Universidad Complutense de Madrid

Facultad de Ciencias Físicas

Vacuum polarisation and regular gravitational collapse

Polarización de vacío y colapso gravitatorio regular

Valentin Boyanov Savov

Supervisors:

Luis Javier Garay Elizondo,
Carlos Barceló Serón,
Raúl Carballo Rubio

Universidad Complutense de Madrid

Facultad de Ciencias Físicas

Tesis Doctoral

Vacuum polarisation and regular gravitational collapse

Polarización de vacío y colapso gravitatorio regular

Memoria para optar al grado de Doctor en Física

presentada por

Valentin Boyanov Savov111email<EMAIL_ADDRESS><EMAIL_ADDRESS>
Directores:

Luis Javier Garay Elizondo,
Carlos Barceló Serón,
Raúl Carballo Rubio

“First, the matter of thought exists in itself, while though in itself is
empty […] Second, the matter object in itself is something complete, which
does not require thought at all, while thought is something to be completed by
it […] Third, the object and thought are each in a sphere separate from the
other […] thought does not leave itself when receiving and adapting to matter,
but rather just modifies itself […] These are the fallacies which bar the
gateway into philosophy, and must be overcome before entry.”

\- G. W. F. Hegel, Science of Logic (1812).

###### Abstract

It is the goal of this work to revisit and revise the problem of black hole
(BH) formation and evolution in semiclassical gravity—a theory in which
spacetime is treated classically, while matter admits a quantum description,
coupling to gravity through an expectation value of a stress-energy tensor
operator. Particularly, we analyse the vacuum expectation value of this
operator for a test scalar field in spherically symmetric spacetimes in which
trapped regions either form or are close to forming. First, we look at the
magnitude of potential corrections to the spacetime evolution in the vicinity
of outer horizons formed by collapsing matter in different dynamical regimes.
We find that when the matter approaches an adiabatic collapse regime while
close to forming a trapped region (i.e. close to crossing its Schwarzschild
radius), the vacuum energy tends to grow unboundedly. This relates to the
Boulware state divergence at horizons, which in turn can be related to the
existence of static horizonless BH mimicker solutions to the semiclassical
Einstein equations. This suggests that the growing vacuum energy in slow
collapse regimes may stabilise the matter into a final horizonless
configuration.

We then look at how the dynamics of such horizonless ultracompact objects can
lead to the emission of Hawking radiation (without the formation of trapped
surfaces). We find that an oscillatory movement of its surface results in the
emission of bursts of radiation, while a slow collapse which asymptotically
approaches the crossing of the Schwarzschild radius can result in a thermal
spectrum akin to that of BHs, but with a modified temperature. We also analyse
the causal structure of the latter family of spacetimes (in which the
formation of a trapped surface is approached asymptotically), finding that
they posses an event horizon and either a Cauchy horizon, or two separate
future (null and timelike) infinity regions in the interior and exterior.

Finally, we look at the vacuum energy content in the vicinity of a BH inner
horizon, a region of spacetime which is classically known to amplify
perturbations in a highly non-linear manner. On a classical level, the
evolution of the inner horizon in the presence of generic perturbations
present in the astrophysical medium leads to the so-called “mass inflation
instability”, wherein curvature around and below the initial position of the
inner horizon grows exponentially, while the inner horizon itself tends to
approach the origin. On a semiclassical level, we find a negative ingoing flux
of energy, akin to the one which drives Hawking evaporation at the outer
horizon. However, backreaction from this flux seems to indeed be amplified,
growing exponentially and quickly overcoming of the Planck scale suppression
suffered by semiclassical dynamics. Indeed, classical mass inflation and the
semiclassical effect (which we dub inner horizon inflation) act in opposite
ways on the inner horizon: one pushing it inwards and the other outwards.
Analyses comparing the two effects suggest that the semiclassical one may
dominate at late times, making it possible for the trapped region to disappear
from the inside out, on a time scale much shorter than the Hawking evaporation
time. As an aside, we find that the techniques used for analysing the
semiclassical stability of the inner horizon can also be applied to other
geometries with similar causal properties. In particular, we look at the
geometry of a warp drive spacetime, and use its causal structure to argue that
an instability previously found in 2-dimensional models can be avoided in
higher dimensions.

To complete the picture of BH objects in semiclassical gravity, we propose
that, if trapped regions indeed disappear on short timescales, then the
ultracompact objects observed astrophysically may indeed be horizonless BH
mimickers, formed from slowly collapsing matter, the initial conditions for
which are obtained after the dissipation from one or several iteration of
trapped region formation, inner horizon inflation and recollapse.

Resumen

El objetivo de este trabajo es reexaminar y revisar el problema de la
formación y evolución de agujeros negros en gravedad semiclásica, una teoría
en la que el espaciotiempo se trata de forma clásica, mientras que la materia
admite una descripción cuántica, acoplándose a la gravedad a través de un
valor esperado de un operador tensor de energía-momento. En particular,
analizamos el valor esperado en vacío de este operador para un campo escalar
de prueba en espaciotiempos esféricamente simétricos en los que se forman o
están a punto de formarse regiones atrapadas. En primer lugar, examinamos la
magnitud de las potenciales correcciones a la evolución del espaciotiempo en
las proximidades de horizontes externos formados por materia en colapso en
diferentes regímenes dinámicos. Encontramos que cuando la materia se aproxima
a un régimen de colapso adiabático mientras está cerca de formar una región
atrapada (es decir, cerca de cruzar su radio de Schwarzschild), la energía de
vacío tiende a crecer ilimitadamente. Esto se relaciona con la divergencia
presente en el estado de Boulware en horizontes, que a su vez se puede
relacionar con la existencia de soluciones de las ecuaciones semiclásicas de
Einstein de objetos estáticos sin horizonte capaces de imitar
observacionalmente a los agujeros negros. Esto sugiere que la creciente
energía de vacío en los regímenes de colapso lento puede estabilizar la
materia en una configuración final sin horizonte.

Después, estudiamos cómo la dinámica de tales objetos ultracompactos sin
horizonte puede conducir a la emisión de radiación de Hawking (sin la
formación de superficies atrapadas). Encontramos que un movimiento oscilatorio
de su superficie da lugar a la emisión de ráfagas de radiación, mientras que
un colapso lento que se aproxime asintóticamente al cruce del radio de
Schwarzschild puede dar lugar a un espectro térmico similar al de los agujeros
negros, pero con una temperatura modificada. También analizamos la estructura
causal de esta última familia de espaciotiempos (en los que la formación de
una superficie atrapada se aproxima asintóticamente), encontrando que poseen
un horizonte de sucesos y, o bien un horizonte de Cauchy, o bien una
separación en dos de la región asintótica futura (de género tiempo y nulo)
entre el interior y el exterior.

