$\tau=\sqrt{1-\frac{2m}{r_{\Delta}}}\,\Delta v\;,$ (2.2)

where $v$ is the standard advanced inertial time at ${\mathfs{I}}^{-}$. The
quantity $q$ is maximized for

$\begin{split}r_{\Delta}=\frac{7}{3}m.\end{split}$ (2.3)

This means that the quantum Region III+tIII extends macroscopically outside
the BH horizon. The bouncing time is defined to be the value of $\Delta v$ for
which the nonclassicality parameter (linear in $\Delta v$) becomes of order
unity. This happens for

$\Delta v\equiv v_{\Delta}-v_{s}\sim\tau\sim m^{2}\;.$ (2.4)

Due to the time symmetry of the construction, the observer at $r_{\Delta}$
sees the entire bouncing process happening in a proper time $\tau_{\rm
tot}=2\tau\sim m^{2}$. This time scale is very important in what follows and
is argued to produce possible experimental observations [Barrau and Rovelli,
2014, Barrau et al., 2014, Barrau et al., 2016b].

#### 2.2 Semiclassical Stability

The question any classical ansatz spacetime has to be confronted with is
whether it admits a physically reasonable quantum state for the test fields
living on it. This requirement represents the first step toward addressing the
problem of _back-reaction_. More precisely, in those regions where we can
trust the validity of QFT in curved spacetime one expects the quantum dynamics
to be well approximated by the semiclassical Einstein’s equation

$G_{ab}(g_{ab})=8\pi\,\langle\,T_{ab}(g_{ab})\rangle\;,$ (2.5)

where $\langle\,T_{ab}(g_{ab})\,\rangle$ represents the expectation value of
the stress-energy tensor of the quantum matter fields propagating on the
metric $g_{ab}$.

The most famous example is the effect of Hawking evaporation on a black hole
background [Hawking, 1974, Hawking, 1975]. The original computation has been
made in the fixed background approximation, completely neglecting the back-
reaction. However, this leads to an infinite amount of radiated energy from
the hole, clearly in contradiction with energy conservation. Intuitively, one
expects the energy radiated to be balanced by a reduction of the Bondi mass of
the black hole, leading to the evaporation of the hole and consequently the
well-known loss of information paradox [Hawking, 1976]. There are both
analytical and numerical works indicating some general features of the
evaporation problem [Hajicek and Israel, 1980, Bardeen, 1981, Parentani and
Piran, 1994, Massar, 1995b]; nevertheless, a complete description remains
unsolved even in the semiclassical regime of equation (2.5).

Indeed, the complete backreaction problem could be framed in a formal
approximation procedure where one starts by evaluating $\langle
T_{ab}(g_{ab}^{0})\rangle$ on a seed background $g_{ab}^{0}$, and then inserts
the result into semiclassical Einstein equations (2.5) in other to find a new
metric $g_{ab}^{1}$: the first-order quantum corrected background metric.
Iterating the process one can try to find higher-order corrected line elements
eventually converging to a consistent solution $g_{ab}$ of equation (2.5).
Every single step is in general a really difficult task to achieve and the
final convergence is not even guaranteed.

Fortunately, for the present analysis it will be sufficient to solve a much
simpler problem. Indeed, the classical initial background
$g_{ab}^{0}$—solution of the classical Einstein equations—is a good zeroth
approximation of the quantum dynamics only if the quantum corrections coming
from $\langle T_{ab}(g_{ab}^{0})\rangle$ are small in semiclassical regions.
This stability of the seed background under the effects of the propagation of
quantum test fields living on it will be called quantum-stability property. In
the following of this Section, we will compute $\langle
T_{ab}(g_{ab}^{0})\rangle$ for the model of reference [Haggard and Rovelli,
2015] and show that it diverges in Region tII. The quantum-stability property,
therefore, is not satisfied by the fireworks model.

The computation of $\langle\,T_{ab}(g_{ab}^{0})\rangle$ on a given unperturbed
geometry can be already a very difficult task. In fact, there is in general
uncertainties related to the choice of the appropriate physical state for the
quantum fields and, at the same time, one needs to appeal to renormalization
techniques to eliminate usual $UV$ divergences of QFT in a way that is
consistent with general covariance [Wald, 1995]. Both issues are more subtle
and difficult when the background spacetime is not flat. However, the great
symmetry of our example and its direct relationship with the well-studied
Schwarzschild geometry will allow us to make very precise statements.

#### 2.3 Analytic Calculation in the $1+1$ Setting

In this section we use spherical symmetry and we neglect back-scattering as
well as the influence of modes other than $s$-modes. This allows for an
effective description in terms of a $1+1$ theory. These simplifications make
possible the analytic computation of effects that qualitatively remain valid
in the $3+1$ framework. More precisely, we show that the computation of
$\langle\,T_{ab}(g_{ab}^{0})\rangle$ in the framework of the fireworks
background presents a divergent behaviour. Quantum fields are represented by a
single massless scalar $\phi$ satisfying the Klein-Gordon equation

$g_{0}^{ab}\nabla_{a}\nabla_{b}\phi=0$ (2.6)

with $g_{0}^{ab}$ the background geometry of the fireworks model in the $r-t$
space. In more detail, the metric in Region II+tII is given by

$ds_{0}^{2}=-\left(1-\frac{2m}{r}\right)dvdu,$ (2.7)

where $v=t+r_{*}$ and $u=t-r_{*}$, with $t$ the Killing parameter and

$r_{*}=r+2m\log\left(\frac{r}{2m}-1\right).$ (2.8)

In Region I the metric is

$ds_{0}^{2}=-dvdu_{in},$ (2.9)

where $u_{in}=t_{M}-r$ and $v=t_{M}+r$ and $t_{M}$ is the inertial Minkowski
time defined by an observer at the center of the shell. The explicit relation
between $u_{in}$ and Schwarzschild coordinates can be computed from the
matching conditions that follow from demanding continuity of the metric across
the shell, namely

$u=u_{\rm in}-4m\log\left(\frac{v_{s}-u_{\rm in}-4m}{4m}\right).$ (2.10)

The state representing gravitational collapse. The fireworks model describes
the physics of a collapsing shell that would classically lead to the formation
of a spherical black hole spacetime. This physical situation imposes clear-cut
constraints on the initial conditions of the quantum state of the field
$\phi$. On the one hand, the state for the in-modes of the quantum fields on
${\mathfs{I}}^{-}$ must not be substantially excited. In other words, aside
from the zeroth order matter distribution defining the collapsing shell that
will lead to the formation of the trapped regions in the future, no
substantial amount of energy momentum of $\phi$ is poured in from
${\mathfs{I}}^{-}$.333In Appendix A.3 we study the contrasting situation where
an infinite amount of radiation is sent from infinity: the Hartle-Hawking
state. This is translated into the demand that the in-modes of the quantum
field on ${\mathfs{I}}^{-}$ must be in the vacuum state. A similar boundary
condition must hold also for the out-modes in the flat interior of the
collapsing shell (Region I). Small perturbations of these conditions could be
admitted yet, and this would not change the conclusions that will follow.

These two conditions are satisfied by the so-called vacuum in-state $\ket{in}$
[Unruh, 1976], defined as the unique vacuum state of the Fock space where
positive frequencies are defined with respect to the mode expansion of
solutions of (2.6) of the form

$\phi_{in}=e^{i\omega v},\qquad\phi_{out}=e^{i\omega u_{in}}\;.$ (2.11)

This state corresponds to the required physical condition that there is no
incoming radiation from ${\mathfs{I}}^{-}$ as well as no outgoing radiation
from inside the shell. This state represents the idealized physical situation
one wants to describe in the context of gravitational collapse.

The region of applicability. There is uncertainty on the features of the
quantum fields in the future domain of dependence of Region III+tIII as the
effective $1+1$ geometry is expected not to capture all the physics of the
dynamics of the field through that part of the spacetime. Therefore, all of
the components of $\langle\,T_{ab}(g_{ab}^{0})\rangle$ that we want to compute
can be used to describe the energy momentum expectation value only in Region I
and in the portion of Region II in the past of $\Sigma_{\mathcal{E}\to\Delta}$
union the null outgoing ray $u=u_{\Delta}$ starting at $\Delta$ and reaching
${\mathscr{I}}^{+}$.

Nevertheless, whatever might be the dynamics in the strong quantum region, we
expect to be able to predict without uncertainties at least some of the
components of $\langle\,T_{ab}(g_{ab}^{0})\rangle$ for those points to the
future of the horizon ${\mathfs{H}}_{+}$. A closer look shows that, due to the
decoupling of in and out modes for a conformal theory in the present $1+1$
context, the component $\langle\,T_{vv}(g_{ab}^{0})\rangle$ is independent of
the features of the quantum Region III+tIII. Both
$\langle\,T_{uu}(g_{ab}^{0})\rangle$ and $\langle\,T_{uv}(g_{ab}^{0})\rangle$,
on the other hand, will be modified by quantum gravity effects. In those
regions of applicability, the computation comes out to be a standard
computation [Davies et al., 1976, Birrell and Davies, 1984], well illustrated
for instance in [Fabbri and Navarro-Salas, 2005].

With these preliminary considerations stated, we are now ready to compute the
expectation value of the energy momentum tensor in the vacuum in-state defined
on the background geometry of the fireworks spacetime. In the region of
interest, and for $\langle\,T_{vv}(g_{ab}^{0})\rangle$ we can simply import
the results from the standard calculation on a background given by the
gravitational collapse of a shell of mass $m$. Following for instance [Fabbri
and Navarro-Salas, 2005], see Appendix A.3, the components of the covariant
quantum stress-energy tensor are given by

$\begin{split}\bra{in}T_{uu}\ket{in}&=\frac{\hbar}{24\pi}\left[-\frac{m}{r^{3}}+\frac{3}{2}\frac{m^{2}}{r^{4}}-\frac{8m}{(u_{in}-v_{s})^{3}}-\frac{24m^{2}}{(u_{in}-v_{s})^{4}}\right]\\\
\bra{in}T_{vv}\ket{in}&=\frac{\hbar}{24\pi}\left[-\frac{m}{r^{3}}+\frac{3}{2}\frac{m^{2}}{r^{4}}\right]\\\
\bra{in}T_{uv}\ket{in}&=-\frac{\hbar}{24\pi}\left(1-\frac{2m}{r}\right)\frac{m}{r^{3}}\;.\end{split}$
(2.12)

While the above equations seem to show that
$\langle\,T_{ab}(g_{ab}^{0})\rangle$ is finite everywhere, they do not. The
problem is that the Eddington-Finkelstein coordinates used to compute them are
not well defined at the trapping horizons: the modes are infinitely
oscillating there. A clear analysis of the divergence behavior of the tensor
$\langle\,T_{ab}(g_{ab}^{0})\rangle$ can be achieved by using good coordinates
close to the trapping horizons. The expectation value of the energy momentum
tensor in our state can be shown to be regular in whole Region II, see for
instance [Fabbri and Navarro-Salas, 2005]. What about Region tII?

Only $\langle\,T_{vv}(g_{ab}^{0})\rangle$ is relevant for the rest of our
analysis: as mentioned above, indeed, it is the only component of the energy
momentum tensor for which (2.12) can be trusted in the future of
${\mathfs{H}}_{+}$ independently of the unknown geometry of Region III+tIII. A
suitable choice of good coordinates are the Minkowski null coordinates
$(u,v_{\rm out})$ in terms of which the metric in Region tI takes the form

$ds_{0}^{2}=-dudv_{out}.$ (2.13)

Continuity of the metric across the outgoing shell implies

$v=v_{\rm out}-4m\log\left(\frac{u_{s}-v_{\rm out}-4m}{4m}\right).$ (2.14)

Since, by definition, $\langle\,T_{ab}(g_{ab})\rangle$ is covariant, one finds

$\bra{in}T_{v_{\rm out}v_{\rm out}}\ket{in}=\left(\frac{dv}{dv_{\rm
out}}\right)^{2}\bra{in}T_{vv}\ket{in}=\left(\frac{u_{s}-v_{\rm
out}}{u_{s}-v_{\rm out}-4m}\right)^{2}\bra{in}T_{uu}\ket{in}\;.$ (2.15)

In these coordinates and on the outgoing shell, $u_{s}-v_{\rm out}=2r$. The
above quantity diverges at the white hole trapping horizon $r=2m$ (which in
the patchwork construction of [Haggard and Rovelli, 2015] is close to
${\mathfs{H}}_{+}$) as $(r-2m)^{-2}$. This divergence of $\langle
T_{ab}\rangle$ is, as we have just shown, explicit in the simplified $1+1$
context.444In the same way one can show that all the components of the
renormalized energy momentum tensor remain finite at the future horizon (close
to ${\mathfs{H}}_{-}$). However, it is a general feature that remains valid in
the physical $3+1$ context. Some references where explicit calculations are
given are [Christensen and Fulling, 1977, Candelas, 1980, Balbinot et al.,
1999]. All this is implied by the very general result implying that the
Hartle-Hawking state is the only globally nonsingular state—satisfying the
Hadamard condition that implies the regularity of $\langle T_{ab}\rangle$—on
the maximally extended Schwarzschild spacetime which is invariant under
Killing time translations [Wald, 1995].

We conclude that in the vacuum in-state the expectation value of energy-
momentum tensor diverges at the trapping horizon $r=2m$ close to
${\mathfs{H}}_{+}$. However, this horizon is outside the region of validity of
our calculation as defined above: it is completely inside the future domain of
dependence of the quantum Region III+tIII.555One can try to interpolate the
black hole patch with the white hole one by an effective metric, see for
example [Barceló et al., 2015]. This is however not relevant for our
discussion. Nonetheless, the would-be-divergent component is still
problematic. The reason is that the trapping horizon and ${\mathfs{H}}_{+}$
get exponentially close to each other along the generators of
${\mathfs{H}}_{+}$.

More precisely, let us call $r_{s}$ the value of the radius at the
intersection of ${\mathfs{H}}_{+}$ and the outgoing shell; see Fig. 2.1. From
the integration of the null geodesic equation, one finds

$\begin{split}r_{s}&=2m\left(1+W\left[\frac{r_{\Delta}-2m}{2m}\exp\left\\{\frac{r_{\Delta}-2m}{2m}-\frac{\Delta
u}{4m}\right\\}\right]\right)\end{split}$ (2.16)

where $W[x]$ is the Lambert function and $\Delta u=u_{s}-u_{\Delta}$. Clearly,
$r_{s}$ represents the closest point to the past horizon for which we can
trust the expression of $\bra{in}T_{v_{\rm out}v_{\rm out}}\ket{in}$ given in
equation eq. (2.15). Consequently, it also gives the largest possible value of
that component of the energy momentum tensor. At that point we have

$\begin{split}\bra{in}T_{v_{\rm out}v_{\rm
out}}\ket{in}\rvert_{r_{s}}&=\frac{\hbar}{24\pi}\left(\frac{r_{s}}{r_{s}-2m}\right)^{2}\left[-\frac{m}{r_{s}^{3}}+\frac{3}{2}\frac{m^{2}}{r_{s}^{4}}\right]\\\
&\sim-\frac{\hbar}{192\pi}\left(\frac{\exp\left\\{\Delta
v/(4m)-(r_{\Delta}-2m)/(2m)\right\\}}{r_{\Delta}-2m}\right)^{2}\;,\end{split}$
(2.17)

where we used the fact that $r_{s}\to 2m$ and that, by construction,

$\Delta u=\Delta v.$ (2.18)

Demanding the quantum energy-momentum tensor to be sub-planckian everywhere,
we can find a relation between the two parameters of the models, namely
$r_{\Delta}$ and $\Delta v$. In fact, ($\hbar=1$)

$\lvert\bra{in}T_{v_{\rm out}v_{\rm out}}\ket{in}\rvert<1$ (2.19)

implies

$\left(\frac{r_{\Delta}}{2m}-1\right)e^{\frac{r_{\Delta}}{2m}-1}>\frac{1}{\sqrt{768\pi}\,}me^{\frac{\Delta
v}{4m}}\quad\Rightarrow\quad\frac{r_{\Delta}}{2m}-1>W\left[\frac{1}{\sqrt{768\pi}\,m}e^{\frac{\Delta
v}{4m}}\right]\,.$ (2.20)

The longer the lifetime $\Delta v$ of the hole, the more the quantum region
must extend out of the classical horizon (as parametrized by $r_{\Delta}$) in
order for the stress-energy tensor to be subplanckian along
${\mathfs{H}}_{+}$. In particular, if, as estimated in [Haggard and Rovelli,
2015], $r_{\Delta}=\frac{7}{3}m$ (see eq. (2.3)), condition (2.20) implies

$\Delta v\lesssim m\log m\;.$ (2.21)

That is, if we do not want trans-Planckian behaviors of the renormalized
quantum stress-energy tensor, the lifetime of the hole has to be so short that
the model would already be ruled out by present observations. For instance,
the characteristic time $\tau=m\log(m)$ would be of about $10$ minutes for the
central supermassive black hole in our Milky Way. For the same black hole one
could try to tune the parameter $r_{\Delta}$ to allow a lifetime of order
$m^{2}$; however, a simple look at equation (2.20) shows that this would imply
extending the quantum region outside of the horizon to include almost the
whole of the observable universe.

