horizons in the former. The extremal limit maintains the correspondence.

Linear perturbations of flat Minkowski spacetime in terms of conformally
invariant matter models, allow us to consider and prove suitable analogues of
the laws of black hole mechanics. When quantum effects are considered, thermal
properties make the classical mechanical laws amenable to a suitable
thermodynamical interpretation, where entropy variations are equal to $1/4$ of
the conformal area changes of the horizon in Planck units. The near horizon
and near bifurcate surface features of the geometry of the radial MCKF have
the same structure of the stationarity Killing field for static black holes.
The Minkowski vacuum state is the analogue of the Hartle-Hawking thermal state
from the particle interpretation that is natural to the MCKF.

This work represents another simple setting where the relationship between
thermality, gravity and geometry is manifest in the semiclassical framework.
It gives a simple and complete example in which thermal properties analogous
to those of black holes are manifest in flat spacetime. It improves the
standard analogy given by the study of gravity perturbations and quantum field
theory of the Rindler wedge. On a more speculative perspective, we think that
even when the interpretation of temperature, energy, and area entering the
thermodynamical relations is subtle, this simple example could shed some light
into a more fundamental description of the link between black hole entropy and
(quantum) geometry. But this is something we will investigate in the future.

###  Chapter 4 Light Cone Black Holes

_This Chapter overlaps with the first draft of a paper in preparation_ [De
Lorenzo and Perez, 2018a]. It contains all the main results of the latter.
Additional details, connections with other works in the literature and minor
results will appear in the final published version.

The boundary of the causal complement of a spherical ball of radius
$r_{\scriptscriptstyle H}$ at time $t$ in Minkowski spacetime is a bifurcate
conformal Killing horizon [Herrero and Morales, 1999]. The associated
conformal Killing vector field becomes null on the light cones of the two
events that intersect the sphere defined by the boundary of the ball 111These
events are the past and future ‘centers’ of the ball itself.. Such null
surfaces separate the whole of Minkowski in regions where the conformal
Killing field $\xi^{a}$ is either timelike or spacelike. These regions are in
direct correspondence with the different regions defined by the outer and
inner horizons of non-extremal Reissner-Nordstrom black holes. When
$r_{\scriptscriptstyle H}\to 0$ some regions collapse and the features of the
conformal Killing field now correspond to those of extremal Reissner-Nordstrom
black holes.

Minkowski Conformal Killing Fields (MCKFs) admit a conformally invariant
[Jacobson and Kang, 1993] definition of surface gravity
$\kappa_{\scriptscriptstyle SG}$

$\nabla_{a}(\xi\cdot\xi)\,\hat{=}-2{\kappa_{\scriptscriptstyle SG}}\xi_{a}\,,$
(4.1)

where $\hat{=}$ means that the equality holds at the Killing horizon. All four
laws of black hole thermodynamics have a suitable version for the conformal
killing horizons defined by MCKFs: The surface gravity
$\kappa_{\scriptscriptstyle SG}$ is constant on the horizon and it is
associated to a mathematical notion of temperature (conformal temperature)

${T=\frac{\kappa_{\scriptscriptstyle SG}}{2\pi}={\rm constant}}.$ (4.2)

Under linear perturbations induced by conformally invariant matter fields the
current $J_{a}=\delta T_{ab}\xi^{b}$ is conserved, namely

$\nabla^{a}J_{a}=\nabla^{a}(\delta T_{ba}\xi^{b})=0.$ (4.3)

The previous equation can be used to establish a suitable version of the first
law for MCKF

${\delta M=\frac{\kappa_{\scriptscriptstyle SG}}{8\pi}\,{\delta A}+\delta
M_{\infty}\,,}$ (4.4)

where

$\delta M=\int_{\Sigma}J_{a}d\Sigma^{b}$ (4.5)

is the conformally invariant mass of the perturbation evaluated at an initial
Cauchy surface $\Sigma$, $\delta M_{\infty}$ is the conformal mass flow at
${\mathfs{I}}^{+}$, and $\delta A$ is a conformally invariant notion defined
as

$\delta A\equiv\int\limits_{H^{+}}\frac{\kappa}{{\kappa_{\scriptscriptstyle
SG}}}\,\delta\theta dS^{2}dv\,.$ (4.6)

Here $\delta\theta$ is the first order perturbation of the expansion of the
generator of the horizon, $v$ is the advanced Minkowski time (a natural affine
parameter for the generators), $dS^{2}$ the flat background area measure of
the spherical cross section $v=$constant of the unperturbed light cone, and
$\kappa$ is defined by

$\xi^{a}\nabla_{a}\xi^{b}{\,\hat{=}\,}\kappa\,\xi^{b}\,.$ (4.7)

Unlike $\kappa_{\scriptscriptstyle SG}$, the function $\kappa$ is not constant
and is not conformally invariant. Finally, provided that $\delta T_{ab}$
satisfies the weak energy condition, which is equivalent to the strong one for
conformally invariant matter, the second law holds, namely

${\delta A\geq 0.}$ (4.8)

In the ‘extremal’ limit $r_{\scriptscriptstyle H}\to 0$ the temperature (4.2)
goes to zero as well as the area of the bifurcate sphere $A=4\pi
r_{\scriptscriptstyle H}^{2}$. We can interpret this as a form of third law of
thermodynamics. This, plus equations (4.2), (4.4), and (4.8), are the analog
of the laws of black hole mechanics for the light cones in flat spacetimes
that define the conformal Killing horizons associated to MCKFs [De Lorenzo and
Perez, 2018b]–see previous Chapter.

The previous formal analogy between the properties of MCKFs and thermodynamics
of black holes captures the basic mathematical features of the latter on a
background with trivial gravitational field. However, the reverse side of it
is that the various conformal invariant notions entering the laws have no
clear physical meaning: $\delta M$ is not an energy measured by any real
physical observer, the conformal temperature is not the one detected by any
physical thermometer, and $\delta A$ is not really the change of the geometric
area of the bifurcating sphere. Perhaps the most disturbing aspect of the
previous analogy is precisely the definition of $\delta A$ entering (4.4) with
its non-obvious geometric meaning due to the fact that
$\kappa_{\scriptscriptstyle SG}/\kappa\not=1$ for conformal Killing fields in
general and the MCKFs in particular.

Nevertheless, the previous limitation can be resolved if one performs a
conformal transformation sending $({\mathbb{R}}^{4},\eta_{ab})$ to a model
spacetime $(M,\tilde{g}_{ab})$ with $\tilde{g}_{ab}=\omega^{2}\eta_{ab}$ so
that $\xi^{a}$ becomes a genuine Killing field. The associate conformal
bifurcate horizons will be mapped into bifurcate Killing horizons in $g_{ab}$.
If $\delta T_{ab}$ comes from a conformally invariant matter model, then
$\delta\tilde{T}_{ab}=\omega^{-2}\delta T_{ab}$ the conservation of the
associated current $\tilde{J}_{a}$ holds in the new spacetime. Equations
(4.2), (4.4), and (4.8) remain true in the target spacetime with identical
numerical values for a given perturbation. However, all the quantities
involved acquire the standard physical and geometric meaning that they have in
the context of black holes.

The question we want to explore here is what are the generic global features
of the spacetimes obtained by the previous procedure. Which are the cases
where these spacetimes represent black holes? What are they in the other
cases? Even when demanding that the procedure does not break spherical
symmetry there is clearly an infinite number of possibilities. Indeed, if
$\omega_{1}$ is a solution, $\omega_{2}$ is a new solution as long as
$\xi(\omega_{2}/\omega_{1})=0$. We will see that the generic global features
can be made apparent in a small number of representative cases. The simplest
case corresponds to $\omega=\alpha/r^{2}$ that reproduces the Bertotti-
Robinson solution [Bertotti, 1959, Robinson, 1959] of Einsteins-Maxwell
theory–Section 4.3. Such solution has been know to encode the near horizon
geometry of close-to-extremal and extremal Reissner-Nordstrom black holes.
Another representative example is the de Sitter realization where the
bifurcating horizons correspond to intersecting cosmological horizons (there
is no black hole in this case)–see Section 4.4.1. Weakly asymptotically
(Anti)-de Sitter black hole realizations are also presented–Section 4.4.2–,
together with a more exotic asymptotically flat spacetime with Killing
horizons but no black holes–Section 4.5.

Radial MCKF with conformal Killing horizons associated to light cones
bifurcating at a sphere generalize to arbitrary dimensions. As long as the
matter perturbing the geometry is conformally invariant, the generalization of
equations (4.2), (4.4), and (4.8) is also valid. The Bertotti-Robinson
representation, which in arbitrary dimensions is given by $AdS_{2}\times
S_{d-2}$, remains the simplest one. For $d=2$ the light cone black hole
corresponds to the Jackiw-Teitelboim solution [Jackiw and Teitelboim, 1984].

#### 4.1 Radial Conformal Killing Fields in
Minkowski Spacetime

Consider Minkwoski spacetime in spherical coordinates

$\begin{split}ds^{2}_{\scriptscriptstyle
M}=\eta_{\mu\nu}dx^{\mu}dx^{\nu}&=-dt^{2}+dr^{2}+r^{2}dS^{2}\\\
&=-dvdu+\frac{(v-u)^{2}}{4}dS^{2}\end{split}$ (4.9)

where $dS^{2}$ is the unit-sphere metric, while $v=t+r$ and $u=t-r$ the
standard Minkowskian null coordinates. The conformal group in four dimensional
Minkowski spacetime ${\mathbb{M}}^{4}$ is isomorphic to the group $SO(5,1)$.
Any generator defines a Conformal Killing Field in Minkowski spacetime (MCKF),
namely a vector field $\xi$ along which the metric $\eta_{ab}$ changes only by
a conformal factor:

${\mathfs{L}}_{\xi}\,\eta_{ab}=\nabla_{a}\xi_{b}+\nabla_{b}\xi_{a}=\frac{\psi}{2}\eta_{ab}$
(4.10)

with

$\psi=\nabla_{a}\xi^{a}\,.$ (4.11)

The most general radial MCKF can be written (up to Poincarè transformations)
as

$\begin{split}\xi^{\mu}\frac{\partial}{\partial
x^{\mu}}&=\frac{1}{r_{\scriptscriptstyle O}^{2}-r_{\scriptscriptstyle
H}^{2}}\left((t^{2}+r^{2}-r_{\scriptscriptstyle
H}^{2})\frac{\partial}{\partial{t}}+2\,t\,r\,\frac{\partial}{\partial
r}\right)\\\ &=\frac{1}{r_{\scriptscriptstyle O}^{2}-r_{\scriptscriptstyle
H}^{2}}\left((v^{2}-r_{\scriptscriptstyle H}^{2})\,\frac{\partial}{\partial
v}+(u^{2}-r_{\scriptscriptstyle H}^{2})\,\frac{\partial}{\partial
u}\right)\end{split}$ (4.12)

where $r_{\scriptscriptstyle H}$ and $r_{\scriptscriptstyle O}$ are two
constants defining the family of Killing field. They are defined to be
respectively the value of the radius of the sphere where the Killing field is
zero $\xi|_{r=r_{\scriptscriptstyle H}}=0$ and the radius of the sphere on the
surface $t=0$ where the Killing field is normalised,
$\xi\cdot\xi|_{r=r_{\scriptscriptstyle O}}=-1$. Indeed the norm is given by

$\xi\cdot\xi=-\frac{(v^{2}-r_{\scriptscriptstyle
H}^{2})(u^{2}-r_{\scriptscriptstyle H}^{2})}{(r_{\scriptscriptstyle
O}^{2}-r_{\scriptscriptstyle H}^{2})^{2}}$ (4.13)

and at $t=0$, we have $v=-u=r$. This also shows that the causal behaviour of
$\xi$ changes from being timelike to spacelike, passing through being null
along two constant $u=u_{\pm}$ and two constant $v=v_{\pm}$ rays. This
behaviour divides Minkowski spacetime in six regions, Fig. 4.1.

Figure 4.1: The most general radial conformal Killing field $\xi$ in Minkowski
spacetime divided the latter in six regions. The field $\xi$ is spacelike in
the shaded regions, timelike elsewhere. It becomes null on the lightcones
separating the regions. It vanishes at the tips of the ligthcones and at their
intersection.

Consider now any spacetime conformally related to Minkowski

$g_{ab}=\omega^{2}\,\eta_{ab}\,.$ (4.14)

In such a spacetime, the conformal Killing field $\xi$ remains so. Indeed

${\mathfs{L}}_{\xi}g_{ab}={\mathfs{L}}_{\xi}(\omega^{2}\eta_{ab})=\left(\frac{\psi}{2}+\xi^{a}\partial_{a}\,(\log\,\omega^{2})\right)g_{ab}.$
(4.15)

In particular, there exist conformal transformations such that $\xi$ becomes a
proper Killing field. From the above equation, it follows that those are given
by conformal factors satisfying

$\frac{\psi}{2}+\xi^{a}\partial_{a}\,(\log\omega^{2})=0$ (4.16)

Let us define the following coordinates transformation

$\begin{split}\tau&=\frac{(r_{\scriptscriptstyle O}^{2}-r_{\scriptscriptstyle
H}^{2})}{4\,r_{\scriptscriptstyle H}}\log\frac{(u-r_{\scriptscriptstyle
H})(v-r_{\scriptscriptstyle H})}{(u+r_{\scriptscriptstyle
H})(v+r_{\scriptscriptstyle H})}\\\ x&=\frac{2(r_{\scriptscriptstyle
H}^{2}-u\,v)}{v-u}.\end{split}$ (4.17)

The Minkowski metric becomes

$ds_{\scriptscriptstyle M}^{2}=\frac{r^{2}}{x_{\scriptscriptstyle
0}^{2}}\left(-\frac{x^{2}-x_{\scriptscriptstyle BH}^{2}}{x_{\scriptscriptstyle
0}^{2}}\,d\tau^{2}+\frac{x_{\scriptscriptstyle
0}^{2}}{x^{2}-x_{\scriptscriptstyle BH}^{2}}\,dx^{2}+x_{\scriptscriptstyle
0}^{2}\,dS^{2}\right)\,.$ (4.18)

where

$x_{\scriptscriptstyle BH}\equiv{2\,r_{\scriptscriptstyle H}},\ \ \
x_{\scriptscriptstyle 0}^{2}\equiv r_{\scriptscriptstyle
O}^{2}-r_{\scriptscriptstyle H}^{2},$ (4.19)

and where $r$ is now a function of $\tau$ and $x$. The outer and inner
horizons are located respectively at $x_{\pm}=\pm x_{\scriptscriptstyle BH}$.
Minkowskian $\mathscr{I}^{\pm}$ are located at finite $x$, while $i^{0}$ at
$x\to+\infty$. Finally, the origin $u=v$ corresponds to $x\to{\rm
sign}(r_{\scriptscriptstyle H}^{2}-t^{2})\,\infty$. In these coordinates, our
radial MCKF reduces to

$\xi^{\mu}\frac{\partial}{\partial x^{\mu}}=\frac{\partial}{\partial\tau}\,,$
(4.20)

simplifying Eq. (4.16) as

$\frac{\partial}{\partial\tau}(\log\omega^{2})=-\frac{\psi}{2}\,.$ (4.21)

The general solution to the above equation is

$\log(\omega^{2})=-\int\frac{\psi}{2}\,d\tau+F(x,\theta,\varphi)$ (4.22)

where $F(x,\theta,\phi)$ is a general dimension-less function of
$(x,\theta,\phi)$. The function $\psi$ can be explicitly computed and the
integral performed, giving

$\log(\omega^{2})=\log\left(\frac{x_{\scriptscriptstyle
0}^{2}}{r^{2}}\right)+F(x,\theta,\varphi)\,,$ (4.23)

where again the Minkowskian radial coordinate $r$ is a function of $\tau$ and
$x$. Redefining for convenience $F\equiv-\log\,P^{2}$, we find

$\omega^{2}=\frac{x_{\scriptscriptstyle
0}^{2}}{r^{2}}\,\frac{1}{P^{2}(x,\theta,\varphi)}\,.$ (4.24)

The above equation defines an infinite family of conformal transformations of
Minkowski spacetime, such that the target spacetime admits a genuine Killing
field corresponding to MCKF. Each member of the family is defined by a choice
of the function $P(x,\theta,\phi)$. The metric of such spacetimes is given by

$ds^{2}=\omega^{2}ds_{\scriptscriptstyle
M}^{2}=\frac{1}{P^{2}(x,\theta,\varphi)}\left(-\frac{x^{2}-x_{\scriptscriptstyle
BH}^{2}}{x_{\scriptscriptstyle 0}^{2}}\,d\tau^{2}+\frac{x_{\scriptscriptstyle
0}^{2}}{x^{2}-x_{\scriptscriptstyle BH}^{2}}\,dx^{2}+x_{\scriptscriptstyle
0}^{2}\,dS^{2}\right)\,.$ (4.25)

As expected, it does not depend on the Killing time $\tau$. Cleary, any
additional coordinates transformation that does not depend on $\tau$ sends the
metric in an equivalent $\tau$-independent form. In Appendix A.7 some
interesting example is presented. In particular, it is shown the precise
relation between the above coordinates and the one presented in Eq. (3.50) [De
Lorenzo and Perez, 2018b].