Por último, examinamos el contenido de energía de vacío en las proximidades
del horizonte interno de un agujero negro, una región del espaciotiempo
clásicamente conocida por su amplificación no lineal de perturbaciones. A
nivel clásico, la evolución del horizonte interno en presencia de
perturbaciones genéricas presentes en el medio astrofísico conduce a la
inestabilidad llamada ”inflación de masa”, debido a la que la curvatura
alrededor y por debajo de la posición inicial del horizonte interno crece
exponencialmente, mientras que el propio horizonte interno tiende a acercarse
al origen. A nivel semiclásico, encontramos un flujo de energía entrante
negativo, similar al responsable de la evaporación de Hawking del horizonte
externo. Sin embargo, el efecto que tiene este flujo sobre la geometría parece
ser amplificado, creciendo exponencialmente y superando rápidamente la
supresión Planckiana que sufre la dinámica semiclásica. De hecho, la inflación
de masa clásica y el efecto semiclásico (que denominamos inflación del
horizonte interno) actúan de forma opuesta sobre el horizonte interno: uno lo
empuja hacia dentro y el otro hacia fuera. Los análisis que comparan ambos
efectos sugieren que el semiclásico puede dominar en tiempos tardíos, haciendo
posible que la región atrapada desaparezca desde dentro hacia fuera, en una
escala temporal mucho más corta que el tiempo de evaporación de Hawking.
Adicionalmente, observamos que las técnicas utilizadas para analizar la
estabilidad semiclásica del horizonte interno también pueden aplicarse a otras
geometrías con propiedades causales similares. En particular, examinamos la
geometría de un espaciotiempo de warp drive y utilizamos su estructura causal
para argumentar que una inestabilidad encontrada previamente en modelos
bidimensionales puede evitarse en dimensiones más altas.

Para completar la imagen de los agujeros negros en gravedad semiclásica,
proponemos que, si las regiones atrapadas desaparecen en escalas de tiempo
cortas, los objetos ultracompactos observados astrofísicamente pueden ser
imitadores de agujeros negros que no tienen horizonte, formados a partir de
materia que colapsa lentamente, cuyas condiciones iniciales se obtienen tras
la disipación de una o varias iteraciones de formación de regiones atrapadas,
inflación del horizonte interno y recolapso.

Acknowledgements

I would like to express my gratitude to my supervisors, Carlos, Raúl and Luis,
for teaching me most of what I know about black holes and quantum fields, and
for all their work in our joint research endeavours. Also, I would like to
thank Vitor Cardoso and Niayesh Afshordi for the hospitality and guidance they
provided during my research visits.

I also thank my friends and colleagues, both in Madrid and in Granada, who
have made my work and everyday life a lot more enjoyable during these past few
years. Also, I thank my family, as well as my sister’s cats (Fluff no1 and
Fluff no2), for continuing to provide a home whenever I need it. Also, shout-
out to the “Empanadas Margarita” shop near the Genil river in Granada, who
make the best Argentinian empanadas (and possibly the best food) I have ever
had. Last but not least, I would like to thank Kentaro Miura for the
inspiration he has given to so many people, including myself.

I acknowledge financial support from the Spanish Government through the
fellowship FPU17/04471, as well as the projects PID2020-118159GB-C43 and
PID-2020-118159GB-C44.

###### Contents

  1. Introduction
    1. $\mathscr{I}$-1. Preliminaries to quantisation in curved spacetimes
      1. $\mathscr{I}$-1.1. Flat spacetime quantisation
      2. $\mathscr{I}$-1.2. Curved spacetime quantisation
    2. $\mathscr{I}$-2. Semiclassical gravity
      1. $\mathscr{I}$-2.1. RSET in spherically symmetric spacetimes
      2. $\mathscr{I}$-2.2. Quantisation and renormalisation in 1+1 dimensions
      3. Vacuum states and thermal particle fluxes
      4. $\mathscr{I}$-2.3. Semiclassical Einstein equations
    3. $\mathscr{I}$-3. Black holes in classical and semiclassical gravity
      1. $\mathscr{I}$-3.1. Classical BH formation and evolution
      2. $\mathscr{I}$-3.2. Semiclassical BHs: the standard picture and beyond
  2. I Semiclassical effects near the outer horizon
    1. 1 Oscillating shell model and radiation emission
      1. 1.1 Spherical thin-shell geometries
        1. 1.1.1 Interpretation of the terms in $g$ and $h$
      2. 1.2 Oscillating shells and semiclassical effects
        1. 1.2.1 Resonance between in-crossing and out-crossing effects
        2. 1.2.2 Semiclassical effects
      3. 1.3 Summary and discussion
    2. 2 Horizon formation from slowly collapsing matter
      1. 2.1 Conformal factor at the horizon
      2. 2.2 RSET at the horizon for the “in” vacuum
      3. 2.3 Energy density, flux and pressure perceived by a free-falling observer at the horizon
      4. 2.4 Summary and discussion
    3. 3 Asymptotic horizon formation
      1. 3.1 The geometries
        1. 3.1.1 Static configurations
        2. 3.1.2 Including time-dependence
      2. 3.2 Causal structure
        1. 3.2.1 Causal structure for finite $g$
        2. 3.2.2 Causal structure with non-vanishing $f$
        3. 3.2.3 Diverging $g$ in finite time
      3. 3.3 Energy conditions
      4. 3.4 Hawking temperature
      5. 3.5 Conclusions
  3. II Inner horizon: instabilities and evolution
    1. 4 Classical inner horizon instability: a model of mass inflation
      1. 4.1 Mass inflation with thin shells
      2. 4.2 Geometry inside the mass-inflated region
      3. 4.3 Summary and conclusions
    2. 5 Semiclassically corrected inner horizon evolution
      1. 5.1 Backreaction on a BH with a static inner horizon
        1. 5.1.1 Geometric model and RSET
        2. 5.1.2 Perturbed Einstein equations
      2. 5.2 Dynamical horizons and self-consistent solutions
        1. 5.2.1 RSET around dynamical horizons
        2. 5.2.2 Perturbed equations and Hawking evaporation
        3. 5.2.3 Inner horizon expansion for a simple background
        4. 5.2.4 Collapsing matter: singularity or bounce
      3. 5.3 Conclusions
    3. 6 Interlude: blueshift instabilities in the horizon structure of warp drive spacetimes
      1. 6.1 Warp drive and the semiclassical instability
      2. 6.2 Geodesics and stability in 2+1 dimensions
        1. 6.2.1 Movement on the central axis
        2. 6.2.2 Other unstable points?
        3. 6.2.3 Vicinity of the unstable points and vacuum energy divergence
        4. 6.2.4 Numerical analysis of non-stationary configurations: further stabilising the warp drive
      3. 6.3 Conclusions
    4. 7 Classical mass inflation vs semiclassical inner horizon inflation
      1. 7.1 Series expansion around horizon formation
      2. 7.2 Time-integrable example
      3. 7.3 Conclusions and discussion
    5. Conclusions: the evolution of black holes in semiclassical gravity
    6. A RSET in 1+1 dimensions
      1. A.1 Geodesic point-splitting
      2. A.2 Renormalisation
      3. A.3 Trace anomaly
    7. B Oppenheimer-Snyder collapse
    8. C Geodesics approaching the point of divergent radial stretching

### Introduction

A black hole (BH) in classical general relativity (GR) is a finite region of
spacetime in which the future of all causal trajectories is disconnected from
that of the rest of the universe. The geometric description of its inner
region reveals a complex structure even in the classical theory [1, 2, 3, 4],
though this is all hidden behind an event horizon from which nothing can
escape. Incidentally, no information to confirm the existence of this inner
region as described by GR can escape either, making its study within this
theory a purely mathematical exercise.

This is no longer the case when quantum corrections are considered. As Hawking
showed [5], when one takes into account the presence of quantum fields in a
curved spacetime [6, 7] with an event horizon, the mass of the BH will slowly
be reduced, while an equivalent amount of energy will be emitted at infinity—a
process known as Hawking evaporation. As the mass is depleted, the BH outer
horizon would slowly shrink and reveal (i.e. bring into causal contact with
the outside universe) the innermost regions of the object. This is the first
indication that a complete description of BHs is physically necessary,
including an appropriate regularisation or dynamical avoidance of classical
singularities.

