$ is the coordinate running along the affinely parametrized geodesics
$\partial_{r}$ that takes us away from the horizon.

  * •

The affinely parametrized null generator on the horizon is given by
$\partial_{v}$ and $v$ is the coordinate along those null generators.

  * •

Constant $v$-slices of the horizon is a co-dimension two hyper-surface ${\cal
H}_{v}$ spanned by $(d-2)$ spatial coordinates $x^{i}$.

  * •

Lower-case Latin indices: $\\{i,j,\cdots,i_{1},j_{1},\cdots\\}$ are used for
coordinates on ${\cal H}_{v}$.

  * •

The full space-time coordinates are $x^{\mu}=\\{r,v,x^{i}\\}$ and coordinates
on the horizon $\cal H$ are $\\{v,x^{i}\\}$.

  * •

The intrinsic metric on the horizon is $h_{ij}$ and the associated covariant
derivative is $\nabla_{i}$.

##  Appendix B More details regarding the boost weight of a covariant tensor

In this appendix, we will provide additional details regarding some of the
results which were used in the paper.

###  B.1 Equivalence between different ways of evaluating the Boost weights

We will first try to show the equivalence between the definitions of Boost
Weight mentioned in §2.3. We will collect them here for reference:

${\cal A}\rightarrow\widetilde{\cal A}=\lambda^{w}{\cal
A}\,,~{}~{}\text{under}~{}~{}\\{r\rightarrow\tilde{r}=\lambda~{}r,~{}v\rightarrow\tilde{v}=\lambda^{-1}v\\}~{}~{}\Rightarrow~{}~{}\text{boost
weight of ${\cal A}$ is $w$}\,.$ (150) $\begin{split}{\cal
L}_{\xi}S_{\alpha_{1}\alpha_{2}\cdots\alpha_{k}}=\,\left[w+(v\partial_{v}-r\partial_{r})\right]S_{\alpha_{1}\alpha_{2}\cdots\alpha_{k}}\,\end{split}$
(151)

where $\xi=v\partial_{v}-r\partial_{r}$ and we identify the quantity $w$ as
the boost weight of the covariant tensor
$S_{\alpha_{1}\alpha_{2}\cdots\alpha_{k}}$ as

$\begin{split}w\,\equiv\,&\text{net boost weight of a covariant tensor (with
all indices lowered)}\\\ =\,&\text{number of lower $v$ indices - number of
lower $r$ indices.}\end{split}$ (152)

To prove the equivalence, we consider a generic tensor component with $m_{v}$
lower $v$ indices and $m_{r}$ lower $r$ indices : $S_{v\dots vr\dots
r}(r,v,x^{i})$ where we have explicitly indicated the functional dependence on
the space-time variables. Let us consider an infinitesimal Boost
transformation:

$\tilde{r}=(1+\epsilon)r\hskip 28.45274pt\tilde{v}=(1-\epsilon)v\hskip
28.45274pt\text{where}\,\,\,\epsilon\,\,\,\text{is infinitesimal}$ (153)

Under the above transformation, the covariant tensor component $S_{v\dots
vr\dots r}$ transforms as follows:

$\begin{split}S_{\tilde{v}\dots\tilde{v}\tilde{r}\dots\tilde{r}}(\tilde{r},\tilde{v},x^{i})&=\left(\dfrac{\partial
v}{\partial\tilde{v}}\right)^{m_{v}}\left(\dfrac{\partial
r}{\partial\tilde{r}}\right)^{m_{r}}S_{v\dots vr\dots r}(r,v,x^{i})\\\
&=(1+\epsilon)^{m_{v}}(1-\epsilon)^{m_{r}}S_{v\dots vr\dots r}(r,v,x^{i})\\\
&=(1+(m_{v}-m_{r})\epsilon)\,\,S_{v\dots vr\dots
r}(r,v,x^{i})+O(\epsilon^{2})\end{split}$ (154)

Now we will extract the Lie Derivative along the Integral curves of this
particular coordinate transformation:

$\begin{split}&S_{\tilde{v}\dots\tilde{v}\tilde{r}\dots\tilde{r}}(r+\epsilon
r,v-\epsilon v,x^{i})=(1+(m_{v}-m_{r})\epsilon)\,\,S_{v\dots vr\dots
r}(r,v,x^{i})+O(\epsilon^{2})\\\
&S_{\tilde{v}\dots\tilde{v}\tilde{r}\dots\tilde{r}}(r,v,x^{i})+\epsilon
r\partial_{r}S_{\tilde{v}\dots\tilde{v}\tilde{r}\dots\tilde{r}}(r,v,x^{i})-\epsilon
v\partial_{v}S_{\tilde{v}\dots\tilde{v}\tilde{r}\dots\tilde{r}}(r,v,x^{i})+O(\epsilon^{2})\\\
&=(1+(m_{v}-m_{r})\epsilon)\,\,S_{v\dots vr\dots
r}(r,v,x^{i})+O(\epsilon^{2})\\\
&S_{\tilde{v}\dots\tilde{v}\tilde{r}\dots\tilde{r}}(r,v,x^{i})-S_{v\dots
vr\dots r}(r,v,x^{i})=(m_{v}-m_{r})\epsilon S_{v\dots vr\dots r}(r,v,x^{i})\\\
&\hskip 56.9055pt+\epsilon v\partial_{v}S_{v\dots vr\dots
r}(r,v,x^{i})-\epsilon r\partial_{r}S_{v\dots vr\dots
r}(r,v,x^{i})+O(\epsilon^{2})\\\ &\implies\mathcal{L}_{\xi}S_{v\dots vr\dots
r}(r,v,x^{i})=(m_{v}-m_{r})S_{v\dots vr\dots r}(r,v,x^{i})\\\ &\hskip
56.9055pt+v\partial_{v}S_{v\dots vr\dots r}(r,v,x^{i})-r\partial_{r}S_{v\dots
vr\dots r}(r,v,x^{i})\end{split}$ (155)

where in the final step we have defined
$\xi^{\mu}\partial_{\mu}=v\partial_{v}-r\partial_{r}$ and the coordinate
transformation is generated along the integral curves of this vector field.
Thus, we have proved the equivalence between the two definitions of Boost
Weight of the tensor component $S_{v\dots vr\dots r}$ which is given by
$w=m_{v}-m_{r}$. The generalization of the above equivalence for tensors of
the form $S_{v\dots vij\dots}(r,v,x^{i})$ and $S_{r\dots rij\dots}(r,v,x^{i})$
is straightforward once we realise that the Boost transformation doesn’t
change the coordinates $x^{i}$. Thus, for $S_{v\dots vij\dots}(r,v,x^{i})$, we
essentially have $m_{r}=0$ and for $S_{r\dots rij\dots}(r,v,x^{i})$, we have
$m_{v}=0$.