##### 4.1.1 The Hartle-Hawking temperature

By the standard Wick rotation $\tau\to-i\bar{\tau}$ the metric becomes
Euclidean

$ds^{2}=\frac{1}{P^{2}}\left(\frac{x^{2}-x_{\scriptscriptstyle
BH}^{2}}{x_{\scriptscriptstyle
0}^{2}}\,d\bar{\tau}^{2}+\frac{x_{\scriptscriptstyle
0}^{2}}{x^{2}-x_{\scriptscriptstyle BH}^{2}}\,dx^{2}+x_{\scriptscriptstyle
0}^{2}\,dS^{2}\right)\,.$ (4.26)

The apparent singularity at $x=+x_{\scriptscriptstyle BH}$ can be removed in
the usual way by introducing a new set of coordinates

$\begin{split}\rho^{2}&=\frac{x_{\scriptscriptstyle
BH}^{2}}{x_{\scriptscriptstyle 0}^{2}}(x^{2}-x_{\scriptscriptstyle BH}^{2})\\\
\varphi&=\frac{x_{\scriptscriptstyle BH}}{x_{\scriptscriptstyle
0}^{2}}\,\tau\,,\end{split}$ (4.27)

in which the metric becomes

$ds^{2}=\frac{1}{P^{2}}\left(\rho^{2}d\varphi^{2}+\frac{x_{\scriptscriptstyle
0}^{2}}{x_{\scriptscriptstyle 0}^{2}+\rho^{2}}\
d\rho^{2}+x_{\scriptscriptstyle 0}^{2}dS^{2}\right)\,.$ (4.28)

Assuming that $P$ is non vanishing at $\rho=0$, the previous metric would have
a conical singularity at $x=+x_{\scriptscriptstyle BH}$ unless
$0\leq\varphi\leq 2\pi$. The quantum state of fields compatible with this
topology of the Euclidean continuation is a thermal state with temperature

$T_{\scriptscriptstyle HH}=\frac{1}{2\pi}\frac{x_{\scriptscriptstyle
BH}}{x_{\scriptscriptstyle 0}^{2}}=\frac{r_{\scriptscriptstyle
H}}{\pi(r_{\scriptscriptstyle O}^{2}-r_{\scriptscriptstyle H}^{2})}$ (4.29)

which is the exactly Hartle-Hawking temperature found in [De Lorenzo and
Perez, 2018b], Eq. (3.96). Even when the local physical temperature will
depend on the function $P$, the above temperature–which we termed conformal
temperature–is conformally invariant and it is related to the conformally
invariant notion of surface gravity defined in the introduction via the
standard relation $T_{\scriptscriptstyle HH}=\kappa_{\scriptscriptstyle
SG}/(2\pi)$. In the following sections we will study the spacetime
realizations corresponding to different choices of the function
$P(x,\theta,\varphi)$.

#### 4.2 Light Cone Black Holes

The causal structure of a spacetime $g_{ab}$ is easily readable once the
Carter-Penrose diagram for $g_{ab}$ is found. In our case the procedure to
find it is straightforward. Indeed, we already know [Wald, 1984] that the
coordinate transformation

$\begin{split}T+R&=2\,\arctan\left(\frac{v}{r_{\scriptscriptstyle
H}}\right)\\\ T-R&=2\,\arctan\left(\frac{u}{r_{\scriptscriptstyle
H}}\right)\,.\end{split}$ (4.30)

is such that the Minkowski metric $\eta_{ab}$ becomes conformally related to
the Einstein Universe metric $g_{ab}^{\scriptscriptstyle EU}$, i.e.

$\begin{split}g_{ab}^{\scriptscriptstyle EU}&=\Omega^{2}_{{\scriptscriptstyle
M}}\,\eta_{ab}\\\
-dT^{2}+dR^{2}+\sin^{2}R\,dS^{2}&=\Omega^{2}_{{\scriptscriptstyle
M}}(-dt^{2}+dr^{2}+r^{2}dS^{2})\end{split}$ (4.31)

with

$\Omega^{2}_{{\scriptscriptstyle M}}=\frac{4r_{\scriptscriptstyle
H}^{2}}{(v^{2}+r_{\scriptscriptstyle H}^{2})(u^{2}+r_{\scriptscriptstyle
H}^{2})}\,.$ (4.32)

From the transformation (4.30) one can see that the Minkwoski spacetime covers
only a portion of the Einstein’s Universe spacetime. Such portion gives the
Carter-Penrose diagram of $({\mathbb{R}}^{4},\eta_{ab})$. In particular, one
can notice that infinite physical distances in $({\mathbb{R}}^{4},\eta_{ab})$
are mapped into finite distances in the Einstein’s Universe, by the fact that
$\Omega^{2}_{{\scriptscriptstyle M}}$ vanishes for $u,v\to\pm\infty$.

Any conformally flat spacetime $g_{ab}=\omega^{2}\,\eta_{ab}$ will be
conformally mapped to the Einstein Universe by the same coordinate
transformation:

$g_{ab}^{\scriptscriptstyle EU}=\Omega^{2}_{{\scriptscriptstyle
M}}\,\eta_{ab}=\frac{\Omega^{2}_{{\scriptscriptstyle
M}}}{\omega^{2}}\,g_{ab}\equiv\Omega^{2}\,g_{ab}\,.$ (4.33)

Using (4.24), the conformal factor $\Omega$ mapping the generic metric (4.25)
to the Einstein’s universe is found to be

$\Omega^{2}=\frac{4\,r^{2}}{(v^{2}+r_{\scriptscriptstyle
H}^{2})(u^{2}+r_{\scriptscriptstyle
H}^{2})}\,P^{2}(x,\theta,\varphi)=\frac{(v-u)^{2}}{(v^{2}+r_{\scriptscriptstyle
H}^{2})(u^{2}+r_{\scriptscriptstyle H}^{2})}\,P^{2}(x,\theta,\varphi)\,.$
(4.34)

As for Minkowski, vanishing of $\Omega$ implies infinite distances in the
physical spacetime $g_{ab}$, defining therefore the boundary of it.

Let us now analyze different interesting choices of the function
$P(x,\theta,\varphi)$.

#### 4.3 The Bertotti-Robinson Realization

The most interesting realization is also the simplest one:
$P(x,\theta,\varphi)=1$. From eq. (4.24), one can see that this spacetime is
simply found dividing the Minkowski metric by $r^{2}$. Such an apparently
simple operation has striking consequences. The metric is

$ds^{2}=-\frac{x^{2}-x_{\scriptscriptstyle BH}^{2}}{x_{\scriptscriptstyle
0}^{2}}\,d\tau^{2}+\frac{x_{\scriptscriptstyle
0}^{2}}{x^{2}-x_{\scriptscriptstyle BH}^{2}}\,dx^{2}+x_{\scriptscriptstyle
0}^{2}\,dS^{2}\,.$ (4.35)

The Ricci and the Kretschmann scalars come out to be

$\begin{split}\mathbf{R}&=0\\\
\mathbf{R}_{abcd}\mathbf{R}^{abcd}&=\frac{8}{x_{\scriptscriptstyle
0}^{4}}\,.\end{split}$ (4.36)

Since the metric is diagonal, we can easily define a tetrad
$\mathbf{e}_{a}{}^{I}$ as

$\begin{split}\mathbf{e}_{\mu}^{0}\,dx^{\mu}&=\sqrt{-g_{\tau\tau}}\,dt\\\
\mathbf{e}_{\mu}^{1}\,dx^{\mu}&=\sqrt{g_{xx}}\,dr\\\
\mathbf{e}_{\mu}^{2}\,dx^{\mu}&=\sqrt{g_{\theta\theta}}\,d\theta\\\
\mathbf{e}_{\mu}^{3}\,dx^{\mu}&=\sqrt{g_{\varphi\varphi}}\,d\varphi\,.\end{split}$
(4.37)

In this tetrad, the Einstein tensor is diagonal and given by

$\mathbf{G}_{IJ}=\mathbf{G}_{ab}\,\mathbf{e}^{a}_{I}\,\mathbf{e}^{b}_{J}=\frac{1}{x_{\scriptscriptstyle
0}^{2}}{\rm diag}(1,-1,1,1)\,.$ (4.38)

The metric (4.35) can be also found as a solution to the Einstein-Maxwell
equations for a the vector potential given by

$\mathbf{A}=\frac{x}{x_{\scriptscriptstyle 0}}dt\,,$ (4.39)

from which the electromagnetic tensor

$\mathbf{F}=d\mathbf{A}=\frac{1}{x_{\scriptscriptstyle 0}}dx\wedge dt\,.$
(4.40)

The solution is static and spherically symmetric with a constant radial
electric field whose flux defines the charge of the spacetime

$Q=\frac{1}{8\pi}\int_{S}\mathbf{\epsilon}_{abcd}\mathbf{F}^{cd}=x_{\scriptscriptstyle
0}\,.$ (4.41)

The stress-energy tensor satisfies the weak, strong and dominant energy
conditions. The spacetime is topologically $AdS_{2}\times S^{2}$. Such
solution is known in the literature as the Bertotti-Robinson spacetime
[Bertotti, 1959, Robinson, 1959]. Its Carter-Penrose diagram is depicted in
Figure 4.2. The geometry is everywhere regular. There are no singularities,
despite the presence of trapped surfaces and the fact that the usual energy
conditions are satisfied. Singularity theorems–see for instance [Wald,
1984]–are avoided due to the fact that the spacetime is not globally
hyperbolic, which rules out some versions of the theorems, and the generic
null geodesic congruence condition is not satisfied 222Null geodesic violating
the null generic geodesic condition are those generating ${\mathfs{I}}^{\pm}$
in Minkowski spacetime, which now pass through the bulk of the RB solution.,
which rules out those that do not require global hyperbolicity.

Figure 4.2: The causal structure of the Bertotti-Robinson spacetime. The two
spherical dimensions are suppressed, so that each point represent a sphere.
The Killing field is spacelike in the shaded regions and timelike elsewhere.
The boundary is timelike and no singularities are present. Grey lines and
numbers show how the 6 regions of Minkowski spacetime, see Fig. 4.1, are
conformally mapped into the bulk of the target spacetime.

##### 4.3.1 Close to extremal Reissner-Nordstrom
near-horizon geometry

The Bertotti-Robinson solution is known to correspond to the near horizon
geometry of a Reissner-Nordstrom (RN) black hole close to extremality–see for
instance [Fabbri and Navarro-Salas, 2005]. This fact in turn provides a simple
interpretation to the laws of light cone mechanics [De Lorenzo and Perez,
2018b]–Chapter 4–in terms of the standard laws of black hole thermodynamics.
The RN metric is given by

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\left(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\right)dt^{2}+\left(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\right)^{-1}dr^{2}+r^{2}dS^{2}$
(4.42) $\displaystyle=$
$\displaystyle-\frac{(r-r_{+})(r-r_{-})}{r^{2}}dt^{2}+\frac{r^{2}}{(r-r_{+})(r-r_{-})}dr^{2}+r^{2}dS^{2},$
(4.43)

and the associated electromagnetic field by

$\mathbf{A}=-\frac{Q}{r}dt,$ (4.44)

where $r_{\pm}=M\pm\sqrt{M^{2}-Q^{2}}$. The near extremal case corresponds to
$M=Q+\delta M$ with $\delta M^{2}\ll Q^{2}$, for which the near horizon metric
and electromagnetic field is obtained by expanding in the new coordinate $x$
defined by $r=x_{\scriptscriptstyle 0}+x$. The leading order gives the metric
(4.35) and electromagnetic field (4.39) with $x_{\scriptscriptstyle 0}=Q$ and
$x_{\scriptscriptstyle BH}=\sqrt{2\,Q\,\delta m}$. We can relate the physical
parameter or the RN solution to the parameters of the MCKF in flat spacetime
via

$Q^{2}={r_{\scriptscriptstyle O}^{2}-r_{\scriptscriptstyle H}^{2}}\qquad{\rm
and}\qquad\delta m=\frac{2r_{\scriptscriptstyle H}^{4}}{(r_{\scriptscriptstyle
O}^{2}-r_{\scriptscriptstyle H}^{2})^{\frac{3}{2}}}\,.$ (4.45)

This shows that the limit $r_{\scriptscriptstyle H}\to 0$ corresponds exactly
the extremal limit of the RN solution. On the RN side the temperature goes to
zero and the bifurcating sphere goes away to infinity. On the Minkowski side,
the radius of the bifurcating sphere $r_{\scriptscriptstyle H}$ shrinks to
zero and the conformal Killing horizon becomes the light cone of a single
event.

#### 4.4 De Sitter and Asymptotically $dS$ and $AdS$
Realizations

Another interesting case arises by sending a constant $x=x_{*}$ Killing
observer to infinity. From the discussion under Eq. (4.34), this is achieved
by choosing a function $P(x,\theta,\phi)$ which vanishes at $x=x_{*}$. The
simplest choice admitting a regular differential structure at infinity is

$P(x)=\frac{x_{*}-x}{x_{\scriptscriptstyle 0}}\,.$ (4.46)

The corresponding metric is therefore

$ds^{2}=\frac{x_{\scriptscriptstyle
0}^{2}}{(x_{*}-x)^{2}}\left(-\frac{x^{2}-x_{\scriptscriptstyle
BH}^{2}}{x_{\scriptscriptstyle 0}^{2}}\,d\tau^{2}+\frac{x_{\scriptscriptstyle
0}^{2}}{x^{2}-x_{\scriptscriptstyle BH}^{2}}\,dx^{2}+x_{\scriptscriptstyle
0}^{2}\,dS^{2}\right)\,.$ (4.47)

In the new coordinate

$X^{2}=\frac{x_{\scriptscriptstyle 0}^{4}}{(x_{*}-x)^{2}}$ (4.48)

the metric takes the simple form

$ds^{2}=-F(X)\,d\tau^{2}+\frac{1}{F(X)}\,dX^{2}+X^{2}\,dS^{2}$ (4.49)

with

$F(X)=1-\frac{2\,x_{*}}{x_{\scriptscriptstyle
0}^{2}}\,X+\frac{(x_{*}^{2}-x_{\scriptscriptstyle
BH}^{2})}{x_{\scriptscriptstyle 0}^{4}}\,X^{2}\,.$ (4.50)

The observer sent to infinity corresponds now to $X\to+\infty$. The Ricci
scalar is

$\mathbf{R}=-12\left(\frac{x_{*}^{2}-x_{\scriptscriptstyle
BH}^{2}}{x_{\scriptscriptstyle 0}^{4}}+\frac{x_{*}}{x_{\scriptscriptstyle
0}^{2}\,X}\right)$ (4.51)

which tends to a constant as $X\to+\infty$

$4\,\Lambda=\lim_{X\to+\infty}\mathbf{R}=-12\,\frac{x_{*}^{2}-x_{\scriptscriptstyle
BH}^{2}}{x_{\scriptscriptstyle 0}^{4}}\,.$ (4.52)

Such constant is positive if $x_{*}$ is chosen in between the horizons and
negative elsewhere. Moreover, for $x_{*}\neq 0$, it diverges as $X$ approaches
zero

$\mathbf{R}|_{X\to 0}=-12\,\frac{x_{*}}{x_{\scriptscriptstyle
0}^{2}\,X}+O(1)\,.$ (4.53)

As in the previous case, we can define a diagonal tetrad as in (4.37), which
gives the diagonal Einstein’s tensor

$\begin{split}\mathbf{G}_{00}&=-3\,\frac{x_{*}^{2}-x_{\scriptscriptstyle
BH}^{2}}{x_{\scriptscriptstyle 0}^{4}}+4\,\frac{x_{*}}{x_{\scriptscriptstyle
0}^{2}\,X}\\\ \mathbf{G}_{11}&=3\,\frac{x_{*}^{2}-x_{\scriptscriptstyle
BH}^{2}}{x_{\scriptscriptstyle 0}^{4}}-4\,\frac{x_{*}}{x_{\scriptscriptstyle
0}^{2}\,X}\\\ \mathbf{G}_{22}&=3\,\frac{x_{*}^{2}-x_{\scriptscriptstyle
BH}^{2}}{x_{\scriptscriptstyle 0}^{4}}-2\,\frac{x_{*}}{x_{\scriptscriptstyle
0}^{2}\,X}\\\ \mathbf{G}_{33}&=3\,\frac{x_{*}^{2}-x_{\scriptscriptstyle
BH}^{2}}{x_{\scriptscriptstyle 0}^{4}}-2\,\frac{x_{*}}{x_{\scriptscriptstyle
0}^{2}\,X}\,.\end{split}$ (4.54)

The metric can therefore be interpreted as a solution to the Einstein’s
equation with a cosmological constant given by (4.52) and a stress-energy
tensor given by

$\mathbf{T}_{IJ}=\frac{2\,x_{*}}{x_{\scriptscriptstyle
0}^{2}\,X}\,\text{diag}(2,-2,-1,-1)\,.$ (4.55)

The global as well as the local nature of these spacetimes depends on the
explicit value of $x_{*}$.

##### 4.4.1 De Sitter realization, $x_{*}=0$

For $x_{*}=0$, $\mathbf{T}_{ab}=0$, the Ricci scalar is non-diverging, and the
Einstein tensor (4.54) corresponds to that of a positive cosmological constant
term $\Lambda\,g_{ab}$ with

$\Lambda=3\frac{x_{\scriptscriptstyle BH}^{2}}{x_{\scriptscriptstyle
0}^{4}}\,.$ (4.56)

For this choice of $x_{*}$, the metric (4.49) is manifestly that of de Sitter
spacetime in terms of static coordinates. The bifurcating Killing horizon
corresponds to the union of a past and future cosmological horizons
intersecting at the bifurcating sphere, as shown in the Carter-Penrose diagram
in Figure 4.3.

Figure 4.3: The causal structure of the de Sitter spacetime. The two spherical
dimensions are suppressed, so that each point represent a sphere. The Killing
field is spacelike in the shaded regions and timelike elsewhere. The boundary
is spacelike and no singularities are present. Grey lines and numbers show how
the 6 regions of Minkowski spacetime depicted in Fig. 4.1 are conformally
mapped into the bulk of the target spacetime.

##### 4.4.2 Asymptotically (Anti)-de Sitter realizations

If $x_{*}<0$ one has a positive cosmological constant for
$|x_{*}|>|x_{\scriptscriptstyle BH}$ and a negative one for
$|x_{*}|<|x_{\scriptscriptstyle BH}$ with a $\mathbf{T}_{ab}$ violating all
the standard energy conditions. In these cases the metric is asymptotically
$dS$ and $AdS$ respectively. In the $AdS$ case the decay rate to the
asymptotic geometry is slower with respect to the one imposed by the standard
reflecting boundary conditions [Ashtekar and Das, 2000] 333We thank A.
Ashtekar for clarifications about this point.. This implies that a well
defined notions of conserved charges at infinity is not possible. The
spacetime is therefore weakly asymptotically $AdS$.

If $x_{*}>0$, instead, one has a positive asymptotic cosmological constant for
$x_{*}<x_{\scriptscriptstyle BH}$ and a negative one for
$x_{*}>x_{\scriptscriptstyle BH}$. Now $\mathbf{T}_{ab}$ satisfies the weak
($\rho\geq 0,\rho+p_{i}\geq 0$) and the dominant ($\rho\geq|p_{i}|$) energy
condition but not the strong one ($\rho+p_{i}\geq 0,\rho+\sum_{i}p_{i}\geq
0$). For negative asymptotic cosmological constant, there are black hole
regions, plus inner and outer Killing horizons. The boundary is again weakly
$AdS$. The new feature with respect to the Betrotti-Robinson realization is
the appearance of a time like curvature singularity at $X=0$. There is no
black hole region in the $dS$ realization and the time like curvature
singularities at $X=0$ remain. The Carter-Penrose diagrams corresponding to
these cases are shown in Figure 4.4.