#### 2.4 Asymmetric Fireworks

The issues presented in the previous section constrain the white hole lifetime
to be much shorter than the one defined in the original paper. Similar
constraints can be found from simple classical considerations.666Personal
communications with Eugenio Bianchi and Matteo Smerlak. In all cases the
problems are related to the instability due to the presence of a white hole
horizon: infinite blueshift of perturbations that are well behaved at
${\mathfs{I}}^{-}$. Our argument is related to those classical instabilities
if we replace the concept of perturbations by quantum fluctuations in the in-
vacuum. However, an important point is that, in all cases, the constraints
concern the lifetime of the white hole horizon only. The lifetime of the black
hole horizon (which is the one constrained by observations) can be freely set
without running into the present type of instabilities.

This can be easily seen from eq. (2.16). The relevant parameter for our
discussion is the $\Delta u$ that we identified with $\Delta v$, due to the
choice made originally in [Haggard and Rovelli, 2015] to place the point
$\Delta$ on the surface $t=0$. Discarding the identification (2.18) and
following exactly the same procedure, the crucial bound in eq. (2.21) now
becomes

$\Delta u\lesssim m\log m\;.$ (2.22)

A possible way out, therefore, is to abandon the time-symmetric nature of the
bounce in the original form of the fireworks model. More precisely, to avoid
the time-symmetric condition $\Delta u=\Delta v$ one can modify the
construction of the spacetime (Section 2.1) by choosing the outgoing bouncing
shell to come out at a retarded time $U_{s}$ different from $-V_{s}$. The
resulting spacetime, depicted in Fig. 2.2, differs from the original one as if
the point $\Delta$ has been moved away from the $t=0$ surface along a curve
$r=r_{\Delta}$.

Figure 2.2: Geometry of a asymmetric bouncing scenario. The parameter $\Delta
v\sim m^{2}$ captures both the accumulation time for quantum gravitational
effect outside the horizon and the lifetime of the hole. The parameter $\Delta
u$, on the other hand, represents the lifetime of the white hole and it is
forced by the arguments presented in the text to be of order $m\log m$.

In particular, one can choose the value of $U_{s}$ such that the quantum
stability requirement, expressed by eq. (2.22), is satisfied. Moreover, the
analysis of the nonclassicality parameter presented at the end of Section 2.1
is still precisely valid, and so are eq. (2.3) and (2.4). The accumulation of
quantum gravitational effects outside the horizon that allows the black-hole-
to-white-hole transition has not been modified, and the above instabilities
are removed simply by shortening the lifetime of the white hole horizon.

In Section 2.6 we will largely discuss the nature and the consequences of time
asymmetry introduced in our modification of the model. Here we just want to
emphasize that the lifetime of the whole process (from collapse to
annihilation) remains of the order of $m^{2}$ as in the original model, much
shorter than the $m^{3}$ time scale predicted by Hawking evaporation.777In
doing this simple comparison between time scales we are making a little abuse
of notation. For a more precise statement, see the precise analysis reported
in Appendix A.2. This implies that the nature of the time asymmetry is not a
dissipative effect due to the Hawking evaporation as one could intuitively
expect: the energy radiated after a time of the order of $m^{2}$ is just of
the order of the Planck mass $m_{P}$. The Hawking effect is negligible and the
processes discussed here are basically nondissipative.

#### 2.5 Black-Hole-to-White-Hole Instability

The modification proposed also removes another related type of instability
studied in [Eardley, 1974, Lake, 1978, Blau, 1989, Barrabès et al., 1993, Ori
and Poisson, 1994]. The idea is the following. Since a white hole is
attractive, any small perturbation of ambient matter will be accelerated
toward it. At the same time, since no matter can cross the white hole horizon,
after a sufficiently long time, a macroscopic mass will be accreted onto an
arbitrarily thin shell close to the horizon and will produce, when interacting
with any object coming out from the white hole, a new collapse into a future
singularity.

Figure 2.3:  Left: The Dray-’t Hooft geometry. Four Schwarzschild patches with
different masses are glued together along null shells which intersect at a
radius $r_{s}$. The condition for the geometries to be glued smoothly
generates a relation between the four masses and $r_{s}$. Right: The death of
a white hole. A white hole emits all its mass $m$ along a massive null shell
at the retarded time $u_{s}$. A small massive perturbation $\nu$ is sent at
the advanced time $v_{\Delta}$ into the white hole geometry and interact with
the outgoing shell at $r=r_{s}$. At the interaction point we have a Dray-’t
Hooft geometry with $m=m$, $\mathcal{M}=0$, $M=m+\nu$. The last mass $\mu$ is
uniquely determined by the constraint which leads to eq. (2.23):
$\mu=r_{s}/(r_{s}-2m)\nu$. If $r_{s}$ is lower then $2\mu$, the emerging shell
is captured in the future black hole horizon of the new geometry generated by
the interaction and cannot escape to infinity. The white hole is dead
recollapsing into a black hole.

The interaction between any small matter perturbation of mass $\nu$ sent for
instance along the null geodesic $v=v_{\Delta}$ and the outgoing mass $m$
shell at $r=r_{s}$ can be described by a Dray-’t Hooft geometry [Dray and ’t
Hooft, 1985] (Fig. 2.3-Left). The spacetime for $v>v_{\Delta}$ and $u>u_{s}$
is a Schwarzschild geometry with mass $\mu$, given as a function of the
initial parameter $m$, $\nu$ and the radius $r_{s}$ by

$\mu=\frac{r_{s}}{r_{s}-2m}\,\nu\;.$ (2.23)

It is clear now that if $r_{s}<2\mu$ the outgoing shell is captured inside the
new black hole horizon and cannot escape to infinity: any small perturbation
$\nu$ interacting with the bouncing shell will cause the system to recollapse
into a black hole of mass $\mu$; see Fig. 2.3-Right. Thus, no fireworks can be
seen from infinity. The model is, however, still valid if $r_{s}>2\mu$ or
equivalently if

$r_{s}>2(m+\nu).$ (2.24)

From equation (2.16) we get

$\frac{\Delta
u}{4m}<\left(\frac{r_{\Delta}-2(m+\nu)}{2m}\right)\log\left(\frac{{r_{\Delta}}-2m}{2\nu}\right)\;,$
(2.25)

and assuming $\nu\ll m$ we find again

$\Delta u\lesssim m\log\left(\frac{m}{\nu}\right)\;.$ (2.26)

The tunneling process is strongly unstable under perturbations if $\Delta u$,
the lifetime of the white hole, is bigger then of order $m\log m$. This
argument could surely be discussed together with the other issues that have
forced us to consider an asymmetric bouncing scenario, but presented in this
way the different time scales involved become clear. This has been extensively
discussed in a recent paper by Barceló et al. [Barceló et al., 2016] in the
context of the original symmetric model. The asymmetric modification that we
have introduced here also cures this instability.

#### 2.6 Smashing Watches

In this section we want to discuss the physical consequences of the
introduction of a time asymmetry in the model. The bouncing process can be
described by a quantum field in the $\ket{in}$ vacuum state on
${\mathfs{I}}^{-}$ evolving into a final state $\ket{out}$ on
${\mathfs{I}}^{+}$. Both states represent an idealized flat initial geometry
with an infinitely diluted, but sharply defined, spherical shell carrying mass
$m$. More precisely, from the point of view of an observer at infinity, the in
and the out classical data are just equivalent.

On the other hand, the semiclassical analysis of the dynamics of the state
$\ket{in}$ across the spacetime tells us that the state in the future must be
very different from what it was in the past. We have actually shown that the
components of $\langle\,T_{ab}(g_{ab}^{0})\rangle$ are perfectly smooth for
$u<u_{\Delta}$ while they are dangerously diverging in some regions to the
future $u>u_{\Delta}$. These divergences can be cured by modifying the
background in consistency with this time asymmetry. We have achieved this by
shortening the lifetime of the white hole in Section 2.4; see Figure 2.2.

Nevertheless, in doing so we have preserved the equivalence of the past and
future classical data. The point we want to emphasize here is that the time-
asymmetric nature of the inner spacetime needed to avoid instabilities should
imply strong modifications also in the classical final _out_ description of
the model, that can be very different from the simple mean field approximation
proposed by the fireworks model.

One can illustrate the point in terms of the nonclassicality parameter $q$,
eq. (2.1). Recall that the idea is that quantum effects accumulate from
$v=v_{s}$ along the world line of a stationary observer at $r_{\Delta}$ until
the quantity $q$ becomes of order one at $v=v_{\Delta}$. This happens after a
time of the order of $m^{2}$. Let us now run the process backward in time.
This inverse process is still a bounce now described by an initial state given
by $\ket{out}$ evolving into $\ket{in}$. Its dynamics is given by the time
reversal of the original one. However, as $q$ only knows about the local
geometry, one finds that for the reverse process $q$ is far from unity at
$\Delta$. This means that something must be very different for the later
observer; something else (not explicitly stated in the model of Figure 2.2)
must contribute to the nonclassicality so that it builds up very much quicker
in the inverse process.

If correct, the cause of the shortening of time scales in the future of the
bounce must be found in the details of the quantum state of the system beyond
the mean-field approximation implicitly used when proposing a background
geometry. Notice that the future observer is exposed to quantum gravitational
effects coming from the would-be-singularity—whatever replaces the singularity
predicted by the classical theory, i.e. Region III+tIII. These effects must be
important enough to drastically reduce the lifetime of the white hole from
$m^{2}$ to $m\log m$.

But then if these quantum gravitational effects are so strong, why should we
trust a semiclassical description at all in the vicinity of the white hole?
Why should the spacetime become classical again so quickly with the mass $m$
entirely carried by a spherical bouncing shell? It is hard to address these
questions without a full quantum dynamical treatment.

Nevertheless, the standard collapse process strongly suggests irreversibility
already at the classical level. Gravitational collapse is like breaking a
watch. This can be intuitively seen, from the classical point of view, by
considering the standard spacetime depicting the gravitational collapse of a
spherical shell (put the diagram on the right of Figure 2.3 upright). Initial
states given by the shell plus smooth matter and geometry perturbation at
${\mathfs{I}}^{-}$ are special, they are ‘low-entropy’ states representing our
‘watches’. They come in different types depending on the details of the
initial state. This states are bound to evolve into very complicated final
states: smashed watches. This is clear from the fact that only a very precise
fine tuning of the features of the state at ${\mathfs{I}}^{+}\cup i_{+}\cup H$
would evolve backwards to our nice watch at ${\mathfs{I}}^{-}$ (those final
states are measure zero in the phase space of possible final states).

The previous irreversibility mechanism becomes even more apparent if quantum
gravity is brought into the discussion. Everything that crosses the horizon
${\mathfs{H}}_{-}$ will end up at the would-be-singularity exciting degrees of
freedom that were not available at low energies.

Figure 2.4: Geometry of a general asymmetric scenario. The region where the
semiclassical analysis breaks down is shaded. The metric outside is isomorphic
to Schwarzschild with mass $m$.

The phase space regions available for these falling degrees of freedom can
become dramatically larger with the potential effect of further increasing the
irreversibility of the overall process. Concretely, as the shell approaches
$r=0$ more and more degrees of freedom get excited: from known standard model
degrees of freedom (quark-gluon plasma phase, Hagedorn exponential growth of
available degrees of freedom, etc.) to beyond standard model degrees of
freedom and all the way down to Planck scale. At that ultimate fundamental
level, in an approach like LQG, quantum geometries are degenerate: the phase
space of available ‘geometries’ at the Planck scale includes a huge number of
configurations (microstates) which are simply overlooked in the low energy
coarse graining associated with the semiclassical background geometry proposed
to describe the process.888These microstates are responsible for black hole
entropy in LQG [Barbero G. and Perez, 2015] and have been argued to provide a
simple, natural resolution of Hawking’s information loss paradox in [Perez,
2015] in the more conservative framework where Hawking evaporation is the main
quantum effect for BHs with $m\gg 1$ [Ashtekar and Bojowald, 2005]. All this
implies the type of irreversibility proper to systems that satisfy the second
law of thermodynamics.

In view of all this we find no reason to discard scenarios where the spacetime
does not become semiclassical so quickly to the future of the bounce and where
the initial mass $m$ shell dissolves into a quantum substance after the
bounce. The details can only be described in the context of full quantum
gravity. This very uncertain state of affairs is represented in Figure 2.4.

#### 2.7 Conclusions

We have explored certain instabilities of the fireworks scenario proposed in
[Haggard and Rovelli, 2015] and have proposed a simple way to resolve them.
These instabilities are all associated with the presence of a white hole
trapping horizon that is sufficiently long lived. General considerations
demand the gravitational collapse (even in the fast scenario of fireworks
where Hawking radiation does not play an important dynamical role) to be time-
asymmetric and it is precisely by allowing such asymmetry that the
instabilities are resolved. In this way the black hole phase lasts a time of
order $m^{2}$ followed by an extremely fast explosion where the mass $m$ is
radiated back to infinity in a time shorter than $m\log(m)$ in Planck units
($10^{-4}s$ for a solar mass BH, $10^{-9}s$ for a lunar mass BH). The same
considerations of the irreversible nature of the gravitational collapse lead
to uncertainties in the description of the details of this late bounce. A more
precise (not yet available) quantum gravity description of the dynamics across
the would-be-singularity could shed light on these details. It is possible
that, despite these uncertainties, the scenarios discussed here could lead to
some generic observable phenomenology (for instance the $m\log(m)$ explosion
scale). We leave this question to the experts.

##

Part II

Spacetime and Thermodynamics

In the introduction to the previous Part, I have briefly mentioned that the
debate around the information paradox is crucially related to what is the deep
meaning of the Bekenstein-Hawking entropy. In this introductory section to the
second part of the manuscript I would like to discuss what is the BH entropy,
how it arises and its importance for the physics of gravitation, as well as
giving an overview on the different approaches to its explanation. To do so,
it is useful to abandon Planck units for a while and keep all the constants of
Nature explicit.

The idea of assigning an entropy to BHs was first introduced by Jacob
Bekenstein [Bekenstein, 1972, Bekenstein, 1973]. His proposal was grounded on
previous results that suggested an analogy between a BH and a thermodynamical
system. A first result is the _no-hair theorem_ [Carter, 1971, Hawking, 1972],
stating that any stationary axisymmetric BH can by described, no matter how it
has been formed, by means of only three parameters: the mass $M$, the angular
momentum $J$ and the charge $Q$. This is reminiscent of a description of a
system with only macroscopic thermodynamical variable such as energy,
temperature and pressure. A second result is Hawking’s _area law theorem_
[Hawking, 1972], showing that the area of the horizon can never decrease when
classical processes, such as matter satisfying the weak energy falling into
the hole, are considered. Again a reminiscence of the second law of
thermodynamics for which the entropy of a system can only increase. If an
external observer releases a box of particles into the horizon of a BH, the
initial entropy of the box becomes inaccessible to him. Given the no-hair
theorem, moreover, the BH can be described by only three parameters, and there
is no way for the external observer to determine its interior entropy. He
might therefore conclude that the entropy of the Universe has decreased, in
contradiction with the second law of thermodynamics. The problem is resolved
if one assigns an entropy to the BH so that it compensates the loss of the
entropy of the box [Bekenstein, 1972, Bekenstein, 1973]. Inspired by the area
law theorem, Bekenstein suggested to assign a dimension-less entropy to the BH
proportional to its horizon area

$S_{\scriptscriptstyle BH}=\eta\frac{A}{\ell_{\scriptscriptstyle P}^{2}}\,,$
(2.27)

where $\eta$ is a dimension-less parameter. The Planck area at the denominator
was introduced only on dimensional grounds. At the same time he proposed a
_generalised second law (GSL)_ which states that the sum of the BH entropy and
the total exterior world entropy does not decrease. The BH-thermodynamics
analogy was then completed in [Bardeen et al., 1973], where the four laws of
BH mechanics were precisely staten.

  * _0th law_ :

The surface gravity $\kappa$ of the horizon is constant on the horizon itself.

  * _1st law_ :

For two stationary black holes differing only by small variations in the
parameters $M$, $J$, and $Q$,

$c^{2}\,\delta M=\frac{c^{2}\,\kappa}{8\pi\,G}\delta
A+\Omega_{\scriptscriptstyle H}\,\delta J+\Phi_{\scriptscriptstyle H}\delta Q$
(2.28)

where $\Omega_{\scriptscriptstyle H}$ and $\Phi_{\scriptscriptstyle H}$ are
respectively the angular velocity and the electric potential at the horizon.

  * _2nd law_ :

The area of the horizon can never decrease

$\delta A\geq 0\,.$ (2.29)

  * _3rd law_ :

It is impossible by any procedure to reduce the surface gravity $\kappa$ to
zero in a finite number of steps.

For the derivation of the four laws I refer to the original paper [Bardeen et
al., 1973], to [Wald, 1995] for an alternative physical derivation of the
first law, as well as to Chapter 3, where the latter is generalised to light
cones in Minkowski spacetime. The zeroth and the first law identify (a
multiple of) the surface gravity $\kappa$ as the temperature, reinforce the
analogy area-entropy, and designate the change in charge and angular momentum
as work terms on the system.