###  B.2 Structure of covariant tensors in the stationary background solution

We will now show that if $\xi$ is a Killing vector for our space-time, then
the Lie derivative of any covariant tensor evaluated on that space-time will
vanish. We will consider the above generic tensor component $S_{v\dots vr\dots
r}$ which has $m_{v}$ number of lower $v$ indices and $m_{r}$ number of lower
$r$ indices. Assume that once the tensor component is evaluated in our gauge,
it has $m^{(1)}_{v}$ factors of $r$ and $m^{(2)}_{v}$ factors of
$\partial_{v}$’s such that $m^{(1)}_{v}+m^{(2)}_{v}=m_{v}$. Thus, a Boost
weight of $w=m_{v}-m_{r}$ implies that the tensor component takes the
following form

$S_{v\dots vr\dots
r}=r^{m^{(1)}_{v}}(\partial_{v})^{m^{(2)}_{v}}(\partial_{r})^{m_{r}}Y(rv)$
(156)

where $Y$ is a function of the product $rv$ so that it has Boost weight zero.
Using this, we further simplify the tensor component:

$\begin{split}S_{v\dots vr\dots
r}&=r^{m_{v}^{(1)}}\left(\partial_{v}\right)^{m_{v}^{(2)}}\left(\partial_{r}\right)^{m_{r}}Y(rv)\\\
&=r^{m_{v}^{(1)}}\left(\partial_{v}\right)^{m_{v}^{(2)}}\left(v^{m_{r}}\left[\left(d\over
dz\right)^{m_{r}}Y(z)\right]_{z=rv}\right)\\\
&=r^{m_{v}^{(1)}}\sum_{m=0}^{k}{{}^{k}C}_{m}\left[\left(\partial_{v}\right)^{m}v^{m_{r}}\right]\left(\partial_{v}\right)^{m_{v}^{(2)}-m}\left[\left(d\over
dz\right)^{m_{r}}Y(z)\right]_{z=rv}\text{where}~{}k=Min(m_{r},m_{v}^{(2)})\\\
&=r^{m_{v}^{(1)}+m_{v}^{(2)}}\sum_{m=0}^{k}{{}^{k}C}_{m}\left[\left(\partial_{v}\right)^{m}v^{m_{r}}\right]r^{-m}\left[\left(d\over
dz\right)^{m_{r}+m_{v}^{(2)}-m}Y(z)\right]_{z=rv}\\\
&=r^{m_{v}}\sum_{m=0}^{k}{{}^{k}C}_{m}\left[m_{r}!\over(m_{r}-m)!\right]v^{m_{r}-m}r^{-m}\left[\left(d\over
dz\right)^{m_{r}+m_{v}^{(2)}-m}Y(z)\right]_{z=rv}\\\
&=r^{m_{v}}v^{m_{r}}\sum_{m=0}^{k}{{}^{k}C}_{m}\left[m_{r}!\over(m_{r}-m)!\right](rv)^{-m}\left[\left(d\over
dz\right)^{m_{r}+m_{v}^{(2)}-m}Y(z)\right]_{z=rv}\\\
&=r^{m_{v}}v^{m_{r}}~{}\tilde{Y}(rv)\end{split}$ (157)

where

$\tilde{Y}(rv)=\sum_{m=0}^{k}{{}^{k}C}_{m}\left[m_{r}!\over(m_{r}-m)!\right](rv)^{-m}\left[\left(d\over
dz\right)^{m_{r}+m_{v}^{(2)}-m}Y(z)\right]_{z=rv},~{}k=Min(m_{r},m_{v}^{(2)})$

We use the final result of 157 to evaluate the Lie derivative of the tensor
component:

$\begin{split}\mathcal{L}_{\xi}S_{v\dots vr\dots
r}&=(m_{v}-m_{r}+v\partial_{v}-r\partial_{r})(r^{m_{v}}v^{m_{r}}\tilde{Y}(rv))\\\
&=(m_{v}-m_{r}+m_{r}-m_{v})r^{m_{v}}v^{m_{r}}\tilde{Y}(rv)=0\end{split}$ (158)

where in the final step we have used the fact that
$(v\partial_{v}-r\partial_{r})\tilde{Y}(rv)=0$. This vanishing of the Lie
derivative on our stationary space-time was possible only after we have
explicitly worked out the $\partial_{r}$’s and $\partial_{v}$’s in terms of
factors of $r$’s and $v$’s.

##  Appendix C A brief review of the Iyer-Wald procedure in component notation

We re-express the main results of Iyer:1994ys  in component notation.

  * •

Lemma 2.1: Type of Lagrangians considered

We consider Lagrangians which are diffeomorphism covariant in the sense of
eq.(209). It has been proved that (Iyer:1994ys, ) such Lagrangians can always
be re-expressed in the following form

$L=L(g_{\mu\nu},R_{\mu\nu\rho\sigma},D_{\alpha_{1}}R_{\mu\nu\rho\sigma},\dots,D_{(\alpha_{1}}\dots
D_{\alpha_{m})}R_{\mu\nu\rho\sigma},\phi,D_{\alpha_{1}}\phi,\dots,D_{(\alpha_{1}}\dots
D_{\alpha_{l})}\phi)$ (159)

where $\phi$ denotes the collection of all matter fields 424242We have
suppressed the index structure of the matter sector.. In (Iyer:1994ys, ), the
Lagrangian is viewed as an $d$ form $\mathbf{L}$ given by
$\mathbf{L}_{\alpha_{1}\dots\alpha_{d}}=\epsilon_{\alpha_{1}\dots\alpha_{d}}L$
434343Here $\epsilon$ is canonical the volume form (i.e) $\epsilon_{b_{1}\dots
b_{d}}=\sqrt{-g}\varepsilon_{b_{1}\dots b_{d}}$ where $\varepsilon_{b_{1}\dots
b_{d}}$ is the Levi Civita symbol . The variation of this Lagrangian is given
by

$\delta\mathbf{L}=\mathbf{E}^{\mu\nu}\delta
g_{\mu\nu}+\mathbf{E}_{\phi}\delta\phi+d\mathbf{\Theta}$ (160)

where
$\mathbf{E}^{\mu\nu}_{\alpha_{1}\dots\alpha_{d}}=\epsilon_{\alpha_{1}\dots\alpha_{d}}E^{\mu\nu}$
($E^{\mu\nu}$ is the equation of motion for the metric field) and
$\mathbf{E}_{\phi}=\epsilon_{\alpha_{1}\dots\alpha_{d}}E_{\phi}$ ($E_{\phi}$
is the Equation of motion for the matter fields). $\mathbf{\Theta}$, an $d-1$
form is the total derivative term called the Symplectic Potential. By taking
the Hodge dual we can revert to component notation using the following
identity which can be proven straightforwardly

$d\mathbf{\Theta}=d(\epsilon_{\mu\alpha_{2}\dots\alpha_{d}}\Theta^{\mu})=\epsilon_{\alpha_{1}\dots\alpha_{d}}D_{\mu}\Theta^{\mu}$
(161)

Thus the variation of the Lagrangian becomes

$\begin{split}\delta(\epsilon_{\alpha_{1}\dots\alpha_{d}}L)=\epsilon_{\alpha_{1}\dots\alpha_{d}}E^{\mu\nu}\delta
g_{\mu\nu}+\epsilon_{\alpha_{1}\dots\alpha_{d}}E_{\phi}\delta\phi+\epsilon_{\alpha_{1}\dots\alpha_{d}}D_{\mu}\Theta^{\mu}\\\
\implies\delta(\sqrt{-g}L)=\sqrt{-g}E^{\mu\nu}\delta
g_{\mu\nu}+\sqrt{-g}E_{\phi}\delta\phi+\sqrt{-g}D_{\mu}\Theta^{\mu}\end{split}$
(162)