On the classical side, the GR description of the evolution of a BH interior
has gone a long way since the first model of spherical collapse [8]. It is now
known that the region of large curvature begins much further out than what
might be expected from looking at the Schwarzschild solution [4]; that the
central singularity may well have a chaotic oscillatory character [3], and
that another oscillatory singularity (albeit a milder one) develops at finite
radii over a Cauchy horizon [9].

Although many works on semiclassical BH physics still treat the interior of
these objects as essentially the Schwarzschild solution (particularly, many of
those which analyse the issue of information loss, e.g. [10, 11]), there have
been analyses which partially incorporate a more realistic version of a
classical BH interior [12, 13, 14, 15, 16, 17]. Still, a self-consistent
semiclassical solution of gravitational collapse which incorporates all the
inherent complexity of this problem does not yet exist. While it may well be
the case that only a full quantum description of gravity can provide us with a
consistent picture of BH spacetimes as a whole (see e.g. [18, 19, 20]), the
semiclassical approach used by Hawking still has a wide range of potential
applications, especially pertaining to BHs, which have not yet been fully
explored.

It is the goal of this thesis to bring together and expand upon the different
aspects of semiclassical BH analysis: from effects around the formation of
trapped regions, to their short- and long-term evolution under backreaction,
as well as the relation of this evolution to semiclassically-sustained static
horizonless BH mimickers [21, 22]. We work in asymptotically flat spacetimes,
and test these effects with a quantum massless scalar field in the “in” vacuum
state, defined as the Minkowski vacuum at past null infinity. We also work in
spherical symmetry, and our main tool for probing semiclassical corrections is
the renormalised stress-energy tensor (RSET) of this massless field,
calculated in the Polyakov approximation [23], which makes use of dimensional
reduction in the angular variables and a quantisation in a 1+1 dimensional
spacetime to approximate the radial and temporal components of the RSET, much
like an $s$-wave approximation.

We begin by studying the magnitude of semiclassical corrections in several
spacetimes which represent a collapse of matter in different dynamical
regimes. Building on previous work in this direction [24], we find that this
magnitude is large enough for backreaction to be relevant to the evolution
when the collapse is slowed down or reversed just before a trapped region is
formed [25]. This can be understood in terms of the fact that if the
background dynamics is stopped before a trapped region forms, the “in” vacuum
state quickly relaxes to the static Boulware state, which approaches a
divergent energy contribution at the horizon [26, 27, 28]. Particularly, we
use a toy model with a spherical thin shell of matter to explore three
different dynamical regimes: a small oscillation of the matter surface just
above the Schwarzschild radius, an asymptotic approach toward this radius, and
a crossing of this radius (and trapped region formation) at arbitrarily low
speeds. In the oscillating case, we find a series of bursts of outgoing
radiation emitted when the shell is closest to crossing its gravitational
radius but bounces back, separated by periods of approximately thermal
emission during the rest of each oscillation cycle. In the case of horizon
crossing at low speeds, we find large values in all components of the RSET,
tending to a divergence in the zero limit of the speed parameter, suggesting
backreaction would become extremely large when matter collapses slowly,
possibly leading to the formation of horizonless static ultracompact objects
[21, 22].

In the case of asymptotic approach toward the formation of an apparent
horizon, we find an exact thermal emission of Hawking-like radiation, although
at a temperature lower than that of Hawking radiation of a Schwarzschild BH of
the same mass. In light of this intriguing result, we subsequently analysed
the causal structure of these spacetimes [29], finding that an asymptotic
tendency in time toward the formation of a trapped region, if continued
indefinitely, is in fact sufficient to generate an event horizon, since the
expansion of outgoing radial light rays can tend to zero sufficiently quickly
for their overall radial displacement to be finite in infinite time. The
temperature of the quantum emission of these objects depends not only on the
surface gravity of the asymptotically formed apparent horizon, but also on the
details of their dynamical approach toward this formation. This discrepancy
leads to an exponential growth of the RSET components in the vicinity of the
would-be horizon, suggesting that these periods of thermal BH behaviour could
only last for a short time in a full self-consistent semiclassical evolution
of objects without trapped surfaces.

We then turn our attention to semiclassical effects in standard scenarios of
BH formation, where matter collapses quickly and the RSET is initially very
small in comparison to the classical energy in the system. We perturbatively
analyse backreaction, focusing in particular on the dynamics of the trapped
region. Classically, when a trapped region forms, it usually does not extend
all the way to the origin (at least initially). This is the case when a BH has
electric charge or angular momentum [1], or when an effectively classical
regular central region is present [30, 31]. The lower boundary of the trapped
region is known as the inner apparent horizon, and it has some unique
geometric properties. On the one hand, geodesic observers which approach it
experience a slow-down of their proper time parameter with respect to the
outside universe, making their trajectories extendable past the bounds of the
initial universe through what is known as a Cauchy horizon (as beyond it,
Cauchy initial data is no longer sufficient to determine the evolution). On
the other hand, small energy perturbations in its vicinity have a
disproportionately large effect on the growth of spacetime curvature. This
latter property in particular has lead to the discovery of the mass inflation
instability, wherein generic decaying classical perturbations present in the
astrophysical medium lead to an exponential growth of curvature (and, in
spherical symmetry, of the locally defined Misner-Sharp mass) [2, 32, 9].

We first look at the effect that backreaction from the RSET has on a
classically static inner horizon, as well as on an inner horizon which moves
to the origin prior to the formation of a spacelike singularity (as e.g.
happens in the Oppenheimer-Snyder model [8]). We find that the amplifying
effect the inner horizon has on perturbations makes it so the seemingly
negligible RSET can lead to drastic changes in the evolution of the trapped
region [33]. Particularly, since the RSET violates energy positivity
conditions [34], it actually pushes the inner horizon outwards, tending to
reduce the size of the trapped region from the inside. Our perturbative
analysis reveals that the initial tendency of this inner horizon semiclassical
correction is exponential in time, making it clear that, on the one hand, a
quasi-stationary approximation such as the one used to analyse Hawking
evaporation would not be adequate, and on the other, that the time scale
involved in inner-horizon-related dynamics is much shorter than the Hawking
evaporation time. This latter conclusion in particular leads to the
possibility that the whole semiclassical picture of BHs should be revised, and
full attention should be placed on how the inner horizon evolve in realistic
scenarios which involve both classical and semiclassical perturbations.

In an attempt to address this issue, we construct a toy model for mass-
inflation-inducing classical perturbations based on spherical thin shells
interacting with a generic (spherical) BH with an inner horizon. We analyse
when and how mass inflation is triggered from this interaction, and then use
these backgrounds to once again perform a perturbative semiclassical
backreaction analysis [35]. The classical and semiclassical perturbations
basically have opposite effects: the former pushing the inner horizon inwards,
while the latter pushing it outwards; we find that initially, the classical
evolution continues, but after a short time (linear in the mass) the
semiclassical outward push tends to dominate. This is further corroborated by
the asymptotic analyses performed in [14, 15, 16, 17], where it is found that
semiclassical backreaction at the Cauchy horizon, if such a horizon were to
form, would dominate over classical mass inflation, forming a stronger
curvature singularity. Extrapolating from our analysis, we argue that even in
the presence of mass inflation, the inner horizon may still end up having an
outward movement, extinguishing the trapped region from the inside and
changing the lifetime of BHs to an extremely short one. If a full numerical
analysis of the semiclassical Einstein equations were to indeed lead to this
outcome, then the extremely compact dark astrophysical objects observed [36,
37] may likely turn out to be horizonless BH mimickers, such as those obtained
semiclassically [21, 22].