(a) Weakly asymptotically AdS.

(b) Asymptotically dS.

Figure 4.4: The causal structure of the different spacetime realisations when
an observer is sent to infinity. The two spherical dimensions are suppressed,
so that each point represent a sphere. The Killing field is spacelike in the
shaded regions and timelike elsewhere. Grey lines and numbers show how the 6
regions of Minkowski spacetime depicted in Fig. 4.1 are conformally mapped
into the bulk of the target spacetime.

#### 4.5 An Asymptotically Flat Realisation

The inner horizons of the Bertotti-Robinson realization of Section 4.3 can
become the boundary of the spacetime via a particular choice of
$P(x,\theta,\varphi)$ in (4.25). Such realization was already exhibited in [De
Lorenzo and Perez, 2018b]–see Appendix A.6–to illustrate some aspects of the
light cone thermodynamical laws. In this case the metric is

$g_{ab}=\frac{16\,r_{\scriptscriptstyle H}^{4}}{(u-r_{\scriptscriptstyle
H})^{2}(v+r_{\scriptscriptstyle H})^{2}}\,\eta_{ab}\,.$ (4.57)

In Appendix A.7, a coordinate transformation is found such that the metric
looks

$ds^{2}=\frac{1}{\Delta}\left(-(1-2\,z)\,d\bar{\tau}^{2}+\frac{1}{1-2\,z}\,dz^{2}+z^{2}\,dS^{2}\right)\,,$
(4.58)

where the new coordinates are dimensionless and
$\Delta=\frac{4r_{\scriptscriptstyle H}^{2}}{x_{\scriptscriptstyle 0}^{4}}$.
In these coordinates the horizon is located at $z=1/2$. Moreover, $z$ is
positive and greater then $1/2$ outside, and decreases to zero at the
Minkowskian $i^{0}$ and origin. Inside the horizon, on the other hand,
increases from $z=1/2$ to $z\to\infty$, the latter corresponding to the inner
horizon.

Using the technique of Section 4.2, we can draw its Carter-Penrose diagram and
study the properties of its boundary. The conformal factor $\Omega^{2}$
mapping (4.58) to the Einstein’s Universe
$g_{ab}=\Omega^{2}\,g_{ab}^{\scriptscriptstyle EU}$ is given by

$\begin{split}\Omega^{2}&=\frac{(u-r_{\scriptscriptstyle
H})^{2}(v+r_{\scriptscriptstyle H})^{2}}{4r_{\scriptscriptstyle
H}^{2}\,(u^{2}+r_{\scriptscriptstyle H}^{2})(v^{2}+r_{\scriptscriptstyle
H}^{2})}\\\ &=\frac{(1-\sin U)(1+\sin V)}{4r_{\scriptscriptstyle H}^{2}}\\\
&=\left(\frac{\cos T+\sin R}{2\,r_{\scriptscriptstyle
H}}\right)^{2}\end{split}$ (4.59)

where $U$ and $V$ are the Einstein’s Universe null coordinates. The boundary
$\mathscr{I}$ is given by the condition $\Omega=0$, and therefore

$\mathscr{I}:\,\qquad T-R=U=\frac{\pi}{2}\qquad{\rm and}\qquad
T+R=V=-\frac{\pi}{2}\,.$ (4.60)

which is equivalent to $u=r_{\scriptscriptstyle H}$ and
$v=-r_{\scriptscriptstyle H}$, or simply $z\to\infty$. The boundary is made of
two constant $U$ or $V$ surfaces, being therefore null.

The gradient of the conformal factor is found to be

$\big{(}\tilde{\nabla}_{\mu}\omega\big{)}\,dx^{\mu}=-\frac{\sin
T}{2r_{\scriptscriptstyle H}}\,dT+\frac{\cos R}{2r_{\scriptscriptstyle
H}}\,dR$ (4.61)

which is non-zero at $\mathscr{I}$. The Ricci tensor in the diagonal tetrad is

$\mathbf{R}_{IJ}=\mathbf{R}_{ab}\,\mathbf{e}^{a}_{I}\mathbf{e}^{b}_{J}=\frac{2\Delta}{z}{\rm
diag}(-1,1,2,2)$ (4.62)

which vanishes in a neighbourhood of $\mathscr{I}$, i.e. for $z\to\infty$. Our
spacetime fulfils all the conditions of the definition of conformally flatness
[Frauendiener, 2004]. Finally, the Ricci scalar is

$\mathbf{R}=\frac{12\,\Delta}{z}$ (4.63)

showing a curvature singularity at $z=0$. The resulting causal structure is
shown in Fig. 4.5.

Figure 4.5: The causal structure of the conformally flat realization. The two
spherical dimensions are suppressed, so that each point represent a sphere.
The grey lines and numbers show how the 6 regions of Minkowski spacetime
depicted in Fig. 4.1 are conformally mapped into the bulk of the target
spacetime.

For this last case we made explicit the analysis for the construction of the
Carter-Penrose diagram. The very same strategy is used for the previous cases
as well, but, for brevity, we preferred not to present explicitly.

#### 4.6 Conclusions

In Chapter 4, a mathematical analogy of the laws of black hole thermodynamics
has been proven for light cones in Minkowski spacetime [De Lorenzo and Perez,
2018b]. This has been possible by observing that intersecting light cones are
bifurcating conformal Killing horizons for the most general radial conformal
Killing field $\xi$. The causal behavior of $\xi$ closely resemble the causal
behavior of the Killing field of a Reissner-Nordstrom black hole–see Fig. 3.1.
Such conformal stationarity allowed us to prove the laws of light cone
thermodynamics, which however describe the properties of notions which have no
direct physical meaning in flat space. At the same time, they all are
conformally invariant, acquiring therefore the standard geometric meaning in
conformally flat spacetime where the conformal Killing field becomes a genuine
Killing field. In this Chapter we have studied the properties of the spacetime
satisfying this condition. The most interesting case is the simplest one. It
turns out that the conformal Killing horizon structure of light cones in
Minkowski spacetime is the conformal partner of the Killing horizon structure
of the Bertotti-Robinson spacetime. The latter is known to encode the near
horizon geometry of close-to-extremal and extremal charged black holes. This
result completely clarifies the nature of light cone thermodynamics, that can
now be seen as arising from a conformal transformation of the standard laws of
BH thermodynamics. This in turn strengthen our initial claim that light cones
in Minkowski spacetime encode, in a suitable sense, the main properties of BH
horizons, thus providing an intriguing analogue of BHs in a spacetime with
trivial curvature. The analogy is more strict and direct than the one usually
considered between near horizon BH geometry and Rindler spacetime. As
explained in Section 3.1, indeed, the second analogy is strictly valid only
for the $(t,r)$ plane of a BH spacetime, or in the infinite area limit. The BH
and Rindler horizons, indeed, have different topologies, being respectively
$S^{2}\times\mathbb{R}$ and $\mathbb{R}^{2}\times\mathbb{R}$. Additionally,
the Rindler wedge cannot be seen as the region outside the horizon, since it
lies itself in the domain of dependence of the latter. This in turn implies
that no finite energy flux can escape the Rindler horizon, and no notion of
asymptotic observer can be defined. These difficulties are not present in the
light cone case presented in these last two Chapters. The light cone topology
is $\mathbb{R}\times S^{2}$ as for the BH case, and energy can be sent to
infinity without crossing the horizon from the complement of the diamond,
Region II in Fig. 4.1, which therefore plays the role of the outside region.
The analogy is indeed so strict that for conformally invariant matter the
light cone structure is indistinguishable from the near horizon geometry of a
close-to-extremal Reissner-Nordstrom black hole. Other interesting conformally
flat spacetime where the conformal Killing horizon structure becomes a proper
Killing one have been presented.

Possible applications of these intriguing analogy are currently under
investigation.

###  Chapter 5 Spacetime Thermodynamics with Contorsion

_This Chapter overlaps with a paper in preparation_ [De Lorenzo et al., 2018].

In a famous paper [Jacobson, 1995], Ted Jacobson proposed that Einstein
equations could have a thermodynamical origin, compatible with the
thermodynamical interpretation of the laws of black hole mechanics [Bardeen et
al., 1973]. His argument, based on a geometric interpretation of Clausius
relation, has been later extended to include non-equilibrium terms and higher
derivative gravity theories [Eling et al., 2006, Chirco and Liberati, 2010,
Guedens et al., 2012], and more recently to spacetimes with non-propagating
torsion, namely Einstein-Cartan first-order gravity [Dey et al., 2017]. This
last paper motivates the study presented here. The main difficulty of
extending Jacobson’s idea to Einstein-Cartan gravity is that there are two
sets of independent field equations to be derived: the torsion equations as
well as the Einstein equations. The authors of [Dey et al., 2017] show that it
is possible to derive the latter ser for a special type of torsion, and by
identifying the torsional terms as a non-equilibrium contribution to Clausius
relation. The torsion equations are not derived, and whether they can have
also a thermodynamical origin is left as an open question. In our paper we
also do not provide a derivation of the torsional equations, but we show that
if they hold, the tetrad Einstein equations can be derived without the need of
non-equilibrium terms nor restrictions on torsion. The technical result that
allows us to achieve this is the identification of the conserved energy-
momentum tensor.

The last point is crucial: in Einstein-Cartan theory, there is no conserved
energy-momentum tensor that appears as source of the field equations.
Nonetheless, if one restricts to invertible tetrads (and this appears
necessary to connect with the metric theory and the familiar notions used in
Jacobson’s argument), the connection can always be written as a Levi-Civita
one plus a contorsion tensor. Using this well-known decomposition, the tetrad
Einstein equations can be written as the Levi-Civita Einstein tensor on the
left hand side, and a torsion dependent effective energy-momentum tensor
$T^{\scriptscriptstyle eff}$ on the right hand side. By taking the Levi-Civita
covariant derivate of both sides, the left one vanishes due to Bianchi’s
identities. This in turn implies the vanishing of the right hand side,
allowing to identify the conserved energy-momentum tensor also in the presence
of torsion. For the thermodynamical argument, on the other hand, one needs to
identify a conserved energy-momentum tensor without using the field equations,
since these are to be derived. The first result of our paper is to show that
the conservation of $T^{\scriptscriptstyle eff}$ in the Einstein-Cartan theory
can be derived _without_ using the tetrad field equations. The proof is simple
although rather lengthy, and best done using differential forms. It follows
from the Noether identities of the theory, and requires the matter and torsion
field equations to be satisfied.

Our second result is to use this conserved energy-momentum tensor and the
contorsion description to show that the tetrad Einstein equations can be
derived from the Clausius relation with the same assumptions and hypothesis of
the metric case [Jacobson, 1995], without the need of the non-equilibrium
terms and the restrictions on torsion used in [Dey et al., 2017]. This is
possible because the starting point of Jacobson’s argument, a Killing horizon
associated with a locally boosted observer, is a notion which is insensitive
to the presence of torsion. In particular, the generators of the Killing
horizons follow the Levi-Civita geodesic equation. This turns out to suffice
to recover the tetrad Einstein equations from the equilibrium Clausius
relation, since the torsion terms are identified by the effective energy-
momentum tensor. A further advantage of our derivation is that it includes
also the Immirzi term in the Einstein-Cartan theory. Our results build on the
discussion of [Dey et al., 2017]. Our title is motivated by this paper, and
meant to stress the role that the contorsion decomposition plays in the
derivation.

To complete our discussion, we also look at the laws of black hole mechanics
in the presence of torsion. The zeroth law is unaffected, and it can be proven
exactly as in the metric case, provided that the energy conditions are imposed
on eff. The first law on the other hand depends on torsion. We consider here
the ‘physical process’ version of the first law [Wald, 1995], which is closely
related to Jacobson’s argument run backwards. Using the same contorsion
decomposition as before, the formal expression of the first law is unchanged,
but the quantities appearing depend on torsion through the effective energy-
momentum tensor. The second law has a more marginal dependence, in the sense
that torsion simply enters the inequalities on the energy conditions required.

Finally we give in one Appendix a brief comparison of two slightly different
versions of Jacobson’s argument [Jacobson, 1995, Guedens et al., 2012], and
present an alternative derivation technically closer to the first law.

We use metric signature with mostly plus, and natural units $G=c=\hbar=1$.

#### 5.1 Einstein-Cartan Field Equations and Matter Sources

Let us begin by briefly reviewing the field equations of Einstein-Cartan
theory and the contorsion decomposition. We refer the reader to [Hehl et al.,
1995] for more details, and to the Appendix A.8 for definitions and notation.
We consider the following first-order action,

$S_{\scriptscriptstyle EC}(e,\omega)=\frac{1}{16\pi\,}\int
P_{IJKL}\Big{(}e^{I}\wedge e^{J}\wedge
F^{KL}(\omega)-\frac{\Lambda}{6}e^{I}\wedge e^{J}\wedge e^{K}\wedge
e^{L}\Big{)},$ (5.1)

where

$P_{IJKL}:=\frac{1}{2\gamma}(\eta_{IK}\eta_{JL}-\eta_{IL}\eta_{JK})+\frac{1}{2}\epsilon_{IJKL},$
(5.2)

and $\gamma$ is the Immirzi parameter. We restrict attention to invertible,
right-handed tetrads. The action is then equivalent to first-order general
relativity 111Sometimes called Einstein-Palatini general relativity because
proving its equivalence to general relativity uses the Palatini identity.

$S_{\scriptscriptstyle
EP}(g,\Gamma)=\frac{1}{16\pi\,}\int[\sqrt{-g}(g^{\mu\rho}g^{\nu\sigma}R_{\mu\nu\rho\sigma}(\Gamma)-2\Lambda)+\frac{1}{\gamma}\tilde{\epsilon}^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}]d^{4}x,$
(5.3)

with initially independent metric and connections, which are related to the
fields of 5.1 by the familiar formulas

$g_{\mu\nu}=e_{\mu}^{I}e_{\nu}^{J}\eta_{IJ},\qquad\Gamma^{\rho}_{\mu\nu}=e_{I}^{\rho}D_{\mu}e^{I}_{\nu}:=e_{I}^{\rho}(\partial_{\mu}e^{I}_{\nu}+\omega^{IJ}_{\mu}e_{J\nu}).$
(5.4)

We collectively denote the matter fields as $\psi$, and consider a general
matter Lagrangian $L_{m}(e,\omega,\psi):={\cal L}_{m}(e,\omega,\psi)d^{4}x$.
Varying the matter action we have

$\displaystyle\delta S_{m}=\int\delta L_{m}=\int\left(2\tau^{\mu}{}_{I}\delta
e^{I}_{\mu}+\sigma^{\mu}{}_{IJ}\delta\omega^{IJ}_{\mu}+E_{m}\delta\psi\right)e\,d^{4}x,$
(5.5)

where $E_{m}$ denotes the matter field equations, and we defined the source
terms

$\displaystyle\tau^{\mu}{}_{I}:=\frac{1}{2e}\frac{\delta{\mathcal{L}}_{m}}{\delta
e^{I}_{\mu}}=-\frac{1}{2e}\frac{\delta{\mathcal{L}}_{m}}{\delta
e^{\nu}_{J}}e^{\nu}_{I}e^{\mu}_{J}=:\tau^{J}{}_{\nu}e^{\nu}_{I}e^{\mu}_{J},\qquad\sigma^{\mu}{}_{IJ}=\frac{1}{e}\frac{\delta{\mathcal{L}}_{m}}{\delta\omega^{IJ}_{\mu}}.$
(5.6)

The sign choice in the definition of $\tau$ is not universal in the
literature. We picked it this way in analogy with the metric energy-momentum
tensor $T_{\mu\nu}^{\Gamma}$,

$T^{\Gamma}_{\mu\nu}:=-\frac{2}{\sqrt{-g}}\frac{\delta{\mathcal{L}}_{m}(g,\Gamma)}{\delta
g^{\mu\nu}}=-\frac{1}{e}\frac{\delta{\mathcal{L}}_{m}(e,\omega)}{\delta
e^{I(\mu}}e^{I}_{\nu)}=2\tau^{I}{}_{(\mu}e_{\nu)I},$ (5.7)

which coincides with the one of general relativity in the absence of torsion.

The field equations obtaining varying 5.1 and the matter action are

$\displaystyle G^{\mu}{}_{I}(e,\omega)+\Lambda
e^{\mu}_{I}+\frac{1}{2\gamma}\epsilon^{\mu\nu\rho\sigma}e^{\alpha}_{I}R_{\alpha\nu\rho\sigma}(e,\omega)=16\pi\,\tau^{\mu}{}_{I},$
(5.8a) $\displaystyle
P_{IJKL}\epsilon^{\mu\nu\rho\sigma}e^{K}_{\nu}T^{L}_{\rho\sigma}=-16\pi\,\sigma^{\mu}_{IJ}.$
(5.8b)

Here

$G^{\mu}{}_{I}(e,\omega):=\frac{1}{4}\epsilon_{IJKL}\epsilon^{\mu\nu\rho\sigma}e_{\nu}^{J}F_{\rho\sigma}^{KL}(\omega)=G^{\mu\nu}(e,\omega)e_{\nu
I}$ (5.9)

is the first-order Einstein tensor, the Riemann tensor and curvature are
related by $R_{\mu\nu\rho\sigma}(e,\omega)=e_{\mu I}e_{\nu
J}F^{IJ}_{\rho\sigma}(\omega)$, and $T^{I}:=d_{\omega}e^{I}$ is the torsion.
The first set 5.8a contains the ten Einstein equations, plus six redundant
equations. Although $G_{\mu\nu}(e,\omega)$ is not symmetric a priori, it is
easy to show that the Noether identity associated with invariance of the
action under internal Lorentz transformations (see 5.31a below) implies that
the equations for $G^{\mu}{}_{[I}e_{J]\mu}$ are automatically satisfied. The
relevant content of 5.8a is therefore just its symmetric part, which in turn
gives the tetrad Einstein equations

$G_{\mu\nu}(e,\omega)+\Lambda
g_{\mu\nu}+\frac{1}{2\gamma}\epsilon_{(\mu}{}^{\lambda\rho\sigma}R_{\nu)\lambda\rho\sigma}(e,\omega)=8\pi\,T^{\Gamma}_{\mu\nu},$
(5.10)

or equivalently as functions of $(g,\Gamma)$ via 5.4.