  * •

Lemma 3.1: General form of $\Theta^{\mu}$

We can write Eqn (23) of Iyer:1994ys  explicitly as

$\mathbf{\Theta}_{\alpha_{2}\dots\alpha_{d}}=2\left(E^{\mu\nu\rho\sigma}_{R}D_{\sigma}\delta
g_{\nu\rho}\right)\epsilon_{\mu\alpha_{2}\dots\alpha_{d}}+\mathbf{\Theta^{\prime}}_{\alpha_{2}\dots\alpha_{d}}$
(163)

where

$E^{\mu\nu\rho\sigma}_{R}=\dfrac{\partial L}{\partial
R_{\mu\nu\rho\sigma}}-D_{\alpha_{1}}\dfrac{\partial L}{\partial
D_{\alpha_{1}}R_{\mu\nu\rho\sigma}}+\dots+(-1)^{m}D_{(\alpha_{1}}\dots
D_{\alpha_{m})}\dfrac{\partial L}{\partial D_{(\alpha_{1}}\dots
D_{\alpha_{m})}R_{\mu\nu\rho\sigma}}$ (164)

Writing $\mathbf{\Theta}$ and $\mathbf{\Theta^{\prime}}$ which are $d-1$ forms
as a Hodge duals of one-forms $\Theta_{\mu}$ and $\Theta^{\prime}_{\mu}$
respectively,

$\epsilon_{\mu\alpha_{2}\dots\alpha_{d}}\Theta^{\mu}=2\left(E^{\mu\nu\rho\sigma}_{R}D_{\sigma}\delta
g_{\nu\rho}\right)\epsilon_{\mu\alpha_{2}\dots\alpha_{d}}+\epsilon_{\mu\alpha_{2}\dots\alpha_{d}}\Theta^{\prime\mu}$
(165)

Now we can contract with $\epsilon^{\beta\alpha_{2}\dots\alpha_{d}}$ 444444The
contraction of two volume form tensors is given by ($\delta^{\alpha}_{\beta}$
is the Kronecker delta)
$\epsilon^{\mu_{1}\mu_{2}\dots\mu_{p}\alpha_{1}\dots\alpha_{d-p}}\,\epsilon_{\mu_{1}\mu_{2}\dots\mu_{p}\beta_{1}\dots\beta_{d-p}}=(-1)^{s}\,p!\,(d-p)!\,\delta^{[\alpha_{1}}_{\beta_{1}}\dots\delta^{\alpha_{d-p}]}_{\beta_{d-p}}$
to get

$\Theta^{\beta}=2E^{\beta\nu\rho\sigma}_{R}D_{\sigma}\delta
g_{\nu\rho}+\Theta^{\prime\beta}$ (166)

To find $\Theta^{\prime\beta}$, we write Eqn(24) of Iyer:1994ys  in component
notation. In Eqn(24) of Iyer:1994ys , since $\mathbf{\Theta^{\prime}}$ is an
$d-1$ form proportional to $\epsilon$, by inspection, it is clear that
$\mathbf{S}^{\mu\nu}(\phi)$,
$\mathbf{T_{i}}(\phi)^{\mu\nu\rho\sigma\alpha_{1}\dots\alpha_{i}}$ and
$\mathbf{U_{i}}(\phi)^{\alpha_{1}\dots\alpha_{i}}$ are also $d-1$ forms
proportional to $\epsilon$. Thus, we can write Eqn(24) of Iyer:1994ys  as

$\begin{split}\epsilon_{\mu\alpha_{2}\dots\alpha_{d}}\Theta^{\prime\mu}=S^{\gamma\mu\nu}(\phi)\epsilon_{\gamma\alpha_{2}\dots\alpha_{d}}\delta
g_{\mu\nu}+\sum_{i=0}^{m-1}T_{i}(\phi)^{\gamma\mu\nu\rho\sigma\beta_{1}\dots\beta_{i}}\epsilon_{\gamma\alpha_{2}\dots\alpha_{d}}\delta
D_{(\beta_{1}}\dots D_{\beta_{i})}R_{\mu\nu\rho\sigma}\\\
+\sum_{i=0}^{l-1}U_{i}(\phi)^{\gamma\beta_{1}\dots\beta_{i}}\epsilon_{\gamma\alpha_{2}\dots\alpha_{d}}\delta
D_{(\beta_{1}}\dots D_{\beta_{i})}\phi\end{split}$ (167)

Thus contracting with $\epsilon^{\beta\gamma_{2}\dots\gamma_{d}}$,

$\begin{split}\Theta^{\prime\beta}=S^{\beta\mu\nu}(\phi)\delta
g_{\mu\nu}&+\sum_{i=0}^{m-1}T_{i}(\phi)^{\beta\mu\nu\rho\sigma\alpha_{1}\dots\alpha_{i}}\delta
D_{(\alpha_{1}}\dots D_{\alpha_{i})}R_{\mu\nu\rho\sigma}\\\
&+\sum_{i=0}^{l-1}U_{i}(\phi)^{\beta\alpha_{1}\dots\alpha_{i}}\delta
D_{(\alpha_{1}}\dots D_{\alpha_{i})}\phi\end{split}$ (168)

which is exactly eq.(24) of (Iyer:1994ys, ) written in Component Notation.

  * •

The basic algorithm to compute the Noether charge

First we compute the Noether current $d-1$ form associated to a vector field
$\zeta^{\mu}$ which is defined as follows

$\mathbf{J}_{\alpha_{2}\dots\alpha_{d}}=\mathbf{\Theta}_{\alpha_{2}\dots\alpha_{d}}(\psi,\mathcal{L}_{\zeta}\psi)-\zeta\cdot\mathbf{L}_{\alpha_{2}\dots\alpha_{d}}=\epsilon_{\beta\alpha_{2}\dots\alpha_{d}}(\Theta^{\beta}-\zeta^{\beta}L)$
(169) 454545Where $\psi$ denotes the collection of all fields.