Under this hypothesis, we conjecture that the full picture of BHs in
semiclassical gravity consists in the following. Initially, a trapped region
is formed by quickly collapsing classical matter, followed by an equally quick
outward inflation of the inner horizons and disappearance of this trapped
region. After some energy is dissipated from this process, the dispersed
classical matter recollapses under its own gravity. With possibly several
iterations of this process, enough energy is dissipated for the initial
recollapse conditions to be those of “slow collapse” in which the Boulware-
like terms in the RSET are ignited, leading to a final relaxation into a
semiclassical ultracompact horizonless BH mimicker.

This thesis is based on several publications [25, 29, 38, 33, 35, 39, 40]. The
structure is as follows. The remainder of the Introduction chapter presents a
brief overview of the theory of quantum fields in curved spacetimes,
explaining the ambiguities in quantising on non-trivial backgrounds and how
they can translate into different possible values for the RSET, and thus to
different self-consistent evolutions of spacetime. We briefly show how
quantisation is performed and how the RSET is calculated in a 1+1 dimensional
spacetime, on the one hand, because the simplicity allows for analytical
results, and on the other, because we then use the result to approximate the
RSET for spherically-symmetric 3+1 dimensional spacetimes. A short review of
how this theory has been applied to the study of BHs thus far is also
presented.

In part I we study the magnitude of semiclassical effects in the vicinity of
horizon formation for collapse scenarios with different classical dynamics. In
Chapter 1 we start by looking at the model of an oscillating thin shell which
periodically approaches crossing its own Schwarzschild radius only to bounce
back outwards each time. We find periodic emission of nearly thermal Hawking-
like radiation, separated by non-thermal bursts, along with large values of
the RSET. In Chapter 2 we use the same model, but allow the matter surface to
cross the Schwarzschild radius, and do so at arbitrarily low speeds. We find
that nearly static classical matter close to crossing this radius results in
large values of the RSET. In Chapter 3 we analyse a matter distribution which
collapses so slowly that it only approaches the formation of a trapped surface
asymptotically in time. We find an emission of thermal Hawking-like radiation
along with a growing value of the RSET, as well as some interesting causal
behaviours.

In part II we study semiclassical corrections to BHs with an inner apparent
horizon. In Chapter 4 we begin by reviewing the classical behaviour of such a
horizon, focusing in particular on the instability under perturbations known
as “mass inflation”. We present a simple model which contains the essential
characteristics necessary to reproduce this instability, while also being
analytically solvable. On the other hand, in Chapter 5 we focus on the
semiclassically sourced evolution of an inner horizon in the absence of
classical perturbations, obtaining the result that trapped regions have a
tendency to evaporate not only form the outside, but also from the inside. In
Chapter 6 we take a brief detour, wherein we apply the conceptual results of
the inner horizon analysis to determine the semiclassical stability of a
different class of spacetimes: the warp drive. Finally, in Chapter 7 we return
to BHs and put together the classical and semiclassical perturbations in order
to see what the evolution of a generic inner horizon may be. We find a
tendency for classical evolution to dominate initially, but for semiclassical
effects to become relevant after a very short timescale (of the order of the
BH mass in geometric units).

In the Conclusions we summarise our results and link them together to list the
possibilities of what the ultimate fate of BHs in semiclassical gravity could
be.

We use the metric signature $(-,+,+,+)$. Greek letters $\mu,\nu$, etc. are
used for spacetime indices in 3+1 dimensions, and Latin letters $a,b$, etc.
for indices in 1+1 dimensions, unless otherwise stated. In general we use
natural units, where $c=1$, $G=1$ and $\hbar=1$, except when writing the
semiclassical Einstein equations, where having $\hbar$ written explicitly is
useful for comparison between the classical and semiclassical terms (to make
the dimensionality of the constant clear we in fact use the square of the
Planck length, $l_{\rm p}^{2}$, rather than $\hbar$).

####  $\mathscr{I}$-1. Preliminaries to quantisation in curved spacetimes

The quantisation of a field in Minkowski spacetime follows a well established
procedure (see e.g. [41]), the physical validity of which is shown by a myriad
of experimental results. However, this procedure relies heavily on the
symmetries present in flat spacetime (i.e. Poincaré invariance), making its
generalisation to curved spacetimes a complicated task. We begin by briefly
explaining the method of this generalisation and its inherent ambiguities,
following in part the discussion presented in [6].

Here and throughout the thesis, we make use of a massless real scalar $\phi$
minimally coupled to gravity. Given that we will work in spherical symmetry,
the quantisation of this field is sufficient to probe generic effects from the
presence of quantised matter. Its action is

$S_{\phi}=\int d^{4}x\sqrt{-g}\,\mathcal{L}_{\phi}=-\frac{1}{2}\int
d^{4}x\sqrt{-g}\,\nabla^{\mu}\phi\nabla_{\mu}\phi,$ ($\mathscr{I}$.1)

where $g$ is the determinant of the metric tensor $g_{\mu\nu}$, and
$\mathcal{L}_{\phi}$ is the Lagrangian density. The equation of motion which
determines the dynamics of this field, known as the Klein-Gordon equation, is
simply

$\nabla^{\mu}\nabla_{\mu}\phi=0,$ ($\mathscr{I}$.2)

where $\nabla^{\mu}$ is the covariant derivative compatible with $g_{\mu\nu}$.

#####  $\mathscr{I}$-1.1. Flat spacetime quantisation

In flat spacetime, using Cartesian coordinates adapted to an inertial observer
we have $\nabla_{\mu}=\partial_{\mu}$. We define the Klein-Gordon product,

$(\phi_{1},\phi_{2})_{\rm
KG}=-i\int_{\mathbf{\Sigma}_{t}}\phi_{1}\overset{\text{\tiny$\leftrightarrow$}}{\partial}_{t}\phi_{2}^{*}\,d\mathbf{\Sigma}_{t},$
($\mathscr{I}$.3)

where $\mathbf{\Sigma}_{t}$ is a $t=$ const. hypersurface, $t$ being $x^{0}$,
and
$f\overset{\text{\tiny$\leftrightarrow$}}{\partial}_{t}g=f\partial_{t}g-(\partial_{t}f)g$.
For pair of solutions to the Klein-Gordon equation, this constitutes a pseudo-
scalar product (as the norm it generates is not positive-definite), the values
of which are independent of the particular choice of $t=$ const. slice. A
basis of solutions to the Klein-Gordon equation which is orthonormal with
respect to the product ($\mathscr{I}$.3) are the normalised plane waves

$\begin{split}u_{\mathbf{k}}&=\frac{1}{\sqrt{2\omega(2\pi)^{3}}}e^{-i\omega
t+i\mathbf{k}\cdot\mathbf{x}}\quad\text{with norm
}\delta(\mathbf{k}-\mathbf{k}^{\prime}),\\\
v_{\mathbf{k}}=u_{\mathbf{k}}^{*}&=\frac{1}{\sqrt{2\omega(2\pi)^{3}}}e^{i\omega
t-i\mathbf{k}\cdot\mathbf{x}}\quad\text{with norm
}-\delta(\mathbf{k}-\mathbf{k}^{\prime}),\end{split}$ ($\mathscr{I}$.4)

where $\omega=|\mathbf{k}|$ is the frequency and $\mathbf{k}$ is the wave
number, $\mathbf{x}$ being the 3-vector of the spatial Minkowski coordinates.
A generic solution can be expanded in terms of this basis as

$\phi=\sum_{\mathbf{k}}(a_{\mathbf{k}}u_{\mathbf{k}}+a_{\mathbf{k}}^{*}u_{\mathbf{k}}^{*}).$
($\mathscr{I}$.5)

Formally, if the domain in $\mathbf{x}$ is infinite, then $\mathbf{k}$ becomes
continuous and should be integrated over rather than summed, here and in the
expressions below.