As the treatment of [Bardeen et al., 1973] is made within purely classical GR,
no $\hbar$ enters the game. Moreover, let me quote a very explicative passage
of the original paper

> _It should however be emphasised that $\kappa$ and $A$ are distinct from the
> temperature and entropy of the black hole. In fact the effective temperature
> of a black hole is absolute zero. One way of seeing this is to note that a
> black hole cannot be in equilibrium with black body radiation at any non-
> zero temperature, _because no radiation could be emitted from the hole_
> whereas some radiation would always cross the horizon into the black hole._
> [Bardeen et al., 1973]

This view drastically changed with Hawking’s result that black holes radiate
particles as a black body at temperature

$T_{\scriptscriptstyle H}=\frac{\hbar\,\kappa}{2\pi c\,k_{\scriptscriptstyle
B}}$ (2.30)

would do. When the above expression is inserted into the first law one finds

$\frac{c^{2}}{k_{\scriptscriptstyle B}}\delta M=T_{\scriptscriptstyle
H}\frac{c^{3}{}\delta A}{4\,\hbar\,G}+\frac{\Omega_{\scriptscriptstyle
H}}{k_{\scriptscriptstyle B}}\,\delta J+\frac{\Phi_{\scriptscriptstyle
H}}{k_{\scriptscriptstyle B}}\delta Q$ (2.31)

where both sides have now dimensions of a temperature. This equation allows to
identify a dimension-less entropy for a black hole as

$S_{\scriptscriptstyle
BH}=\frac{c^{3}A}{4G\,\hbar}=\frac{A}{4\ell_{\scriptscriptstyle P}^{2}}\,.$
(2.32)

Bekenstein’s idea became a fact of semi-classical gravity: BHs have an entropy
which is proportional to the area of the horizon. The proportionality constant
$\eta$ in (2.27) is fixed to be $1/4$. The above expression in Planck units
becomes the familiar $S_{\scriptscriptstyle BH}=A/4$.

Let me emphasise three interesting features. First of all,
$S_{\scriptscriptstyle BH}$ is huge! For a solar mass black hole it turns out
to be $10^{18}$ times the thermal entropy of our burning sun. Second, imagine
dropping a mass of $1\,Kg$ into the supermassive BH at the centre of our
galaxy. Its entropy change in this process is of the same order of (again) the
thermal entropy of the sun, and, more importantly, it does not depend on the
initial entropy of the mass that fell in. These observations give a first idea
about the mysterious nature of such entropy. Third, $S_{\scriptscriptstyle
BH}$ scales with the 2-dimensional area, contrary to the 3-dimensional volume
scaling of the entropy of standard thermodynamical systems.

As for the information paradox described in the introduction to Part I, a
complete understanding of $S_{\scriptscriptstyle BH}$ would probably need a
complete theory of quantum gravity. A first way to see this is the following:
imagine to divide the area of the horizon in triangular cells of area $s$. One
has therefore $N=A/s$ number of cells. By for instance colouring such cells
black or white, one can encode information on it. The maximum number of
combinations is $2^{N}$, therefore allowing the encoding of $\sim N=A/s$ bits
of information. If this information were to be lost, this would result in a
entropy $S\sim A/s$. This shows that to match the BH entropy, the dimension of
the cells needs to be $s=\ell_{\scriptscriptstyle P}^{2}$, therefore appealing
to quantum gravity for its explanation. Another way is to push the analogy
with thermodynamics further and ask the central question: can this entropy be
thought as a counting of the total number of possible microscopic states
corresponding to the same macroscopic parameters $M$, $J$ and $Q$? The
definition of micro-states for a geometric objet such a BH seems to point
again toward a quantum description of spacetime, and therefore a theory of QG.
Once more, “black holes as a gateway to the quantum”. And once more, since a
full QG theory is not available, it is crucial to try to find as more insights
as possible from semi-classical reasonings, in the very spirit of this thesis
999More recently, a new mathematically elegant and clean derivation of the
first law identified $S_{\scriptscriptstyle BH}$ with the Noether charge
associated with the diffeomorphism generated by the horizon Killing field
[Wald, 1993]. Such an additional classical identification may provide useful
insights for a more fundamental statistical explanation of the entropy..

The debate around the nature of the BH entropy presents two main conceptually
different voices. On one side is the proposal that $S_{\scriptscriptstyle BH}$
counts in some way the number of micro-states of the BH that can communicate
with the world outside the horizon [Jacobson et al., 2005]. This idea, which I
would call _horizon interpretation_ , is realised in two different ways. In
the first approach, $S_{\scriptscriptstyle BH}$ is considered to be given by
the entanglement entropy (EnEn) of quantum fields across the horizon [Bombelli
et al., 1986]. Being related to the Ultra-Violet (UV) structure of fields in
the proximity of the entangling surface, the EnEn is proportional to the area
of the latter. The entropy of the BH is therefore interpreted as a semi-
classical feature which does not need to be necessarily related to quantum
gravitational degrees of freedom. Nonetheless, the EnEn is a UV divergent
quantity; to match the BH entropy, therefore, one has to evoke a UV cutoff at
Planck scale, bringing QG back in play. Moreover, the EnEn is also
proportional to the number of different field species which exist in Nature,
at odd with the fact that $S_{\scriptscriptstyle BH}$ seems to have a
universal character [Solodukhin, 2011]. The other manifestation of the horizon
interpretation is that $S_{\scriptscriptstyle BH}$ counts the quantum
geometrical degrees of freedom sitting on the horizon surface. This view is
supported by computations in Loop Quantum Gravity, where an explicit counting
of the vast degeneracy of quantum surface states compatible with the horizon
of a BH can be performed [Perez, 2017]. The incompleteness of LQG, as well as
approximations and some ambiguities, make these computations inconclusive.

On the other side of the debate one finds the idea of interpreting the fact
that $S_{\scriptscriptstyle BH}$ scales as in a lower dimensional system as an
indication of some new fundamental aspects of Nature. This led to the
formulation of the so called _holographic principle_ [Susskind, 1995]. It is a
postulated new principle of Nature, at the level of the equivalence, the gauge
or the uncertainty principles. It roughly states that there exists a
fundamental description of the classical physical world in terms of a hologram
on a lower dimensional screen. More precisely, it is a tentative way of rising
up to a fundamental principle the so called _entropy bounds_. The first of
such bounds was proposed by Bekenstein in [Bekenstein, 1981] with the aim of
proving the generalised second law. Indeed, If it were possible to have matter
systems with arbitrarily large entropy at a given mass and size, the GSL would
be violated simply by throwing such matter into a BH. The entropy bound
roughly states that there exists a maximum value of the entropy that a matter
system can have, which is proportional to its energy $E$ times the radius $R$
of the smallest sphere that fits around the matter system: $S\lesssim E\,R$.
In this way the GSL is safe. The bound is corroborated by some examples
[Bekenstein, 1981], but it has been disproven by several counterexamples. The
simplest one is to observe that a collapsing spherical star of given initial
entropy and mass will always reach a small enough radius for the bound to be
violated. Several tentative refinements eventually brought to a formulation of
the bound that it has not been disproven yet, namely Bousso’s covariant
formulation.

> Let $A(B)$ be the area of an arbitrary co-dimension two spatial surface $B$
> (which need not be closed). A co-dimension one hypersurface $L$ is called a
> _light-sheet_ of $B$ if $L$ is generated by light rays which begin at $B$,
> extend orthogonally away from $B$, and have non-positive expansion,
> $\theta\leq 0$ everywhere on $L$. Let $S$ be the entropy on any light-sheet
> of $B$. Then $S\leq A(B)/4$. […] The light-sheet construction is well-
> defined in the limit where geometry can be described classically. It is
> conjectured to be valid for all physically realistic matter systems.
> [Bousso, 2002]

Explaining in full details the Bousso’s bound goes clearly beyond the aim of
this Section, and I refer to [Bousso, 2002] and to a series of enlightening
on-line lectures [Bousso, 2011] for more details. At the same time, the
Bousso’s bound is the most structured basis of the holographic principle,
which in turn takes on a dominant role in the debate around the questions
about BH physics presented in this thesis. I would like therefore to clarify
some points, which I believe are important to better understand the different
views on the issues.

First, reading the statement for the first time one could ask why is the
entropy on (the elegant but not obvious geometric notion of) light-sheets
considered, instead of, for instance, the more natural entropy on any
spacelike surface enclosed in $B$. The answer is that tentatives of defining
entropy bounds using other surfaces have been disproved by simple counter
examples, as for instance the one reported above for the Bekenstein’s bound.
The geometric definition of light-sheets allows to avoid those problems. For a
hint on how light-sheets look like, see the highly symmetric examples in Fig.
2.5.

Figure 2.5: The light-sheets are easily identified in highly symmetric
situations. (a) A surface $B$ of a sphere in flat spacetime has two light-
sheets given by the past and the future ingoing light cones shining from $B$.
In other terms, the boundaries of the future and past Cauchy development of
$B$. The entropy on any light sheet in this case is therefore the same as the
entropy contained in the 3-dimensional sphere. This is not the case in general
curved geometries, as for instance in (b) a spacetime describing a spherically
symmetric collapsing star forming a BH. The light sheet for three different
spherical 2D-surfaces are depicted. For a sphere $B_{\rm out}$ outside the
horizon, the two light-sheets are the analogue for the case in flat spacetime
and again the entropy on them is the same as for the 3-dimensional sphere
enclosed. For a sphere $B_{\rm in}$ in the interior part of the black hole, on
the other hand, the light-sheets are the two future directed light cones. In
this picture $B_{\rm in}$ is the outer boundary of the collapsing star at a
given time. The entropy of matter on the out-going one is zero, thus different
from the entropy of the enclosed 3-d sphere. The in-going light-sheet, on the
other hand, intersect all the collapsing matter. If imagined at later times,
at some point such light-sheet would not intersect all the matter anymore,
terminating at the singularity. This indicates how the Bousso’s bound avoids
the simple counterexample to the Bekenstein’s bound reported in the main text.
See [Bousso, 2002] for more details. Finally the most interesting surface
depicted is the one on the horizon, $B_{\scriptscriptstyle H}$. The light-
sheets associated to it are three: the two pieces of horizon emanating from
it, past and future, and the in-going light cone terminating either at the
regular centre of the star, or at the singularity depending on the cross-
section.

On the other hand, identifying them in general curved spacetimes and for
general surfaces $B$ is usually hard if not impossible. Due to this
difficulty, the Bousso’s bound has been shown to hold only in highly symmetric
situations, and no general proof is available. Second, as written in the
statement, the definition of light-sheets is possible only for classical
spacetimes and the entropy which is bounded is the entropy given by classical
matter degrees of freedom on it. It is valid therefore for a non-evaporating
black hole formed by a collapsing mass. In spherical symmetric case, there are
three light-sheets associated with a cross-section $B$ of the event horizon at
a given time: the pieces of event horizon to the past and to the future of
$B$, and the ingoing spherical light front reaching the singularity, see Fig.
2.5. The piece of horizon to the past of $B$ is crossed by the matter that
formed the BH. The Bousso’s bound tells us that the entropy of such matter
cannot be bigger than one-fourth the area of the horizon. The future piece
does not encounter any matter, and therefore is trivial. The ingoing light
front might, depending on the cross section, encounter all the matter, part of
the matter to then meet the singularity, or just go straight to the
singularity with no matter in it 101010Notice that the validity of the
construction of such light-sheet is not ensured close to the singularity,
since it is not a well defined classical region.. The entropy of matter on
these last two light-sheets is therefore clearly less or equal to the one in
the first one. No more information is gained by looking at them. To this
point, the bound only gives us information about the entropy of the matter
felt into the hole, but it does not give any hint about the nature of the BH
entropy in the spirit of the discussion above. It does so when the Bousso’s
bound is erected to the holographic principle:

> The area of any surface $B$ measures the information content of an
> underlying theory describing all possible physics on the light-sheets of
> $B$. [Bousso, 2002]

Given this postulate, the BH entropy acquires now the meaning of measure of
all possible degrees of freedom an underlying quantum theory of everything may
provide. This includes eventual quantum gravitational effects at the
singularity, correlations between fields outside and inside the horizon, and
so on. Thus providing an _interior interpretation_ of $S_{\scriptscriptstyle
BH}$, in contrast with the horizon interpretation discussed above. Coming now
from a fundamental principle, moreover, this interpretation can be applied to
cross-sections of the horizon of an evaporating BH, with important
consequences regarding the information paradox. The proposed scenarios arising
from this line of though are discussed at the end of the introduction to Part
I. To conclude, let me mention that there are no explicit realisations of the
holographic principle yet, except for the so called _$AdS$ /CFT
correspondence_ [Maldacena, 1999]. Roughly speaking, in few particular
realisations of String Theory on asymptotically high-dimensional
$A$nti-$d$e$S$itter ($AdS$) spacetimes–such as $AdS_{5}\times S^{5}$–, one can
find a slicing of the spacetime such that the state of strings on each slide
is fully described by data not exceeding $A$ bits, where $A$ is the area of
the boundary of the slice. On this boundary one can define an auxiliary lower
dimensional supersymmetric Conformal Field Theory (CFT)–such as $N=4$ super
Yang-Mills–which generates the unitary evolution of boundary data from slice
to slice, and therefore of the bulk spacetime. The idea that such
correspondence might be true also in more physical situations has earned a
huge consensus in the String Theory community, bringing to conjecture the so
called _gauge/gravity duality_ , and to the original paper [Maldacena, 1999]
being the most cited one in the field of high energy physics. However, even
the less ambitious $AdS$/CFT correspondence has not yet been formulated in
sufficient detail and with sufficient precision to make a clear argument.

I hope this discussion helps clarifying what are the basis on which the two
main different perspectives (with their sub-classes) about the nature of
$S_{\scriptscriptstyle BH}$ rely. None of them gives yet a definitive answer.
Additional classical and semiclassical investigations in the spirit of this
thesis may help choosing between them, or maybe finding a completely new
paradigm.

The ideas discussed above are not the only ones that have arisen from the
suggestive relation between thermodynamics and black holes. In Chapter 5, this
thesis also deals with the so called _thermodynamics of spacetime_ approach
originated in [Jacobson, 1995]. The basic idea is to turn the logic around, in
the following sense. The thermodynamical properties of BHs, with the
consequent proportionality between area and entropy, are derived from the
Einstein’s equations. In [Jacobson, 1995], instead, the relation between area
and entropy is taken as an assumption (among others), and the Einstein’s
equations are derived from the equilibrium Clausius’ relation $\delta S=\delta
Q/T$ between heat $Q$, entropy $S$ and temperature $T$. One of the assumptions
can be relaxed by mean of a non-equilibrium description in which the
generalised Clausius’ relation is used $\delta S=\delta Q/T+\delta S_{i}$
[Eling et al., 2006, Chirco and Liberati, 2010]. In this more general setting
$\delta S_{i}$ represents the irreversible part of the entropy due to non-
equilibrium heat fluxes associated with internal degrees of freedom. In
Chapter 5 the framework is discussed to derive the Einstein-Cartan’s equations
describing a theory of gravity with non-vanishing torsion. In doing so, subtle
aspects of both the equilibrium and non-equilibrium settings are discussed,
and the ambiguity of some assumption underlined. Therefore I refer the reader
to that final Chapter for more details. Here I would only like to conclude
this introductory section with a brief observation. Even though the derivation
is based on many assumptions and it is in my opinion not completely clean, the
underlying idea is quite interesting. From a non-perturbative quantum
gravitational perspective, indeed, one expects that a full QG theory will
provide quantum states describing discrete quantum geometries, and a dynamics
given by a sort of quantum version of the Einstein’s equation. In LQG, for
instance, the dynamics is provided by the quantum version of the Hamiltonian
constraint, which classically is equivalent to the Einstein’s equation. An
interesting point commended in [Jacobson, 1995] is that the result can be
interpreted as suggesting that there may actually not be quantum version of
the Einstein’s equation, since they will arise only as an equation of state
for the thermodynamical equilibrium description of fundamental degrees of
freedom.

###  Chapter 3 Light Cone Thermodynamics

_This Chapter completely overlaps with the published paper_ [De Lorenzo and
Perez, 2018b].

#### 3.1 The Results in a Nutshell

Classical Black Holes behave in analogy with thermodynamical systems [Bardeen
et al., 1973]. According to general relativity they satisfy the four laws of
black hole mechanics. The surface gravity $\kappa_{\scriptscriptstyle SG}$ of
a stationary black hole is constant on the horizon: the zeroth law. Under
small perturbations, stationary black holes—which are characterized by a mass
$M$, an angular momentum $J$, and a charge $Q$—satisfy the first law

$\delta M=\frac{\kappa_{\scriptscriptstyle SG}}{8\pi}\delta A+\Omega\delta
J+\Phi\delta Q,$ (3.1)

where $\Omega$, $\Phi$, and $A$ are the angular velocity, electrostatic
potential, and area of the stationary (Killing) horizon. The Hawking area
theorem [Hawking and Ellis, 1973]

$\Delta A\geq 0$ (3.2)

is regarded as the second law. The third law—expected to be valid from the
cosmic censorship conjecture [Penrose, 1969]—corresponds to the statement that
extremal black holes, for which $\kappa_{\scriptscriptstyle SG}=0$, cannot be
obtained from a non extremal one by a finite sequence of physical processes.
When quantum effects are considered these analogies become facts of
semiclassical gravity. Stationary black holes radiate particles in a thermal
spectrum with temperature $T=\kappa_{\scriptscriptstyle SG}/(2\pi)$ [Hawking,
1974, Hawking, 1975] so that the first term in (3.1) can be interpreted as a
heat term expressed in terms of changes in the black hole entropy $S=A/4$ in
Planck units. The area law (3.2) is promoted to the generalized second law
[Bekenstein, 1973]. In the quantum realm another statement that could be
associated to the third law is that extremal black holes should have vanishing
entropy, an argument for which can be found in [Hawking et al., 1995].