It is often convenient to write the field equations using the language of
differential forms, as we did in the action 5.1. To that end, we use the Hodge
dual $\star$ mapping $p$-forms to $(4-p)$-forms (see Appendix A.8 for
conventions). This allows us to define the Einstein 3-from

$\star\\!G_{I}(\omega):=-\frac{1}{2}\epsilon_{IJKL}e^{J}\wedge
F^{KL}(\omega),$ (5.11)

where the opposite sign with respect to 5.9 is a consequence of Lorentzian
signature, and equivalently the dual source forms $\star\tau_{I}$ and
$\star\sigma_{IJ}$. The field equations 5.8 then read

$\displaystyle\star\\!G_{I}(\omega)+\Lambda\star\\!e_{I}-\frac{1}{\gamma}e^{J}\wedge
F_{IJ}(\omega)=16\pi\,\star\\!\tau_{I},$ (5.12a) $\displaystyle
P_{IJKL}\,e^{K}\wedge T^{L}=8\pi\,\star\\!\sigma_{IJ}.$ (5.12b)

##### 5.1.1 The contorsion tensor

Although connections form an affine space with no preferred origin, the
presence of an invertible tetrad suggests a natural origin: the Levi-Civita
connection $\omega_{\mu}^{IJ}(e)$ associated with the tetrad. We can then
always decompose an arbitrary connection into Levi-Civita plus a contorsion
tensor $C_{\mu}^{IJ}$ as

$\omega_{\mu}^{IJ}=\omega_{\mu}^{IJ}(e)+C_{\mu}^{IJ}.$ (5.13)

Torsion and curvature are related to the contorsion as follows:

$\displaystyle T^{I}=C^{IJ}\wedge e_{J},$ (5.14) $\displaystyle
F^{JK}(\omega)=F^{JK}(e)+d_{\omega(e)}C^{JK}+C^{JM}\wedge
C_{M}{}^{K}=F^{JK}(e)+d_{\omega}C^{JK}-C^{JM}\wedge C_{M}{}^{K},$ (5.15)

where $d_{{\omega(e)}}$ is the exterior derivative with respect to the Levi-
Civita connection. Plugging this decomposition into the field equations we
find

$\displaystyle\star\\!G_{I}(e)+\Lambda\star\\!e_{I}=16\pi\,\star\\!\tau_{I}+P_{IJKL}(d_{\omega(e)}C^{JK}+C^{JM}\wedge
C_{M}{}^{K}),$ (5.16) $\displaystyle P_{IJKL}e^{K}\wedge C^{LM}\wedge
e_{M}=8\pi\,\star\\!\sigma_{IJ}.$ (5.17)

The fact that the field equations for the Einstein-Cartan theory can be
recasted as in 5.16 is the source of an old debate in the literature about the
role of torsion [Hehl and Weinberg, 2007]: if we forget about the notion of
affine parallel transport defined by $\omega^{IJ}$, and use simply the one
defined by $\omega^{IJ}(e)$ in the sector of invertible tetrads, then the
theory is indistinguishable from ordinary metric theory with some non-minimal
matter coupling. The non-minimality is captured by the effective energy-
momentum tensor sourcing 5.16, i.e.

$\star\\!\tau^{\scriptscriptstyle
eff}_{I}:=\star\tau_{I}+\frac{1}{16\pi\,}P_{IJKL}(d_{\omega(e)}C^{JK}+C^{JM}\wedge
C_{M}{}^{K}).$ (5.18)

While we take no stand in the debate, we will heavily use this fact in the
thermodynamic discussion below. Before getting there, we need to review in the
next Section the relation between the conservation of the energy-momentum
tensor and the Bianchi identities.

For convenience of the reader, we report the relation between torsion and
contorsion in tensor language,

$\displaystyle
T^{\rho}{}_{\mu\nu}:=e^{\rho}_{I}\,T^{I}{}_{\mu\nu}=-2C_{[\mu,\nu]}{}^{\rho}=2\Gamma^{\rho}_{[\mu\nu]},$
(5.19) $\displaystyle
C_{\mu,\nu\rho}=\frac{1}{2}T_{\mu,\nu\rho}-T_{[\nu,\rho]\mu},$ (5.20)
$\displaystyle C_{(\mu,\nu)\rho}=T_{(\mu,\nu)\rho}.$ (5.21)

Here and in the following we use a comma between indices to bundle up those
with special symmetry properties. Derivative operators will always be
explicitly written. The Einstein equations 5.8a read

$\displaystyle G_{\mu\nu}(e)+\Lambda g_{\mu\nu}=8\pi\,T^{\scriptscriptstyle
eff}_{\mu\nu},$ (5.22) $\displaystyle T^{\scriptscriptstyle
eff}_{\mu\nu}=2\tau^{I}{}_{(\mu}e_{\nu)I}+\frac{1}{16\pi\,}\Big{(}6g_{\alpha(\mu}\delta^{\alpha\rho\sigma}_{\nu)\gamma\delta}-\frac{2}{\gamma}g_{\gamma(\mu}\epsilon_{\nu)\delta}{}^{\rho\sigma}\Big{)}\Big{(}\overset{\scriptscriptstyle
e}{\nabla}{}_{\rho}C_{\sigma,}{}^{\gamma\delta}+C_{\rho,}{}^{\gamma\lambda}C_{\sigma,\lambda}{}^{\delta}\Big{)}.$
(5.23)

We refrained from expanding the completely antisymmetric
$\delta^{\alpha\rho\sigma}_{\nu\gamma\delta}$ since no useful simplification
occurs.

#### 5.2 Noether Identities and Conservation Laws

The gravity action 5.1 is invariant under internal Lorentz transformations

$\delta_{\lambda}e^{I}=\lambda^{I}{}_{J}e^{J},\qquad\delta_{\lambda}\omega^{IJ}=-d_{\omega}\lambda^{IJ},$
(5.24)

as well as diffeomorphisms,222Note that the Lie derivatives 5.25 are not
gauge-covariant objects. It is often convenient to consider the linear
combination of transformations
$L_{\xi}=\pounds_{\xi}+\delta_{\omega\lrcorner\xi}$ which is covariant.

$\displaystyle\delta_{\xi}e^{I}=\pounds_{\xi}e^{I}=de^{I}\lrcorner\xi+d(e^{I}\lrcorner\xi)=d_{\omega}e^{I}\lrcorner\xi+d_{\omega}(e^{I}\lrcorner\xi)-(\omega^{I}{}_{J}\lrcorner\xi)e^{J},$
(5.25a)
$\displaystyle\delta_{\xi}\omega^{IJ}=\pounds_{\xi}\omega^{IJ}=d\omega^{IJ}\lrcorner\xi+d(\omega^{IJ}\lrcorner\xi)=F^{IJ}\lrcorner\xi+d_{\omega}(\omega^{IJ}\lrcorner\xi).$
(5.25b)

Specializing the variation of the action 5.1 to 5.24 and 5.25 respectively,
and integrating by parts, one obtains the following Noether identities,333To
obtain 5.27a, we used the identity A.144 below. For the reader’s convenience,
we report the identities also in the more common $\gamma$-less case,
$\displaystyle\star\\!G_{[I}\wedge
e_{J]}=-\frac{1}{2}\epsilon_{IJKL}e^{K}\wedge d_{\omega}T^{L},\qquad$
$\displaystyle d_{\omega}\star\\!G_{I}=-\frac{1}{2}\epsilon_{IJKL}T^{J}\wedge
F^{KL}.$ (5.26)

$\displaystyle P_{IJKL}e^{K}\wedge F^{LM}\wedge e_{M}=P_{IJKL}e^{K}\wedge
d_{\omega}T^{L},$ (5.27a) $\displaystyle d_{\omega}(P_{IJKL}e^{J}\wedge
F^{KL})=P_{IJKL}T^{J}\wedge F^{KL}.$ (5.27b)

These are nothing but contracted forms of the Bianchi identities
$d_{\omega}F^{IJ}=0$, $d_{\omega}T^{I}=F^{IJ}\wedge e_{J}$. Using the field
equations 5.12 in 5.27 one finds additional relations for the matter sources,

$\displaystyle d_{\omega}\star\\!\sigma_{IJ}=2\star\\!\tau_{[I}\wedge e_{J]},$
(5.28a) $\displaystyle d_{\omega}\star\\!\tau_{I}=\frac{1}{2}F^{JK}\lrcorner
e_{I}\wedge\star\sigma_{JK}+T^{J}\lrcorner e_{I}\wedge\star\tau_{J}.$ (5.28b)

These matter Noether identities can also be derived without reference to the
field equations 5.12: they follow from invariance of the matter action 5.5
under 5.24 and 5.25, on-shell of the matter field equations. See [Hehl and
McCrea, 1986, Hehl et al., 1995, Barnich et al., 2016] for more details.

Recall now that, in the metric formalism, invariance of the matter Lagrangian
under diffeomorphisms guarantees the conservation of the energy-momentum
tensor,

$\delta_{\xi}L_{m}=d(L_{m}\lrcorner\xi)\quad\Rightarrow\quad\nabla_{\mu}T^{\mu\nu}=0,$
(5.29)

on-shell of the matter field equations. In the first-order formalism with
tetrads, the energy-momentum tensor does not appear immediately in the field
equations: the closest object we have is the source $\tau$ of the tetrad
Einstein equations 5.8a. This quantity is however not conserved, as we can see
from 5.28b, whose right-hand side does not vanish on-shell. Nevertheless,
although $\tau$ is not conserved, it is easy to identity an effective energy-
momentum tensor which is conserved, thanks to the contorsion decomposition
5.16. If we take the Levi-Civita exterior derivative $d_{{\omega(e)}}$ on both
sides of 5.16, the left-hand side vanishes identically. This in turns implies
the vanishing of the right-hand side, which gives a local conservation law

$d_{{\omega}(e)}\tau^{\scriptscriptstyle eff}_{I}=0$ (5.30)

valid _also in the presence of torsion_. Equivalently in terms of tensors, the
object with vanishing (Levi-Civita) divergence is $T^{\scriptscriptstyle
eff}_{\mu\nu}$ as defined in 5.23, and can be bona-fide considered as the
genuine energy-momentum tensor of the theory.444Another way to identify this
conserved object is to solve the torsion equation – which in the case of
Einstein-Cartan is simply algebraic since torsion does not propagate, and plug
the solution back into the action. Varying the resulting matter action with
respect to the tetrad then immediately gives the effective energy-momentum
tensor 5.23.  This equation provides the basis of energy conservation in
Einstein-Cartan theory.

This fact is well-known in the literature [Hehl, 1976], and the discussion
above on the effective energy-momentum tensor should also be closely related
to what Hehl calls the Freund superpotential in [Böhmer and Hehl, 2018].

For later purposes, we are interested in whether it is possible to derive the
conservation law 5.30 _without_ using the tetrad Einstein equations. This is a
bit of a strange question if one starts from an action principle, but it is
crucial to Jacobson’s thermodynamical argument, where this is not the case. We
could not find the answer to this question in the literature, which turns out
to be affirmative. The result is the following:

Proposition 1: _The matter Noether identities 5.28 on-shell of the matter and
torsion field equations imply the conservation law for the effective energy-
momentum tensor 5.30. _

The proof is a somewhat lengthy exercise in algebraic identities, and we leave
it to Appendix 2. We also looked for a stronger result, namely whether 5.30
also holds without imposing the torsion equation, but we did not succeed. The
proof in the Appendix shows explicitly the step in which we use the torsion
field equation. It should be emphasized, however, that the obstacle we see in
proving such a stronger result is technical but not conceptual.

In tensorial language, the Noether identities for a generic gauge and diff-
invariant Lagrangian density $\cal L$ read (see e.g. [Barnich et al., 2016])

$\displaystyle
D_{\mu}\frac{\delta{\mathcal{L}}}{\delta\omega^{IJ}_{\mu}}+\frac{\delta{\mathcal{L}}}{\delta
e^{[I}_{\mu}}e_{J]}^{\mu}=0,$ (5.31a)
$\displaystyle\frac{\delta{\mathcal{L}}}{\delta\omega^{IJ}_{\mu}}F^{IJ}_{\nu\mu}(\omega)+\frac{\delta{\mathcal{L}}}{\delta
e^{I}_{\mu}}T_{\nu\mu}^{I}-e_{\nu}^{I}D_{\mu}\frac{\delta{\mathcal{L}}}{\delta
e^{I}_{\mu}}=0,$ (5.31b)

on-shell of the matter field equations. For the Lagrangian density in 5.1,
these give respectively contractions of the algebraic and differential Bianchi
identities,

$\displaystyle
2R_{[\mu\nu]}=-\nabla_{\rho}T^{\rho}{}_{\mu\nu}-2\nabla_{[\mu}T^{\rho}{}_{\nu]\rho}+T^{\rho}{}_{\rho\sigma}T^{\sigma}{}_{\mu\nu},$
(5.32)
$\displaystyle\nabla_{\nu}G^{\nu}{}_{\mu}=T^{\rho}{}_{\mu\sigma}R^{\sigma}{}_{\rho}-\frac{1}{2}T^{\nu}{}_{\rho\sigma}R^{\rho\sigma}{}_{\mu\nu},$
(5.33)

from the $\gamma$-less terms, and

$\displaystyle\epsilon^{\alpha\nu\rho\sigma}R_{\mu\nu\rho\sigma}=\epsilon^{\alpha\nu\rho\sigma}(\nabla_{\nu}T_{\mu,\rho\sigma}+T_{\mu,\lambda\nu}T^{\lambda}{}_{\rho\sigma}),$
(5.34)
$\displaystyle\epsilon^{\alpha\beta\rho\sigma}\nabla_{\beta}R_{\mu\nu\rho\sigma}=\epsilon^{\alpha\beta\rho\sigma}T^{\lambda}{}_{\beta\rho}R_{\mu\nu\sigma\lambda}$
(5.35)

for the part in $1/\gamma$. As for the matter action,

$\displaystyle D_{\mu}(e\sigma^{\mu}_{IJ})=-2e\tau^{\mu}{}_{[I}e_{J]\mu},$
(5.36) $\displaystyle
D_{\mu}(e\tau^{\mu}{}_{I})=ee^{\mu}_{I}\left(\frac{1}{2}F_{\mu\nu}^{JK}\sigma^{\nu}_{JK}+T^{J}_{\mu\nu}\tau^{\nu}{}_{J}\right).$
(5.37)

#### 5.3 Einstein Equations from Thermodynamics

We now come to the main motivation for our paper: show that Proposition 1
allows us to run Jacobson’s argument with the usual equilibrium assumptions.
To better appreciate our point, let us briefly recall the key steps of the
metric case, referring the reader to [Jacobson, 1995] for more details.

##### 5.3.1 The metric case

Consider an arbitrary metric $g_{\mu\nu}$ on a manifold, a point $P$ and a
neighbourhood sufficiently small for spacetime to be approximately flat.
Denote by $\xi^{\mu}$ the future-pointing (approximate) Killing vector
generating a Rindler horizon $\cal H$ within the approximately flat region,
with bifurcating surface $\cal B$ through the point $P$. This is by
construction hypersurface orthogonal, null at the horizon but not outside, and
vanishing at $\cal B$:

$\xi^{2}\stackrel{{\scriptstyle\cal
H}}{{=}}0,\qquad\partial_{\mu}\xi^{2}=:-2\,\kappa\,\xi_{\mu},\qquad\xi^{\mu}\stackrel{{\scriptstyle\cal
B}}{{=}}0.$ (5.38)

Since it is Killing, it is also geodesic,

$\xi^{\nu}\overset{\scriptscriptstyle
e}{\nabla}{}_{\nu}\xi^{\mu}=-\frac{1}{2}\partial_{\mu}\xi^{2}=\kappa\,\xi^{\mu}.$
(5.39)

The inaffinity $\kappa$ can be proven to be constant on the horizon, and
referred to as the horizon surface gravity. For a Rindler horizon, constancy
of $\kappa$ follows immediately from the vanishing of the Riemann
tensor.555For a stationary black hole horizon, this is the content of the
zeroth law of black hole mechanics, which requires to impose the dominant
energy condition to be shown. See [Bardeen et al., 1973] for the proof. It is
useful to introduce an affine parameter $\lambda$ along the null geodesics,
with origin at the point $P$. It can be easily shown that

$\xi^{\mu}=-\lambda\,\kappa\,l^{\mu},\qquad
l^{\mu}\partial_{\mu}=\partial_{\lambda}.$ (5.40)

Figure 5.1: _The set-up thermodynamical derivation of Einstein’s equation as
proposed in[Jacobson, 1995]. Local flatness allows to consider approximate
Rindler observers $\xi^{\mu}$ around any point $P$ of a given spacetime. The
associate Rindler horizon has bifurcate surface $\mathcal{B}$ passing through
$P$. The system is perturbed by a small flux of matter crossing the past
horizon and entering the left wedge. For the derivation to be valid, an
infinite family of $\xi^{\mu}$ is actually considered, one per each
direction._

Given this geometric set-up, the first step of Jacobson’s argument is to
associate to the Rindler horizon its Unruh temperature:

$(i)\qquad T=\frac{\kappa}{2\pi},\qquad\kappa={\rm constant}.$ (5.41)

Next, three assumptions are made: first, that there is an energy flux through
the horizon in the near past of $P$, see Fig. 5.1, given by a _conserved_
energy-momentum tensor $T_{\mu\nu}$:

$(ii)\qquad\Delta U:=\int_{\cal H}T_{\mu\nu}\xi^{\mu}l^{\nu}d\lambda
d^{2}S=-\kappa\int_{\cal H}T_{\mu\nu}l^{\mu}l^{\nu}\lambda d\lambda
d^{2}S,\qquad\nabla_{\mu}T^{\mu}{}_{\nu}=0,$ (5.42)

where we used 5.40 and the constancy of $\kappa$. This energy flux will be
interpreted thermodynamically as a heat flux, $\Delta U=\Delta Q$. Second
assumption, that there is a notion of entropy variation associated to the
horizon, which is (universally, i.e. independently of the matter state)
proportional to the area variation:

$(iii)\qquad\Delta S=\eta\,{\Delta A}=\eta\int_{\cal H}\theta d\lambda
d^{2}S,$ (5.43)

where $\theta$ is the expansion of horizon. This is controlled by the
Raychadhuri equation for $l^{\mu}$,

$\frac{d\theta}{d\lambda}=-\frac{\theta^{2}}{2}-\sigma_{\mu\nu}\sigma^{\mu\nu}-R_{\mu\nu}l^{\mu}l^{\nu}.$
(5.44)

The final, technical assumption made in [Jacobson, 1995] is that at $P$ one
can take $\theta=\sigma_{\mu\nu}=0$, and approximate the solution of the
Raychadhuri equation simply by $\theta=-\lambda
R_{\mu\nu}l^{\mu}l^{\nu}+O(\lambda^{2})$. Using this approximation,

$\Delta S=-\eta\int_{\cal H}\lambda\,R_{\mu\nu}l^{\mu}l^{\nu}d\lambda d^{2}S.$
(5.45)

Finally, we observe that using $(i-iii)$ and the approximation 5.45, the
Clausius first law of thermodynamics $\Delta Q=T\Delta S$ implies

$\int_{\cal
H}\left(\frac{2\pi}{\eta}\,T_{\mu\nu}-R_{\mu\nu}\right)l^{\mu}l^{\nu}\lambda
d\lambda d^{2}S=0.$ (5.46)

Since this is valid for an arbitrary direction of the Killing boost and at any
point, we can remove the integral. The Einstein equations (with an
undetermined cosmological constant) then follow by imposing the conservation
law $\nabla^{\mu}T_{\mu\nu}=0$. The Newton constant is determined by
$G=1/(4\eta)$.