The basis ($\mathscr{I}$.4) allows us to divide solutions into two subspaces:
one comprised of solutions which can be obtained from an expansion solely in
$u_{\mathbf{k}}$’s, which we will denote by $S_{\oplus}$, and one of solutions
which are obtained from $v_{\mathbf{k}}$’s, which will be denoted by
$S_{\ominus}$. The $S_{\oplus}$ functions are referred to as positive-
frequency solutions, and the $S_{\ominus}$ ones as negative-frequency
solutions. Any solution can be expressed as $s=s_{+}+s_{-}$, where $s_{+}\in
S_{\oplus}$ and $s_{-}\in S_{\ominus}$. Note that while all positive-frequency
solutions have positive norm, the reverse is not true: a solution with an
overall positive norm can have negative-frequency components. In other words,
the product ($\mathscr{I}$.3) is not sufficient to define a unique positive-
and negative-frequency subspace separation. This will play a key role in the
ambiguity of quantisation in curved spacetimes.

The quantisation of this field is performed by promoting $\phi$ to an operator
$\hat{\phi}$ which satisfies the equal time commutation relations

$\begin{split}[\hat{\phi}(t,\mathbf{x}),\hat{\phi}(t,\mathbf{x}^{\prime})]&=0,\\\
[\hat{\pi}(t,\mathbf{x}),\hat{\pi}(t,\mathbf{x}^{\prime})]&=0,\\\
[\hat{\phi}(t,\mathbf{x}),\hat{\pi}(t,\mathbf{x}^{\prime})]&=i\delta^{3}(\mathbf{x}-\mathbf{x}^{\prime}),\end{split}$
($\mathscr{I}$.6)

where $\hat{\pi}=\partial_{t}\hat{\phi}$ is the canonical conjugate of the
field. The field operator can now be expanded in terms of the same mode basis,
but the coefficients in the expansion become operators themselves,

$\hat{\phi}=\sum_{\mathbf{k}}(\hat{a}_{\mathbf{k}}u_{\mathbf{k}}+\hat{a}_{\mathbf{k}}^{\dagger}u_{\mathbf{k}}^{*}).$
($\mathscr{I}$.7)

The operators $\hat{a}_{\mathbf{k}}$ and $\hat{a}^{\dagger}_{\mathbf{k}}$ are
known as particle annihilation and creation operators, which act on the
Hilbert space which spans all quantum states. Thanks to the orthonormality of
the modes ($\mathscr{I}$.4), the relations ($\mathscr{I}$.6) translate into
the simple commutation relations for $\hat{a}_{\mathbf{k}}$ and
$\hat{a}^{\dagger}_{\mathbf{k}}$,

$\begin{split}[\hat{a}_{\mathbf{k}},\hat{a}_{\mathbf{k}^{\prime}}]&=0,\\\
[\hat{a}_{\mathbf{k}}^{\dagger},\hat{a}_{\mathbf{k}^{\prime}}^{\dagger}]&=0,\\\
[\hat{a}_{\mathbf{k}},\hat{a}_{\mathbf{k}^{\prime}}^{\dagger}]&=\delta_{\mathbf{kk}^{\prime}},\end{split}$
($\mathscr{I}$.8)

which are akin to those of a harmonic oscillator. We can then define the
vacuum state as the one annihilated by all the $a_{\mathbf{k}}$ operators,

$\hat{a}_{\mathbf{k}}\ket{0}=0,\quad\forall\mathbf{k}.$ ($\mathscr{I}$.9)

A state of $n$ particles with momentum $\mathbf{k}$ is then obtained by acting
$n$ times on $\ket{0}$ with the creation operator
$\hat{a}_{\mathbf{k}}^{\dagger}$. A quantum field can thus be thought of as an
infinite assembly of quantum harmonic oscillators.

Ultimately, the plane-wave basis used for the flat spacetime quantisation is,
in a sense, a natural choice, given the symmetry of the background: the total
number of particles in this quantisation is Poincaré invariant, though their
momentum is of course relative. In other words, if the whole procedure were
carried out in any other set of inertial Minkowski coordinates, the resulting
vacuum state would be the same, and the momentum of particle states would be
related by a boost.

#####  $\mathscr{I}$-1.2. Curved spacetime quantisation

If we attempt to repeat the same procedure in an arbitrary curved spacetime
background, then we quickly run into problems. In a globally hyperbolic
spacetime [42], we can define a covariant generalisation of the Klein-Gordon
product,

$(\phi_{1},\phi_{2})_{\rm
KG}=-i\int_{\mathbf{\Sigma}_{t}}\phi_{1}\overset{\text{\tiny$\leftrightarrow$}}{\nabla}_{\mu}\phi_{2}^{*}\,n^{\mu}\sqrt{-h}\,d\mathbf{\Sigma}_{t},$
($\mathscr{I}$.10)

where $t$ is any time function and $\mathbf{\Sigma}_{t}$ is again a $t=$
const. surface, $n^{\mu}$ is a unit vector orthogonal to this surface, and $h$
is the determinant of the induced metric on the surface. For solutions to the
Klein-Gordon equation, this product is invariant under changes of the time
function $t$. Any solution $s$ therefore has a specific well-defined norm, and
orthonormal bases, such as the one used for quantisation in flat spacetime,
can be constructed. However, there are infinitely many such bases, which
generally have a different separation into $s_{+}$ and $s_{-}$ sectors, (which
determine the separation between creation and annihilation variables and thus
define the vacuum state).

Quantising the field in a curved spacetime brings with it two difficulties.
The first is a practical one: obtaining explicit expressions for any basis of
solutions to ($\mathscr{I}$.2) is generally a very challenging task. Only in a
small handful of spacetimes can any basis be written in terms of known
functions. The second difficulty is a conceptual one: for generic spacetimes,
there is no a priori indication as to which one of the infinitely many bases
should be chosen for quantisation [43, 6, 7], different choices generally
leading to physically inequivalent results.