Thus taking the Hodge dual we get the following expression for Noether current

$J^{\mu}=\Theta^{\mu}(\psi,\mathcal{L}_{\zeta}\psi)-\zeta^{\mu}L$ (170)

Given the Noether current $d-1$ form $\mathbf{J}$, we can define the Noether
charge $d-2$ form as $\mathbf{J}=d\mathbf{Q}$ on-shell. We write this $d-2$
form as the Hodge dual of a $2$ form

$\mathbf{Q}_{\alpha_{3}\dots\alpha_{d}}=\dfrac{1}{2}\epsilon_{\beta\gamma\alpha_{3}\dots\alpha_{d}}Q^{\beta\gamma}$
(171)

Using the relation

$d\left(\dfrac{1}{2}\epsilon_{\beta\gamma\alpha_{3}\dots\alpha_{d}}Q^{\beta\gamma}\right)=D_{\gamma}Q^{\beta\gamma}\epsilon_{\beta\alpha_{2}\dots\alpha_{d}}$
(172)

We can see that $\mathbf{J}=d\mathbf{Q}$ essentially amounts to

$J^{\mu}=\Theta^{\mu}-\zeta^{\mu}L=D_{\nu}Q^{\mu\nu}\hskip 28.45274pt(on-
shell)$ (173)

To see how this works out in practice, let us consider a pure gravity theory
with no matter sector. The variation of such a Lagrangian is given by

$\delta(\sqrt{-g}L)=\sqrt{-g}E^{\mu\nu}\delta
g_{\mu\nu}+\sqrt{-g}D_{\mu}\Theta^{\mu}$ (174)

We have performed a similar analysis in §2.5. Suppose we consider the
variation under the diffeomorphism generated by $\zeta^{\mu}$ (i.e.) $\delta
g_{\mu\nu}=\mathcal{L}_{\zeta}g_{\mu\nu}=D_{\mu}\zeta_{\nu}+D_{\nu}\zeta_{\mu}$,
the variation of the Lagrangian is given by

$\delta(\sqrt{-g}L)=\sqrt{-g}D_{\mu}(\zeta^{\mu}L)$ (175)

due to the diffeomorphism covariance eq.(209). The variation of the Lagrangian
thus provides

$\sqrt{-g}D_{\mu}(\zeta^{\mu}L)=\sqrt{-g}E^{\mu\nu}\mathcal{L}_{\zeta}g_{\mu\nu}+\sqrt{-g}D_{\mu}\Theta^{\mu}(\mathcal{L}_{\zeta}g_{\alpha\beta})$
(176)

Thus we do some manipulations in the above expression as follows:

$\begin{split}D_{\mu}(\zeta^{\mu}L)=&2E^{\mu\nu}D_{\mu}\zeta_{\nu}+D_{\mu}\Theta^{\mu}\\\
D_{\mu}(\zeta^{\mu}L)=&D_{\mu}(2E^{\mu\nu}\zeta_{\nu})-2(D_{\mu}E^{\mu\nu})\zeta_{\nu}+D_{\mu}\Theta^{\mu}\\\
D_{\mu}(\Theta^{\mu}-\zeta^{\mu}L)=&D_{\mu}J^{\mu}=-D_{\mu}(2E^{\mu\nu}\zeta_{\nu})\\\
\implies
J^{\mu}=&\Theta^{\mu}-\zeta^{\mu}L=-2E^{\mu\nu}\zeta_{\nu}+D_{\nu}Q^{\mu\nu}\end{split}$
(177)

In the third step, we have used the Generalized Bianchi Identity
$(D_{\mu}E^{\mu\nu}=0)$. In the final step, when integrating out the covariant
derivatives, it must be clear that there must be a piece $D_{\nu}Q^{\mu\nu}$
locally where $Q^{\mu\nu}=-Q^{\nu\mu}$ (we can trivially show that
$D_{\mu}D_{\nu}Q^{\mu\nu}=0$ just by going to a set of Local inertial
coordinates). So we finally have

$\Theta^{\mu}(\mathcal{L}_{\zeta}g_{\alpha\beta})-\zeta^{\mu}L+2E^{\mu\nu}\zeta_{\nu}=D_{\nu}Q^{\mu\nu}$
(178)

Thus, the algorithm for extracting the Noether charge is to calculate
$\Theta^{\mu}(\psi,\delta\psi)$ for $\delta\psi=\mathcal{L}_{\zeta}\psi$ and
use the above equation to find $Q^{\mu\nu}$. This relation was derived in
§2.5, see eq.(49).

  * •

Proposition 4.1: General form of the Noether charge

Proposition 4.1 of (Iyer:1994ys, ) proves that the Noether charge $d-2$ form
can always be expressed in the following form

$\mathbf{Q}=\mathbf{W}_{\rho}(\psi)\zeta^{\rho}+\mathbf{X}^{\rho\sigma}(\psi)D_{[\rho}\zeta_{\sigma]}+\mathbf{Y}(\psi,\mathcal{L}_{\zeta}\psi)+d\mathbf{Z}(\psi,\zeta)$
(179)

where $\psi$ denotes the collection of all the fields of the theory. Since
$\mathbf{Q}$ is an $d-2$ form,
$\mathbf{W}_{\rho}(\psi),\mathbf{X}^{\rho\sigma}(\psi),\mathbf{Y}(\psi,\mathcal{L}_{\zeta}\psi)$
are $d-2$ forms. $\mathbf{Z}(\psi,\zeta)$ is an $d-3$ form. Thus, once again,
we write the $d-2$ forms as Hodge duals of two forms:

$(\mathbf{W}_{\rho}(\psi))_{\alpha_{3}\dots\alpha_{d}}=\dfrac{1}{2}\epsilon_{\mu\nu\alpha_{3}\dots\alpha_{d}}W^{\mu\nu}_{\hskip
9.95863pt\rho}(\psi)$ (180)
$\mathbf{Y}(\psi,\mathcal{L}_{\zeta}\psi)_{\alpha_{3}\dots\alpha_{d}}=\dfrac{1}{2}\epsilon_{\mu\nu\alpha_{3}\dots\alpha_{d}}Y^{\mu\nu}(\psi,\mathcal{L}_{\zeta}\psi)$
(181)

$\mathbf{X}^{\rho\sigma}$ can be chosen to be of the form

$(\mathbf{X}^{\rho\sigma}(\psi))_{\alpha_{3}\dots\alpha_{d}}=-E^{\mu\nu\rho\sigma}_{R}\epsilon_{\mu\nu\alpha_{3}\dots\alpha_{d}}$
(182)

Since $\mathbf{Z}$ is an $d-3$ form, we can write it as a Hodge dual of a
$3-$form

$\mathbf{Z}_{\alpha_{4}\dots\alpha_{d}}=\dfrac{1}{6}\epsilon^{\mu\nu\rho}_{\hskip
11.38092pt\alpha_{4}\dots\alpha_{d}}Z_{\mu\nu\rho}$ (183)

We now use the following result which can be proven straightforwardly:

$d\left(\dfrac{1}{6}\epsilon^{\mu\nu\rho}_{\hskip
11.38092pt\alpha_{4}\dots\alpha_{d}}Z_{\mu\nu\rho}\right)=D_{\mu}Z^{\mu\nu\rho}\epsilon_{\nu\rho\alpha_{3}\dots\alpha_{d}}$
(184)

Collecting all these results, we can rewrite Eqn(52) of Iyer:1994ys  as

$\begin{split}\dfrac{1}{2}\epsilon_{\beta\gamma\alpha_{3}\dots\alpha_{d}}Q^{\beta\gamma}=\dfrac{1}{2}\epsilon_{\beta\gamma\alpha_{3}\dots\alpha_{d}}W^{\beta\gamma}_{\hskip
9.95863pt\rho}\zeta^{\rho}-E^{\beta\gamma\rho\sigma}_{R}\epsilon_{\beta\gamma\alpha_{3}\dots\alpha_{d}}D_{[\rho}\zeta_{\sigma]}\\\
+\dfrac{1}{2}\epsilon_{\beta\gamma\alpha_{3}\dots\alpha_{d}}Y^{\beta\gamma}+D_{\rho}Z^{\beta\gamma\rho}\epsilon_{\beta\gamma\alpha_{3}\dots\alpha_{d}}\end{split}$
(185)