##### 5.3.2 The torsional case

In the Einstein-Cartan theory 5.1 the connection is a priori affine, and
torsion can be present, affecting the geodesic and Raychaudhuri equations. One
may then think that the argument above should be substantially revisited. As
we now show, this is actually not the case. The first observation we make is
that the starting point of Jacobson’s argument, a Killing horizon, is a purely
metric notion:

$\displaystyle 0=\pounds_{\xi}g_{\mu\nu}$
$\displaystyle=\xi^{\alpha}\partial_{\alpha}g_{\mu\nu}+g_{\mu\alpha}\partial_{\nu}\xi^{\alpha}+g_{\nu\alpha}\partial_{\mu}\xi^{\alpha}$
(5.47) $\displaystyle=2\overset{\scriptscriptstyle
e}{\nabla}{}_{(\mu}\xi_{\nu)}=\nabla_{(\mu}\xi_{\nu)}+T_{(\mu}{}^{\rho}{}_{\nu)}\xi_{\rho}.$

Hence by definition, it does not depend on torsion, in spite of the apparent
presence of the latter in the last expression above. The constancy of $\kappa$
on the approximate Rindler horizon also follows like in the metric case from
the vanishing of the metric Riemann tensor. Being Killing and null,
$\xi^{\mu}$ is automatically geodetic with respect to the Levi-Civita
connection (which we recall is always well-defined and at disposal since we
are only interested in the sector of Einstein-Cartan theory with invertible
tetrads), so 5.39 still holds. Hence, we can run most of the argument as in
the metric case. Step $(i)$ is unchanged. For step $(ii)$, we follow
[Jacobson, 1995] and define the energy flux as the integral of the _conserved_
energy-momentum tensor. Proposition 1 identifies this object uniquely as
$T^{\scriptscriptstyle eff}_{\mu\nu}$ defined in 5.23, with its torsional
dependence. Step $(iii)$ is also unchanged: since the generators of the
Killing horizon follow the Levi-Civita geodesics 5.39, the change of the
expansion of the generators is governed by the Raychaudhuri equation 5.44 with
the metric Ricci tensor $R_{\mu\nu}(e)$ appearing on the right-hand side.
Imposing again the equilibrium Clausius relation $\Delta Q=T\Delta S$ with
these $(i-iii)$, and using the same approximation 5.45, we arrive exactly at

$\int_{\cal H}\left(\frac{2\pi}{\eta}\,T^{\scriptscriptstyle
eff}_{\mu\nu}-R_{\mu\nu}(e)\right)l^{\mu}l^{\nu}\lambda d\lambda d^{2}S.$
(5.48)

We conclude that the torsion-full Einstein equations, in the form 5.22, can be
derived à la Jacobson from the equilibrium Clausius relation. No need to
consider a torsion-full Raychaudhuri equation, non-equilibrium terms and
restrictions on torsion, as argued in [Dey et al., 2017] and reviewed in the
next Section. It suffices to use the result of Proposition 1 to identify the
correct energy-momentum tensor.

There is however an important caveat to our procedure: we are assuming the
torsion equations to hold, since we used them to prove Proposition 1. This may
look unsatisfactory, since it is currently not known whether these equations
can be derived from a thermodynamical description. Our logic is that if such a
description of the torsion equations exists, then it is consistent to assume
that they hold when deriving the tetrad Einstein equations. This said, it is
also possible that Proposition 1 holds off-shell of the torsion equations, so
that these are not needed to derive the Einstein equations. Nonetheless, one
would still need to be able to derive the torsion equations from
thermodynamics for the whole framework to make sense. Assuming them to hold
seems thus to us coherent if a complete thermodynamical framework exists.

In any case, the main problem if one does not want to use the conserved
energy-momentum tensor is the ambiguity that one faces in defining it, see
e.g. [Hehl, 1976]. The prescription used by the authors of [Dey et al., 2017],
for instance, is to take simply what would be the source 5.6 of the tetrad EC
equations, namely the derivative of the matter Lagrangian with respect to the
tetrad (or equivalently up to a symmetrization, the metric). Notice however
that this can be tricky in the first-order theory, because one can either work
with $(g,\Gamma)$ or $(g,C)$ as fundamental variables. The field equations are
completely equivalent since the actions are related by a (non-linear) field
redefinition. However the sources differ as

$\displaystyle T^{\Gamma}_{\mu\nu}:=\frac{2}{\sqrt{-g}}\frac{\delta
L_{m}(g,\Gamma)}{\delta g^{\mu\nu}},$ (5.49) $\displaystyle
T^{C}_{\mu\nu}=\frac{2}{\sqrt{-g}}\frac{\delta L_{m}(g,C)}{\delta
g^{\mu\nu}}=T^{\Gamma}_{\mu\nu}+\frac{2}{\sqrt{-g}}\frac{\delta
L_{m}(g,\Gamma)}{\delta\Gamma^{\alpha}_{\beta\gamma}}\frac{\delta\Gamma^{\alpha}_{\beta\gamma}}{\delta
g^{\mu\nu}}.$ (5.50)

Both coincide with the general relativity energy-momentum tensor when torsion
vanishes. This type of ambiguity reminds us that using a conserved energy-
momentum tensor, when available, is always the best choice. We now show how
this ambiguity in turn affects the non-equilibrium approach to deriving the
first-order tetrad equations.

##### 5.3.3 Non-equilibrium approach

A more general setting including a non-vanishing shear has been considered in
[Eling et al., 2006, Chirco and Liberati, 2010]. In this case the presence of
additional terms on the right-most side of 5.43 is incompatible with the
equilibrium Clausius relation. Hence to run Jacobson’s argument one must
assume that there are non-equilibrium terms,

$\Delta Q=T\Delta S+\Delta S_{\scriptscriptstyle non-equi}.$ (5.51)

The interpretation of the shear-squared terms as non-equilibrium is justified
a priori from the horizon tidal heating effect [Chirco and Liberati, 2010].
Notice, however, that the same shear-squared terms enter both the non-
equilibrium contribution and the equilibrium $T\Delta S$ contribution, since
one is still assuming that the entropy variation is proportional to the area
variation. This feature seems to us unusual from a thermodynamical
perspective. In any cases, in the case of the tetrad EC equations, the
application of the non-equilibrium approach is even more problematic, as we
now explain.

We start as before from the observation that a Killing horizon is metric-
geodetic, and use the same approximations leading to the integrated metric
Raychaudhuri equation 5.45, but this time allowing a non-zero shear in 5.44.
Then from 5.51 we obtain

$\frac{2\pi}{\eta}\int_{\cal H}T^{\rm??}_{\mu\nu}l^{\mu}l^{\nu}\lambda
d\lambda d^{2}S=\int_{\cal
H}\Big{(}R_{\mu\nu}(e)l^{\mu}l^{\nu}+\sigma_{\mu\nu}\sigma^{\mu\nu}\Big{)}\lambda
d\lambda d^{2}S+\Delta S_{\scriptscriptstyle non-equi}.$ (5.52)

The delicate point now is how to define the heat flux, namely what
$T^{??}_{\mu\nu}$ needs to be used on the left-hand side of the above
equation. Clearly, the identification of the non-equilibrium terms that will
be needed to obtain the tetrad EC equations 5.22 depends on how we define the
energy-momentum tensor. If, as in the previous Section, the conserved one is
used, the only non-equilibrium term comes from the shear, which can then be
argued for as in the metric theory following [Eling et al., 2006, Chirco and
Liberati, 2010]. This shows how the derivation of the EC tetrad equations from
the conserved energy-momentum tensor and metric Raychaudhuri equation can be
easily extended to the presence of shear.

If we chose instead to define the heat flux via a source tensor like in 5.49,
we would need additional non-equilibrium terms in order to fully reproduce the
Einstein equations 5.22. The crucial point is whether they can be justified a
priori as in the example of the tidal heating, else the construction is
artificial. The authors of [Dey et al., 2017] argue that this is possible, if
$(a)$ we choose $T^{??}_{\mu\nu}=T^{C}_{\mu\nu}$ for the heat flux, and $(b)$
we define the non-equilibrium terms as those arising from the torsion-full
Raychaudhuri equation that include torsion-full derivatives of $l^{\mu}$.
There are three problems that we can see with this construction. First, since,
as already stressed several times, a Killing vector is metric-geodesic, it is
in general not geodesic with respect to the torsion-full connection. Using
5.13, indeed, we see that

$\xi^{\nu}{\nabla}_{\nu}\xi_{\mu}=\kappa\,\xi_{\mu}-C_{\nu,\mu\rho}\,\xi^{\nu}\xi^{\rho}=\kappa\,\xi_{\mu}-T_{\nu,\mu\rho}\,\xi^{\nu}\xi^{\rho}.$
(5.53)

For this reason, the authors of [Dey et al., 2017] restrict torsion to satisfy

$C_{\nu,\mu\rho}\xi^{\nu}\xi^{\rho}=0.$ (5.54)

This condition implies that metric and torsion-full geodesics coincides. Since
the metric and torsion-full geodesic expansion also coincide,666In the
presence of torsion, the displacement of a vector $q^{\mu}$ Lie dragged along
$\xi^{\mu}$ is given by
$\xi^{\nu}\nabla_{\nu}q^{\mu}=B_{\mu\nu}q^{\nu},\qquad
B_{\mu\nu}:=\nabla_{\nu}\xi_{\mu}+T_{\mu,\lambda\nu}\xi^{\lambda}=\overset{\scriptscriptstyle
e}{\nabla}{}_{\nu}\xi_{\mu}+C_{\rho,\mu\nu}\xi^{\rho},$ hence introducing the
usual projector $\perp^{\mu\nu}$ on a 2d space-like surface orthogonal to
$\xi^{\mu}$, we have
$\theta:=\perp^{\mu\nu}B_{\mu\nu}=\overset{\scriptscriptstyle e}{\theta}{}$.
it follows that the torsion-full Raychaudhuri equation is _identical_ to the
metric one. Therefore, it is unclear what one gains from this approach, except
for a restriction on torsion that in the equilibrium approach presented in the
previous Section is not necessary.777Since in order to recover the Einstein
equations we will need to consider arbitrary boost Killing vectors, see
discussion below 5.46, the restriction on torsion should hold for any
$\xi^{\mu}$. This implies that the only non-vanishing part of the torsion
field is its completely antisymmetric irreducible part. A priori it could be
possible to consider a relaxation of 5.54, allowing for a right-hand side
proportional to $l_{\mu}$ rather than vanishing, since this would only
mismatch the inaffinity of metric and torsion-full geodesics. However we don’t
know whether the derivation of [Dey et al., 2017] can be extended to this
case.

Second, the identification of the non-equilibrium contributions as torsion-
full covariant derivatives of $l^{\mu}$ does not appears to us to be well-
grounded a priori: We are not aware of any proof that in a spacetime with
torsion it is the torsion-full shear that gives the tidal heating.
Furthermore, the condition of vanishing initial expansion implies that at the
point $P$ we have
$\nabla_{\mu}l^{\mu}=\perp^{\mu\nu}\\!T_{\mu,\nu\rho}l^{\rho}$, making some
‘non-equilibrium terms’ indistinguishable from terms without derivatives, as
the authors of [Dey et al., 2017] acknowledge in a footnote.

Third, the ambiguity in picking a non-conserved $T^{??}_{\mu\nu}$, as
discussed before. Had we chosen the alternative source $T^{\Gamma}$, which is
also more natural from the perspective of a metric-connection action, the same
identification of non-equilibrium contributions would not work, as it would
miss the terms with covariant derivatives of the contorsion in 5.22.

Summarizing, although the non-equilibrium approach has the advantage of
allowing to relax the assumption of an initial non-vanishing shear [Eling et
al., 2006, Chirco and Liberati, 2010], it is, in our opinion, inherently
ambiguous when applied to first-order gravity.

#### 5.4 On the Laws of Black Hole Mechanics with Torsion

As mentioned in the introduction, Jacobson’s derivation is inspired by the
laws of black hole thermodynamics. Having shown that the derivation works also
in the presence of torsion, at least as far as recovering the tetrad Einstein
equations, the next question we considered is what happens to the these laws.

We have recalled earlier that the surface gravity of the Rindler horizon is
constant simply because the Riemann tensor vanishes. For a general horizon,
constancy of the surface gravity is the zeroth law, and its proof uses the
Einstein equations and the dominant energy conditions. In the presence of
torsion, we can follow the proof with the equations 5.22, and the only
modification is that the dominant energy condition will be a restriction on
the effective energy-momentum tensor.

More interesting is the modification that occurs to the first law. To see
this, let us consider the ‘physical process’ version of the proof [Wald,
1995], in which an initially stationary black hole is perturbed by some matter
falling inside the horizon. For our generalization, we suppose that the in-
falling matter has spin and sources torsion, and that the metric and
connection satisfy the Einstein-Cartan field equations.

As in the metric case, we assume that all matter falls into the black hole,
and that the latter is not destroyed by the process, but settles down to a new
stationary configuration [Wald, 1995, Gao and Wald, 2001]. These assumptions
are motivated by the no-hair theorem and the cosmic censorship conjecture,
which keep their value also in a theory with non-propagating torsion. For
example, it is known that a compact ball of static or slowly spinning torsion-
full Weyssenhoff fluid888This is a single component of torsion (the trace
part) generated by the gradient of a scalar [Griffiths and Jogia, 1982].
admits a solution which satisfies the junction conditions with an external
Schwarzschild or slowly rotating Kerr [Prasanna, 1975, Arkuszewski et al.,
1974].

Following [Wald, 1995], we use the linearized Einstein equation to study the
effect on the horizon geometry caused by the in-falling matter at first order
in perturbation theory,

$g_{\mu\nu}=g_{\mu\nu}^{0}+h_{\mu\nu},\qquad C_{\rho,\mu\nu}=c_{\rho,\mu\nu}.$
(5.55)

Being null and hypersurface orthogonal, the affine horizon generators are
metric geodetic, and their expansion is governed by the Raychaudhuri equation
5.44. The background generators $l^{\mu}$ are proportional to the Killing
generators $\xi^{\mu}$ satisfying $l^{\mu}=-(\lambda\kappa)^{-1}\xi^{\mu}$,
with constant $\kappa$ by the zeroth law. They have vanishing shear and
expansion, giving therefore at first order

$\frac{d}{d\lambda}\delta\theta=-\delta R_{\mu\nu}(h)l^{\mu}l^{\nu}.$ (5.56)

Integrating along the horizon $\cal H$ from the bifurcation surface $\cal B$
to a cut $S_{\infty}$ at future null infinity, we have for the total area
variation

$\Delta A=\int_{\cal H}\delta\theta\,d\lambda d^{2}S=\int_{\cal H}\delta
R_{\mu\nu}(h)l^{\mu}l^{\nu}\,\lambda d\lambda d^{2}S,$ (5.57)

where we integrated by parts and used that $\lambda|_{\cal B}=0$ since
$\xi^{\mu}|_{\cal B}=0$, and that $\theta|_{S_{\infty}}=0$ by the late time
settling down assumption.

In the standard particular case of torsion-less matter with conserved energy-
momentum tensor $T_{\mu\nu}$, we have from the linearized Einstein equations

$\int_{\cal H}\delta R_{\mu\nu}(h)l^{\mu}l^{\nu}\,\lambda d\lambda
d^{2}S=8\pi\int_{\cal H}\delta T_{\mu\nu}(h)l^{\mu}l^{\nu}\,\lambda d\lambda
d^{2}S.$ (5.58)

At this order, we can substitute $l^{\mu}=-(\lambda\kappa)^{-1}\xi^{\mu}$ in
the right-hand side integrand

$-\frac{8\pi}{\kappa}\int_{\cal H}\delta
T_{\mu\nu}(h)\xi^{\mu}l^{\nu}\,\lambda d\lambda
d^{2}S=\frac{8\pi}{\kappa}\int_{\cal H}\delta
T_{\mu\nu}(h)\xi^{\mu}\,dH^{\nu}=\frac{8\pi}{\kappa}(\Delta M-\Omega_{H}\Delta
J),$ (5.59)

where in the first equality we used that fact the future-pointing volume form
on $\cal H$ is $dH_{\mu}=-l_{\mu}d\lambda d^{2}S$, and in the second the
explicit expression
$\xi^{\mu}=\partial_{t}^{\mu}+\Omega_{H}\partial_{\phi}^{\mu}$ as well as the
definitions of $\Delta M$ and $\Delta J$ used in [Wald, 1995]. We conclude
that the linearized Einstein equations imply the first law of perturbations
around a stationary black hole,999To make contact between this ‘physical
process’ version of the first law, and the one in terms of ADM charges, recall
that since we are assuming all matter to be falling in the black hole, the
integral along the horizon equals the integral on a space-like hypersurface
$\Sigma$ extending from $\cal B$ to a 2-sphere $S_{\infty}$ at spatial
infinity $i^{0}$. Using again the Einstein equations and the explicit form of
the conserved Noether current (see [Iyer and Wald, 1994], here $\kappa$ is the
Komar charge and $\Theta$ the Einstein-Hilbert symplectic potential) we find
$\int_{\cal H}\delta T_{\mu\nu}(h)\xi^{\mu}dH^{\nu}=\int_{\Sigma}\delta
T_{\mu\nu}(h)\xi^{\mu}d\Sigma^{\nu}=\int_{\cal
S_{\infty}}(k_{\xi}-\Theta\lrcorner\xi)-\int_{\cal B}k_{\xi}=\Delta
M_{ADM}-\Omega_{H}\Delta J_{ADM},$ where the final result follows from a
standard calculation with
$\xi^{\mu}=\partial_{t}^{\mu}+\Omega_{H}\partial_{\phi}^{\mu}$. See [De Paoli
and Speziale, 2018] for a derivation of the first law with covariant
Hamiltonian methods for Einstein-Cartan theory.