To see that this is the case, let us first write the field operator as

$\hat{\phi}=\sum_{i}(\hat{a}_{i}u_{i}+\hat{a}^{\dagger}_{i}u_{i}^{*}),$
($\mathscr{I}$.11)

where $u_{j}$ is a set of functions which form an orthonormal basis with
respect to the Klein Gordon product, and $u_{i}$ and $u_{i}^{*}$ generalise
the positive- and negative-frequency subspaces of solutions (the range of the
$i$ index would then be half of the number of elements in the basis). The
separation into these two subspaces is unique for each basis of solutions (as
defined by the associated complex structure [43, 7]) and, being directly
related to the definition of the creation and annihilation operators, defines
the vacuum and particle states. If we consider an expansion using an
alternative basis,

$\hat{\phi}=\sum_{i}(\hat{b}_{i}\tilde{u}_{i}+\hat{b}^{\dagger}_{i}\tilde{u}_{i}^{*}),$
($\mathscr{I}$.12)

where $\tilde{u}_{j}$ are also orthonormal, and $\hat{b}$ and
$\hat{b}^{\dagger}$ are our new annihilation and creation operators, then the
annihilation operator of the first quantisation can be written as

$\hat{a}_{i}=\sum_{k}(\alpha_{ki}\hat{b}_{k}+\beta^{*}_{ki}\hat{b}^{\dagger}_{k}),$
($\mathscr{I}$.13)

where

$\alpha_{ki}=(\tilde{u}_{k},u_{i})_{\rm
KG},\quad\beta_{ki}=-(\tilde{u}_{k},u_{i}^{*})_{\rm KG}$ ($\mathscr{I}$.14)

are known as the Bogolyubov coefficients. When $\beta_{ki}\neq 0$, the vacuum
state of one quantisation can appear full of particles for another, and vice
versa. Therefore, some additional criterion appears to be needed in order to
single out a particular basis and vacuum state.

In flat spacetime, this criterion is derived from the symmetries of the
background. The free quantisation which respects these symmetries lies at the
base of the standard model of particle physics. The same criterion can be
extended directly to stationary spacetimes, where the timelike Killing vector
field can be used for a unique separation between the positive- and negative-
frequency subspaces $S_{\oplus}$ and $S_{\ominus}$, and consequently to a set
of equivalent quantisations [43]. Particularly, when the temporal part of the
Klein-Gordon equation is separable, specifying a basis of solutions for this
part already allows for a separation between the two subspaces, and choosing
different spatial components for the remainder of the basis function alone
cannot mix them.

Another case which allows for a natural choice of quantisation is found in
spacetimes which are stationary at least in a region large enough to define
the initial conditions of a field solution. The quantisation is then defined
as the preferred one in the region of symmetry, and then the corresponding
modes are evolved through the subsequently dynamical spacetime. For instance,
this can be done in cosmology if one considers that the scale factor of the
Friedmann–Lemaître–Robertson–Walker geometry is initially constant for some
duration of time [44]. For asymptotically flat spacetimes, a similar approach
can be applied. Particularly, the spacetime need only be flat at past null
infinity in order to provide preferred initial conditions for the modes of a
massless field. We will refer to the vacuum state of this quantisation as the
“in” state [5, 23, 45], and we will use it throughout this thesis for
calculating the RSET probing semiclassical effects in spacetimes of
gravitational collapse.

For generic dynamical spacetimes, however, the choice of modes and vacuum
state remains ambiguous. This is why much of the techniques developed for
working with quantum fields in curves spacetimes, such as the covariant
renormalisation we will discuss below, are formulated in a way which can be
applied for any choice of modes. The matter of how the ambiguity is fixed in
our universe as a whole is, for the semiclassical theory, relegated to a
choice of initial conditions, which can only be determined by the observation
of cosmological deviations from classicality.

####  $\mathscr{I}$-2. Semiclassical gravity

The physical implications of the quantisation ambiguity presented above may
not appear immediately clear, as the discussion so far has been focused on the
dynamics of a single field free of any interaction. In order to endow this
field with a physical meaning, one has to model its interactions, either with
other fields or with some effective detector system. For a field in curved
spacetimes, it turns out that interaction is in fact inevitable, as the
covariant description of its dynamics already couples it (minimally) to
gravity, and thus endows it with a physical role as a source of curvature. In
other words, one may define a field with no interactions to other fields or
detectors, but so long as it has energy it will gravitate, and thus have an
observable effect.

This brings with it an issue of its own: namely, after quantising the field,
it would still have to dictate how the (seemingly) classical spacetime curves.
What would the gravitational field of a superposition of states look like?
Consistently coupling these systems requires either the quantisation of
gravity (allowing for superpositions of gravitational fields and causalities)
[46, 47, 48], or a “classicalisation” of matter when it reaches the energy
scale at which it interacts gravitationally [49, 50, 51]. Despite the fact
that, as of yet, no approach in either direction has been fully successful,
probing the interface between the realms of gravity and quantum matter is
still possible, albeit only approximately and in certain situations.

The theory of semiclassical gravity describes the evolution of a classical
spacetime with a quantum stress-energy tensor source, which is brought to
classicality by means of an expectation value, i.e. the RSET mentioned above.
This theory can be argued to be the leading order contribution in a fully
quantum system of gravity and matter [12, 52], and in general is intuitively
expected to provide a good approximation when quantum matter is not highly
delocalised (c.f. [53, 54, 55, 56]). From a classical standpoint, it is merely
a modified version of Einstein’s theory in which source terms (the right-hand
side of the field equations) have a dependence on the derivatives of the
metric of order higher than one (unlike in GR). From a quantum standpoint, it
reflects the ambiguity in the quantisation (different choices leading to
different spacetime evolutions) and the change in the energy content of the
vacuum and particle states as the background evolves, properly encoding these
characteristics in the source term [6, 7]. In practice, the RSET is usually
computed in the vacuum state of the chosen quantisation, and focus is placed
on the counter-intuitive fact that this state has an energy content even after
renormalisation.

Let us explicitly see how this theory is constructed in the case of the
massless scalar seen above. Starting from the classical theory, when
considering the dynamics of spacetime, the full action of the theory must
include the Einstein-Hilbert term [57], becoming

$S=\int\mathop{d^{4}x}\sqrt{-g}\left(\frac{R}{16\pi}-\frac{1}{2}\nabla^{\sigma}\phi\nabla_{\sigma}\phi\right),$
($\mathscr{I}$.15)

where $R$ is the Ricci scalar (the trace of the Ricci tensor $R_{\mu\nu}$).
The variation of this action with respect to the metric gives the Einstein
equations,

$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=8\pi T_{\mu\nu},$ ($\mathscr{I}$.16)

with the scalar field stress-energy tensor

$T_{\mu\nu}=\nabla_{\mu}\phi\nabla_{\nu}\phi-\frac{1}{2}g_{\mu\nu}\nabla^{\sigma}\phi\nabla_{\sigma}\phi.$
($\mathscr{I}$.17)

Once the field is quantised, a stress-energy tensor operator can be
constructed by substituting $\phi\to\hat{\phi}$ (if the field is real and the
operator $\hat{\phi}$ is self-adjoint, then no additional symmetrisation is
needed).222Alternatively, the operator expectation value can be constructed
from a variational principle of an effective action [6, 58], which provides
alternative methods for renormalisation to the one presented below, giving
equivalent results. The resulting tensor operator, being quadratic in the
field, has a divergence in its expectation value. In flat spacetime, this is
none other than the divergence which is subtracted through “normal ordering”
of the creation and annihilation operators [41], equivalent to simply
subtracting the expectation value in vacuum from the ones in all other states.
In curved spacetimes, rather than directly subtracting the vacuum state value,
a more involved renormalisation procedure is followed, making use of the local
geometric nature of the diverging terms [6, 7], as we will see. Once the
procedure is complete, we are left with a prescription for the RSET which
sources the semiclassical Einstein equations.