All this is naturally interpreted as providing valuable information about the
quantum theory of gravity of which the semiclassical treatment should be a
suitable limit of. In this respect it is useful to have examples of a similar
behaviour in simplified situations. The Fulling-Davies-Unruh thermal
properties associated to quantum field theory in flat spacetimes [Fulling,
1973, Davies, 1975, Unruh, 1976] is often used as an example illustrating, in
a simplified arena, some of the aspects behind the physics of black holes. In
this respect, the Rindler (Killing) horizon associated with a family of
constantly accelerated observers in Minkowski spacetime is taken as an
analogue of the black hole horizon. This analogy is supported further by the
statement that the near horizon geometry of a non-extremal black hole can be
described, in suitable coordinates, by the metric

$ds^{2}=-\kappa_{\scriptscriptstyle
SG}R^{2}dt^{2}+dR^{2}+(r_{\scriptscriptstyle H}^{2}+a^{2})dS^{2}+O(R^{3})$
(3.3)

where $dS^{2}=d\vartheta^{2}+\sin^{2}\vartheta d\varphi^{2}$ is the metric of
the unit two-sphere, $a\equiv J/M$. At least in the $(R,t)$ “plane”, the above
equation matches the 2d Rindler metric where $R=0$ is the location of the
black hole horizon. The thermodynamical properties of Rindler horizons have
been extensively discussed in the literature [Bianchi and Satz, 2013,
Jacobson, 1995, Wall, 2010, Chirco et al., 2010, Padmanabhan, 2010]. However,
strictly speaking, the near horizon geometry is not Rindler due to the
presence of the term $r_{H}^{2}dS^{2}$ in the previous equation that makes the
topology of the horizon $S^{2}\times{\mathbb{R}}$ instead of
${\mathbb{R}}^{3}$, which implies the area of the black hole horizon to be
finite $A=4\pi(r^{2}_{H}+a^{2})$ instead of infinite. Another obvious
difference is that, in contrast with Rindler, the Riemann curvature is non
zero at the black hole horizon. Only in the infinite area limit the local
geometry becomes exactly that of a Rindler horizon. Furthermore, in contrast
with black hole horizons, the Rindler horizon has a domain of dependence that
includes the whole of what one would regard as the outside region. In fact the
Rindler horizon defines a good initial value characteristic surface. More
precisely, any regular initial data for an hyperbolic equation such as a
Klein-Gordon or Maxwell field with support on the corresponding wedge—what
would be the outside—can be encoded in data on the Rindler horizon [Wald,
1984]. An implication of this is that no energy flow can actually escape to
infinity without crossing the Rindler horizon. No notion analogous to the
asymptotic observers outside of the black hole exists when considering the
Rindler wedge and its Killing horizon boundary. A geometric way to stating
this is that the Rindler horizon is given by the union of the past light cone
of a point at ${\mathfs{I}}^{+}$ with the future light cone of a point at
${\mathfs{I}}^{-}$. In this sense the Rindler wedge is better described as a
limiting case of the interior of finite diamonds—see next paragraph—rather
than representing faithfully the outside region of a black hole spacetime. In
this work we show that there exists a more complete analogue of black holes in
Minskowski spacetime.

There is a natural interest in double cone regions in Minkowski spacetime,
also called diamonds, in algebraic quantum field theory [Hislop and Longo,
1982] or in the link between entanglement entropy and Einstein equations
[Jacobson, 2016]. The conformal relationship with the Rindler wedge has been
used in order to define the corresponding modular Hamiltonian and study
thermodynamical properties in [Martinetti and Rovelli, 2003, Martinetti,
2009]. Here we concentrate on the causal complement of the diamond, and show
that it shares several analogies with the exterior region of a stationary
spherically symmetric black hole.

Such flat spacetime regions as the diamond and its complement are directly
related to the geometry of radial Minkowski Conformal Killing vector Fields
(MCKFs). What we shall show is that radial MCKFs can be classified in a
natural correspondence with black holes spacetimes of the Reissner-Nordstrom
(RN) family (i.e. $J=0$). They can be timelike everywhere in correspondence
with the naked singularity case $M^{2}<Q^{2}$ where the stationarity Killing
field is timelike everywhere. But more interestingly, radial MCKFs can become
null, and being surface forming, generate conformal Killing horizons. As we
will show in Section 3.2 these are conformal bifurcate Killing horizons
analogue, in a suitable sense, to the black hole horizons in the RN family.
The results of this paper can be summarised as follows:

  1. 1.

Radial MCKFs define conformal Killing horizons: Radial MCKFs become null on
the light cones of two events on Minkowski spacetime that are separated by a
timelike interval. By means of a Lorentz transformation these two events can
be located on the time axis of an inertial frame. A further time translation
can place the two events in a time reflection symmetric configuration so that
the symmetry $t\to-t$ of Reissner-Nordstrom spacetimes is reproduced.

  2. 2.

They have the same topology as black hole Killing horizons: The topology of
the conformal Killing horizon in Minkowski spacetime is
$S^{2}\times{\mathbb{R}}$ as for the Killing horizons of the RN spacetime.

  3. 3.

These horizons are of the bifurcate type: Radial MCKFs vanish on a
2-dimensional sphere of radius $r_{H}$ and finite area $A=4\pi r_{H}^{2}$ that
is the analogue of the minimal surface where the Killing horizon of the RN
black hole vanishes. The bifurcate surface is the intersection of the two
light cones described above.

  4. 4.

They separate events in spacetime as in the BH case: The global structure of
the radial MCKF is closely analogous to the one of the Killing horizon of the
RN spacetime. More precisely, there are basically the same worth of regions
where the radial MCKF and the RN time translational Killing vector field is
timelike and spacelike respectively. In the non-extremal case there are outer
and inner horizons in correspondence to the non-extremal RN solution. One of
the two asymptotically flat regions of the maximally extended RN spacetime
corresponds to the points in the domain of dependence of the portion of the
$t=0$ hypersurface in Minkowski spacetime inside the bifurcate sphere, namely
the diamond; the other asymptotically flat region corresponds to the domain of
dependence of its causal complement, namely the black hole exterior in our
analogy. There are regions where the radial MCKF becomes spacelike. These too
are in correspondence with regions in the non-extremal RN black hole, namely
the regions between the inner and the outer horizons. In the extremal limit
the regions where the radial MCKF is spacelike, as well as one of the
asymptotic region, disappear and the correspondence with the extremal RN
solution is maintained. All this will be shown in detail in the following
section; the correspondence is illustrated in Figure 3.1.

  5. 5.

They satisfy the zeroth law: The suitably generalized notion of surface
gravity $\kappa_{\scriptscriptstyle SG}$ is constant on the conformal Killing
horizon: the zeroth law. Extremal Killing horizons have
${\kappa_{\scriptscriptstyle SG}}=0$.

  6. 6.

They satisfy the first law: Considering the effects of matter perturbations
described by a conformally invariant matter model, one can show that radial
conformal Killing horizons satisfy the balance law $\delta
M=\kappa_{\scriptscriptstyle SG}\,\delta A/(8\pi)+\delta M_{\infty}$: _the
first law_. Here $\delta M$ is the conformally invariant mass of the
perturbation, $\delta M_{\infty}$ is the amount of conformal energy flowing
out to future null infinity ${\mathfs{I}}^{+}$, and $\delta A$ is what we call
conformal area change. The name stems from the fact that it corresponds to the
change of a geometric notion with the meaning of horizon area in the
appropriate conformal frame.

  7. 7.

They satisfy the second law: In the type of processes considered above and
assuming the usual energy conditions, $\delta A\geq 0$: the second law.

  8. 8.

They have constant (conformal) temperature: When quantum fields are
considered, a constant Hawking-like temperature $T=\kappa_{\scriptscriptstyle
SG}/(2\pi)$ can be assigned to radial MCKFs. In view of this, $\delta S=\delta
A/4$ in Planck units acquires the meaning of entropy variation of the
conformal horizon.

  9. 9.

They satisfy a version of the third law: Extremal radial MCKFs have vanishing
temperature as well as vanishing entropy: the third law.

  10. 10.

Minkowski vacuum is the associated Hartle-Hawking state: The Minkowski vacuum
of any conformally invariant quantum field can be seen as the state of thermal
equilibrium—usually called Hartle-Hawking state—in the Fock space defined with
respect to the MCKF.

  11. 11.

The near MCKF horizon limit matches the expression (3.3) with $a=0$. The
Rindler horizon limit could be obtained by sending the events mentioned in
Item 1 suitably to future $i^{+}$ and past $i^{-}$ timelike infinity
respectively. In that limit $r_{\scriptscriptstyle H}\to\infty$, the near
horizon metric becomes the Rindler metric, and the corresponding radial MCKF
becomes the familiar boost Killing field.

The properties listed above will be discussed in more detail in the sections
that follow. In Section 3.2 we construct general radial MCKFs from the
generators of the conformal group $SO(5,1)$, and explain their geometry. The
causal domains they define and the analogy with black holes shown in Figure
3.1 will be clarified there. The analogue of classical laws of black hole
thermodynamics are shown to hold in a suitable sense for radial MCKFs in
Section 3.3. Finally we show, in Section 3.4, that a semiclassical temperature
can be assigned to radial MCKFs and we discuss the physical meaning of that
temperature.

#### 3.2 Conformal Killing Fields in Minkowski
Spacetime

The conformal group in four dimensional Minkowski spacetime ${\mathbb{M}}^{4}$
is isomorphic to the group $SO(5,1)$ with its 15 generators given explicitly
by [Di Francesco et al., 1997]

$\begin{split}&P_{\mu}=\partial_{\mu}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ {\rm Translations}\\\
&L_{\mu\nu}=\left(x_{\nu}\partial_{\mu}-x_{\mu}\partial_{\nu}\right)\ \ \ \ \
\ \ \ \ \ \ \ \ {\rm Lorentz\ transformations}\\\ &D=x^{\mu}\partial_{\mu}\ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\rm Dilations}\\\
&K_{\mu}=\left(2x_{\mu}x^{\nu}\partial_{\nu}-x\cdot x\,\partial_{\mu}\right)\
\ \ \ \ \ \,{\rm Special\ conformal\ transformations,}\end{split}$ (3.4)

where $f\cdot g\equiv f^{\mu}g_{\mu}$. 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:

${\mathfs{L}}_{\xi}\,\eta_{ab}=\nabla_{a}\xi_{b}+\nabla_{b}\xi_{a}=\frac{\psi}{2}\eta_{ab}$
(3.5)

with

$\psi=\nabla_{a}\xi^{a}\,.$ (3.6)

Consider now the Minkowski metric in spherical coordinates

$ds^{2}=-dt^{2}+dr^{2}+r^{2}dS^{2}\,.$ (3.7)

Then dilations can be written as

$D=r\partial_{r}+t\partial_{t}$ (3.8)

and $K_{0}$ as

$K_{0}=-2tD-(r^{2}-t^{2})P_{0}\,.$ (3.9)

Together with $P_{0}=\partial_{t}$, those are the only generators that do not
contain angular components. Hence the most general radial MCKF has the form

$\begin{split}\xi=&-aK_{0}+bD+cP_{0}\\\
=&(2at+b)D+[a(r^{2}-t^{2})+c]P_{0},\end{split}$ (3.10)

with $a,b,c$ arbitrary constants. Explicitly

$\begin{split}\xi^{\mu}\partial_{\mu}&=\big{[}a(t^{2}+r^{2})+bt+c\big{]}\partial_{t}+r(2at+b)\partial_{r}\\\
&=(av^{2}+bv+c)\partial_{v}+(au^{2}+bu+c)\partial_{u}\,,\end{split}$ (3.11)

where $v=t+r$, $u=t-r$ are the standard null coordinates. The norm of $\xi$ is
easily computed to be

$\xi\cdot\xi=-(av^{2}+bv+c)(au^{2}+bu+c)\,.$ (3.12)

Its causal behaviour, therefore, can be studied introducing the quantity

$\Delta\equiv b^{2}-4ac\,.$ (3.13)

The complete classification of such MCKFs is given in [Herrero and Morales,
1999]. Here we are interested in the case $a\neq 0$, where we have three
different types of behaviour depending on the sign of the parameter $\Delta$.
When $\Delta<0$, the MCKF is timelike everywhere, like the stationarity
Killing field in the Reissner-Nordstrom solutions with a naked singularity
$M^{2}<Q^{2}$. When $\Delta>0$, on the other hand, the MCKF is null along two
constant $u$ and two constant $v$ null hypersurfaces respectively given by

$u_{\pm}=\frac{-b\pm\sqrt{\Delta}}{2a}\ \ \ \ {\rm or}\ \ \ \
v_{\pm}=\frac{-b\pm\sqrt{\Delta}}{2a}\,.$ (3.14)

In other words, the MCKF is null on the past and future light cones of two
points $O^{\pm}$, with coordinates given respectively by
$O^{\pm}:(t=u_{\pm},r=0)$. These two light cones divide Minkowski spacetime
into six regions. In those regions the norm of the MCKF changes going from
timelike to spacelike as depicted on the top right panel of Figure 3.1. The
boundary of these regions are null surfaces generated by the MCKF; they define
conformal Killing horizons. The vector field $\xi$ vanishes at the bifurcate
2-dimensional surface defined by the intersection of the previous null
surfaces, namely at the sphere

$t=t_{H}=-\frac{b}{2a}\,,\qquad r=r_{H}=\frac{\sqrt{\Delta}}{2a}\,.$ (3.15)

The “extremal” case $\Delta=0$ is a limiting case between the other two: the
MCKF is null on the light cones $u_{0}=v_{0}=-b/(2a)$ emanating from a single
point $O$, and timelike everywhere else. This is depicted in the bottom right
panel of Figure 3.1.

Figure 3.1: The Penrose diagram of the Reissner-Nordstrom black hole on the
left compared with the causal structure of the radial CKF in Minkowski
spacetime on the right, in both the non-extremal $\Delta>0$ and extremal
$\Delta=0$ case. The letters $S$ and $T$ designate the regions where the
Killing or conformal Killing fields are spacelike or timelike respectively.
The light cone emanating from the points $O^{\pm}$ (and $O$ in the extremal
case) are the hypersurface where the MCKF is null.

It is interesting to notice that the four regions around the bifurcate sphere
in the non-extremal case are in one-to-one correspondence with the
corresponding four regions around the bifurcate sphere in the case of
stationary black holes of the Reissner-Nordstrom family. The correspondence is
maintained in the extremal limit where the bifurcate sphere degenerates to a
point and the four regions collapse to a single one. In the black hole case
the bifurcate sphere is pushed to infinity and one of the asymptotically flat
regions disappears. In our case the bifurcate sphere is shrunk to a point at
the origin and the region in the interior of the light cones, the diamond,
disappears. The analogy is emphasised in Figure 3.1.

The flow of $\xi$ describes uniformly accelerated observers, with integral
curves being a one parameter family of rectangular hyperbolas given by
[Herrero and Morales, 1999]

$t^{2}-\left(r+\frac{\zeta}{2a}\right)^{2}=\frac{\Delta-\zeta^{2}}{4a^{2}}\,,$
(3.16)

where $\zeta$ is the parameter labeling members of the family. The complete
situation is depicted in Figure 3.2. From the picture it is clear that, seen
from the point of view of the observers that follow the MCKF in Region II, the
boundary of the region is a bifurcate conformal Killing horizon with topology
$S^{2}\times{\mathbb{R}}$. This is the same topology as the one of bifurcate
Killing horizons of stationary black holes in the asymptotically flat
spacetime context.

Figure 3.2: The flow of the radial MCKF, depending on the value of the
parameter $\Delta$, Eq. (3.13). When $\Delta<0$: the MCKF is timelike
everywhere. a) $\Delta>0$: the Minkowski spacetime is divided into 6 different
regions, where the norm of the MCKF changes from being timelike to being
spacelike, through being null along the four light rays $u_{\pm}$, $v_{\pm}$.
b) $\Delta=0$: the MCKF is everywhere timelike except for the two null rays
$u_{0}=v_{0}$ where it is null.

To summarise: radial conformal Killing fields in Minkowski spacetime generate
bifurcate conformal Killing horizons that reproduce the main topological
features of stationary spherically symmetric Killing horizons. This is the
first obvious indication that makes MCKFs interesting for drawing analogies
with black holes. The aim of what follows is to show that this analogy is more
profound and extends very nicely to the thermodynamical properties of black
hole Killing horizons. In Section 3.3, indeed, we will be able to define, in a
suitable sense that will become clear, the four laws of thermodynamics for
bifurcate MCKF horizons.

