supertranslation vector field tangent to $H$ is given by 565656In the Rindler
coordinate system in flat spacetime, we have $u=\frac{1}{\sqrt{2}W}e^{-W\tau}$
and $v=\frac{1}{\sqrt{2}W}e^{+W\tau}$ where $\tau$ is the proper time of the
accelerating observer with proper acceleration $W$. Hence at proper time
$\tau$, she is at $(u(\tau),v(\tau)$, so the Killing boost tangent to the
curve has components $\xi^{u}=\frac{du}{d\tau}=-Wu$ and
$\xi^{v}=\frac{dv}{d\tau}=+Wv$. The difference between this case and the
present analysis is that in flat spacetime $\xi$ is a global Killing field
everywhere spanning an exact Killing horizon.

$\xi=W(v\partial_{v}-u\partial_{u})$ (C.1)

has norm $\xi^{\mu}\xi_{\mu}=-1+O(\frac{1}{W^{2}})$ on $H$, and the norm of
the acceleration vector field is given by

$a^{2}=g_{\mu\nu}\xi^{\rho}\nabla_{\rho}\xi^{\mu}\xi^{\sigma}\nabla_{\sigma}\xi^{\nu}=-W^{2}+O(W)$
(C.2)

In order to get these formulas, the acceleration $W$ must be large compared to
the tidal forces and the gravitational twist, exact relations are given below.
Locally, through the equivalence principle, the observer is uniformly
accelerating with norm $W$ in flat spacetime with a local Killing horizon. If
the acceleration $W$ is large compared the inverse proper time after which the
curvature become non negligible, the observer accelerating along $H$ perceives
a thermal spectrum [93, 94]. Indeed, in quantum theory, the local Unruh
temperature is

$T_{ext}=\frac{W\hbar}{2\pi ck_{B}}$ (C.3)

where the fundamental constants have been reintroduced on purpose. As
$\frac{\hbar}{2\pi ck_{B}}$ is very small, we need large acceleration in order
to get sensitive Unruh temperature, consistently to the approximations we made
in order to write $\xi^{\mu}\xi_{\mu}=-1$ and $a^{\mu}a_{\mu}=-W^{2}$ on $H$.
This temperature should be measured by the thermometer of an observer
accelerating locally along $H$. However we could have chose another curve of
constant local acceleration, and so the local temperature would be different.
In any case, the quantity $W$ is not relevant in the local balance law (3.25),
as it appears as a global multiplicative factor, and we can usually normalize
it to one. Nevertheless, it gives a thermodynamic interpretation of the
formula (3.25) from the point of view of an observer who is locally
acceleration along $H$ and can carry a locally well defined notion of
temperature, if the acceleration $W$ is large enough. We compute exact
relations in the remaining of this section.

The metric near ${\cal N}$ is given by (3.10) and so the inverse metric is
given by

$\displaystyle
g^{-1}=u^{2}(g_{AB}P^{A}P^{B}+F)\partial_{u}\partial_{u}-2\partial_{u}\partial_{v}+2uP^{A}\partial_{u}\partial_{A}+g^{AB}\partial_{A}\partial_{B}$
(C.4)

where $P^{A}=g^{AB}P_{B}$. As we will be interested in constant external
temperature, we will take $W(x^{A})=W$ to simplify the calculations. The norm
of $\xi$ on $H$ is given by

$\xi^{\mu}\xi_{\mu}=-1-\frac{F}{4W^{2}}$ (C.5)

Our first assumption is to set $W^{2}\gg
F=-\frac{1}{2}\partial_{u}^{2}g_{vv}+o(u^{2})$. Hence the acceleration must be
much bigger than the local tidal force. However, in order to have measurable
temperature, we need huge proper acceleration, of the order
$\frac{ck_{B}}{\hbar}$ at least. Hence, this assumption seems to be relevant
for our purpose, and for the following we will assume that
$\xi_{\nu}\xi^{\nu}=-1$. The next step is to compute the acceleration vector
given by