#####  $\mathscr{I}$-2.1. RSET in spherically symmetric spacetimes

To summarise, calculating the RSET of our scalar field involves finding a
basis of solutions to eq. ($\mathscr{I}$.2) corresponding to a physically
reasonable quantisation (e.g. the “in” quantisation in asymptotically flat
spacetimes [23, 45], or adiabatic quantisation in cosmology [59]), then using
the field operator ($\mathscr{I}$.11) to construct the stress-energy tensor
operator through the functional expression ($\mathscr{I}$.17), and computing
its renormalised expectation value in the vacuum state of the chosen
quantisation. This calculation can only be performed analytically in cases
with either a high degree of symmetry, such as a conformally coupled field in
a homogeneous and isotropic cosmological model [60, 52], or for a lower number
of dimensions, such as the 1+1 case [61], or a combination of the two, such as
the 2+1 dimensional BH [62].

For BH spacetimes in 3+1 dimensions, one seems to be left with no other choice
but to attempt to calculate the RSET numerically. Indeed, tremendous progress
has been made in this direction over the past decades [63, 64, 65, 66, 67, 68,
69]. However, the trade-off for the precision which these calculations provide
lies in the fact that they can be performed only on certain fixed backgrounds,
and only in certain quantum vacuum states. The full process of gravitational
collapse and BH evolution in semiclassical gravity cannot as yet be treated
self-consistently with this approach.

A trade-off in the opposite direction can be made by sacrificing the precision
of the result (indeed, only preserving its qualitative character in certain
regions) for a tool which can be applied to calculating the RSET in a variety
of dynamical geometries which represent the formation and evolution of BHs and
stellar objects: the Polyakov approximation [23]. This approximation consists
in dimensionally reducing a spherically symmetric spacetime by integrating out
the angular variables, then quantising in the 1+1 dimensional radial-temporal
sector and calculating the RSET there, and finally applying the result back to
3+1 dimensions.

The line element of a 3+1 dimensional spherically-symmetric geometry can be
written as

$ds^{2}=h_{ab}(t,r)x^{a}x^{b}+r^{2}d\Omega^{2},$ ($\mathscr{I}$.18)

where the indices $a,b$ refer to the temporal and radial dimensions and
$h_{ab}$ is a $2\times 2$ tensor. We can consider a field $\phi$ propagating
on this spacetime for which the $s$-wave contribution is dominant, i.e. which
essentially has a dependence only on the radial and temporal coordinates,
$\phi=\phi(t,r)$. As a functional of the field, the Lagrangian density
$\mathcal{L}_{\phi}$ also loses its angular dependence, and the part of the
action dictating the field dynamics can then be reduced to two dimensions by
integrating out the angular variables

$S_{\phi}=\int\mathop{d^{4}x}\sqrt{-g}\,\mathcal{L}_{\phi}(t,r)=4\pi\int\mathop{d^{2}x}\sqrt{-h}\,r^{2}\,\mathcal{L}_{\phi}(t,r),$
($\mathscr{I}$.19)

where $h$ is the determinant of $h_{ab}$. The relation between the stress-
energy tensor in the four-dimensional theory $T_{\mu\nu}^{(4)}$ and the
stress-energy tensor corresponding to the dimensionally-reduced field dynamics
$T_{ab}^{(2)}$ is then

$T_{\mu\nu}^{(4)}=\frac{\delta_{\mu}^{a}\delta_{\nu}^{b}T_{ab}^{(2)}}{4\pi
r^{2}}.$ ($\mathscr{I}$.20)

We note, however, that the two dimensional theory described by
($\mathscr{I}$.19) is not that of a free field—a part of the four-dimensional
kinetic term takes on the role of a potential after the dimensional reduction,
namely the part responsible for backscattering in the $s$-wave sector.

The Polyakov approximation to the RSET is obtained with two simplifying
assumptions. The first one is that this potential in the equation of motion of
the dimensionally-reduced field can be disregarded. How well this assumption
is justified varies depending on the spacetime in question, but it is worth
noting in particular that the potential is in fact zero both at infinity and
at the horizons of BHs [70]. It therefore may be expected that horizon-related
effects are captured well in this approximation; indeed, Hawking’s result only
changes slightly when one includes backscattering (correcting the black-body
spectrum with grey-body factors) [6]. The second assumption is that the two-
dimensional theory retains at least a qualitative similarity to the full four-
dimensional one even after quantisation and renormalisation. In other words,
that eq. ($\mathscr{I}$.20) can be applied to relate the RSET of the two
dimensional theory with an approximation to the four dimensional RSET. Of
course, this approximation is far from exact, as dimensional reduction and
renormalisation do not commute [71]. Additionally, the Polyakov approximation
becomes less reliable the closer one gets to the origin of spherical
coordinates $r=0$, as it tends to a non-physical divergence there. In spite of
these issues, it works well in providing an analytical expression for a
conserved RSET which captures some of the essential characteristics of quantum
field theory in curved spacetimes, such as the ambiguity in the choice of
vacuum and the violation of energy conditions. For the purposes of this work,
it is important to note that it does capture some well-established horizon-
related effects, such as Hawking evaporation, the Boulware state outer horizon
divergence, and the Hadamard state Cauchy horizon divergence (all of which
will be explained in more detail below) [72, 26, 12, 23]. We will therefore
work with this approximation to probe semiclassical effects near the edges of
trapped regions or, more generally, in regions where the causal structure
approaches a light-trapping behaviour.

#####  $\mathscr{I}$-2.2. Quantisation and renormalisation in 1+1 dimensions

To construct the Polyakov RSET for spherically symmetric spacetimes, we need
to quantise our test field in 1+1 dimensions, isolate the divergence in the
expectation value of the stress-energy tensor operator, subtract this
divergence in a covariant manner (i.e. with counterterms proportional to
tensor quantities), and finally plug the result into the approximation
($\mathscr{I}$.20). Luckily, in 1+1 dimensions the system is simple enough for
this whole procedure to be done analytically, even while keeping the spacetime
and the choice of quantum modes arbitrary. This fact in particular allows not
only the easy calculation of the RSET on fixed backgrounds, but also to
determine the background at the same time as the Polyakov RSET in a
semiclassically self-consistent manner.

In 1+1 dimensions, a spacetime metric has only one physical degree of freedom,
and therefore takes on a conformally flat form in appropriate sets of
coordinates. For the calculation of the RSET, it is convenient to work in
double null coordinates,

$ds^{2}=-C(u,v)\,du\,dv,$ ($\mathscr{I}$.21)

where $C$ is the conformal factor. The Klein-Gordon equation for the minimally
coupled scalar ($\mathscr{I}$.2) in this coordinate system takes the form

$\phi_{,uv}=0,$ ($\mathscr{I}$.22)

where the comma indicates partial differentiation. Note that this equation is
conformally invariant, and therefore has the same form as in flat spacetime
for any function $C$. Analogously to the flat spacetime case, a basis of
solutions which is orthonormal with respect to the Klein-Gordon product
($\mathscr{I}$.10) is given by the right- and left-moving plane waves

$\phi_{\omega}(u)=\frac{1}{\sqrt{4\pi\omega}}e^{-i\omega
u},\quad\phi_{\omega}(v)=\frac{1}{\sqrt{4\pi\omega}}e^{-i\omega v},$
($\mathscr{I}$.23)

and their complex conjugates, with $\omega$ being a positive real number (the
frequency of the wave). Quantization can be performed by defining the field
operator