##### 3.2.1 Introducing two geometric scales defining the
radial MCKF

Here we associate the parameters $a,b$, and $c$ in (3.10) with geometric
notions. First we set $b=0$ by means of a time translation $t\to t-b/(2a)$. In
this way the points $O^{\pm}$ are placed on the time axis in the future and
the past of the origin at equal timelike distance, and the distributions of
regions become $t$-reflection symmetric. We make this choice from now on.
Notice in addition that the parameter $a$ has dimension $[length]^{-2}$, while
$c$ is dimensionless. We can therefore rewrite those constants in terms of two
physical length scales. The first one is the radius of the bifurcate sphere
$r=r_{\scriptscriptstyle H}$, Eq. (3.15):

$r_{\scriptscriptstyle H}^{2}=\frac{\Delta}{4a^{2}}=-\frac{c}{a}\,.$ (3.17)

There is also another natural geometric scale associated to the radius of the
sphere $r_{\scriptscriptstyle O}$ at $t=0$ where we demand $\xi$ to be
normalized. Such sphere represents the ensemble of events where the MCKF can
be associated with the orbits of observers. We call this sphere the observers
sphere. The normalization condition at $r_{\scriptscriptstyle O}$ is the
analogue of the normalization condition for the stationarity Killing vector
field at infinity in asymptotically flat stationary spacetimes, e.g.
stationary black holes, or the selection of a special observer trajectory when
normalizing the boost Killing field in the Rindler wedge. Therefore, we demand
the condition $\xi\cdot\xi|_{t=0,r=r_{\scriptscriptstyle O}}=-1$ which,
together with Eq. (3.17), allows to determine both $a$ and $c$ as a function
of $r_{\scriptscriptstyle H}$ and $r_{\scriptscriptstyle O}$. Explicitly one
finds

$a=\frac{1}{r_{\scriptscriptstyle O}^{2}-r_{\scriptscriptstyle H}^{2}}\,,\ \ \
\ \ c=-\frac{r_{\scriptscriptstyle H}^{2}}{r_{\scriptscriptstyle
O}^{2}-r_{\scriptscriptstyle H}^{2}}\,.$ (3.18)

The radial conformal Killing field takes then the form

$\begin{split}\xi^{\mu}\partial_{\mu}&=\frac{1}{r_{\scriptscriptstyle
O}^{2}-r_{\scriptscriptstyle H}^{2}}\Big{[}(t^{2}+r^{2}-r_{\scriptscriptstyle
H}^{2})\partial_{t}+2tr\,\partial_{r}\Big{]}\\\
&=\frac{v^{2}-r_{\scriptscriptstyle H}^{2}}{r_{\scriptscriptstyle
O}^{2}-r_{\scriptscriptstyle
H}^{2}}\,\partial_{v}+\frac{u^{2}-r_{\scriptscriptstyle
H}^{2}}{r_{\scriptscriptstyle O}^{2}-r_{\scriptscriptstyle
H}^{2}}\,\partial_{u}\,;\end{split}$ (3.19)

its norm becomes

$\xi\cdot\xi=-\frac{(v^{2}-r_{\scriptscriptstyle
H}^{2})(u^{2}-r_{\scriptscriptstyle H}^{2})}{(r_{\scriptscriptstyle
O}^{2}-r_{\scriptscriptstyle H}^{2})^{2}}\,;$ (3.20)

the parameter $\Delta$

$\Delta=\frac{4r_{\scriptscriptstyle H}^{2}}{(r_{\scriptscriptstyle
O}^{2}-r_{\scriptscriptstyle H}^{2})^{2}}\,,$ (3.21)

which implies

$u_{\pm}=v_{\pm}=\pm r_{\scriptscriptstyle H}\,.$ (3.22)

From equation (3.19) we clearly see that $\xi$ vanishes at the bifurcate
sphere $r=r_{\scriptscriptstyle H}$ and that $\xi=\partial_{t}$ at the
observers sphere $r=r_{\scriptscriptstyle O}$; both spheres are defined to be
on the $t=0$ surface. The vector field vanishes also at $O^{\pm}$. These two
length scales completely determine the radial MCKF forming conformal Killing
horizons.

#### 3.3 Light Cone Thermodynamics

In this central section of the paper, we will formulate the laws of
thermodynamics for the bifurcate conformal Killing horizon generated by the
radial MCKF. The horizon is defined by the two pieces of light cones meeting
at the bifurcate sphere of radius $r_{H}$. It is the boundary of the causal
complement of the diamond, Region II: the analogue of the exterior region of a
stationary black hole spacetime. In Figure 3.3, the $S^{2}\times{\mathbb{R}}$
topology of the horizon, together with the structure of Region II, is
emphasised.

Figure 3.3: A $(2+1)$ dimensional diagram depicting the regions of interest in
Minkowski spacetime. The two cones truncated at the sphere of radius
$r=r_{\scriptscriptstyle H}$ represent the bifurcate conformal Killing
horizon. They meet future and past null infinity on spherical cross-sections,
represented by the two bigger rings. The central ring is the bifurcate sphere
$r=r_{H}$, where $\xi=0$. The horizon is therefore a sphere with radius
growing at the speed of light. In the center, one can also see the Cauchy
development of the bifurcate sphere, _the diamond_ , Region I. Shaded in
yellow is Region II, the region representing the outside of the horizon.
Geometrically, it is the Cauchy development of the complement of Region I. The
remaining part of Minkowski spacetime is occupied by Regions III to VI, which,
to simplify the picture, are not clearly depicted here.

As already mentioned, see Eq. (3.5), a MCKF $\xi$ satisfies

${\mathfs{L}}_{\xi}\,\eta_{ab}=\nabla_{a}\xi_{b}+\nabla_{b}\xi_{a}=\frac{\psi}{2}\eta_{ab}$
(3.23)

with

$\psi=\nabla_{a}\xi^{a}\,.$ (3.24)

A conformal Killing horizon $H$ is defined as the surface where the MCKF is
null, $\xi\cdot\xi=0$. Therefore, the gradient of $\xi\cdot\xi$ must be
proportional to the normal to the horizon $\xi$. The proportionality factor
defines the surface gravity ${\kappa_{\scriptscriptstyle SG}}$ 111It is easy
to check that ${\kappa_{\scriptscriptstyle SG}}$ is invariant under conformal
transformations $\eta_{ab}\to g_{ab}=\Omega^{2}\eta_{ab}$ [Jacobson and Kang,
1993]. Under such transformations we have
$\nabla_{a}(g_{bc}\xi^{b}\xi^{c})\hat{=}\nabla_{a}(\Omega^{2}\eta_{bc}\xi^{b}\xi^{c})\hat{=}\Omega^{2}\nabla_{a}(\eta_{bc}\xi^{b}\xi^{c})\hat{=}{\kappa_{\scriptscriptstyle
SG}}\Omega^{2}\eta_{ab}\xi^{b}\hat{=}{\kappa_{\scriptscriptstyle
SG}}g_{ab}\xi^{b},$ (3.25) where we used that $\xi\cdot\xi=0$ on the horizon.
via the equation

$\nabla_{a}(\xi\cdot\xi)\,\hat{=}-2{\kappa_{\scriptscriptstyle
SG}}\eta_{ab}\xi^{b}\,.$ (3.26)

The symbol $\hat{=}$ stands for relations valid only at the horizon; we will
use this notation whenever stressing such property is necessary. The CKF is
also geodesic at the horizon [Sultana and Dyer, 2004, Dyer and Honig, 1979],
so that one can define the function $\kappa$ as

$\xi^{a}\nabla_{a}\xi^{b}{\,\hat{=}\,}\kappa\,\xi^{b}\,.$ (3.27)

Thus $\kappa$ is the function measuring the failure of $\xi$ to be an affine
geodesic on the horizon. While for Killing horizons
$\kappa={\kappa_{\scriptscriptstyle SG}}$, for conformal Killing horizons the
following relation is valid:

${\kappa_{\scriptscriptstyle SG}}=\kappa-\frac{\psi}{2}\,.$ (3.28)

We will now use these relations in the special case of the MCKF defined in the
previous sections.

##### 3.3.1 The zeroth law

It is immediate to show that, for a general CKF, the quantity
${\kappa_{\scriptscriptstyle SG}}$ is Lie dragged along the field itself
[Sultana and Dyer, 2004, Dyer and Honig, 1979]; namely

${\mathfs{L}}_{\xi}{\kappa_{\scriptscriptstyle SG}}=0\,.$ (3.29)

In our case, by spherical symmetry, this implies that
${\kappa_{\scriptscriptstyle SG}}$ is actually constant on the horizon $H$
222It can however be shown for a general CKF under some assumptions [Sultana
and Dyer, 2004].. This proves _the zeroth law_ of light cone thermodynamics.

##### 3.3.2 The first law

Let us now define the energy-momentum current

$J^{a}=T^{ab}\xi_{b}\,.$ (3.30)

Conformally invariant field theories satisfy $T^{a}_{\;a}=0$ on shell [Di
Francesco et al., 1997]. For these theories, the current $J^{a}$ is conserved.
Indeed

$\nabla_{a}J^{a}=\frac{\psi}{4}\;T^{a}_{\;a}=0\,.$ (3.31)

In such cases the current defines a conserved charge

$M=\int_{\Sigma}T_{ab}\,\xi^{a}d\Sigma^{b}$ (3.32)

with $d\Sigma^{a}$ being the volume element of a general Cauchy hypersuface
$\Sigma$.

We want now to study the analogue of what is called the “physical process
version” [Hawking and Hartle, 1972, Gao and Wald, 2001] of the first law of
black hole thermodynamics, which is also valid for more general bifurcate
Killing horizons [Amsel et al., 2008] 333There are works in the literature
where the mechanical laws for conformal Killing horizons are investigated from
the purely Hamiltonian perspective [Chatterjee and Ghosh, 2015b, Chatterjee
and Ghosh, 2015a]. The strategy used in our specific and simple flat spacetime
example seems more transparent for a deeper geometric insight.. This will
define the _first law of light cone thermodynamics_. Let us therefore consider
the process in which a small amount $\delta M$ of such “energy” 444See
Subsection 3.4.2 for a discussion on the meaning of such a conserved quantity.
passes through what plays the role of the future horizon $H^{+}$, namely the
future light cone $u=u_{-}$. The passage of the matter will perturb the
horizon. We will show that, at first order in linearised gravity, there is a
balance law relating $\delta M$ with the conformal area change of the horizon
$H^{+}$ (see below). The crucial difference in proving this relation with
respect to the black hole case is that here the cross-sectional area of the
horizon is changing along the affine geodesic generators of the light cone,
even when no perturbation is considered. More technically, if we take the
advanced time $v=t+r$ as an affine parameter along the null generators
$\ell=\partial_{v}$ of $H^{+}$ in the flat background geometry, then one can
show that the expansion $\theta$ is

$\theta=\frac{1}{r}=\frac{2}{v-u}\,.$ (3.33)

By definition, the expansion is the rate of change of the cross-sectional area
with respect to the parameter [Poisson, 2004], namely,

$\theta=\frac{1}{dS}\frac{\partial dS}{\partial v}\,.$ (3.34)

In our case $dS=1/4\,(v-u)^{2}\sin\vartheta d\vartheta d\varphi=$ and the two
above equations are indeed in agreement. The non vanishing of those
quantities, therefore, implies that the area of the horizon is constantly
increasing even if no flux of energy is considered. Consequently, we need to
distinguish the two different changes in area: the background one that we call
$dS_{0}$, and the one induced by the passage of the perturbation that we call
$dS_{1}$. When a generic flux of energy is considered, we can write the
expansion as

$\theta=\frac{1}{dS_{0}+dS_{1}}\frac{\partial}{\partial v}(dS_{0}+dS_{1})\,.$
(3.35)

In the approximation in which $dS_{1}/dS_{0}\ll 1$, we can expand it up to
first order and find

$\begin{split}\theta&=\frac{1}{dS_{0}}\frac{\partial dS_{0}}{\partial
v}+\frac{1}{dS_{0}}\left(\frac{\partial dS_{1}}{\partial
v}-\frac{1}{dS_{0}}\frac{\partial dS_{0}}{\partial
v}dS_{1}\right)+O\left(\frac{dS_{1}}{dS_{0}}\right)^{2}\\\
&\equiv\theta_{0}+\theta_{1}+O\left(\frac{dS_{1}}{dS_{0}}\right)^{2},\end{split}$
(3.36)

where $\theta_{0}$ is the unperturbed expansion given by Eqs. (3.33)-(3.34)
and where we defined the perturbation

$\theta_{1}\equiv\frac{1}{dS_{0}}\left(\frac{\partial dS_{1}}{\partial
v}-\theta_{0}dS_{1}\right)=\frac{\partial}{\partial
v}\left(\frac{dS_{1}}{dS_{0}}\right)\,.$ (3.37)

Moreover, by the Raychauduri equation one has that the variation of the
expansion is connected with the flux of energy as [Wald, 1984]

$\ell(\theta)=-\frac{1}{2}\theta^{2}-\sigma_{ab}\sigma^{ab}+\omega_{ab}\omega^{ab}-8\pi
T_{ab}\ell^{a}\ell^{b}\,;$ (3.38)

where $\sigma_{ab}$ and $\omega_{ab}$ are the shear and twist tensors
respectively, and $\ell(f)=\partial_{v}f$ for any function $f$. Using Eq.
(3.36) and the fact that, in this case $\omega^{0}_{ab}=\sigma^{0}_{ab}=0$, we
get

$\ell(\theta_{0})+\ell(\theta_{1})=-\frac{1}{2}\theta_{0}{}^{2}-\theta_{0}\theta_{1}-8\pi\delta
T_{ab}\ell^{a}\ell^{b}+O(\theta_{1})^{2}\,,$ (3.39)

where $\delta T_{ab}$ represents a small energy perturbation justifying the
use of the perturbed equation (3.36) 555One may wonder whether $\ell$ is still
an affinely parametrised generator of the horizon in the perturbed spacetime.
This would be the case if the $\ell=du$ continues to be a null one-form for
the metric $g_{ab}=\eta_{ab}+\delta g_{ab}$. Using the gauge symmetry of
linearized gravity $\delta g_{ab}^{\prime}=\delta
g_{ab}+2\nabla^{\scriptscriptstyle(0)}_{(a}v_{b)}$ (for a vector field
$v^{a}$), the condition $g^{ab}du_{a}du_{b}=0$ is equivalent to $\delta
g^{uu}+\ell(\ell\cdot v)=0$, which can be solved for the gauge parameter $v$.
. Since $\theta_{0}$ is, by definition, the solution of the unperturbed
Raychauduri equation—Eq. (3.38) with $T_{ab}=0$—, the above equation reduces
at first order to

$\ell(\theta_{1})+\theta_{0}\theta_{1}=-8\pi\delta T_{ab}\ell^{a}\ell^{b}\,.$
(3.40)

The last elements we need before starting the proof of the first law are the
following: first notice from Eq. (3.19) that, on the horizon, our MCKF $\xi$
is parallel to $\ell$, namely

$\xi\,\hat{=}\,\alpha\ell\,,$ (3.41)

with $\alpha=(v^{2}-r_{\scriptscriptstyle H}^{2})/(r_{\scriptscriptstyle
O}^{2}-r_{\scriptscriptstyle H}^{2})$. Substituting the above equation into
definition (3.27), one can show that

$\kappa\,\hat{=}\,\ell(\alpha)=\partial_{v}\alpha\,.$ (3.42)

Now we are ready to prove the first law of light cone thermodynamics. We
consider a generic perturbation $\delta M$ of conformally invariant energy
defined by (3.32). Since this quantity is conserved, we can choose the Cauchy
surface to integrate over in the more convenient way. For our purposes, we
choose the union of the future horizon $H^{+}$ with the piece of
${\mathfs{I}}^{+}$ contained in Region II, the latter being
$\Sigma_{\infty}\equiv\rm{II}\cap{\mathfs{I}}^{+}$. Eq. (3.32) therefore
becomes

$\begin{split}\delta M&=\int\limits_{H^{+}}\delta
T_{ab}\,\xi^{a}d\Sigma^{b}_{0}+\int\limits_{\Sigma_{\infty}}\delta
T_{ab}\xi^{a}d\Sigma_{\infty}^{a}=\\\ &=\int\limits_{H^{+}}\alpha\delta
T_{ab}\,\ell^{a}\ell^{b}\,dS_{0}dv+\delta M_{\infty}\\\
&=-\frac{1}{8\pi}\int\limits_{H^{+}}\alpha\Big{(}\partial_{v}(\theta_{1})+\theta_{0}\theta_{1}\Big{)}dS_{0}dv+\delta
M_{\infty}\\\
&=-\frac{1}{8\pi}\int\limits_{H^{+}}\alpha\Big{(}\partial_{v}(\theta_{1})dS_{0}+\partial_{v}(dS_{0})\theta_{1}\Big{)}dv+\delta
M_{\infty}\\\
&=-\frac{1}{8\pi}\left[\oint_{H^{+}(v)}\alpha\,\theta_{1}dS_{0}\Bigg{|}_{v_{+}}^{\infty}-\int\limits_{H^{+}}\kappa\,\theta_{1}dS_{0}dv\right]+\delta
M_{\infty}\\\ &=\frac{{\kappa_{\scriptscriptstyle
SG}}}{8\pi}\int\limits_{H^{+}}\frac{\kappa}{{\kappa_{\scriptscriptstyle
SG}}}\,\theta_{1}dS_{0}dv+\delta M_{\infty}\end{split}$ (3.43)

To go from the first to the second line, we have used the fact that the
unperturbed surface element of the horizon is given by
$d\Sigma_{0}^{a}=\ell^{a}dS_{0}dv$, the fact that we defined the energy flux
at infinity as

$\delta M_{\infty}\equiv\int\limits_{\Sigma_{\infty}}\delta
T_{ab}\xi^{a}d\Sigma_{\infty}^{b}\,,$ (3.44)

and the proportionality between $\xi$ and $\ell$, Eq. (3.41). The third line
follows from the perturbed Raychauduri equation (3.40). Using the definition
of $\theta_{0}$ (3.34) we obtain the fourth line. Integrating by parts and
using (3.42) leads to line five. The boundary term vanishes from the fact that
$\alpha(r_{H})=0$, i.e. $\xi^{a}=0$ at the bifurcate surface, and that we
choose initial condition at infinity in the usual teleological manner [Gao and
Wald, 2001], namely $\theta_{1}(\infty)=0$.