$\Delta M=\frac{\kappa}{8\pi}\Delta A+\Omega_{H}\Delta J.$ (5.60)

For torsion-generating matter, we can follow exactly the same procedure, the
only difference being that we use the Einstein-Cartan equations 5.22 with the
conserved effective energy-momentum tensor on the right-hand side. The first
law follows as before but with new mass and angular momentum variations

$\Delta M-\Omega_{H}\Delta J=\int_{\cal H}\delta T^{\scriptscriptstyle
eff}_{\mu\nu}(h)\xi^{\mu}\,dH^{\nu}$ (5.61)

determined by the torsion-dependent $T^{\scriptscriptstyle eff}_{\mu\nu}$.
This is consistent with the results of [Arkuszewski et al., 1974] mentioned
above, where the mass of the external Schwarzschild has a torsion contribution
from an effective energy density profile of the static Weyssenhoff fluid
compatible with the formula above.

Following the same approach of treating the effect of torsion as an effective
energy-momentum tensor, we can conclude that also the second law of black hole
mechanics is still valid, provided the required restrictions on the energy-
momentum tensor of matter [Bardeen et al., 1973] are applied to the effective
tensor 5.23.

The third law remains as elusive as in the metric case, and we do not discuss
it here.

#### 5.5 Conclusions

Motivated by the analysis of [Dey et al., 2017], we looked at one aspect of
conservation laws in Einstein-Cartan theory. In the sector of invertible
tetrads, where one can choose to split the connection into the Levi-Civita one
plus a contorsion tensor, it is possible to identify a conserved energy-
momentum tensor for matter even without using Einstein equations. Thanks to
this result, we were able to reproduce Jacobson’s thermodynamical argument
[Jacobson, 1995], and derive the (tetrad) Einstein equations from the
equilibrium Clausius relation. Our derivation is much simpler than the one
proposed in [Dey et al., 2017], and does not require any restriction on
torsion. On the other hand, like in [Dey et al., 2017], we are only able to
derive the tetrad Einstein equations from a thermodynamical argument, and not
the torsion equations. This remains the crucial open question in order to
truly extend Jacobson’s argument to theories with independent metric and
connection.

The set-up we use is the same of [Jacobson, 1995], in particular the initial
shear and expansion vanish, which is argued to be in line with a notion of
thermodynamical equilibrium. An initial non-vanishing shear can be nonetheless
allowed and interpreted as a non-equilibrium term [Eling et al., 2006, Chirco
and Liberati, 2010]. Non-equilibrium terms are also necessary to extend
Jacobson’s argument to derive the field equations of modified theories of
gravity with higher order terms [Guedens et al., 2012]. In the Einstein-Cartan
formalism, higher-order terms typically introduce propagating torsion, see
e.g. [Tseytlin, 1982]. While in our paper we showed that the equilibrium
Clausius relation is enough to derive the Einstein-Cartan equations (with
vanishing initial shear), in which torsion is non propagating, an extension to
higher-order tetrad-connection theories with propagating torsion will likely
require non-equilibrium terms. It could be interesting if the dissipation
present in this case would be associated with dissipation of energy to the
torsional degrees of freedom. From this perspective, as well as the
perspective of possibly recovering the torsion field equations from a
thermodynamical argument, it could be intriguing to consider existing
condensed matter models in which dissipating lattice defects introduce torsion
[Kröner, 1981].

### Addenda to Part I

#### A.1 Extremal Surfaces have Vanishing Mean
Extrinsic Curvature

In the notation of Section 1.1, the mean extrinsic curvature is defined by

$K=\nabla_{\alpha}n^{\alpha}=h^{ab}e^{\alpha}_{a}e^{\beta}_{b}\nabla_{\alpha}n_{\beta}$
(A.62)

where $\nabla$ is the covariant derivative in $g_{\alpha\beta}$ and
$n_{\alpha}$ is the normal to $\Sigma$. The Levi-Civita connections of
$g_{AB}$ and $\tilde{g}_{AB}$ are related by:

$\Gamma^{B}{}_{AC}=\tilde{\Gamma}^{B}{}_{AC}-\frac{2}{r}\left(\delta_{Cr}\,\delta^{B}_{A}+\delta_{Ar}\,\delta^{B}_{C}-g^{Br}g_{AC}\right)$
(A.63)

For the calculation that follows, keep in mind the following:
$e^{\alpha}_{\phi}=\delta^{\alpha}_{\phi}$ ,
$e^{\alpha}_{\theta}=\delta^{\alpha}_{\theta}$, $h^{\phi\phi}=g^{\phi\phi}$ ,
$h^{\theta\theta}=g^{\theta\theta}$,
$h^{\lambda\lambda}=(g_{AB}e^{A}_{\lambda}e^{B}_{\lambda})^{-1}$ ,
$n_{\alpha}e^{\alpha}_{a}=0$. Also, notice that $n_{\alpha}$ and
$e^{\alpha}_{\lambda}$ can be replaced by $n_{A}$ and $e^{A}_{\lambda}$ when
contracted since they have vanishing angular components.

We then have

$\displaystyle-K$ $\displaystyle=$
$\displaystyle-h^{ab}e^{\alpha}_{a}e^{\beta}_{b}\nabla_{\alpha}n_{\beta}$
(A.64) $\displaystyle=$ $\displaystyle
n_{B}(h^{ab}e^{A}_{a}\nabla_{\alpha}e^{B}_{b})$ $\displaystyle=$
$\displaystyle
n_{B}(g^{\phi\phi}\Gamma^{B}{}_{\phi\phi}+g^{\theta\theta}\Gamma^{B}{}_{\theta\theta}+h^{\lambda\lambda}e^{A}_{\lambda}\nabla_{A}\,e^{B}_{\lambda})$
$\displaystyle=$ $\displaystyle
n_{B}(g^{\phi\phi}\Gamma^{B}{}_{\phi\phi}+g^{\theta\theta}\Gamma^{B}{}_{\theta\theta}+\frac{2}{r}h^{\lambda\lambda}g^{Br}g_{AC}e^{A}_{\lambda}e^{C}_{\lambda})$
$\displaystyle=$ $\displaystyle
n_{B}g^{Br}(-g^{\phi\phi}\frac{g_{\phi\phi,r}}{2}-g^{\theta\theta}\frac{g_{\theta\theta,r}}{2}+\frac{2}{r})$
$\displaystyle=$ $\displaystyle
n_{B}g^{Br}(-\frac{1}{r}-\frac{1}{r}+\frac{2}{r})$ $\displaystyle=$
$\displaystyle 0$

where we used $\Gamma^{B}{}_{\phi\phi}=-\frac{1}{2}g^{Br}g_{\phi\phi,r}$,
$\Gamma^{B}{}_{\theta\theta}=-\frac{1}{2}g^{Br}g_{\theta\theta,r}$ and that
the surfaces are defined as $\Sigma\sim\gamma\times S^{2}$ with $\gamma$ a
solution of eq. (1.5).

#### A.2 Lifetimes

In this Appendix we compare natural time scales that appear in the collapse
models. As recalled in Section 2.1, there is the time scale, introduced in
[Haggard and Rovelli, 2015], defined as the proper time $\tau$ that an
observer seating just outside the horizon at $r_{\Delta}$ has to wait in order
for allow quantum gravitational effects to pile up until $q=1$ in that region.
This time scale, of order $m^{2}$, is referred to as the lifetime of the black
hole in [Haggard and Rovelli, 2015]. However, when one talks about _lifetime_
in black hole physics, one would rather refer to the retarded time elapsed at
${\mathfs{I}}^{+}$ between an initial $u_{0}$ (roughly defined by detection of
the first Hawking quantum), and the complete evaporation of the hole $u_{s}$
(in our case). More precisely $u_{0}$ can be identified with the retarded time
at which the entanglement entropy at ${\mathfs{I}}^{+}$ starts departing
significantly from zero. The results of [Bianchi et al., 2015] show that this
happens for the retarded time corresponding to the collapsing shell shrinking
to $r=3m$. We can write

$\tau_{\rm life}=u_{s}-u_{0}=\Delta u+(u_{\Delta}-u_{0}).$ (A.65)

The second term can be calculated from the diagrams (the result is the same in
different models); the result is

$\tau_{\rm life}=\Delta u+\Delta v+\frac{4}{3}m+4m\log(3).$ (A.66)

This means that, to leading scaling order, the lifetime defined in this way
coincides with the one used in [Haggard and Rovelli, 2015]. It is driven by
$\Delta v$ when it is chosen to scale with $m$ faster than linearly. In the
present models we have $\tau_{\rm life}\sim\tau\sim m^{2}$ if $\Delta v\sim
m^{2}$.

#### A.3 The Hartle-Hawking state

In this Appendix we recall the basic formulae (used in the main text) that
allow to compute the renormalized expectation value of the energy momentum
tensor in $1+1$ dimensions. Moreover, we compute for completeness the analog
of the Hartle-Hawking quantum state in the fireworks background. This state
leads to a regular expectation value of the energy momentum tensor in the
semiclassical part of the spacetime. It has the well-known thermal properties
outside the collapsing shell. However, this state does not represent the
physics of gravitational collapse as it does not satisfies the vacuum boundary
conditions neither at ${\mathfs{I}}^{-}$ nor inside the collapsing shell as
the following calculation shows.

To do this, let us first recall some basic relations [Fabbri and Navarro-
Salas, 2005]. Any $1+1$ spacetime is conformally flat and can therefore be
written as

$ds^{2}=-e^{2\rho}dx_{+}dx_{-}$ (A.67)

for some function $\rho$ and a double null coordinate system $x_{\pm}$. The
mean value of the covariant stress-energy tensor on some state $\ket{\Psi}$
can be defined to be

$\bra{\Psi}T_{\pm\pm}\ket{\Psi}=-\frac{\hbar}{12\pi}\big{(}(\partial_{\pm}\rho)^{2}-\partial^{2}_{\pm}\rho\big{)}+\bra{\Psi}:T_{\pm\pm}:\ket{\Psi}$
(A.68)

where $:T_{\pm\pm}:$ is the normal ordered stress-energy tensor. The off-
diagonal term is given by

$\bra{\Psi}T_{+-}\ket{\Psi}=-\frac{\hbar}{12\pi}\partial_{+}\partial_{-}\rho.$
(A.69)

While $\bra{\Psi}T_{\mu\nu}\ket{\Psi}$ is covariant under a coordinate
transformation $x_{\pm}\to\xi_{\pm}$, the normal ordered stress tensor
transforms as

$\bra{\Psi}:T_{\xi_{\pm}\xi_{\pm}}:\ket{\Psi}=\bra{\Psi}:T_{x_{\pm}x_{\pm}}:\ket{\Psi}-\frac{\hbar}{24\pi}\big{\\{}x_{\pm},\xi_{\pm}\big{\\}}$
(A.70)

where

$\big{\\{}x_{\pm},\xi_{\pm}\big{\\}}=\frac{d^{3}x_{\pm}/d\xi_{\pm}^{3}}{dx_{\pm}/d\xi_{\pm}}-\frac{3}{2}\left(\frac{d^{2}x_{\pm}/d\xi_{\pm}^{2}}{dx_{\pm}/d\xi_{\pm}}\right)^{2}$
(A.71)

is the Schwarzian derivative.

The terms that are independent of the state $\ket{\Psi}$ are vacuum
polarization contributions stemming from the conformal anomaly. For example,
by identifying $x_{+}\to v$ and $x_{-}\to u$ in the Schwarzschild region with
metric (2.7), they become:

$\begin{split}-\frac{\hbar}{12\pi}\big{(}(\partial_{\pm}\rho)^{2}-\partial^{2}_{\pm}\rho\big{)}&=\frac{\hbar}{24\pi}\left[-\frac{m}{r^{3}}+\frac{3}{2}\frac{m^{2}}{r^{4}}\right]\\\
-\frac{\hbar}{12\pi}\partial_{+}\partial_{-}\rho&=-\frac{\hbar}{24\pi}\left(1-\frac{2m}{r}\right)\frac{m}{r^{3}}.\end{split}$
(A.72)

The in-state

The $\ket{in}$ state is defined with respect to the mode expansion in terms of

$\phi_{in}=e^{i\omega v},\qquad\phi_{out}=e^{i\omega u_{in}}\;.$ (A.73)

Inside the collapsing shell this state coincides with the Minkowski vacuum:
the vacuum polarization vanishes and the normal ordered contribution vanishes.
Outside the collapsing shell we have

$\begin{split}\bra{in}T_{uu}\ket{in}&=\frac{\hbar}{24\pi}\left[-\frac{m}{r^{3}}+\frac{3}{2}\frac{m^{2}}{r^{4}}\right]-\frac{\hbar}{24\pi}\big{\\{}u_{in},u\big{\\}}\\\
\bra{in}T_{vv}\ket{in}&=\frac{\hbar}{24\pi}\left[-\frac{m}{r^{3}}+\frac{3}{2}\frac{m^{2}}{r^{4}}\right]\\\
\bra{in}T_{uv}\ket{in}&=-\frac{\hbar}{24\pi}\left(1-\frac{2m}{r}\right)\frac{m}{r^{3}}\;,\end{split}$
(A.74)

where we have explicitly written the vacuum polarization terms (A.72). Using
equation (2.10) one can compute the Schwarzian derivative term and obtain
(2.12).

The Hartle-Hawking-like state

Take the vacuum state $\ket{H}$ of the Fock space where positive frequencies
are defined with respect to the mode expansion of solutions of (2.6) of the
form

$\phi_{in}=e^{i\omega V},\qquad\phi_{out}=e^{i\omega U}\;,$ (A.75)

where $U$ and $V$ are the Kruskal coordinates for the black hole geometry. We
compute are the components of the covariant stress-energy tensor of this state
in the Minkowski patch of the spacetime defining the inside of the collapsing
shell (at least the one connected with the Schwarzschild one without touching
the quantum region) which is described by the metric

$ds^{2}=-dvdu_{\rm in}\;.$ (A.76)

Outside the collapsing shell. Outside the collapsing shell and all over its
classical chronological future one has

$\begin{split}\bra{H}T_{uu}\ket{H}&=\bra{H}T_{vv}\ket{H}=\frac{\hbar}{768\pi
m^{2}}\left(1-\frac{2m}{r}\right)^{2}\left[1+\frac{4m}{r}+\frac{12m^{2}}{r^{2}}\right]\\\
\bra{H}T_{uv}\ket{H}&=-\frac{\hbar}{24\pi}\left(1-\frac{2m}{r}\right)\frac{m}{r^{3}}\end{split}$
(A.77)

Notice that these are well behaved in regular coordinates at the past horizon;
see (2.15). At large $r\to\infty$ we recover the energy momentum tensor of a
thermal bath

$\begin{split}\bra{H}T_{uu}\ket{H}&=\bra{H}T_{vv}\ket{H}=\frac{\hbar}{768\pi
m^{2}}\\\ \bra{H}T_{uv}\ket{H}&=0.\end{split}$ (A.78)

Inside the collapsing shell. In the Minkowski patch of the spacetime the first
term on the right-hand side of eq. (A.68) is zero and, moreover, by definition
the state $\ket{H}$ is such that
$\bra{H}:T_{UU}:\ket{H}=\bra{H}:T_{VV}:\ket{H}=\bra{H}:T_{UV}:\ket{H}=0$.
Therefore we find

$\begin{split}\bra{H}T_{u_{\rm in}u_{\rm
in}}\ket{H}&=-\frac{\hbar}{24\pi}\big{\\{}U,u_{\rm in}\big{\\}}\\\
&=\frac{\hbar}{768\pi m^{2}}\left(1-\frac{8m}{(u_{\rm
in}-v_{s})}+\frac{48m^{2}}{(u_{\rm in}-v_{s})^{2}}\right)\\\
\bra{H}T_{vv}\ket{H}&=-\frac{\hbar}{24\pi}\big{\\{}V,v\big{\\}}\\\
&=\frac{\hbar}{768\pi m^{2}}\\\ \bra{H}T_{u_{\rm in}v}\ket{H}&=0\end{split}$
(A.79)

where we used the matching conditions

$\begin{split}u=-4m\log\left(-\frac{U}{4m}\right)&=u_{\rm
in}-4m\log\left(\frac{v_{s}-u_{\rm in}-4m}{4m}\right)\\\
V&=4m\exp\left(\frac{v}{4m}\right)\;.\end{split}$ (A.80)

For large $r\to\infty$ we recover the thermal fluid in (A.78). The collapsing
shell in this state is initially filled up with radiation at hawking
temperature. Due to the contraction of the shell one gets a divergence of the
energy momentum tensor when the shell crosses the origin at $u_{in}=v_{s}$.

### Addenda to Part II

#### A.4 Coordinate Transformations

The radial Conformal Killing Field in Minkowski spacetime $\xi$ naturally
divides the space in six regions. For each of these regions, there exists a
coordinate transformation
$(t,r,\vartheta,\varphi)\to(\tau,\rho,\vartheta,\varphi)$ adapted to the MCKF
in the sense that $\xi(\tau)=-1$. In this Appendix, we write down such
transformations explicitly.

##### The non-extremal case $\Delta\not=0$.

Region I (the diamond). The coordinate transformation is given by [Haggard,
2013]

$\begin{split}t&=\frac{\sqrt{\Delta}}{2a}\frac{\sinh(\tau\sqrt{\Delta})}{\cosh(\rho\sqrt{\Delta})+\cosh(\tau\sqrt{\Delta})}\\\
r&=\frac{\sqrt{\Delta}}{2a}\frac{\sinh(\rho\sqrt{\Delta})}{\cosh(\rho\sqrt{\Delta})+\cosh(\tau\sqrt{\Delta})}\end{split}$
(A.81)

with $-\infty<\tau<+\infty$ and $0<\rho<+\infty$.