$\xi^{\nu}\nabla_{\nu}\xi^{\mu}=a^{\mu}$ (C.6)

The computation gives

$\displaystyle a^{v}$ $\displaystyle=W^{2}v(1-\frac{F}{2W^{2}})$
$\displaystyle a^{u}$ $\displaystyle=W^{2}u(1-\frac{1}{4W^{2}}\frac{\partial
F}{\partial\ln{v}}+\frac{1}{2W^{2}}P^{A}\frac{\partial
P_{A}}{\partial\ln{v}}+\frac{F}{W^{2}}-\frac{P^{2}}{2W^{2}}+\frac{1}{4W^{3}}-\frac{1}{4W^{4}}P^{A}\partial_{A}F)$
$\displaystyle a^{A}$
$\displaystyle=W^{2}(\frac{P^{A}}{4W^{4}}F+\frac{g^{AB}}{2W^{2}}\frac{\partial
P_{B}}{\partial\ln{v}}+\frac{1}{8W^{4}}g^{AB}\partial_{B}F-\frac{P^{A}}{W^{2}})$
(C.7)

Then, the norm of the acceleration is given by

$\displaystyle a^{2}$
$\displaystyle=2g_{uv}a^{v}a^{u}+2g_{Av}a^{v}a^{A}+g_{vv}a^{v}a^{v}+g_{AB}a^{A}a^{B}$
$\displaystyle=-W^{2}(1-\frac{F}{2W^{2}})(1-\frac{1}{4W^{2}}\frac{\partial
F}{\partial\ln{v}}+\frac{1}{2W^{2}}P^{A}\frac{\partial
P_{A}}{\partial\ln{v}}+\frac{F}{W^{2}}-\frac{P^{2}}{2W^{2}}+\frac{1}{4W^{3}}-\frac{1}{4W^{4}}P^{A}\partial_{A}F)$
$\displaystyle+W^{2}(1-\frac{F}{2W^{2}})(\frac{P^{2}}{4W^{4}}F+\frac{P^{A}}{2W^{2}}\frac{\partial
P_{A}}{\partial\ln{v}}+\frac{1}{8W^{4}}P^{A}\partial_{A}F-\frac{P^{2}}{W^{2}})-u^{2}v^{2}W^{2}(1-\frac{F}{2W^{2}})F$
$\displaystyle+W^{4}g_{AB}(\frac{P^{A}}{4W^{4}}F+\frac{g^{AC}}{2W^{2}}\frac{\partial
P_{C}}{\partial\ln{v}}+\frac{1}{8W^{4}}g^{AC}\partial_{C}F-\frac{P^{A}}{W^{2}})(\frac{P^{B}}{4W^{4}}F+\frac{g^{BC}}{2W^{2}}\frac{\partial
P_{C}}{\partial\ln{v}}+\frac{1}{8W^{4}}g^{BC}\partial_{C}F-\frac{P^{B}}{W^{2}})$
$\displaystyle=-W^{2}+O(W)$ (C.8)

The last equality of (C.8) makes sense only if the physical quantities
$F=-\frac{1}{2}\partial_{u}^{2}g_{vv}$ and $P_{A}=\partial_{u}g_{Av}$ verify
$\frac{F}{W^{2}}\ll 1$ (the same condition as in (C.5) in order to have the
norm of $\xi$ equal to $-1$) and $\frac{P_{A}}{W}\ll 1$. Furthermore, the
derivatives of $F$ and $P_{A}$ with respect to the ”time” $\ln{v}$ and the
angular coordinates $A$ must also be very small compared to $W^{2}$ and $W$
respectively. Hence, we have to consider large enough accelerations, much
larger than the local tidal forces and gravitational twist and their
variations. However, as already noticed, we need significant accelerations in
order to get a non infinitesimal Unruh temperature. If these conditions are
satisfied, then C.8 gives us