We define the conformal area change $\delta A$ of the horizon as

$\delta A\equiv\int\limits_{H^{+}}\frac{\kappa}{{\kappa_{\scriptscriptstyle
SG}}}\,\theta_{1}dS_{0}dv=\frac{8\pi}{{\kappa_{\scriptscriptstyle
SG}}}{\int\limits_{H^{+}}\delta T_{ab}\,\xi^{a}d\Sigma^{b}},$ (3.45)

in terms of which the first law follows

$\delta M=\frac{\kappa_{\scriptscriptstyle SG}}{8\pi}\,{\delta A}+\delta
M_{\infty}\,.$ (3.46)

Some remarks are in order concerning the interpretation of equation (3.45) and
(3.46):

First notice that the definition of $\delta A$ reduces to the standard
expression when $\xi$ is a Killing field. Indeed, in that case
$\kappa={\kappa_{\scriptscriptstyle SG}}$ and the unperturbed area of the
horizon is constant, i.e. $\theta_{0}=0$. From the definition (3.37), (3.45)
is therefore the change in area of the Killing horizon.

Now we argue that, in a suitable sense, (3.45) retains the usual
interpretation in the case of the expanding ($\theta_{0}\not=0$) conformal
Killing horizon associated to a MKCF when perturbed with conformal invariant
matter. Under such circumstances, it can be verified that all the quantities
appearing in the first law (3.46) are conformal invariant; this follows
directly from the conformal invariance of ${\kappa_{\scriptscriptstyle SG}}$
(see Footnote 1), and that of the flux density $\delta
T_{ab}\,\xi^{a}d\Sigma^{b}$ when conformal matter is considered 666This a
consequence of the fact that under a transformation
$g^{\prime}_{ab}=\Omega^{2}g_{ab}$ the energy momentum tensor of conformally
invariant matter $\delta T_{ab}$ transforms as $\delta
T^{\prime}_{ab}=\Omega^{-2}\delta T_{ab}$ [Wald, 1984], and the volume element
transforms as $d\Sigma^{\prime a}=\Omega^{2}d\Sigma^{a}$, see Eq. (3.63)..
Therefore, $\delta A$ is conformally invariant and this is the key for its
geometric interpretation. To see this one can conformally map Minkowski
spacetime to a new spacetime $g_{ab}=\Omega^{2}\eta_{ab}$ where $\xi$ becomes
a bonafide Killing vector field. Under such conformal transformation
$\kappa\to\kappa_{\scriptscriptstyle SG}$ and $\theta_{0}\to 0$, and thus the
conformal invariant quantity $\delta A$ acquires the standard meaning of
horizon area change, thus justifying its name. We show an explicit realization
of such conformal map in Appendix A.6.

Finally, as $\xi$ diverges at ${\mathfs{I}}^{+}$ one might be worried that
$\delta M_{\infty}$ might be divergent. However, for massless fields the
peeling properties of $T_{ab}$ are just the right ones for $\delta M_{\infty}$
to be convergent. Indeed this follows from the fact that

$T_{uu}=\frac{T_{uu}^{0}}{v^{2}}+O(v^{-3})\ \ \ \ \ \ \
T_{uv}=\frac{T_{uv}^{0}}{v^{4}}+O(v^{-5})\,,$

and the form of $\xi$ given in (3.19). For a massless scalar field this is
shown in [Wald, 1984]; for Maxwell fields this can be seen in [Adamo et al.,
2012]. All this is expected from the fact that the current (3.30) is
conserved.

##### 3.3.3 The second law

The quantity $\delta A$ is strictly positive in the context of first order
perturbations of Minkowski spacetimes. This follows directly from the first
law and the assumption that the conformal matter satisfies the energy
condition $T_{ab}\ell^{a}\ell^{b}\geq 0$. It is the standard manifestation of
the attractive nature of gravity in its linearized form. Needless is to say
that this version of the second law is somewhat trivial in comparison with the
very general area theorem for black hole [Hawking and Ellis, 1973], as well as
generic Killing [Chruśiel et al., 2001], horizons.

##### 3.3.4 The third law

In our context the third law is valid in a very concrete and strict fashion.
In the limit of extremality, $r_{H}\to 0$, the surface gravity
$\kappa_{\scriptscriptstyle SG}\to 0$ and, the analogue of the entropy, the
area $A$, goes to zero as well. This version of the third law is the analogue
of the statement that at zero temperature the entropy vanishes, which is only
true for systems with non degenerate ground states. No dynamical-process
version of the third law appears to make sense in our context. This might
resonate at first sight with the statement [Hawking et al., 1995] that
extremal BHs must have vanishing entropy, but the similarity is only in
appearance as the area of the bifurcate sphere remains non-vanishing in the BH
case.

#### 3.4 Quantum Effects: “Hawking radiation” and
Conformal Temperature

When the laws of black hole mechanics where discovered, they were thought as a
mere analogy with the one of thermodynamics. It is only after Hawking’s
discovery of semiclassical radiation [Hawking, 1974, Hawking, 1975] that they
assumed a proper status of laws of black hole thermodynamics. In the previous
Section, we established the equivalent of the early analogy for the case of
light cones in Minkowski spacetime and their gravitational perturbation. In
what follows, we show that, also in this case, a semiclassical computation can
be performed to give a thermodynamical meaning to those laws. In a suitable
sense, radial MCKFs can be assigned a temperature

$T=\frac{{\kappa_{\scriptscriptstyle SG}}}{2\pi}\,.$ (3.47)

Thus the first law (3.46) becomes

$\delta M=T\delta S+\delta M_{\infty}\,.$ (3.48)

with

$T=\frac{{\kappa_{\scriptscriptstyle SG}}}{2\pi}\ \ \ \ \ \ {\rm and}\ \ \ \ \
\ \delta S=\frac{\delta A}{4}\,,$ (3.49)

exactly as for stationary black holes.

To do so, let us start by noticing that for each region Minkowski spacetime is
divided into by our radial MCKF $\xi$, there exists a coordinate
transformation $(t,r,\vartheta,\varphi)\to(\tau,\rho,\vartheta,\varphi)$
adapted to the MCKF in the sense that $\xi(\tau)=-1$. The explicit maps are
written in Appendix A.4. Here we report the one for the region of interest,
namely Region II. It reads [Haggard, 2013]

$\begin{split}t&=\frac{\sqrt{\Delta}}{2a}\frac{\sinh(\tau\sqrt{\Delta})}{\cosh(\rho\sqrt{\Delta})-\cosh(\tau\sqrt{\Delta})}\\\
r&=\frac{\sqrt{\Delta}}{2a}\frac{\sinh(\rho\sqrt{\Delta})}{\cosh(\rho\sqrt{\Delta})-\cosh(\tau\sqrt{\Delta})}\,,\end{split}$
(3.50)

with $0\leq\rho<+\infty$ and $|\tau|<\rho$. Defining the null coordinates
$\bar{v}=\tau+\rho$ and $\bar{u}=\tau-\rho$, the following relation with
Minkowskian $u$ and $v$ is valid [Brown et al., 1982]:

$\begin{split}v&=t+r=-\frac{\sqrt{\Delta}}{2a}\coth\frac{\bar{u}\sqrt{\Delta}}{2}\\\
u&=t-r=-\frac{\sqrt{\Delta}}{2a}\coth\frac{\bar{v}\sqrt{\Delta}}{2}\,,\end{split}$
(3.51)

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
(3.7) becomes

$ds^{2}=\Omega^{2}\left(-d\tau^{2}+d\rho^{2}+\Delta^{-1}\sinh^{2}(\rho\sqrt{\Delta})dS^{2}\right)$
(3.52)

where the conformal factor takes the value

$\Omega=\frac{\Delta/2a}{\cosh(\rho\sqrt{\Delta})-\cosh(\tau\sqrt{\Delta})}\,.$
(3.53)

As anticipated, the metric depends on the coordinate $\tau$ only through the
conformal factor. The vector $\partial_{\tau}$, therefore, is a conformal
Killing field for the Minkowski spacetime which can be shown to coincide with
Eq. (3.19). Explicitly

$\begin{split}\xi^{a}\partial_{a}=\partial_{\tau}&=\left(av^{2}-\frac{\Delta}{4a}\right)\partial_{v}+\left(au^{2}-\frac{\Delta}{4a}\right)\partial_{u}\\\
&=(av^{2}+c)\,\partial_{v}+(au^{2}+c)\,\partial_{u}\\\
&=\big{(}a(t^{2}+r^{2})+c\big{)}\partial_{t}+2art\,\partial_{r}\,.\end{split}$
(3.54)

##### 3.4.1 Bogoliubov transformations

Consider now a scalar field $\phi$ evolving in Minkowski space. We will define
a vacuum state $\ket{0}$ and its corresponding Fock space $\mathcal{F}$ using
the notion of positive frequency compatible with the notion of energy entering
the first law Eq. (3.46). Making more precise what we anticipated in the first
lines of this section, the remarkable result is that the standard Minkowski
vacuum state is seen, in the Fock space $\mathcal{F}$, as a thermal state at
the constant conformal temperature

$T=\frac{{\kappa_{\scriptscriptstyle SG}}}{2\pi}\,.$ (3.55)

The term conformal temperature is used because the state of the radiation
looks thermal in terms of time translation notion associated to the conformal
Killing time. See Section 3.4.2 for a detailed discussion, where the
relationship between this notion of temperature and the physical temperature
measured by a thermometer is also addressed.

Let us start by defining a meaningful notion of Fock space related to our
conformally static observers. As for the discussion in the previous sections,
Eq. (3.46) holds only for conformally invariant matter models. For
concreteness, here we consider a conformally invariant scalar field $\phi$
satisfying the conformally coupled Klein-Gordon (KG) equation

$\left(\Box^{2}-\frac{1}{6}R\right)\,\phi=0\,,$ (3.56)

where $\Box=g_{ab}\nabla^{a}\nabla^{b}$, with $\nabla^{a}$ the covariant
derivative with respect to a general metric $g_{ab}$, and $R$ the Ricci
curvature scalar. The previous equation is conformally invariant in the sense
that under a conformal transformation $g_{ab}\to g^{\prime}_{ab}=C^{2}g_{ab}$
solutions of (3.56) defined in terms of $g_{ab}$ are mapped into solutions of
the same equation in terms of $g^{\prime}_{ab}$ by the rule
$\phi\to\phi^{\prime}=C^{-1}\phi$ [Wald, 1984, Birrell and Davies, 1984].

As Eq. (3.52) shows, the complement of the diamond in Minkowski space is
conformally related to a region of a static Friedmann-Robertson-Walker (FRW)
spacetime with negative spatial curvature $k=-|\Delta|$; see Appendix A.5 for
further details. The FRW Killing field $\partial_{\tau}$ corresponds to the
Minkowski conformal Killing field in Eq. (3.54). The strategy is therefore to
find a complete set of solutions $U_{i}(x)$ of the KG equation in the static
FRW spacetime, to deduce the one in our region using conformal invariance;
here $i$ is a generic index labeling the modes. The Klein-Gordon equation
(3.56) in the FRW spacetime under consideration reads

$\begin{split}0&=\left(\Box^{2}-\frac{1}{6}R\right)U_{i}(x)\\\
&=\left[\frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}\partial^{\mu}\right)-\frac{1}{6}R\right]U_{i}(x)\\\
&=\left[-\partial_{\tau}^{2}+\frac{1}{\sinh^{2}(\rho\sqrt{\Delta})}\left(\partial_{\rho}\sinh^{2}(\rho\sqrt{\Delta})\,\partial_{\rho}+\frac{\Delta}{\sin\vartheta}\partial_{\vartheta}\sin\vartheta\,\partial_{\vartheta}+\frac{\Delta}{\sin^{2}\vartheta}\partial^{2}_{\varphi}\right)+\Delta\right]U_{i}(x)\,,\end{split}$
(3.57)

where $g=\det g_{ab}$ and we have used the fact that $R=-6\Delta$. One can
solve the previous equation by the ansatz

$U^{\ell
m}_{\omega}(x)=\exp(-i\omega\tau)\frac{R_{\leftrightarrow\omega}^{\ell}(\rho)}{\sinh(\rho\sqrt{\Delta})}Y^{\ell
m}(\vartheta,\varphi)\,,$ (3.58)

which after substitution in (3.57) gives

$\left(\partial^{2}_{\rho}+\omega^{2}-\frac{\ell(\ell+1)\,\Delta}{\sinh^{2}(\rho\sqrt{\Delta})}\right)R^{\ell}_{\leftrightarrow\omega}(\rho)=0\,,$
(3.59)

where $\leftrightarrow$ denotes the two possible solutions: out-going modes
will be denoted by a right arrow ($\rightarrow$) while in-going modes by a
left arrow ($\leftarrow$). Notice that we have substituted the generic index
$i$ with the more specific $\omega$, $\ell$ and $m$. A complete set of
solutions to this equation is given in [Birrell and Davies, 1984]. These modes
are positive frequency modes with respect to the notion of time translation
defined by the Killing time $\tau$, and they are orthonormal with respect to
the Klein-Gordon scalar product, namely

$\begin{split}(U_{\leftrightarrow\omega^{\prime}}^{\ell^{\prime}m^{\prime}},U_{\leftrightarrow\omega}^{\ell
m})&=-i\int_{\Sigma}\left(U_{\leftrightarrow\omega}^{\ell
m}\partial_{a}\bar{U}_{\leftrightarrow\omega^{\prime}}^{\ell^{\prime}m^{\prime}}-\bar{U}_{\leftrightarrow\omega^{\prime}}^{\ell^{\prime}m^{\prime}}\partial_{a}U_{\leftrightarrow\omega}^{\ell
m}\right)d\Sigma^{a}\\\
&=\delta_{\leftrightarrow}\delta^{\ell\ell^{\prime}}\,\delta^{mm^{\prime}}\,\delta(\omega,\omega^{\prime})\,,\end{split}$
(3.60)

where $\delta_{\leftrightarrow}$ means that outgoing modes are orthogonal to
ingoing ones. As said at the beginning of the subsection, due to conformal
invariance the set of modes 777Such modes are the “sphere modes” considered in
[Haggard, 2013]. defined by

$u_{\leftrightarrow\omega}^{\ell
m}(x)=\Omega^{-1}(x)U_{\leftrightarrow\omega}^{\ell
m}(x)=\Omega^{-1}(x)\,e^{-i\omega\tau}\frac{R_{\leftrightarrow\omega}^{\ell}(\rho)}{\sinh(\rho\sqrt{\Delta})}Y^{\ell
m}(\vartheta,\varphi)$ (3.61)

with $\Omega(x)$ given by Eq. (3.53), are a complete set of solutions of the
Klein-Gordon equation in our region of interest, the complement of the diamond
in Minkowski space. Moreover, they satisfy

$\begin{split}(u_{\leftrightarrow\omega^{\prime}}^{\ell^{\prime}m^{\prime}},u_{\leftrightarrow\omega}^{\ell
m})&=(\Omega^{-1}\,U_{\leftrightarrow\omega}^{\ell
m},\Omega^{-1}\,U_{\leftrightarrow\omega^{\prime}}^{\ell^{\prime}m^{\prime}})\\\
&=-i\int_{\Sigma}\left[\Omega^{-1}\,U_{\leftrightarrow\omega}^{\ell
m}\partial_{a}\Big{(}\Omega^{-1}\bar{U}_{\leftrightarrow\omega^{\prime}}^{\ell^{\prime}m^{\prime}}\Big{)}-\Omega^{-1}\bar{U}_{\leftrightarrow\omega^{\prime}}^{\ell^{\prime}m^{\prime}}\partial_{a}\Big{(}\Omega^{-1}\,U_{\leftrightarrow\omega}^{\ell
m}\Big{)}\right]d\Sigma_{\rm\scriptscriptstyle II}^{a}\\\
&=-i\int_{\Sigma}\Big{[}\Omega^{-2}\Big{(}U_{\leftrightarrow\omega}^{\ell
m}\partial_{a}\bar{U}_{\leftrightarrow\omega^{\prime}}^{\ell^{\prime}m^{\prime}}-\bar{U}_{\leftrightarrow\omega^{\prime}}^{\ell^{\prime}m^{\prime}}\partial_{a}\,U_{\leftrightarrow\omega}^{\ell
m}\Big{)}\\\
&\phantom{=-i\int_{\Sigma}\Big{[}\Omega^{-2}\Big{(}U_{\leftrightarrow\omega}^{\ell
m}\partial_{a}\bar{U}_{\leftrightarrow\omega^{\prime}}^{\ell^{\prime}m^{\prime}}-\bar{U}_{\leftrightarrow\omega^{\prime}}^{\ell^{\prime}m^{\prime}}}+\Omega^{-1}\bar{U}_{\leftrightarrow\omega^{\prime}}^{\ell^{\prime}m^{\prime}}U_{\leftrightarrow\omega}^{\ell
m}\Big{(}\partial_{a}\Omega^{-1}-\partial_{a}\Omega^{-1}\Big{)}\Big{]}d\Sigma_{\rm\scriptscriptstyle
II}^{a}\\\ &=-i\int_{\Sigma}\left(U_{\leftrightarrow\omega}^{\ell
m}\partial_{a}\bar{U}_{\leftrightarrow\omega^{\prime}}^{\ell^{\prime}m^{\prime}}-\bar{U}_{\leftrightarrow\omega^{\prime}}^{\ell^{\prime}m^{\prime}}\partial_{a}\,U_{\leftrightarrow\omega}^{\ell
m}\right)d\Sigma^{a}\\\
&=(U_{\leftrightarrow\omega^{\prime}}^{\ell^{\prime}m^{\prime}},U_{\leftrightarrow\omega}^{\ell
m})\,.\end{split}$ (3.62)

Here $\Sigma$ is a Cauchy surface shared by the two conformally related
spacetimes; $d\Sigma^{a}$ and $d\Sigma_{\rm\scriptscriptstyle II}^{a}$ are the
volume elements of $\Sigma$ in the static FRW spacetime and in the complement
of the diamond respectively. The above result is given by the fact that the
two volume elements are related by 888This is generically true for any
hypersurface $\Sigma$ shared between two conformally related spacetimes
$g^{\prime}_{ab}=C^{2}g_{ab}$. Indeed, if $n^{a}$ is the unit normal to
$\Sigma$ with respect to $g_{ab}$, then $n^{\prime a}=C^{-1}n^{a}$ is the unit
normal to $\Sigma$ with respect to $g^{\prime}_{ab}$. The 3-dimensional volume
elements, at the same time, are related by $\sqrt{h^{\prime}}=C^{3}\sqrt{h}$.
It follows that $d\Sigma^{\prime a}=C^{2}d\Sigma^{a}$.