$\begin{split}v&=t+r=\frac{\sqrt{\Delta}}{2a}\tanh\frac{\bar{v}\sqrt{\Delta}}{2}\\\
u&=t-r=\frac{\sqrt{\Delta}}{2a}\tanh\frac{\bar{u}\sqrt{\Delta}}{2}\,.\end{split}$
(A.82)

where we have defined the null coordinates $\bar{v}=\tau+\rho$ and
$\bar{u}=\tau-\rho$. The Minkowski metric (3.7) in the new coordinates reads

$ds^{2}=\Omega_{\rm
I}^{2}\left(-d\tau^{2}+d\rho^{2}+\Delta^{-1}\sinh^{2}(\rho\sqrt{\Delta})dS^{2}\right)$
(A.83)

with

$\Omega_{\rm
I}=\frac{\Delta/2a}{\cosh(\rho\sqrt{\Delta})+\cosh(\tau\sqrt{\Delta})}.$
(A.84)

Regions II (the causal complement of the diamond), III and IV. Region II, III
and IV can be described by the same coordinate transformation given by
[Haggard, 2013]

$\begin{split}t&=\frac{\sqrt{\Delta}}{2a}\frac{\sinh(\tau\sqrt{\Delta})}{\cosh(\rho\sqrt{\Delta})-\cosh(\tau\sqrt{\Delta})}\\\
r&=\frac{\sqrt{\Delta}}{2a}\frac{\sinh(\rho\sqrt{\Delta})}{\cosh(\rho\sqrt{\Delta})-\cosh(\tau\sqrt{\Delta})}\,,\end{split}$
(A.85)

with $-\infty<\tau<+\infty$ and $0\leq\rho<+\infty$. Region II is now given by
the restriction $|\tau|<\rho$, Region III by $\tau>0$ and $\tau>\rho$, while
Region IV by $\tau<0$ and $|\tau|>\rho$. In this case we have

$\begin{split}v&=t+r=-\frac{\sqrt{\Delta}}{2a}\coth\frac{\bar{u}\sqrt{\Delta}}{2}\\\
u&=t-r=-\frac{\sqrt{\Delta}}{2a}\coth\frac{\bar{v}\sqrt{\Delta}}{2}\,.\end{split}$
(A.86)

The metric (3.7) is now

$ds^{2}=\Omega_{\rm
II}^{2}\left(-d\tau^{2}+d\rho^{2}+\Delta^{-1}\sinh^{2}(\rho\sqrt{\Delta})dS^{2}\right)$
(A.87)

with

$\Omega_{\rm
II}=\frac{\Delta/2a}{\cosh(\rho\sqrt{\Delta})-\cosh(\tau\sqrt{\Delta})}.$
(A.88)

For Region II, given the above mentioned restrictions on the coordinate, we
have $\bar{u}\in(-\infty,0)$ and $\bar{v}\in(0,+\infty)$. This is the
transformation used in Section 3.4.

Region V. In the upper of the two regions where $\xi$ is spacelike, the
coordinate transformation can be found to be

$\begin{split}t&=\frac{\sqrt{\Delta}}{2a}\frac{\cosh(\tau\sqrt{\Delta})}{\sinh(\rho\sqrt{\Delta})+\sinh(\tau\sqrt{\Delta})}\\\
r&=\frac{\sqrt{\Delta}}{2a}\frac{\cosh(\rho\sqrt{\Delta})}{\sinh(\rho\sqrt{\Delta})+\sinh(\tau\sqrt{\Delta})}\end{split}$
(A.89)

with $0<\tau<+\infty$ and $0<\rho<+\infty$. The double null coordinates are
here given by

$\begin{split}v&=t+r=\frac{\sqrt{\Delta}}{2a}\coth\frac{\bar{v}\sqrt{\Delta}}{2}\\\
u&=t-r=\frac{\sqrt{\Delta}}{2a}\tanh\frac{\bar{u}\sqrt{\Delta}}{2}\,.\end{split}$
(A.90)

Region VI. Finally, for Region VI we have

$\begin{split}t&=\frac{\sqrt{\Delta}}{2a}\frac{\cosh(\tau\sqrt{\Delta})}{\sinh(\rho\sqrt{\Delta})-\sinh(\tau\sqrt{\Delta})}\\\
r&=\frac{\sqrt{\Delta}}{2a}\frac{\cosh(\rho\sqrt{\Delta})}{\sinh(\rho\sqrt{\Delta})-\sinh(\tau\sqrt{\Delta})}\end{split}$
(A.91)

with $-\infty<\tau<0$ and $0<\rho<+\infty$. This gives

$\begin{split}v&=t+r=-\frac{\sqrt{\Delta}}{2a}\tanh\frac{\bar{u}\sqrt{\Delta}}{2}\\\
u&=t-r=-\frac{\sqrt{\Delta}}{2a}\coth\frac{\bar{v}\sqrt{\Delta}}{2}\,.\end{split}$
(A.92)

In both last two cases, the metric (3.7) becomes

$ds^{2}=\Omega_{V/VI}^{2}\left(-d\tau^{2}+d\rho^{2}+\Delta^{-1}\cosh^{2}(\rho\sqrt{\Delta})dS^{2}\right)$
(A.93)

where, for Region V

$\Omega_{\rm
V}=\frac{\Delta/2a}{\sinh(\rho\sqrt{\Delta})+\sinh(\tau\sqrt{\Delta})}\,,$
(A.94)

and for Region VI

$\Omega_{\rm
VI}=\frac{\Delta/2a}{\sinh(\rho\sqrt{\Delta})-\sinh(\tau\sqrt{\Delta})}\,.$
(A.95)

##### The extremal case $\Delta=0$

In the $\Delta=0$ case, we have only Region II, III and IV and $\xi$ is
everywhere timelike. The coordinate transformation in this extremal case can
be obtained from the previous one by taking the limit $\Delta\to 0$ in all
expressions. The result is

$\begin{split}t=&\frac{\tau}{a\left(\tau^{2}-\rho^{2}\right)}\\\
r=&\frac{\rho}{a\left(\rho^{2}-\tau^{2}\right)}\end{split}$ (A.96)

with $-\infty<\tau<+\infty$ and $0\leq\rho<+\infty$. Region II is now given by
the restriction $|\tau|<\rho$, Region III by $\tau>0$ and $\tau>\rho$, while
Region IV by $\tau<0$ and $|\tau|>\rho$. In this case we have

$\begin{split}v&=t+r=\frac{1}{a\bar{v}}\\\
u&=t-r=\frac{1}{a\bar{u}},\end{split}$ (A.97)

where, given the above mentioned restrictions on the coordinate, we have
$\bar{u}\in(-\infty,0)$ and $\bar{v}\in(0,+\infty)$. The Minkowski metric in
the new coordinates reads

$ds^{2}=\Omega_{\rm ext}^{2}\left(-d\tau^{2}+d\rho^{2}+\rho^{2}dS^{2}\right)$
(A.98)

with

$\Omega_{\rm ext}=\frac{\rho}{a\left(\rho^{2}-\tau^{2}\right)}.$ (A.99)

This coincides with Eq. (12) in [Herrero and Morales, 1999].

##### Near bifurcate sphere approximation

In the non-extremal case, the bifurcate sphere is located at $\rho\to+\infty$
and $\tau=0$. Eq. (3.50) can therefore be expanded in the approximation
$\rho>>1/\sqrt{\Delta}$. This gives a Rindler-like coordinate transformation

$\begin{split}t\sim-\sqrt{\frac{c}{a}}\;e^{-\rho\sqrt{\Delta}}\cosh(\tau\sqrt{\Delta})\\\
r\sim\sqrt{\frac{c}{a}}\;e^{-\rho\sqrt{\Delta}}\sinh(\tau\sqrt{\Delta})\end{split}$
(A.100)

with the would-be proper distance given by
$D=\sqrt{c/a}\;e^{-\rho\sqrt{\Delta}}$. The above approximation is
inconsistent in the case $\Delta=0$.

#### A.5 Static FRW Spacetime and Region II

The coordinate transformations above show that the Regions I to IV in
Minkowski spacetime are conformally related to pieces of a static FRW
spacetime with negative spatial curvature $k=-|\Delta|$. This fact was used in
the computation of Bogoliubov coefficients in Section 3.4. In this Appendix we
give some more details on the geometry of static FRW spacetime and its
relation with Region II. The static FRW spacetime is a solution to the
Einstein equation with zero cosmological constant and the energy stress tensor
of a perfect fluid satisfying the state equation [Hawking and Ellis, 1973]

$\mu=-3p\,.$ (A.101)

Here $\mu$ and $p$ are the energy density and pressure of the fluid
respectively. The metric takes the form

$ds^{2}=-d\tau^{2}+d\rho^{2}+\Delta^{-1}\sinh^{2}(\rho\sqrt{\Delta})dS^{2}\,,$
(A.102)

with $-\infty<\tau<+\infty$ and $0\leq\rho<+\infty$. As shown for example in
[Candelas and Dowker, 1979, Hawking and Ellis, 1973], there exists a
coordinate transformation that conformally maps this space into the Einstein
static universe. This transformation allows to draw the Penrose diagram for
the static FRW spacetime, which results in a diamond shaped diagram depicted
in Figure A.2 101010In the cited references [Candelas and Dowker, 1979,
Hawking and Ellis, 1973], however, they consider the non-static FRW spacetime
with zero cosmological constant $\Lambda=0$ and zero pressure $p=0$. The
resulting Penrose diagram is therefore slightly different, being only the
upper triangle of the whole diamond of Figure A.2, with a “big bang
singularity” for $\tau=0$..

Figure A.2: The Penrose diagram for the static FRW spacetime of Eq. (A.102).
The shaded region is the one conformally related to Region II in Minkowski
spacetime. Its boundaries are in correspondence with pieces of Minkowskian
future and past null infinities $\mathscr{I}^{\pm}_{M}$, as well as with the
bifurcate conformal Killing horizon $H^{+}\cup H^{-}$. The light grey
hyperbolas are radial flow lines of the field $\partial_{\tau}$, or in another
words $\rho=const$ lines.

The structure is very similar to the one of Minkowski, with past and future
null infinities. There is however one main difference: here spacial infinity
$i^{0}$ is a sphere and not a point as in the Minkowski case.

Region II is (conformally) given by the restriction $|\tau|<\rho$. This
corresponds to the region outside the light cone shining from the origin; this
is the shaded region in Figure A.2. From Eq. (A.86), Minkowski future null
infinity $v\to+\infty$ is mapped into the future light cone $\bar{u}=0$.
Analogously, Minkowski past null infinity $u\to-\infty$ is given by the past
light cone $\bar{v}=0$. In the same way, the future horizon $H^{+}$ located at
$u=u_{-}$ is given by $\bar{v}\to+\infty$, namely the piece of FRW
$\mathscr{I}^{+}$ given by $\bar{u}<0$. The past horizon $v=v_{+}$ is, in a
similar way, mapped into the piece of FRW $\mathscr{I}^{-}$ given by
$\bar{v}>0$. Finally, Minkowskian spacial infinite $i^{0}_{M}$ is mapped into
the point given by the origin, while the bifurcate sphere $r=r_{H}$ at $t=0$
is given by the FRW spacial infinity $i^{0}$. In Figure A.2, the flow lines of
the Killing field $\partial_{\tau}$, namely $\rho=const.$ surfaces are also
plotted. From there, the behaviour in Region II of the conformal Killing field
$\xi$ depicted in Figure 3.2 becomes clear. The flow lines of $\xi$ start
their life on $\mathscr{I}^{-}$ to end on $\mathscr{I}^{+}$.

The above discussion shows also that the complete set of solutions (3.65) to
the Klein-Gordon equation (3.56) is a good complete set of solutions also in
our region of interest. The set considered, indeed, is regular everywhere, at
the origin too, where otherwise there could have been a problem in our
setting.

#### A.6 Another Conformal Mapping of Minkowski

The previous map to a suitable FRW spacetime is useful for the calculations in
Section 3.4. However, it is not the best suited for the geometrical
interpretation due to the fact that the horizon $H$ is mapped to infinity in
the FRW spacetime. Here we construct a new conformal mapping of flat spacetime
where $\xi$ becomes a Killing field, and the horizon is mapped to a genuine
Killing horizon embedded in the bulk of the host spacetime. In Minkowski
spacetime we have

${\mathfs{L}}_{\xi}\eta_{ab}=\frac{\psi}{2}\eta_{ab}$ (A.103)

where explicit calculation yields

$\psi=\nabla_{a}\xi^{a}=8at=\frac{4(u+v)}{r_{\scriptscriptstyle
O}^{2}-r_{\scriptscriptstyle H}^{2}}.$ (A.104)

Under a conformal transformation $g_{ab}=\Omega^{2}\eta_{ab}$ one has

${\mathfs{L}}_{\xi}g_{ab}={\mathfs{L}}_{\xi}(\Omega^{2}\eta_{ab})=\left[\frac{\psi}{2}+2\xi(\log(\Omega))\right]g_{ab}.$
(A.105)

Therefore, in the new spacetime $g_{ab}$, $\xi$ will be a Killing field iff

${\psi}+4\xi(\log(\Omega))=0.$ (A.106)

This equation does not completely fix $\Omega$: if $\Omega$ is a solution,
then $\Omega^{\prime}=\omega\Omega$ is also a solution as long as
$\xi(\omega)=0$. Writing explicitly the previous equation using (3.19) we get:

$u+v+(u^{2}-r_{\scriptscriptstyle
H}^{2})\partial_{u}(\log(\Omega))+(v^{2}-r_{\scriptscriptstyle
H}^{2})\partial_{v}(\log(\Omega))=0.$ (A.107)

It is easy to find solutions of the previous equation by separation of
variables. Assuming that we want to preserve spherical symmetry then we can
write $\Omega(u,v)=V(v)U(u)$ and the previous system becomes

$\displaystyle u+(u^{2}-r_{\scriptscriptstyle H}^{2})\partial_{u}(\log(U))$
$\displaystyle=$ $\displaystyle-\lambda,$ $\displaystyle
v+(v^{2}-r_{\scriptscriptstyle H}^{2})\partial_{v}(\log(V))$ $\displaystyle=$
$\displaystyle\lambda,$ (A.108)

where $\lambda$ is an arbitrary constant. If we choose $\lambda=0$ then the
solution is

$\Omega=\frac{\Omega_{0}}{\sqrt{(u^{2}-r_{\scriptscriptstyle
H}^{2})(v^{2}-r_{\scriptscriptstyle H}^{2})}}.$ (A.109)

By fixing the integration constant $\Omega_{0}=r_{\scriptscriptstyle
O}^{2}-r_{\scriptscriptstyle H}^{2}$, the previous solution corresponds to the
one that maps to the FRW spacetime studied in the previous section. This can
be checked by recalling from (A.102) that the conformal factor mapping to the
FRW spacetime is $1/\sqrt{-\xi\cdot\xi}$ and using (3.20). The Killing vector
$\xi$ in the FRW metric is normalized everywhere.

An alternative solution, which does not send the horizon to infinity, is
obtained by choosing $\lambda=r_{\scriptscriptstyle H}$ which yields

$\Omega_{\scriptscriptstyle BH}=\frac{4r^{2}_{\scriptscriptstyle
H}}{(u-r_{\scriptscriptstyle H})(v+r_{\scriptscriptstyle H})},$ (A.110)

where we have chosen the integration constant so that
$\Omega_{\scriptscriptstyle BH}=1$ at the bifurcate surface
$u=-r_{\scriptscriptstyle H}$ and $v=r_{\scriptscriptstyle H}$. In the new
metric

$g_{ab}=\frac{16r^{4}_{\scriptscriptstyle H}}{(u-r_{\scriptscriptstyle
H})^{2}(v+r_{\scriptscriptstyle H})^{2}}\,\eta_{ab},$ (A.111)

the null surfaces $u=-r_{\scriptscriptstyle H}$ and $v=r_{\scriptscriptstyle
H}$ are Killing horizons with constant cross-sectional area $A=4\pi
r_{\scriptscriptstyle H}^{2}$. These surfaces have the same geometric
properties as black hole horizons which justifies the subindex BH in (A.110).
At the inner horizons $u=r_{\scriptscriptstyle H}$ and
$v=-r_{\scriptscriptstyle H}$ the conformal factor diverges. Therefore, in
contrast with the FRW mapping, only these horizons are pushed to infinity. If
$r_{O}\leq\sqrt{5}r_{H}$, then the Killing field $\xi$ is normalized on a
timelike surface outside the horizon where stationary observers measure time
and energy in agreement with those in the FRW mapping. More details on these
geometries can be found in Chapter 4. The previous conformal map plays a
central role for the interpretation of the first law (3.46) as discussed at
the end of Section 3.3.2.