$|a|=\sqrt{-a^{2}}=W$ (C.9)

on $H$. Of course, (C.5) and (C.9) are norms so they are invariant under a
change of frame. Locally, the observer can always consider that spacetime is
flat, and hence as he is submitted to constant acceleration $W$ for any point
on $H$, and sees locally an Unruh temperature given by (C.3)

##  Appendix D Computation of anomalies

Here we come back on the geometric quantities appearing in the flux (3.25) and
(3.47). An analysis of the anomalies of the different physical quantities
characterizing the intrinsic and extrinsic geometries of the null
hypersurfaces already appears in [95], see also [55], and we will give a brief
summary of the main results here. Let us consider a null hypersurface ${\cal
N}$ located at $u=0$, with normal $n_{\mu}=-f\partial_{\mu}u$ on ${\cal N}$.
If $\xi$ is tangent to ${\cal N}$, we have

$\pounds_{\xi}n_{\mu}=\omega_{\xi}n_{\mu}$ (D.1)

with

$\omega_{\xi}=\xi^{\mu}\partial_{\mu}\ln{f}+\xi_{1}^{u}$ (D.2)

where $\xi^{u}=u\xi_{1}^{u}+O(u^{2})$. We restrict ourselves further to the
diffeomorphisms satisfying (B.2) and preserving the universal structure of the
null hypersurface. They correspond to an infinitesimal rescaling of the
normal, and these are precisely the class III transformations 575757A class
III transformation acts on a Newmann-Penrose null tetrad as
$(n,l,m,\bar{m})\rightarrow(An,A^{-1}l,me^{i\theta},\bar{m}e^{-i\theta})$,
where $n$ is the normal, $l$ an auxiliary vector while $m$ and its complex
conjugate complete the basis. from [81]. Hence, a geometric quantity that is
anomaly free must be class III invariant, because such quantities are
invariant through a rescaling of the normal as in (3.17). It is
straightforward to show from (D.1) and (3.1), (3.3), (3.5), (3.6) and 3.8 that

$\displaystyle\Delta_{\xi}\epsilon_{\cal N}$
$\displaystyle=\omega_{\xi}\epsilon_{\cal N}$
$\displaystyle\Delta_{\xi}\epsilon_{S}$ $\displaystyle=0$
$\displaystyle\gamma^{\alpha}_{\mu}\gamma^{\beta}_{\nu}\Delta_{\xi}\gamma_{\alpha\beta}$
$\displaystyle=0$ $\displaystyle\Delta_{\xi}\theta$
$\displaystyle=-\omega_{\xi}\theta$ $\displaystyle\Delta_{\xi}k_{n}$
$\displaystyle=-\omega_{\xi}k_{n}-n^{\mu}\partial_{\mu}\omega_{\xi}$ (D.3)

Hence, the anomaly of all the relevant physical quantities on the null
hypersurface depend linearly on $\omega_{\xi}$. Now, we compute the anomalies
associated to the diffeomorphisms (3.19). First, we need to compute the
proportionality coefficient $\omega_{\xi}$ appearing in (B.7). Hence

$\displaystyle\omega_{\xi}$ $\displaystyle=-l^{\mu}\pounds_{\xi}n_{\mu}$ (D.4)
$\displaystyle=-Wu\partial_{u}\ln{f}+(T+vW)\partial_{v}\ln{f}+Y^{A}\partial_{A}\ln{f}-W$

We look for the diffeomorphisms $\xi$ such that $\omega_{\xi}=0$. There are
several interesting cases. If $T=W=0$, then (D.4) reduces to

$\omega_{Y^{A}\partial_{A}}=Y^{A}\partial_{A}\ln{f}=0$ (D.5)

so if $f$ is independent of $x^{B}$ all the diffeomorphisms
$\xi=Y^{A}\partial_{A}$ are non anomalous. If $W=Y^{A}=0$, then the anomalous
diffeormorphisms are the one satisfying $\partial_{v}\ln{f}=0$, so $f$ is
independent on $v$. Third case, if $T=Y^{A}=0$, then the equation becomes