$d\Sigma^{a}_{\rm\scriptscriptstyle II}=n_{\rm\scriptscriptstyle
II}^{a}\sqrt{-h_{\rm\scriptscriptstyle
II}}\,d^{3}y=\Omega^{2}n^{a}\sqrt{h}\,d^{3}y=\Omega^{2}d\Sigma^{a}\,,$ (3.63)

where $n^{a}$ is the normal to $\Sigma$, $h$ is the determinant of the
intrinsic metric defining $\Sigma$ it self, and $y^{i}$ are the coordinate
describing the latter. The subscript $\rm II$ indicates objects defined in
Region II of Minkowski spacetime; the same objects without any subscript are
in FRW. Eq. (3.62) shows that the modes $u^{\ell m}_{\omega}$ provide a
complete set of solutions inducing a positive definite scalar product, namely
everything one needs to perform the standard quantisation procedure. Hence,
one can write the field operator in Region II of Minkowski spacetime as

$\begin{split}\phi(x)&=\int_{0}^{+\infty}d\omega\sum_{\ell
m}\Big{(}a_{\leftarrow\omega}^{\ell m}u_{\leftarrow\omega}^{\ell
m}(x)+a_{\leftarrow\omega}^{\ell m\,\dagger}{\bar{u}}_{\leftarrow\omega}^{\ell
m}(x)\Big{)}+\Big{(}a_{\rightarrow\omega}^{\ell m}u_{\rightarrow\omega}^{\ell
m}(x)+a_{\rightarrow\omega}^{\ell
m\,\dagger}{\bar{u}}_{\rightarrow\omega}^{\ell m}(x)\Big{)}\\\
&=\int_{0}^{+\infty}d\omega\ \Omega^{-1}(x)\Big{[}\sum_{\ell
m}\Big{(}a_{\leftarrow\omega}^{\ell m}U_{\leftarrow\omega}^{\ell
m}(x)+a_{\leftarrow\omega}^{\ell m\,\dagger}U_{\leftarrow\omega}^{\ell
m}(x)^{*}\Big{)}\\\ &\phantom{=\int_{0}^{+\infty}d\omega\
\Omega^{-1}(x)\Big{[}\sum_{\ell m}\Big{(}a_{\leftarrow\omega}^{\ell
m}U_{\leftarrow\omega}^{\ell m}(x)+a_{\leftarrow\omega}^{\ell
m\,\dagger}}+\Big{(}a_{\rightarrow\omega}^{\ell m}U_{\rightarrow\omega}^{\ell
m}(x)+a_{\rightarrow\omega}^{\ell
m\,\dagger}{\bar{U}}_{\rightarrow\omega}^{\ell
m}(x)\Big{)}\Big{]}\,,\end{split}$ (3.64)

where $a_{\leftrightarrow\omega}^{\ell m\dagger}$ and
$a_{\leftrightarrow\omega}^{\ell m}$ denote the creation and annihilation
operators in the corresponding modes. The vacuum state $\ket{0}$ defined by
$a_{\leftrightarrow\omega}^{\ell m}\ket{0}=0$ is usually called the _conformal
vacuum_ [Birrell and Davies, 1984]. This state is highly pathological from the
perspective of inertial observers. Indeed, it has vanishing entanglement with
the interior of the diamond and would lead to a divergent energy momentum
tensor at $H^{+}$. More precisely, this is not a Hadamard state. The same
thing happens when one considers the Rindler vacuum defined by the boost
Killing field.

Let us notice now that Eq. (3.59) is simplified in the limit $\rho\to+\infty$.
This limit corresponds, in Region II and for $\tau>0$, to the limit
$\bar{v}\to+\infty$ or more clearly $u\to u_{-}$ with $v$ free to span the
whole range $[v_{+},+\infty)$. That is to say a “near horizon limit” 999In the
bottom part of our region, $\tau<0$, this limit corresponds to $v\to v_{+}$,
while $u$ free to vary. That is to say a near past horizon limit.. In this
limit the last term of Eq. (3.59), the one dependent on $\ell$, can be
neglected. Solutions $R(\rho)$, therefore, do not depend on $\ell$ in such
near horizon approximation and are simply given by $\exp(\pm i\omega\rho)$.
The modes (3.61), consequently, behave as

${u^{\ell m}_{\leftarrow\omega}}(x)+{u^{\ell
m}_{\rightarrow\omega}}(x)\approx\frac{1}{\sqrt{\omega}}\,\frac{e^{-i\omega\bar{u}}+e^{-i\omega\bar{v}}}{\Omega\,\sinh(\rho\sqrt{\Delta})}Y^{\ell
m}(\vartheta,\varphi)=\frac{\sqrt{\Delta}}{\sqrt{\omega}}\,\frac{e^{-i\omega\bar{u}}+e^{-i\omega\bar{v}}}{r}Y^{\ell
m}(\vartheta,\varphi)\,,$ (3.65)

where $r$ is the Minkowskian radial coordinate and we have used definition
(3.50).

Clearly, the solution of the Klein-Gordon equation and the consequent
quantisation of the field can be carried out also in the whole Minkowski
spacetime by considering inertial $r=const$ observers. This defines positive
frequency modes $u^{M}_{\omega}$ with respect to the Killing field
$\partial_{t}$, as well as a decomposition of the field as

$\phi(x)=\int_{0}^{+\infty}d\omega\Big{(}{b^{\ell
m}_{\leftarrow\omega}}{u^{\ell
mM}_{\leftarrow\omega}}(x)+{b_{\leftarrow\omega}^{\ell
m\dagger}}\,{\bar{u}^{\ell mM}_{\leftarrow\omega}}(x)\Big{)}+\Big{(}{b^{\ell
m}_{\rightarrow\omega}}{u^{\ell
mM}_{\rightarrow\omega}}(x)+{b_{\rightarrow\omega}^{\ell
m\dagger}}\,{\bar{u}^{\ell mM}_{\rightarrow\omega}}(x)\Big{)}\,.$ (3.66)

In the limit $r\to+\infty$ the Minkowskian solutions can be approximated by

${u^{\ell mM}_{\leftarrow\omega}}(x)+{u^{\ell
mM}_{\rightarrow\omega}}(x)\approx\frac{1}{\sqrt{\omega}}\,\frac{e^{-i\omega
u}+e^{i\omega v}}{r}Y^{\ell m}(\vartheta,\varphi)\,.$ (3.67)

The standard Minkowski vacuum state $\ket{0}_{M}$ of the Fock space is defined
by $b^{\ell m}_{\leftrightarrow\omega}\ket{0}_{M}=0$. The Minkowski modes are
also orthonormal with respect to the Klein-Gordon scalar product, namely

$(u^{\ell
mM}_{\leftrightarrow\omega},u^{\ell^{\prime}m^{\prime}M}_{\leftrightarrow\omega^{\prime}})=\delta_{{\leftrightarrow}}\delta_{\ell\ell^{\prime}}\delta_{mm^{\prime}}\delta(\omega,\omega^{\prime})\,,$
(3.68)

which is immediately verified for outgoing and infalling modes by integrating
on ${\mathfs{I}}^{+}$ and ${\mathfs{I}}^{-}$ solutions in the form (3.67). The
two different vacua are in general non-equivalent and one vacuum state can be
a highly exited state in the Fock space defined by the other, and viceversa.
This idea is formalised by introducing the so-called Bogoliubov
transformations between the two complete sets of modes $u^{\ell
m}_{\leftrightarrow\omega}$ and $u^{\ell mM}_{\leftrightarrow\omega}$.
Briefly—for more details see for example [Wald, 1984]—, since the two sets are
complete, one can expand one set in terms of the other. From now on we
concentrate on the outgoing modes ($\rightarrow$). We get

$u^{\ell m}_{\rightarrow\omega}=\int d\omega^{\prime}\,\Big{(}\alpha^{\ell
m\omega}_{\ell^{\prime}m^{\prime}\omega^{\prime}}\,u^{\ell^{\prime}m^{\prime}M}_{\rightarrow\omega^{\prime}}+\beta^{\ell
m\omega}_{\ell^{\prime}m^{\prime}\omega^{\prime}}\bar{u}^{\ell^{\prime}m^{\prime}M}_{\rightarrow\omega^{\prime}}\Big{)}\,,$
(3.69)

where the $\alpha^{\ell m\omega}_{\ell^{\prime}m^{\prime}\omega^{\prime}}$ and
$\beta^{\ell m\omega}_{\ell^{\prime}m^{\prime}\omega^{\prime}}$ are called
Bogoliubov coefficients. Taking into account the orthonormality conditions
(3.62)-(3.68) we get

$\alpha^{\ell
m\omega}_{\ell^{\prime}m^{\prime}\omega^{\prime}}=(u^{\ell^{\prime}m^{\prime}M}_{\rightarrow\omega^{\prime}},u^{\ell
m}_{\rightarrow\omega})\quad,\quad\beta^{\ell
m\omega}_{\ell^{\prime}m^{\prime}\omega^{\prime}}=-(\bar{u}^{\ell^{\prime}m^{\prime}M}_{\rightarrow\omega^{\prime}},u^{\ell
m}_{\rightarrow\omega})\,,$ (3.70)

and

$\sum_{\ell^{\prime}\in{\mathbb{N}}}\sum_{m^{\prime}=-\ell^{\prime}}^{\ell^{\prime}}\int
d\omega^{\prime}\,\Big{(}\alpha^{\ell
m\omega}_{\ell^{\prime}m^{\prime}\omega^{\prime}}\alpha^{\ell^{\prime}m^{\prime}\omega^{\prime}}_{\ell^{\prime\prime}m^{\prime\prime}\omega^{\prime\prime}}-\beta^{\ell
m\omega}_{\ell^{\prime}m^{\prime}\omega^{\prime}}\bar{\beta}^{\ell^{\prime}m^{\prime}\omega^{\prime}}_{\ell^{\prime\prime}m^{\prime\prime}\omega^{\prime\prime}}\Big{)}=\delta(\omega,\omega^{\prime\prime})\,.$
(3.71)

Moreover, defining the particle number operator for the mode $(\ell,m,\omega)$
in the $u^{\ell m}_{\rightarrow\omega}$-expansion in the usual form $N^{\ell
m}_{\rightarrow\omega}=a^{\ell m\dagger}_{\rightarrow\omega}a^{\ell
m}_{\rightarrow\omega}$, its expectation value on the Minkowski vacuum can
generically be written as

$\bra{0}N^{\ell
m}_{\rightarrow\omega}\ket{0}=\sum_{\ell^{\prime}\in{\mathbb{N}}}\sum_{m^{\prime}=-\ell^{\prime}}^{\ell^{\prime}}\int
d\omega^{\prime}\,|\beta^{\ell
m\omega}_{\ell^{\prime}m^{\prime}\omega^{\prime}}|^{2}$ (3.72)

This object is what we are mainly interested in. It tells us the expectation
value of the number of excitations defined with respect to the conformal
vacuum $\ket{0}$ that are present in the Minkowski quantum vacuum
$\ket{0}_{M}$. The remarkable fact is that the computation of such object
mimics exactly the one for the Hawking’s particle production by a collapsing
black hole.

Let us choose $\mathscr{I}^{+}$ as the hypersurface over which we perform the
integral for the computation of scalar products at least for the outgoing
modes. $\mathscr{I}^{-}$ would be the choice for the ingoing ones. In order to
be able to use the near horizon approximate solutions (3.65), we introduce a
complete set of outgoing wave packets on $\mathscr{I}^{+}$ localized in
retarded time $\bar{u}$ and near the horizon $\bar{u}\to+\infty$ [Hawking,
1975]; see also [Fabbri and Navarro-Salas, 2005]. Concretely,

$u^{\ell
m;jn}=\frac{1}{\sqrt{\epsilon}}\int_{j\epsilon}^{(j+1)\epsilon}d\omega\,e^{2\pi
i\omega n/\epsilon}\,u^{\ell m}_{\rightarrow\omega}$ (3.73)

with integers $j\geq 0$, $n$, and where

$u^{\ell
m}_{\rightarrow\omega}=\frac{\sqrt{\Delta}}{\sqrt{\omega}}\,\frac{e^{-i\omega\bar{u}}}{r}Y^{\ell
m}(\vartheta,\varphi)\,.$ (3.74)

The wave packets $u^{\ell m;jn}$ are peaked around $\bar{u}\simeq 2\pi
n/\epsilon$ with width $2\pi/\epsilon$. When $\epsilon$ is small, the wave
packet is narrowly peaked about $\omega\simeq\omega_{j}=j\epsilon$ and
localised near the horizon. The facts that, due to spherical symmetry, the
modes (3.65) and (3.67) have exactly the same angular dependence, together
with the fact that, in the region where the wave packets are picked, the
behaviour in $\bar{u}$ and $u$ is independent of $\ell$, tells us that
particle creation will be the same in all angular modes.