#### A.7 Coordinate Transformations [continued]

In [De Lorenzo and Perez, 2018b], a coordinate transformation
$(t,r,\theta,\varphi)\to(\tau,\rho,\theta,\varphi)$ for each of the six
regions Minkowski spacetime is divided into by the radial MCKF has been
presented–see Appendix A.4. The transformation was built in such a way the
radial MCKF reduced to

$\xi^{\mu}\frac{\partial}{\partial x^{\mu}}=\frac{\partial}{\partial\tau}\,.$
(A.112)

Those six transformations can actually be grouped in one single transformation
given by

$\begin{split}\tau&=\frac{r_{\scriptscriptstyle O}^{2}-r_{\scriptscriptstyle
H}^{2}}{4r_{\scriptscriptstyle H}}\log\frac{(u-r_{\scriptscriptstyle
H})(v-r_{\scriptscriptstyle H})}{(u+r_{\scriptscriptstyle
H})(v+r_{\scriptscriptstyle H})}\\\ \rho&=\frac{r_{\scriptscriptstyle
O}^{2}-r_{\scriptscriptstyle H}^{2}}{4r_{\scriptscriptstyle
H}}\log\frac{(u+r_{\scriptscriptstyle H})(v-r_{\scriptscriptstyle
H})}{(u-r_{\scriptscriptstyle H})(v+r_{\scriptscriptstyle H})}\,.\end{split}$
(A.113)

The coordinate $\tau$ is the same used in the main text, eq. (4.17). Defining
as in [De Lorenzo and Perez, 2018b]

$\begin{split}\Delta&=\frac{4r_{\scriptscriptstyle
H}^{2}}{(r_{\scriptscriptstyle O}^{2}-r_{\scriptscriptstyle H}^{2})^{2}}\\\
a&=\frac{1}{r_{\scriptscriptstyle O}^{2}-r_{\scriptscriptstyle
H}^{2}}\end{split}$ (A.114)

Minkowski metric becomes

$ds_{\scriptscriptstyle
M}^{2}=\left(\frac{\Delta/2a}{\cosh(\sqrt{\Delta}\,\tau)+\cosh(\sqrt{\Delta}\,\rho)}\right)^{2}\big{(}-d\tau^{2}+d\rho^{2}+\Delta^{-1}\sinh^{2}(\sqrt{\Delta}\,\rho)\,dS^{2}\big{)}\,.$
(A.115)

The transformation is valid everywhere using the standard definition of the
logarithm of a negative number, namely

$\log(-x)=i\,\pi+\log(x)\qquad x>0\,.$ (A.116)

We can solve eq. (4.16) in these coordinates finding

$\omega^{2}=\left(\frac{\cosh(\sqrt{\Delta}\,\tau)+\cosh(\sqrt{\Delta}\,\rho)}{\Delta/2a}\frac{1}{G_{\rho}(\rho)}\right)^{2}\,,$
(A.117)

which in terms of the Minkowskian double-null coordinates $(u,v)$

$\omega^{2}(u,v)=\frac{4r_{\scriptscriptstyle
H}^{4}}{(u^{2}-r_{\scriptscriptstyle H}^{2})(v^{2}-r_{\scriptscriptstyle
H}^{2})}\frac{1}{G^{2}(\rho,\theta,\phi)}\,.$ (A.118)

A conformally flat metric $g_{ab}$ such that the radial MCKF becomes a Killing
field can therefore also be written as

$ds^{2}=\frac{1}{G^{2}_{\rho}(\rho,\theta,\phi)}(-d\tau^{2}+d\rho^{2}+\Delta^{-1}\sinh^{2}(\sqrt{\Delta}\rho)\,dS^{2})\,,$
(A.119)

Choosing the function $G_{\rho}$ to be a normalisation constant given by

$G_{\rho}=\frac{r_{\scriptscriptstyle O}^{2}-r_{\scriptscriptstyle
H}^{2}}{2r_{\scriptscriptstyle H}^{2}}$ (A.120)

one finds

$\omega_{\scriptscriptstyle FRW}=\frac{r_{\scriptscriptstyle
O}^{2}-r_{\scriptscriptstyle H}^{2}}{\sqrt{(u^{2}-r_{\scriptscriptstyle
H}^{2})(v^{2}-r_{\scriptscriptstyle H}^{2})}}$ (A.121)

namely the conformal factor of Eq. (A.109). The choice

$G_{\rho}(\rho)=1/4\,e^{-2\sqrt{\Delta}\,\rho}\,,$ (A.122)

instead, gives

$\omega_{\scriptscriptstyle BH}=\frac{4r^{2}_{\scriptscriptstyle
H}}{(u-r_{\scriptscriptstyle H})(v+r_{\scriptscriptstyle H})}$ (A.123)

defined in Eq. (A.110). The above two conformal factors were found in [De
Lorenzo and Perez, 2018b] by separation of variables.

Another interesting coordinates transformation is given by

$\begin{split}\bar{\tau}&=\sqrt{\Delta}\,\tau\\\
\rho&=\frac{1}{2\sqrt{\Delta}}\log(1-2z)\,.\end{split}$ (A.124)

which implies

$z=\frac{1}{2}\left(1-\frac{(u+r_{\scriptscriptstyle
H})(v-r_{\scriptscriptstyle H})}{(u-r_{\scriptscriptstyle
H})(v+r_{\scriptscriptstyle H})}\right)\,.$ (A.125)

The metric (A.119) takes the following Schwarzschild-like form

$ds^{2}=\frac{\Delta^{-1}}{G^{2}_{z}(z,\theta,\phi)}\left(-(1-2\,z)\,d\bar{\tau}^{2}+\frac{1}{1-2\,z}\,dz^{2}+z^{2}\,dS^{2}\right)\,,$
(A.126)

where $G_{z}^{2}$ is a new function encoding the ambiguity in the conformal
transformations, and the new coordinates are dimensionless. In these
coordinates the horizon is located at $z=1/2$. $z$ is positive and greater
then $1/2$ outside, and decreases to zero at the Minkowskian $i^{0}$ and
origin. Inside the horizon, on the other hand, increases from $z=1/2$ to
$z\to\infty$, the latter corresponding to the inner horizon. In Section 4.5,
these coordinates are used in the simplest case $G_{z}=1$.

#### A.8 Conventions

We take $\underset{\widetilde{}}{\epsilon}{}_{\mu\nu\rho\sigma}$ as the
completely antisymmetric spacetime density with
$\underset{\widetilde{}}{\epsilon}{}_{0123}=1$, and
$\tilde{\epsilon}^{\mu\nu\rho\sigma}\underset{\widetilde{}}{\epsilon}{}_{\mu\nu\rho\sigma}=-4!$.
It is related to the volume 4-form by

$\epsilon:=\frac{1}{4!}\epsilon_{\mu\nu\rho\sigma}dx^{\mu}\wedge
dx^{\nu}\wedge dx^{\rho}\wedge
dx^{\sigma},\qquad\epsilon_{\mu\nu\rho\sigma}:=\sqrt{-g}\,\underset{\widetilde{}}{\epsilon}{}_{\mu\nu\rho\sigma}.$
(A.127)

We define the Hodge dual in components as

$(\star\omega^{(p)})_{\mu_{1}..\mu_{4-p}}:=\frac{1}{p!}\omega^{(p)}{}^{\alpha_{1}..\alpha_{p}}\epsilon_{\alpha_{1}..\alpha_{p}\mu_{1}..\mu_{4-p}}.$
(A.128)

For the internal Levi-Civita density $\epsilon_{IJKL}$ we refrain from adding
the tilde. We use the same convention, ${\epsilon}_{0123}=1$, so the tetrad
determinant is

$e=-\frac{1}{4!}\epsilon_{IJKL}\tilde{\epsilon}^{\mu\nu\rho\sigma}e_{\mu}^{I}e_{\nu}^{J}e_{\rho}^{K}e_{\sigma}^{L},$
(A.129)

and we take $e>0$ for a right-handed tetrad.

Curvature and torsion are defined by

$F^{IJ}(\omega)=d\omega^{IJ}+\omega^{IK}\wedge\omega_{K}{}^{J},\qquad
T^{I}(e,\omega)=d_{\omega}e^{I},$ (A.130)

where $d_{\omega}$ is the covariant exterior derivative, whose components we
denote by $D_{\mu}$, to distinguish them from the spacetime covariant
derivative $\nabla_{\mu}$ with affine connection $\Gamma^{\rho}_{\mu\nu}$. The
relation between the connections on the fiber and on the tangent space is
given by

$D_{\mu}e_{\nu}^{I}=\Gamma^{\rho}_{\mu\nu}e_{\rho}^{I},\qquad\omega^{IJ}_{\mu}=e_{\nu}^{I}{\nabla}_{\mu}e^{\nu
J}$ (A.131)

for $\omega$ and $\Gamma$ general affine connections, plus the metricity
condition $D_{\mu}\eta^{IJ}=0$. The compatibility of the internal covariant
derivative and the tetrad means that
$D_{\mu}f^{I}=e^{I}_{\nu}\nabla_{\mu}f^{\nu}$ and so on.

The commutators of the covariant derivatives satisfy:

$\displaystyle[D_{\mu},D_{\nu}]f^{I}=F^{I}{}_{J\mu\nu}(\omega)f^{J},$ (A.132)
$\displaystyle[D_{\mu},D_{\nu}]f=-T^{\rho}{}_{\mu\nu}(e,\omega)\partial_{\rho}f,$
(A.133)
$\displaystyle[\nabla_{\mu},\nabla_{\nu}]f^{\rho}=R^{\rho}{}_{\sigma\mu\nu}(\Gamma)f^{\sigma}-T^{\sigma}{}_{\mu\nu}\nabla_{\sigma}f^{\rho},$
(A.134)

where

$\displaystyle
R_{\rho\sigma\mu\nu}(\Gamma)=e_{I\rho}e_{J\sigma}F^{IJ}_{\mu\nu}(\omega)\qquad
T^{\rho}{}_{\mu\nu}(\Gamma)=e^{\rho}_{I}\,T^{I}{}_{\mu\nu}(\omega).$ (A.135)

Finally, torsion and contorsion are related by

$\displaystyle
T^{\rho}{}_{\mu\nu}:=e^{\rho}_{I}\,T^{I}{}_{\mu\nu}(e,C)=-2C_{[\mu,\nu]}{}^{\rho}=2\Gamma^{\rho}_{[\mu\nu]}\quad\Leftrightarrow\quad
C_{\mu,\nu\rho}=\frac{1}{2}T_{\mu,\nu\rho}-T_{[\nu,\rho]\mu}.$ (A.136)

Both torsion and contorsion have spinorial decomposition
${\bf(\tfrac{3}{2},\tfrac{1}{2})\oplus(\tfrac{1}{2},\tfrac{3}{2})\oplus(\tfrac{1}{2},\tfrac{1}{2})\oplus(\tfrac{1}{2},\tfrac{1}{2})},$
which corresponds to three irreducible components under Lorentz
transformations (since the latter include parity). They can be defined as
follows [Hehl et al., 1976],

$\displaystyle
C^{\mu,\nu\rho}=\bar{C}^{\mu,\nu\rho}+\frac{2}{3}g^{\mu[\rho}\check{C}^{\nu]}+\epsilon^{\mu\nu\rho\sigma}\hat{C}_{\sigma},$
(A.137) $\displaystyle
g_{\mu\nu}\bar{C}^{\mu,\nu\rho}=0=\epsilon_{\mu\nu\rho\sigma}\bar{C}^{\mu,\nu\rho},\qquad\check{C}^{\mu}:=C_{\nu,}{}^{\mu\nu},\qquad\hat{C}_{\sigma}:=\frac{1}{6}\epsilon_{\sigma\mu\nu\rho}C^{\mu,\nu\rho}.$
(A.138)

#### A.9 Index Jugglers

In this Appendix we prove Proposition 1, namely that the matter Noether
identities 5.28 on-shell of the matter field equations, plus the torsion field
equation 5.12b, imply the conservation law for the effective energy-momentum
tensor 5.30, reported here for convenience

$d_{\omega(e)}\left[\star\tau_{I}+\frac{1}{16\pi\,}P_{IJKL}(e^{J}\wedge
d_{\omega(e)}C^{KL}+e^{J}\wedge C^{KM}\wedge C_{M}{}^{L})\right]=0,$ (A.139)

namely,

$d_{\omega(e)}\star\\!\tau_{I}=\frac{1}{8\pi\,}P_{IJKL}\Big{(}e^{J}\wedge
C^{K}{}_{M}\wedge F^{ML}(e)+e^{J}\wedge d_{\omega(e)}C^{KM}\wedge
C_{M}{}^{L}\Big{)}.$ (A.140)

To prove this identity, we start from 5.28b. On the left-hand side, we split
the connection into Levi-Civita plus contorsion, see 5.13, obtaining

$d_{\omega}\star\\!\tau_{I}=d_{\omega(e)}\star\\!\tau_{I}-(C^{JK}\lrcorner
e_{I})e_{K}\wedge\star\tau_{J}+T^{J}\lrcorner e_{I}\wedge\star\tau_{J}$
(A.141)

where we used

$T^{I}=C^{IJ}\wedge e_{J}\quad\rightarrow\quad C_{I}{}^{J}=-(C^{JK}\lrcorner
e_{I})e_{K}+T^{J}\lrcorner e_{I}.$ (A.142)

In the second term of the right-hand side of A.141 we use the second Noether
identity 5.28a, whereas the last term cancels the corresponding one on the
right-hand side of 5.28b, which then reads

$\displaystyle d_{\omega(e)}\star\\!\tau_{I}$
$\displaystyle=\frac{1}{2}F^{JK}(\omega)\lrcorner
e_{I}\wedge\star\sigma_{JK}-\frac{1}{2}(C^{JK}\lrcorner
e_{I})d_{\omega}\star\\!\sigma_{JK}$
$\displaystyle=\frac{1}{16\pi\,}\left[F^{JK}(\omega)\lrcorner
e_{I}\wedge-(C^{JK}\lrcorner e_{I})d_{\omega}\right]P_{JKLM}e^{L}\wedge
C^{M}{}_{N}\wedge e^{N}$
$\displaystyle=\frac{1}{16\pi\,}\left[\big{(}F^{JK}(e)+d_{\omega(e)}C^{JK}\big{)}\lrcorner
e_{I}\wedge-(C^{JK}\lrcorner e_{I})d_{{\omega}(e)}\right]P_{JKLM}e^{L}\wedge
C^{M}{}_{N}\wedge e^{N}.$ (A.143)

In the second equality above we eliminated the torsion source using the
corresponding field equation 5.17. In the third equality we expanded the
curvature using the contorsion, see 5.15, and observed that the piece
quadratic in $C$ cancels the contorsion part of the exterior derivative in the
last term.

Having performed these simplifications, our goal is to show the equivalence of
the right-hand sides of A.140 and A.143. This will follow from the equivalence
of the terms with the Riemann tensor $F^{IJ}(e)$, and the equivalence of the
terms involving the Levi-Civita exterior derivatives. Both are consequences of
trivial algebraic symmetries. Let us show them one by one. We notice in
advance the following useful cycling identities:

$\displaystyle P_{IJKL}e^{K}\wedge F^{LM}\wedge e_{M}=-P_{ABC[I}e^{A}\wedge
F^{BC}\wedge e_{J]},$ (A.144) $\displaystyle P_{IJKL}F^{KM}\wedge
C_{M}{}^{L}=-P_{ABC[I}F^{AB}\wedge C^{C}{}_{J]},$ (A.145)

which are easy to check.

To show the equivalence of the terms with the curvature, we start hooking a
cotetrad vector field on a trivially vanishing 5-form,

$\displaystyle 0$ $\displaystyle=\Big{(}P_{JKLM}F^{JK}(e)\wedge e^{L}\wedge
C^{M}{}_{N}\wedge e^{N}\Big{)}\lrcorner e_{I}$
$\displaystyle=P_{JKLM}F^{JK}(e)\lrcorner e_{I}\wedge e^{L}\wedge
C^{M}{}_{N}\wedge e^{N}+P_{JKIM}F^{JK}(e)\wedge C^{M}{}_{N}\wedge e^{N}$
$\displaystyle\quad-P_{JKLM}(C^{M}{}_{N}\lrcorner e_{I})F^{JK}(e)\wedge
e^{L}\wedge e^{N}+P_{JKLM}F^{JK}(e)\wedge e^{L}\wedge C^{M}{}_{I}.$ (A.146)

Of these four terms, the third vanishes identically: its $1/\gamma$ part
directly through the algebraic Bianchi identities for the Riemann tensor, the
other part because of the antisymmetry in the $LP$ indices. The second and
fourth terms recombine giving the left-hand side of A.145, hence A.9 gives

$2P_{IJKL}\,F^{KM}(e)\wedge C_{M}{}^{L}\wedge
e^{J}=P_{JKML}\,(F^{JK}(e)\lrcorner e_{I})\wedge e^{L}\wedge C^{M}{}_{N}\wedge
e^{N},$ (A.147)

which proves the equality of the curvature terms of A.140 and A.143.

The equivalence of the $d_{\omega(e)}C$ terms follows analogously. We hook the
following 5-form,

$\displaystyle 0$ $\displaystyle=\Big{(}P_{JKLM}C^{JK}\wedge e^{L}\wedge
d_{\omega(e)}C^{M}{}_{N}\wedge e^{N}\Big{)}\lrcorner e_{I}$
$\displaystyle=P_{JKLM}(C^{JK}\lrcorner e_{I})e^{L}\wedge
d_{\omega(e)}C^{M}{}_{N}\wedge e^{N}-P_{JKIM}C^{JK}\wedge
d_{\omega(e)}C^{M}{}_{N}\wedge e^{N}$ $\displaystyle\quad+P_{JKLM}C^{JK}\wedge
e^{L}\wedge d_{\omega(e)}C^{M}{}_{N}\lrcorner e_{I}\wedge
e^{N}+P_{JKLM}C^{JK}\wedge e^{L}\wedge d_{\omega(e)}C^{M}{}_{I}.$ (A.148)

Using an identity like A.145, the second and fourth term give

$P_{JKM[I}C^{JK}\wedge d_{\omega(e)}C^{M}{}_{N]}\wedge e^{N}=-2P_{IJKL}\wedge
e^{J}\wedge d_{\omega(e)}C^{K}{}_{M}\wedge C^{ML}.$ (A.149)

For the third term we have

$\displaystyle P_{JKLM}C^{JK}\wedge e^{L}\wedge
d_{\omega(e)}C^{M}{}_{N}\lrcorner e_{I}\wedge
e^{N}=-P_{JKLM}d_{\omega(e)}C^{MN}\lrcorner e_{I}\wedge e^{L}\wedge
C^{JK}\wedge e_{N}$ (A.150)
$\displaystyle\qquad=P_{JKLM}d_{\omega(e)}C^{JK}\lrcorner e_{I}\wedge
e^{L}\wedge C^{M}{}_{N}\wedge e^{N},$

which follows from a similar cycling identity as before. Hence, A.9 gives

$\displaystyle 2P_{IJKL}\wedge e^{J}\wedge d_{\omega(e)}C^{K}{}_{M}\wedge
C^{ML}$ $\displaystyle=P_{JKLM}d_{\omega(e)}C^{JK}\lrcorner e_{I}\wedge
e^{L}\wedge C^{M}{}_{N}\wedge e^{N}$ (A.151)
$\displaystyle\quad+P_{JKLM}(C^{JK}\lrcorner e_{I})e^{L}\wedge
d_{\omega(e)}C^{M}{}_{N}\wedge e^{N},$

which proves precisely the equivalence between the $d_{\omega(e)}C$ terms in
A.140 and A.143.

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