$v\partial_{v}\ln{f}-u\partial_{u}\ln{f}=1$ (D.6)

The solutions of D.6 are given by

$f=c_{1}(x^{A})v-\frac{c_{2}(x^{A})}{u}$ (D.7)

but $c_{2}=0$ because $f$ must be defined on the null hypersurface ${\cal N}$
located at $u=0$. Hence, as expected from (D.1), the property of anomaly
freedom does not rely only on the diffeomorphism $\xi$ but also on the chosen
normal $n$. Therefore, when we choose the normal to be
$n^{\mu}=v(\frac{\partial}{\partial v})^{\mu}$ as in section 3, we have

$\omega_{W(v\partial v-u\partial_{u})+Y^{A}\partial_{A}}=0$ (D.8)

but $\omega_{T\partial_{v}}\neq 0$. We would have obtained a different result
with another choice of normal, as $n^{\mu}=\partial_{v}^{\mu}$ for instance,
for which $\omega_{T\partial_{v}+Y^{A}\partial_{A}}=0$ but
$\omega_{W(v\partial v-u\partial_{u})}\neq 0$. Hence, from (D.3), we
understand that we can replace $\delta_{\xi}\theta_{n}$ by
$\pounds_{\xi}\theta_{n}$ for instance only for Weyl supertranslations and
superrotations and disregarding the affine supertranslation, i.e we consider
only the non anomalous diffeomorphisms

$\xi=W(v\partial_{v}-u\partial_{u})+Y^{A}(x^{B})\partial_{A}$ (D.9)

in accordance to what we stated in the main text.

##  Appendix E Dynamical entropy of the 3D light cone

This is an illustrative example of a system going through a succession of
equilibrium states. Let consider the null hypersurface ${\cal N}$ spanned by
outgoing light rays starting from one point in flat spacetime in dimension
$D=3$. We still set $n^{\mu}=v\partial_{v}^{\mu}$. It is well known that there
exists no black hole solution in flat spacetime in dimension $D=3$ because the
Weyl tensor vanishes, even if such solutions exist for negative cosmological
constant, as the BTZ black hole [96, 97]. However, we can still study the
gravitational flux through ${\cal N}$ and the gravitational charges cross
sections. In dimension $D=3$, there is no shear and so there is no
gravitational flux, i.e $\Theta^{D}=\Theta^{Y}=0$. Furthermore, the charges of
both prescriptions are equal. We can compute the charges for a Weyl
supertranslation $\xi=2\pi(v\partial_{v}-u\partial_{u})$, and define in
consequence the entropy

$S=Q_{\xi}^{D}=Q_{\xi}^{Y}=\frac{1}{4}(A-v\frac{dA}{dv})$ (E.1)

The entropy variation on any portion of the null hypersurface ${\cal N}$ is
entirely given by the entropy creation term

$\frac{1}{2\pi}\Delta S=\int_{\Delta{\cal
N}}T_{\mu\nu}\xi^{\mu}n^{\nu}\epsilon_{\cal N}=\frac{1}{2\pi}S_{c}$ (E.2)

that is positive if the null energy conditions are imposed, as usual. Hence,
on the outgoing light cone, the charge vanishes near $v=0$, but increases as
soon as some matter crosses it. During this process, spacetime is not flat.
However, after some matter entered ${\cal N}$, spacetime becomes flat again
and the charges do not vary anymore. The entropy of the new stationary state
is just given by

$S=2\pi\int_{\Delta{\cal N}}T_{\mu\nu}n^{\mu}n^{\nu}\epsilon_{\cal N}\geq 0$
(E.3)

However, this non vanishing charge is not due the local geometry of the null
hypersurface, as spacetime is flat in $D=3$ in the absence of matter and
cosmological constant. Hence, it accounts for the matter which crossed ${\cal
N}$ in the past. Hence, at any time $v$ at which the charge is stationary (no
matter flux) the charge gives us the total matter flux that entered ${\cal N}$
since $v=0$, but does not give any precision on the history of the physical
process.

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