We can thus, contract with $\epsilon^{\mu\nu\alpha_{3}\dots\alpha_{d}}$ to
obtain

$Q^{\mu\nu}=W^{\mu\nu\rho}\zeta_{\rho}-2E^{\mu\nu\rho\sigma}_{R}D_{[\rho}\zeta_{\sigma]}+Y^{\mu\nu}+2D_{\rho}Z^{\mu\nu\rho}$
(186)

We can absorb the factor of 2 accompanying $Z^{\mu\nu\rho}$ into the
definition of $Z^{\mu\nu\rho}$ since this ambiguity arises as $d\mathbf{Z}$ in
$\mathbf{Q}$. Thus En(51) of (Iyer:1994ys, ) explicitly written in component
notation is given by

$Q^{\mu\nu}=W^{\mu\nu\rho}(\psi)\zeta_{\rho}-2E^{\mu\nu\rho\sigma}(\psi)D_{[\rho}\zeta_{\sigma]}+Y^{\mu\nu}(\psi,\mathcal{L}_{\zeta}\psi)+D_{\rho}Z^{\mu\nu\rho}$
(187)

It should be noted that $Y^{\mu\nu}$ and $Z^{\mu\nu\rho}$ are ambiguities in
the definition of the Noether current and charge respectively. The detailed
algorithm mentioned above for calculating the Noether charge essentially
results in the following general form for $Q^{\mu\nu}$

$Q^{\mu\nu}=W^{\mu\nu\rho}(\psi)\zeta_{\rho}-2E^{\mu\nu\rho\sigma}_{R}D_{[\rho}\zeta_{\sigma]}$
(188)

where $E^{\mu\nu\rho\sigma}_{R}$ is given by eq.(164) and
$W^{\mu\nu\rho}=W^{[\mu\nu]\rho}$. This Noether charge is then used to define
an Entropy functional which satisfies the Equilibrium version of the first law
of thermodynamics (Iyer:1994ys, ):

$S_{W}=-2\pi\int_{\Sigma}d^{d-2}x^{i}\sqrt{h}\,\,E^{\mu\nu\rho\sigma}_{R}\epsilon_{\mu\rho}\epsilon_{\nu\sigma}\,,$
(189)

where $x^{i}$ denote the coordinates induced on the horizon, $h=det(h_{ij})$
($h_{ij}$ being the induced metric on the horizon), $\Sigma$ is a co-dimension
$2$ surface on the Black hole horizon and $\epsilon_{\mu\rho}$ being the
corresponding Bi-normal. $E^{\mu\nu\rho\sigma}_{R}$ is given by eq.(164). For
$L=\dfrac{1}{16\pi G}R$, the entropy reduces to $S_{W}=\dfrac{A}{4G}$ which is
the usual Bekenstein-Hawking Entropy.

##  Appendix D Arguments justifying eq.(60)

In this appendix we provide some justifications in support of eq.(60).
According to the definition of $\Theta^{\mu}$ in a general higher derivative
theory of gravity one can write it as a sum of two terms, i.e.
$\Theta\sim\text{Term}_{1}+\text{Term}_{2}$. This will be argued in the sub-
section §3.2.2 (see eq.(76)). One of them, named $\text{Term}_{1}$, has the
following generic structure

$\text{Term}_{1}\sim\left[\text{Tensor-1}\right]~{}\mathcal{L}_{\xi}\left[\text{Tensor-2}\right]\,,$

with appropriate index contractions being implied on the RHS above. On the
other hand, $\text{Term}_{2}$ has a little bit different structure than
$\text{Term}_{1}$, since it involves a covariant derivative of $L_{\xi}$:

$\text{Term}_{2}\sim
D_{\nu}\left[E^{r\mu\nu\rho}~{}\mathcal{L}_{\xi}g_{\mu\rho}\right]|_{r=0}.$

However, if we explicitly evaluate this term at the horizon $(r=0)$ in our
coordinate system we will find that it has the following form:

$\text{Term}_{2}=E^{rijr}(-1+v\partial_{v})\left(\partial_{r}h_{ij}\right)+(1+v\partial_{v})\left[E^{rjkv}\left(\partial_{v}h_{jk}\right)\right].$

A more careful derivation of this can be found in §3.2.2 (see eq.(101),
eq.(102) and eq.(104)). Although note that the term $\partial_{r}h_{ij}$ is
not a covariant tensor component, the operator

$\mathcal{L}_{\xi}=\text{boost weight}+v\partial_{v}$

has a well-defined action on it and we could express this term as
$E^{rijr}~{}\mathcal{L}_{\xi}\left(\partial_{r}h_{ij}\right)$. Since for our
analysis we do not need any details other than boost weight and distribution
of $\partial_{v}$s and $\partial_{r}$s, for our purpose, this term is also
effectively of the same form as $\text{Term}_{1}$. We do not need to process
the second term
$(1+v\partial_{v})\left[E^{rjkv}\left(\partial_{v}h_{jk}\right)\right]$ since
it is already of the structure of equation eq.(60) with ${\cal M}_{(0)}$ set
to zero.

So finally we have to analyze terms of the of the form of $\text{Term}_{1}$.

$\begin{split}&\text{A typical term in $\Theta^{r}$ generated from
$\text{Term}_{1}$}~{}~{}\sim~{}~{}\tilde{t}_{(-k)}~{}(k+1+v~{}\partial_{v})~{}t_{(k+1)}\\\
\Rightarrow~{}&\Theta_{(1)}\sim(k+1)~{}\tilde{t}_{(-k)}t_{(k+1)},~{}~{}~{}\Theta_{(2)}=\tilde{t}_{(-k)}~{}\partial_{v}t_{(k+1)}\end{split}$
(190)

For these terms, eq.(60) reduces to

$\tilde{t}_{(-k)}~{}\partial_{v}t_{(k+1)}-\partial_{v}\left[(k+1)~{}\tilde{t}_{(-k)}~{}t_{(k+1)}\right]=\partial_{v}^{2}~{}(\text{JKM
type terms})$ (191)

Note that for $k=0$ this is trivially true with no JKM type terms.