The surface element of $\mathscr{I}^{+}$ is given by
$d\Sigma^{a}=r^{2}du\,dS^{2}\delta^{a}_{u}$. The Bogoliubov coefficients of
interest can therefore we written as

$\begin{split}\beta^{\ell
m;jn}_{\ell^{\prime}m^{\prime}\omega^{\prime}}&=-(\bar{u}^{\ell^{\prime}m^{\prime}M}_{\rightarrow\omega^{\prime}},u^{\ell
m;jn})\\\ &=i\int_{\mathscr{I}^{+}}du\,dS^{2}\,r^{2}\Big{(}u^{\ell
m;jn}\partial_{u}u^{\ell^{\prime}m^{\prime}M}_{\rightarrow\omega^{\prime}}-u^{\ell^{\prime}m^{\prime}M}_{\rightarrow\omega^{\prime}}\partial_{u}u^{\ell
m;jn}\Big{)}\,.\end{split}$ (3.75)

Since the wave packets vanish for $u\to-\infty$ and for $u>u_{-}$, we can
integrate by part finding

$\beta^{\ell,m;jn}_{\ell^{\prime}m^{\prime}\omega^{\prime}}=2i\int_{-\infty}^{u_{-}}du\,dS^{2}\,r^{2}\,u^{\ell
m;jn}\partial_{u}u^{\ell^{\prime}m^{\prime}M}_{\rightarrow\omega^{\prime}}\,.$
(3.76)

We now need to insert in the equation the explicit form of the modes
(3.73)-(3.74) and the outgoing part of the Minkowskian ones (3.67). However,
before doing that, let us recall that we are working and are mainly interested
in the near horizon limit $\bar{u}\to+\infty$. In this limit, the inverse of
the relation (3.51) between the conformal retarded time $\bar{u}$ and the
Minkowski $u$ simplifies into

$\bar{u}=\frac{2}{\sqrt{\Delta}}\operatorname{arcoth}\left(-\frac{u}{u_{-}}\right)\simeq\frac{1}{\sqrt{\Delta}}\log\left(\frac{u-u_{-}}{2u_{-}}\right)\,.$
(3.77)

So we can write

$\beta^{\ell,m;jn}_{\ell^{\prime}m^{\prime}\omega^{\prime}}=\frac{\sqrt{\Delta}\delta_{\ell,\ell^{\prime}}\delta_{m,m^{\prime}}}{2\pi\sqrt{\epsilon}}\int_{-\infty}^{u_{-}}du\,\int_{j\epsilon}^{\epsilon}d\omega\,e^{2\pi
i\omega
n/\epsilon}\sqrt{\frac{\omega^{\prime}}{\omega}}\,e^{-i\omega\frac{1}{\sqrt{\Delta}}\log\left(\frac{u-u_{-}}{2u_{-}}\right)-i\omega^{\prime}u}\,.$
(3.78)

Defining now $x=u_{-}-u$ we get

$\beta^{\ell,m;jn}_{\ell^{\prime}m^{\prime}\omega^{\prime}}=\frac{\sqrt{\Delta}\delta_{\ell,\ell^{\prime}}\delta_{m,m^{\prime}}}{2\pi\sqrt{\epsilon}}e^{-i\omega^{\prime}u_{-}}\int_{0}^{+\infty}dx\,\int_{j\epsilon}^{\epsilon}d\omega\,e^{2\pi
i\omega
n/\epsilon}\sqrt{\frac{\omega^{\prime}}{\omega}}\,e^{-i\omega\frac{1}{\sqrt{\Delta}}\log\left(\frac{x}{2u_{-}}\right)+i\omega^{\prime}x}\,.$
(3.79)

The integral over the frequency can be performed considering that $\omega$
varies in a small interval around $\omega_{j}$

$\begin{split}\beta^{\ell,m;jn}_{\ell^{\prime}m^{\prime}\omega^{\prime}}&=\frac{\sqrt{\Delta}\delta_{\ell,\ell^{\prime}}\delta_{m,m^{\prime}}}{\pi\sqrt{\epsilon}}e^{-i\omega^{\prime}u_{-}}\sqrt{\frac{\omega^{\prime}}{\omega_{j}}}\int_{0}^{+\infty}dx\,e^{+i\omega^{\prime}x}\,\frac{\sin(\epsilon
L/2)}{L}\,e^{iL\omega_{j}}\\\
&=\frac{\sqrt{\Delta}\delta_{\ell,\ell^{\prime}}\delta_{m,m^{\prime}}}{\pi\sqrt{\epsilon}}e^{-i\omega^{\prime}u_{-}}\sqrt{\frac{\omega^{\prime}}{\omega_{j}}}\;I(\omega^{\prime})\,,\end{split}$
(3.80)

where we have defined

$L(x)=\frac{2\pi
n}{\epsilon}-\frac{1}{\sqrt{\Delta}}\log\left(\frac{x}{2u_{-}}\right)$ (3.81)

and $I(\omega^{\prime})$ as the integral over $x$. The computation of
$\alpha_{jn,\omega^{\prime}}$ gives a similar result

$\begin{split}\alpha^{\ell,m;jn}_{\ell^{\prime}m^{\prime}\omega^{\prime}}&=\frac{\sqrt{\Delta}\delta_{\ell,\ell^{\prime}}\delta_{m,m^{\prime}}}{\pi\sqrt{\epsilon}}e^{i\omega^{\prime}u_{-}}\sqrt{\frac{\omega^{\prime}}{\omega_{j}}}\int_{0}^{+\infty}dx\,e^{-i\omega^{\prime}x}\,\frac{\sin(\epsilon
L/2)}{L}\,e^{iL\omega_{j}}\\\
&=\frac{\sqrt{\Delta}\delta_{\ell,\ell^{\prime}}\delta_{m,m^{\prime}}}{\pi\sqrt{\epsilon}}e^{i\omega^{\prime}u_{-}}\sqrt{\frac{\omega^{\prime}}{\omega_{j}}}\;I(-\omega^{\prime})\,.\end{split}$
(3.82)

Apart from different constants, these objects coincide with the ones defined
in [Fabbri and Navarro-Salas, 2005], and therefore can be solved using exactly
the same techniques and procedure. We refer to the book for details and we
give here only the final result.

The important result is that the relation between
$\alpha^{\ell,m;jn}_{\ell^{\prime}m^{\prime}\omega^{\prime}}$ and
$\beta^{\ell,m;jn}_{\ell^{\prime}m^{\prime}\omega^{\prime}}$ comes out to be

$|\beta^{\ell,m;jn}_{\ell^{\prime}m^{\prime}\omega^{\prime}}|=e^{-\frac{\pi\omega_{j}}{\sqrt{\Delta}}}\;|\alpha^{\ell,m;jn}_{\ell^{\prime}m^{\prime}\omega^{\prime}}|\,.$
(3.83)

Inserting this into Eq. (3.71), one can write

$-\left[1-\exp\left(\frac{2\pi\omega_{j}}{\sqrt{\Delta}}\right)\right]\sum_{\ell^{\prime}\in{\mathbb{N}}}\sum_{m^{\prime}=-\ell^{\prime}}^{\ell^{\prime}}\int_{0}^{+\infty}d\omega^{\prime}|\beta^{\ell,m;jn}_{\ell^{\prime}m^{\prime}\omega^{\prime}}|^{2}=1$
(3.84)

and therefore

$\bra{0}N^{\ell
m}_{\omega_{j}}\ket{0}=\frac{1}{\exp\left(\frac{2\pi\omega_{j}}{\sqrt{\Delta}}\right)-1}\,.$
(3.85)

The above expression coincides with the Planck distribution of thermal
radiation at the temperature

$T=\frac{\sqrt{\Delta}}{2\pi}\,.$ (3.86)

To relate this result to the first law, Eq. (3.46), it is enough to notice
that the explicit value of the conserved quantity ${\kappa_{\scriptscriptstyle
SG}}$ in our case is

${\kappa_{\scriptscriptstyle SG}}=\sqrt{\Delta}\,.$ (3.87)

We have shown what we anticipated at the very beginning of this section: the
light cone $u=u_{-}$ is seen as a (conformal) horizon with an associated
temperature $T$ given by expression (3.47).

The first law can therefore be rewritten as

$\delta M=T\delta S+\delta M_{\infty}\,.$ (3.88)

with

$T=\frac{{\kappa_{\scriptscriptstyle SG}}}{2\pi}\ \ \ \ \ \ {\rm and}\ \ \ \ \
\ \delta S=\frac{\delta A}{4}\,.$ (3.89)

The laws of light cone thermodynamics are now not simply a mere analogy, but
they acquire a precise semiclassical thermodynamical sense, which is better
discussed in the following subsection. This is, to our knowledge, the first
precise implementation of the idea [Sultana and Dyer, 2004, Dyer and Honig,
1979] that the quantity ${\kappa_{\scriptscriptstyle SG}}$ should play the
role of temperature for conformal Killing horizons.

##### 3.4.2 On the meaning of conformal energy and
temperature

In asymptotically flat stationary spacetimes, the time translational Killing
field can be normalized at infinity in order to give the analogue of (3.32)
the physical interpretation of energy as seen from infinity. On the other hand
in our case the vector field $\xi$ is normalized only on the observer sphere
$r=r_{\scriptscriptstyle O}$ and $t=0$. Thus $M$ has not the usual physical
meaning for any observer in Minkowski spacetime. Nevertheless, for conformally
invariant matter the mass $M$ as defined in (3.32) is conformally invariant
(see footnote 6). Using (3.52), and the fact that $\xi$ is actually a
normalized Killing field of the static FRW metric, one can interpret $M$ as
energy in the usual physical manner in that spacetime. This interpretation is
compatible with the notion of frequency we used to compute the Planckian
distribution (3.85). Indeed, the frequency $\omega$ is the one that would be
measured by an observer moving along the Killing field $\partial_{\tau}$ in
the static FRW space. For such notion of frequency $\omega$, the energy quanta
$\varepsilon=\hbar\omega$ correspond to the same physical notion of energy
that defines $M$.

Such interpretation carries over to its thermodynamical conjugate: the
temperature. That is the reason why we call conformal temperature the
temperature appearing in the first law. It carries the physical notion of
temperature, namely the one measured by thermometers, only for observers in
the FRW spacetime where $\xi$ is an actual time translational Killing field.
In this way both energy and temperature have their usual interpretation in a
spacetime that is conformally related to Minkowski.

##### 3.4.3 The Hartle-Hawking-like state

Let us now define a new radial coordinate

$R=\frac{\sqrt{\Delta}}{a}\exp(-\rho\sqrt{\Delta})\,.$ (3.90)

The near horizon limit $\rho\to+\infty$ corresponds now to $R\to 0$. In these
new coordinates, the metric (3.52) can be expanded around $R=0$ finding

$ds_{E}^{2}=-R^{2}d(\sqrt{\Delta}\tau)^{2}+dR^{2}+r_{\scriptscriptstyle
H}^{2}dS^{2}+O({R}{r^{-1}_{H}}\,dR^{2},Rr_{H}dS^{2})\,,$ (3.91)

where $O({R}{r^{-1}_{H}}\,dR^{2},Rr_{H}dS^{2})$ denotes subleading terms of
each component of the metric that do not change the nature of the apparent
singularity present at $R=0$. Notice that the leading order of the local
metric and the topological structure at the point $r=r_{\scriptscriptstyle H}$
are exactly the same as the one in the Reissner-Nordstrom metric, Eq. (3.3).

Moreover, the metric (3.52) can be continued analytically to imaginary
conformal Killing time by sending $\tau\to-i\tau_{E}$. As for the case of
static black holes [Wald, 1984], the result is a real Euclidean metric,
explicitly given by

$ds_{E}^{2}=\Omega_{E}^{2}\left(d\tau_{E}^{2}+d\rho^{2}+\Delta^{-1}\sinh^{2}(\rho\sqrt{\Delta})dS^{2}\right)$
(3.92)

with

$\Omega_{E}=\frac{\Delta/2a}{\cosh(\rho\sqrt{\Delta})-\cos(\tau_{E}\sqrt{\Delta})}.$
(3.93)

Defining again the new coordinate $R$ and carrying out the limit to $R=0$,
which corresponds to the Euclidean analogue of the horizon, we find the
Euclidean version of (3.91)

$ds_{E}^{2}=R^{2}d(\sqrt{\Delta}\tau_{E})^{2}+dR^{2}+r_{\scriptscriptstyle
H}^{2}dS^{2}+O({R}{r^{-1}_{H}}\,dR^{2},Rr_{H}dS^{2})\,.$ (3.94)

The coordinate singularity at $R=0$ can be resolved by defining new
coordinates $X=R\cos(\sqrt{\Delta}\tau_{E})$ and
$Y=R\sin(\sqrt{\Delta}\tau_{E})$. In order to avoid conical singularities one
must identify $\tau_{E}$ with a periodic coordinate such that

$0\leq\tau_{E}\sqrt{\Delta}\leq 2\pi\,.$ (3.95)

This removes the apparent singularity by replacing the first two terms in the
previous metric by the regular $dX^{2}+dY^{2}$ transversal metric. This
periodicity in time is what is used in the black hole case to suggest the
existence of a state—known as the Hartle-Hawking state—of thermal equilibrium
of any quantum field at a temperature given by

$T=\hbar\frac{\sqrt{\Delta}}{2\pi}\,,$ (3.96)

which coincides with the one found in the previous section, Eq. (3.86). This
tells us that the Minkowski vacuum can be regarded as the Hartle-Hawking type
of vacuum of Region II for conformally invariant theories.

Indeed, instead of using the near horizon approximation (and wave packets
peaked there) in the previous section, one could in principle compute the
Bogoliubov coefficients exactly between the Minkowski and conformal Fock
spaces. This should lead to the conclusion that Minkowski vacuum is everywhere
a thermal state with temperature (3.96), as suggested by the previous
analysis. As such computation might be rather involved and in view of keeping
the presentation as simple as possible, one can find additional evidence for
this by computing the expectation value of the normal ordered stress energy
“tensor” $:T_{ab}:$. The quotation marks on the word tensor are because the
object $:T_{ab}:$ does not transform as a tensor under a coordinate
transformation and cannot be interpreted physically as real. In fact the
physical and covariant energy momentum tensor has vanishing expectation value
in the Minkowski vacuum [Wald, 1995].

However, $:T_{ab}:$ can be interpreted as encoding the particle content of the
Minkowski vacuum as seen from the perspective of the MCKF that are of interest
in our analysis. For conformal fields it can be analytically computed in the
$s$-wave approximation $\ell=0$ which reduces the calculation to an effective
2-dimensional system. The 2-dimensional $:T^{\scriptscriptstyle(2)}_{ab}:$ can
be explicitly evaluated via the Virasoro anomaly [Birrell and Davies, 1984,
Fabbri and Navarro-Salas, 2005]. Given two sets of double null coordinates,
like the two we have $(u,v)$ and $(\bar{u},\bar{v})$,
$:T^{\scriptscriptstyle(2)}_{ab}:$ transforms as

$\begin{split}:T^{\scriptscriptstyle(2)}_{\bar{u}\bar{u}}:&=\left(\frac{du}{d\bar{u}}\right)^{2}:T^{\scriptscriptstyle(2)}_{uu}:-\frac{\hbar}{24\pi}\\{u,\bar{u}\\}\\\
:T^{\scriptscriptstyle(2)}_{\bar{v}\bar{v}}:&=\left(\frac{dv}{d\bar{v}}\right)^{2}:T^{\scriptscriptstyle(2)}_{uu}:-\frac{\hbar}{24\pi}\\{v,\bar{v}\\}\end{split}$
(3.97)

where

$\\{x,y\\}=\frac{\dddot{x}}{\dot{x}}-\frac{3}{2}\left(\frac{\ddot{x}}{\dot{x}}\right)^{2}$
(3.98)

is the Schwarzian derivative with dot representing $d/dy$. It is therefore
simple to evaluate the expectation value of this object on the Minkowski
vacuum $\ket{0}_{M}$ in our case. Since ${}_{M}\bra{0}:T_{ab}:\ket{0}_{M}=0$,
we simply have

$\begin{split}{}_{M}\bra{0}:T^{(\scriptscriptstyle
2)}_{\bar{u}\bar{u}}:\ket{0}_{M}=-\frac{\hbar}{24\pi}\\{u,\bar{u}\\}&=\frac{\hbar\Delta}{48\pi}\\\
{}_{M}\bra{0}:T^{\scriptscriptstyle(2)}_{\bar{v}\bar{v}}:\ket{0}_{M}=-\frac{\hbar}{24\pi}\\{v,\bar{v}\\}&=\frac{\hbar\Delta}{48\pi}\,.\end{split}$
(3.99)

The result indicates that the Minkowski state produce a constant ingoing and
outgoing thermal bath at the temperature (3.86) everywhere in Region II, which
is what we expected from a thermal equilibrium state. The near horizon
approximation in the computation of the previous section simplifies the
relation between the two sets of double null coordinates making the
computation analytically simpler, but as discussed above, it should give the
same result everywhere in Region II. As mentioned above, the expectation value
of the covariant stress energy tensor does not coincide with the normal
ordered one. The former is simply vanishing in this case
${}_{M}\bra{0}T_{ab}\ket{0}_{M}=0$.

As a final remark, let us get more insight into the geometry of the Euclidean
continuation (3.92) by writing the coordinate transformation to the flat
Euclidean coordinates $(t_{E},r,\vartheta,\varphi)$ covering
${\mathbb{R}}^{4}$. Defining the angular coordinate
$\alpha_{E}\equiv\tau_{E}\sqrt{\Delta}$ one finds

$\begin{split}t_{E}&=\frac{R\sin(\alpha_{E})}{1-\frac{R}{2r_{H}}\cos(\alpha_{E})+\frac{R^{2}}{4r^{2}_{H}}}\\\
r&=r_{H}\frac{1-\frac{R^{2}}{4r^{2}_{H}}}{1-\frac{R}{2r_{H}}\cos(\alpha_{E})+\frac{R^{2}}{4r^{2}_{H}}}\,.\end{split}$
(3.100)

The bifurcate sphere in the Euclidean continuation corresponds to the sphere
$r=r_{H}$ at $t_{E}=0$. The orbits of the Wick rotated radial conformal
Killing field are orbits of the radial conformal Killing field of
${\mathbb{R}}^{4}$ with fixed points given by the Euclidean shining sphere.
These orbits correspond, on the $(t_{E},r)$ plane, to close loops around the
bifurcate sphere, which degenerate into the $r=0$ line (the Euclidean
$t_{E}$-axis) for $R=2r_{H}$; see Figure 3.4. The coordinates
$(\tau_{E},R,\vartheta,\varphi)$ become singular there. The qualitative
features of the Euclidean geometry of the MCKF is just analogous to that of
the stationarity Killing field in the Euclidean RN solutions.

Figure 3.4: Three dimensional representation of the flow of the conformal
Killing field in the Euclidean spacetime ${\mathbb{R}}^{4}$. The orbits in
this one-dimension-less representation are non-concentric tori around the
bifurcate sphere $r=r_{H}$—here represented as a circle. They degenerate into
the $t_{E}$ axis for $R=2r_{H}$.

#### 3.5 Discussion

We have studied in detail the properties of radial Conformal Killing Fields in
Minkowski spacetime and showed that they present in many respects a natural
analogue of black holes in curved spacetimes. The global properties of radial
MCKFs mimic exactly the causal separation of events in the spacetime of static
black holes, i.e. those in the Reissner-Nordstrom family; see Figure 3.1.
Event Killing horizons in the latter are replaced by conformal Killing