Now extending the arguments in Wall:2015raa  (by transferring $\partial_{v}$ s
to negative boost weight terms after adding total $\partial_{v}$ terms), we
could show that a typical term of positive boost weight $(n+1)$ could always
be expressed as (see eq.(66) in §3.2.1 for more discussion on this)

$\begin{split}&\tilde{T}_{(-k)}~{}\partial_{v}^{n+1+k}T_{(0)}=\partial_{v}^{n+1}\left[\text{JKM
type
terms}\right]+(-1)^{k}\left[{}^{k+n}C_{n}\right]\left[\tilde{T}_{(0)}~{}\partial_{v}^{n+1}T_{(0)}\right]\\\
\text{where}~{}~{}&\tilde{T}_{(0)}=\partial_{v}^{k}~{}\tilde{T}_{(-k)}\end{split}$
(192)

Applying the result eq.(192) to the two terms $\tilde{t}_{(-k)}~{}t_{(k+1)}$
and $\tilde{t}_{(-k)}~{}\partial_{v}~{}t_{(k+1)}$ we find the following
relations

$\begin{split}\tilde{t}_{(-k)}~{}t_{(k+1)}&\sim\tilde{t}_{(-k)}~{}\partial_{v}^{(k+1)}t_{(0)}=\partial_{v}\left[\text{JKM
type
terms}\right]+(-1)^{k}\left[\tilde{t}_{(0)}\partial_{v}t_{(0)}\right]\end{split}$
(193)

by putting $(n=0)$ in eq.(192), and similarly

$\begin{split}&\tilde{t}_{(-k)}~{}\partial_{v}~{}t_{(k+1)}\sim\tilde{t}_{(-k)}~{}\partial_{v}^{(k+2)}t_{(0)}=\partial_{v}^{2}\left[\text{JKM
type
terms}\right]+(-1)^{k}(k+1)\left[\tilde{t}_{(0)}\partial_{v}t_{(0)}\right]\end{split}$
(194)

by putting $(n=1)$ in eq.(192). Equation (191) follows immediately from
equation eq.(193) and eq.(194).

##  Appendix E Detailed proof of Result: $1$ in eq.(66)

A typical component of a covariant tensor with boost weight $w=a+1>0$ will
have the following structure when restricted to the horizon

$t^{(k)}_{(a+1)}|_{r=0}=\tilde{T}_{(-k)}\partial_{v}^{(k+a+1)}T_{(0)}|_{r=0}+\mathcal{O}(\epsilon^{2})\,.$
(195)

Here the subscripts denote the boost weight and as before, for convenience we
have suppressed the index structure for $\tilde{T}$ and $T$.

Our aim is to prove Result:1 which states that $t^{(k)}_{(a+1)}$ could always
be re-expressed as follows

$\begin{split}&t^{(k)}_{(a+1)}=\tilde{T}_{(-k)}\partial_{v}^{(k+a+1)}T_{(0)}\\\
&~{}=\partial_{v}^{a+1}\left[\sum_{m=0}^{k-1}(-1)^{m}\left[{{}^{m+a}C}_{m}\right]\left(\tilde{T}_{(-k+m)}~{}\partial_{v}^{k-m}T_{(0)}\right)\right]+(-1)^{k}\left[{{}^{k+a}C}_{a}\right]\tilde{T}_{(0)}~{}\partial_{v}^{a+1}T_{(0)}\\\
&\text{where}~{}~{}\tilde{T}_{(-k+m)}\equiv\partial_{v}^{m}\tilde{T}_{(-k)}\end{split}$
(196)

###  E.1 The Proof

The basic idea in the proof is to carefully rearrange the $v$ derivatives and
to read off the binomial coefficients appropriately. We use the method of
induction to prove our desired result.

We check that eq.(196) is true for $a=0$:

$\begin{split}t^{(k)}_{(1)}&=~{}\tilde{T}_{(-k)}\partial_{v}^{(k+1)}T_{(0)}\\\
&=~{}\partial_{v}\left[\tilde{T}_{(-k)}\partial_{v}^{(k)}T_{(0)}\right]-\tilde{T}_{(-k+1)}\partial_{v}^{(k)}T_{(0)}\\\
&=~{}\partial_{v}\left[\tilde{T}_{(-k)}\partial_{v}^{(k)}T_{(0)}-\tilde{T}_{(-k+1)}\partial_{v}^{(k-1)}T_{(0)}\right]+\tilde{T}_{(-k+2)}\partial_{v}^{(k-1)}T_{(0)}\\\
&=~{}\cdots\\\
&=~{}\partial_{v}\left[\sum_{m=0}^{k-1}(-1)^{m}\tilde{T}_{(-k+m)}~{}\partial_{v}^{k-m}T_{0}\right]+(-1)^{k}\tilde{T}_{(0)}\partial_{v}T_{(0)}\\\
&=~{}\partial_{v}\left[\sum_{m=0}^{k-1}(-1)^{m}\left[{{}^{m+0}C}_{0}\right]\tilde{T}_{(-k+m)}~{}\partial_{v}^{k-m}T_{0}\right]+(-1)^{k}\left[{{}^{k+0}C}_{0}\right]\tilde{T}_{(0)}\partial_{v}T_{(0)}\end{split}$
(197)

where the fourth step indicates that we must extract all the $k$ derivatives
in the form of total derivative. The final step indicates that $t^{(k)}_{(1)}$
is of the form predicted by eq.(196). Next, we shall show that if the result
eq.(196) is true for $a=n$, then it is also true for $a=n+1$.

The statement for $a=n$ which we assume to be true is given by:

$\begin{split}&t^{(k)}_{(n+1)}=\tilde{T}_{(-k)}\partial_{v}^{(k+n+1)}T_{(0)}\\\
&~{}=\partial_{v}^{n+1}\left[\sum_{m=0}^{k-1}(-1)^{m}\left[{{}^{m+n}C}_{n}\right]\left(\tilde{T}_{(-k+m)}~{}\partial_{v}^{k-m}T_{(0)}\right)\right]+\left[{{}^{k+n}C}_{n}\right]\left(\tilde{T}_{(0)}~{}\partial_{v}^{n+1}T_{(0)}\right)\\\
&\text{where}~{}~{}\tilde{T}_{(-k+m)}=\partial_{v}^{m}\tilde{T}_{(-k)}\end{split}$
(198)

We shall now process
$t^{(k)}_{(n+2)}=\tilde{T}_{(-k)}\partial_{v}^{(k+n+2)}T_{(0)}$

$\begin{split}t^{(k)}_{(n+2)}=&\tilde{T}_{(-k)}\partial_{v}^{(k+n+2)}T_{(0)}\\\
=&~{}\partial_{v}\left[\tilde{T}_{(-k)}\partial_{v}^{(k+n+1)}T_{(0)}\right]-\tilde{T}_{(-k+1)}\partial_{v}^{(k+n+1)}T_{(0)}\\\
=&~{}\cdots\\\
=&~{}\partial_{v}\left[\sum_{m=0}^{k-1}(-1)^{m}~{}\tilde{T}_{(-k+m)}\partial_{v}^{(k+n+1-m)}T_{(0)}\right]+(-1)^{k}~{}\tilde{T}_{(0)}~{}\partial_{v}^{(n+2)}T_{(0)}\end{split}$
(199)

In the first term of the final step, each term within the summation is of the
form $t^{(k-m)}_{(n+1)}$. So we shall apply our previous assumption eq.(198)
here for $a=n$.

$\begin{split}&t^{(k)}_{(n+2)}=\tilde{T}_{(-k)}\partial_{v}^{(k+n+2)}T_{(0)}\\\
=~{}&\partial_{v}\left[\sum_{m=0}^{k-1}(-1)^{m}~{}\tilde{T}_{(-k+m)}\partial_{v}^{(k+n+1-m)}T_{(0)}\right]+(-1)^{k}~{}\tilde{T}_{(0)}~{}\partial_{v}^{(n+2)}T_{(0)}\\\
=~{}&\partial_{v}\sum_{m=0}^{k-1}(-1)^{m}\Bigg{(}\partial_{v}^{n+1}\left[\sum_{p=0}^{k-m-1}(-1)^{p}~{}\left({}^{n+p}C_{n}\right)\tilde{T}_{(-k+m+p)}\partial_{v}^{(k-m-p)}T_{(0)}\right]\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+(-1)^{k-m}\left({}^{k-m+n}C_{n}\right)\tilde{T}_{(0)}\partial_{v}^{n+1}T_{0}\Bigg{)}+(-1)^{k}~{}\tilde{T}_{(0)}~{}\partial_{v}^{(n+2)}T_{(0)}\\\
=~{}&\partial_{v}^{n+2}\bigg{(}\sum_{m=0}^{k-1}\sum_{p=0}^{k-m-1}(-1)^{m+p}\left({}^{n+p}C_{n}\right)\left[\tilde{T}_{(-k+m+p)}\partial_{v}^{(k-m-p)}T_{(0)}\right]\bigg{)}\\\
&+(-1)^{k}\left[1+\sum_{m=0}^{k-1}\left({}^{k-m+n}C_{n}\right)\right]\tilde{T}_{(0)}\partial_{v}^{n+2}T_{0}\\\
=~{}&\partial_{v}^{n+2}\bigg{(}\sum_{m=0}^{k-1}\sum_{p=0}^{k-m-1}(-1)^{m+p}\left({}^{n+p}C_{n}\right)\left[\tilde{T}_{(-k+m+p)}\partial_{v}^{(k-m-p)}T_{(0)}\right]\bigg{)}\\\
&+(-1)^{k}\left[\sum_{m=0}^{k}\left({}^{k-m+n}C_{n}\right)\right]\tilde{T}_{(0)}\partial_{v}^{n+2}T_{0}\end{split}$
(200)

The final result of eq.(200) is almost of the desired form in terms of the
structure of $\partial_{v}$ derivatives, but now we have to transform the
coefficients to the desired form. For this purpose we use the following
identity which is proved at the end of this section:

$\sum_{m=0}^{k}\left({}^{k-m+n}C_{n}\right)={{}^{k+n+1}C_{n+1}}$ (201)

Substituting in eq.(200) we find

$\begin{split}&t^{(k)}_{(n+2)}=\tilde{T}_{(-k)}\partial_{v}^{(k+n+2)}T_{(0)}\\\
=~{}&\partial_{v}^{n+2}\bigg{(}\sum_{m=0}^{k-1}\sum_{p=0}^{k-m-1}(-1)^{m+p}\left({}^{n+p}C_{n}\right)\left[\tilde{T}_{(-k+m+p)}\partial_{v}^{(k-m-p)}T_{(0)}\right]\bigg{)}\\\
&+(-1)^{k}\left[\sum_{m=0}^{k}\left({}^{k-m+n}C_{n}\right)\right]\left(\tilde{T}_{(0)}\partial_{v}^{n+2}T_{0}\right)\\\
=~{}&\partial_{v}^{n+2}\bigg{(}\sum_{m=0}^{k-1}\sum_{p=0}^{k-m-1}(-1)^{m+p}\left({}^{n+p}C_{n}\right)\left[\tilde{T}_{(-k+m+p)}\partial_{v}^{(k-m-p)}T_{(0)}\right]\bigg{)}\\\
&+(-1)^{k}\left[{{}^{k+n+1}C_{n+1}}\right]\tilde{T}_{(0)}\partial_{v}^{n+2}T_{0}\end{split}$
(202)

So the second term in eq.(200) has been cast into the desired form. We will
further process the double sum in the first term. Note that the sum has the
following form $\sim\sum_{m=0}^{k-1}\sum_{p=0}^{k-m-1}g(p)~{}f(m+p)$,
where $g(p)\equiv\left({}^{n+p}C_{n}\right)$ and
$f(m+p)\equiv(-1)^{m+p}\left[\tilde{T}_{(-k+m+p)}~{}\partial_{v}^{(k-m-p)}T_{(0)}\right]$.
Thus, we have

$\begin{split}&\sum_{m=0}^{k-1}\sum_{p=0}^{k-m-1}g(p)~{}f(m+p)\\\
=~{}&\sum_{m=0}^{k-1}\sum_{q=m}^{k-1}g(q-m)~{}f(q)~{}~{}~{}\text{$m$ has been
changed to $q=(m+p)$}\\\
=~{}&\sum_{q=0}^{k-1}\left[\sum_{m=0}^{q}~{}g(q-m)\right]f(q)=\sum_{q=0}^{k-1}\left[\sum_{m=0}^{q}~{}\left({}^{n+q-m}C_{n}\right)\right]f(q)\\\
=~{}&\sum_{q=0}^{k-1}{{}^{n+q+1}C_{n+1}}~{}~{}f(q)\end{split}$ (203)

Substituting eq.(203) in eq.(202) we get the required result eq.(196) that we
set out to prove

$\begin{split}t^{(k)}_{(n+2)}=~{}&\partial_{v}^{n+2}\bigg{(}\sum_{m=0}^{k-1}(-1)^{m}\left({}^{n+1+m}C_{n+1}\right)\left[\tilde{T}_{(-k+m)}\partial_{v}^{(k-m)}T_{(0)}\right]\bigg{)}\\\
&+(-1)^{k}\left[{{}^{k+n+1}C_{n+1}}\right]\left(\tilde{T}_{(0)}\partial_{v}^{n+2}T_{0}\right)\end{split}$
(204)

###  E.2 Proof of the identity eq.(201)

Consider the finite sum

$\begin{split}S(x)=~{}&\sum_{m=0}^{k}(1+x)^{k+n-m}=\sum_{m=0}^{k}\sum_{p=0}^{k+n-m}\left({}^{k-m+n}C_{p}\right)x^{p}\\\
=~{}&(1+x)^{n}\sum_{m=0}^{k}(1+x)^{k-m}=(1+x)^{n}\left[(1+x)^{k+1}-1\over
x\right]\\\ =~{}&\left[(1+x)^{k+n+1}-(1+x)^{n}\over x\right]\end{split}$ (205)

Equating the coefficient of $x^{n}$ on both sides of eq.(205), we get

$\begin{split}\text{Coefficient of $x^{n}$ in $S(x)$}=~{}&\text{Coefficient of
$x^{n+1}$ in $\left[(1+x)^{k+n+1}-(1+x)^{n}\right]$}\\\
\Rightarrow~{}\sum_{m=0}^{k}\left({}^{k-m+n}C_{n}\right)=&~{}~{}{{}^{k+n+1}C_{n+1}}\end{split}$
(206)

Hence proved.

##  Appendix F Variation of a generic covariant tensor under diffeomorphism: a
justification for eq.(75)

Suppose we have a Diffeomorphism Covariant tensor
$S_{\alpha_{1}\dots\alpha_{k}}$ which is a functional of various dynamical
fields $\\{\psi\\}$ (such as
$g_{\mu\nu},R_{\mu\nu\rho\sigma},D_{\alpha_{1}}R_{\mu\nu\rho\sigma},\dots,\phi,D_{\alpha_{1}}\phi,\dots$
), the notion of Diffeomorphism covariance under the vector field
$\xi^{\mu}\partial_{\mu}$ is expressed by the following equation:

$\delta
S_{\alpha_{1}\dots\alpha_{k}}[\delta\psi\rightarrow\mathcal{L}_{\xi}\psi]=\mathcal{L}_{\xi}S_{\alpha_{1}\dots\alpha_{k}}$
(207)

where $\xi^{\mu}\partial_{\mu}=v\partial_{v}-r\partial_{r}$ which is a killing
vector of our background Space-time eq.(17). The reason for this specific
choice will be clear in a moment. It should be emphasized that in the LHS of
eq.((207)), the variation is induced by the variation of the dynamical fields.
More precisely,

$\delta
S_{\alpha_{1}\dots\alpha_{k}}[\delta\psi\rightarrow\mathcal{L}_{\xi}\psi]=\lim_{\epsilon\to
0}\dfrac{S_{\alpha_{1}\dots\alpha_{k}}[\psi+\epsilon\mathcal{L}_{\xi}\psi]-S_{\alpha_{1}\dots\alpha_{1}}[\psi]}{\epsilon}$
(208)

Whereas, in the RHS of eq.((207)), the variation is induced by a space-time
Lie derivative or, in other words, the variation induced by a change of co-
ordinate charts along the flow of the integral curves of
$\xi^{\mu}\partial_{\mu}$ (waldbook, ).Thus, the variation in the LHS affects
only the dynamical fields, whereas the variation in the RHS affects both the
dynamical and background fields if any (Harlow:2019yfa, ). For the equality to
hold, we must make sure that the variations of all the background fields are
set to zero. Hence, the choice of $\xi^{\mu}\partial_{\mu}$ is clear: It is a
symmetry of the only background field present in the problem eq.(17) (i.e.)
$\mathcal{L}_{\xi}g^{eq}_{\mu\nu}=0$ and hence the equality of eq.(207)
holds.We provide a few examples which explicitly demonstrate this.

When one applies this notion of Diffeomorphism covariance to the Lagrangian of
a theory, we get

$\dfrac{\partial
L}{\partial\psi}\mathcal{L}_{\xi}\psi=\mathcal{L}_{\xi}L[\psi]\,,$ (209)

where the LHS is the usual variation of the Lagrangian with respect to the
fields with $\delta\psi=\mathcal{L}_{\xi}\psi$.

##### A scalar field coupled to non-dynamical gravity:

To illustrate this Diffeomorphism covariance, let us consider a simple scalar
field Lagrangian:

$L=g^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi$ (210)

The Lie derivative is given by

$\mathcal{L}_{\xi}L=\xi^{\mu}\partial_{\mu}L$ (211)

The variation of the Lagrangian is however given by

$\delta L=\delta
g^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi+2g^{\mu\nu}\partial_{\mu}\delta\varphi\partial_{\nu}\varphi$
(212)

Substituting $\delta
g_{\mu\nu}=\mathcal{L}_{\xi}g_{\mu\nu}=D_{\mu}\xi_{\nu}+D_{\nu}\xi_{\mu}$ and
$\delta\varphi=\mathcal{L}_{\xi}\varphi=\xi^{\rho}\partial_{\rho}\varphi$, we
get

$\begin{split}\delta
L[\delta\psi\rightarrow\mathcal{L}_{\xi}\psi]&=-(D^{\mu}\xi^{\nu}+D^{\nu}\xi^{\mu})\partial_{\mu}\varphi\partial_{\nu}\varphi+2g^{\mu\nu}\partial_{\mu}(\xi^{\rho}\partial_{\rho}\varphi)\partial_{\nu}\varphi\\\
&=-(\partial^{\mu}\xi^{\nu}+\partial^{\nu}\xi^{\mu}+g^{\mu\rho}\Gamma^{\nu}_{\rho\sigma}\xi^{\sigma}+g^{\nu\rho}\Gamma^{\mu}_{\rho\sigma}\xi^{\sigma})\partial_{\mu}\varphi\partial_{\nu}\varphi\\\
&\,\,\,\,\,\,+2g^{\mu\nu}\partial_{\mu}\xi^{\rho}\partial_{\rho}\varphi\partial_{\nu}\varphi+2g^{\mu\nu}\xi^{\rho}\partial_{\rho}(\partial_{\mu}\varphi)\partial_{\nu}\varphi\\\
&=g^{\mu\nu}\xi^{\rho}D_{\rho}(\partial_{\mu}\varphi\partial_{\nu}\varphi)\\\
&=\xi^{\rho}\partial_{\rho}L=\mathcal{L}_{\xi}L\end{split}$ (213)

where the end result is what you get by computing the Lie derivative directly.

##### Einstein’s theory of gravity:

We now consider a general diffeomorphism invariant theory. Assuming eq.(209),
it has been shown (Iyer:1994ys, ) that such a Lagrangian can be re-expressed
as

$L=L(g_{\mu\nu},R_{\mu\nu\rho\sigma},D_{\alpha_{1}}R_{\mu\nu\rho\sigma},\dots,D_{(\alpha_{1}}\dots
D_{\alpha_{m})}R_{\mu\nu\rho\sigma},\phi,D_{\alpha_{1}}\phi,\dots,D_{(\alpha_{1}}\dots
D_{\alpha_{m})}\phi)$ (214)

where $\phi$ collectively denote the set of matter fields in the theory.
Hence, with this choice of Lagrangian, the variation of the metric
$g_{\mu\nu}$ and $R_{\mu\nu\rho\sigma}$ are treated independently.

To again illustrate this diffeomorphism covariance for a more complicated
theory, consider General relativity with $L=R$:

$\begin{split}\delta L&=2(\delta
g^{\alpha\gamma})g^{\beta\delta}R_{\alpha\beta\gamma\delta}+g^{\alpha\gamma}g^{\beta\delta}\delta
R_{\alpha\beta\gamma\delta}\\\ \implies\delta L[\delta
g_{\mu\nu}\rightarrow\mathcal{L}_{\xi}g_{\mu\nu}]&=2(\mathcal{L}_{\xi}g^{\alpha\gamma})g^{\beta\delta}R_{\alpha\beta\gamma\delta}+g^{\alpha\gamma}g^{\beta\delta}\mathcal{L}_{\xi}R_{\alpha\beta\gamma\delta}\\\
&=\mathcal{L}_{\xi}(g^{\alpha\gamma}g^{\beta\delta}R_{\alpha\beta\gamma\delta})=\mathcal{L}_{\xi}R\end{split}$
(215)

This Diffeomorphism covariance eq.(207) has been used as an important
definition (Iyer:1994ys, ) to prove the first law of black hole thermodynamics
for any diffeomorphism invariant theory.

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