$\hat{\phi}=\sum_{\omega}\left[\hat{a}^{R}_{\omega}\phi_{\omega}(u)+(\hat{a}^{R}_{\omega})^{\dagger}\phi_{\omega}^{*}(u)+\hat{a}^{L}_{\omega}\phi_{\omega}(v)+(\hat{a}^{L}_{\omega})^{\dagger}\phi_{\omega}^{*}(v)\right],$
($\mathscr{I}$.24)

with the creation and annihilation operators for each type of mode satisfying
the non-zero commutation relations

$[\hat{a}^{R}_{\omega},(\hat{a}^{R}_{\omega^{\prime}})^{\dagger}]=[\hat{a}^{L}_{\omega},(\hat{b}^{L}_{\omega^{\prime}})^{\dagger}]=\delta_{\omega\omega^{\prime}}.$
($\mathscr{I}$.25)

The vacuum state $\ket{0}$ is defined by
$\hat{a}^{R}_{\omega}\ket{0}=\hat{a}^{L}_{\omega}\ket{0}=0$. An orthonormal
basis of the standard Fock space containing all particle states can then be
obtained by applying the creation operators $(\hat{a}^{R}_{\omega})^{\dagger}$
and $(\hat{a}^{L}_{\omega})^{\dagger}$ on the vacuum state, with appropriate
normalization.

Since the field operator in eq. $\mathscr{I}$.24 is constructed to be self-
adjoint, its Lagrangian density is the same as that of the real, massless
scalar field, and its corresponding stress-energy tensor operator can be
obtained directly from eq. $\mathscr{I}$.17 with the substitution
$\phi\to\hat{\phi}$. The vacuum expectation value of this tensor can be
obtained by calculating the expectation value
$\bra{0}\nabla_{\mu}\hat{\phi}\nabla_{\nu}\hat{\phi}\ket{0}$. With eq.
$\mathscr{I}$.24 and the commutation relations in eq. $\mathscr{I}$.25, we get

$\begin{split}&\bra{0}\nabla_{u}\hat{\phi}\nabla_{v}\hat{\phi}\ket{0}=0,\\\
&\bra{0}\nabla_{u}\hat{\phi}\nabla_{u}\hat{\phi}\ket{0}=\sum_{\omega}\omega^{2}\phi_{\omega}(u)\phi^{*}_{\omega}(u)=\frac{1}{4\pi}\sum_{\omega}\omega,\\\
&\bra{0}\nabla_{v}\hat{\phi}\nabla_{v}\hat{\phi}\ket{0}=\sum_{\omega}\omega^{2}\phi_{\omega}(v)\phi^{*}_{\omega}(v)=\frac{1}{4\pi}\sum_{\omega}\omega.\end{split}$
($\mathscr{I}$.26)

As the frequency $\omega$ is not bounded from above, the last two expressions
are manifestly divergent, and consequently so is the expectation value of the
stress-energy tensor.

The divergence of the expectation value of operators which are quadratic in
the field is a well known problem from quantum field theory in flat spacetime.
There, the standard procedure is to perform “normal ordering” of the creation
and annihilation operators or, equivalently, to subtract the infinity of the
expectation value in the vacuum state off of the “same type of infinities” in
the expectation values in other states, the result being a finite value. This
procedure works well due to the fact that, in the absence of coupling to
spacetime, only the differences between energy states are measurable, making
the absolute value of energy physically irrelevant. However, in curved
spacetimes this is no longer the case: there is no reason to believe that
zero-point energy would not gravitate. There have been arguments that instead
of a full subtraction, an ultraviolet cutoff should be put on the sums and
integrals of the form ($\mathscr{I}$.26), giving rise to a vacuum energy which
would manifest itself globally in spacetime as a cosmological constant.
However, putting the cutoff at the Planck scale, where the semiclassical
approximation is expected to break down, makes this type of energy much larger
than the observed cosmological constant [73]. The interpretation of this issue
is still an open problem.

Performing a full subtraction of the divergence, rather than putting in an
arbitrary cutoff, seems like the more reasonable choice. However, the
construction of the counter-terms should be done carefully. While it is
possible to directly subtract the whole vacuum expectation value in any given
quantisation, resulting in zero vacuum energy and finite expectation values in
particle states, there are strong indications that this (generally non-
covariant) procedure is not the most reasonable option. Firstly, calculations
of beta coefficients ($\mathscr{I}$.14) between the “preferred” (in terms of
symmetry) vacuum states in different regions of evolving spacetimes suggest
the creation of particles by the gravitational field [59, 5], which should be
taken into account energetically. Secondly, the nature of the divergence turns
out to be such that the counter-terms can be local in curvature tensors,
making them the same for any choice of vacuum state (of Hadamard type, i.e.
stemming from sufficiently regular mode solutions) [6, 7, 74]. After such a
subtraction, the vacuum expectation value of the stress-energy tensor operator
is generally non-zero (though finite) when spacetime is not flat.

One of the most used methods for isolating these divergent terms is known as
covariant point-splitting [74], and it consists of expressing the two-point
correlation function
$\bra{0}\nabla_{\mu}\hat{\phi}(x^{\rho})\nabla^{\prime}_{\nu^{\prime}}\hat{\phi}(x^{\prime\rho})\ket{0}$
as a Laurent series in a geodesic distance parameter which becomes zero when
$x^{\prime\rho}\to x^{\rho}$. The precise method used in [61], where the 1+1
dimensional calculation was originally performed, involves symmetrically
separating the two terms in the two-point function in opposite directions
along an arbitrary geodesic passing through the initial centre point. Both
covariant derivatives are translated to the tangent spaces of the new points
through parallel transport and evaluated there, leaving all terms as functions
of the original point and the derivatives of the geodesic curve, as well as
the small distance parameter. Note that this is slightly different from simply
evaluating the two-point function directly at the two final points, as it
gives a covariant prescription of how the coincidence limit is to be
approached.

We have summarised the calculation of the 1+1 RSET, performed in [61], in
Appendix A. After the covariant subtraction of the divergence, the RSET of
this theory becomes

$\expectationvalue{T_{\mu\nu}}=\frac{1}{48\pi}Rg_{\mu\nu}+\Theta_{\mu\nu},$
($\mathscr{I}$.27)

where the tensor $\Theta_{\mu\nu}$ is defined through its value in the null
coordinates of the mode basis as

$\Theta_{\mu\nu}=-\frac{\sqrt{C}}{12\pi}\text{diag}\left(\partial_{u}^{2}\frac{1}{\sqrt{C}},\partial_{v}^{2}\frac{1}{\sqrt{C}}\right)_{\mu\nu}.$
($\mathscr{I}$.28)

The first term in the RSET ($\mathscr{I}$.27) is expressed in terms of
curvature tensors, and is thus independent of the particular choice of modes
and vacuum state of the quantisation. The second term, $\Theta_{\mu\nu}$, is
the one which encodes the information about the chosen vacuum state, and is
also the one which can become significant even in regions of low curvature.
The RSET is conserved covariantly, making it an appropriate source term for
the semiclassical Einstein equations. It also reduces to zero for the
Minkowski quantisation of flat spacetime, making this spacetime a solution of
the semiclassical equations as well.

##### Vacuum states and thermal particle fluxes

As we discussed above, the quantisation defined in ($\mathscr{I}$.24) depends
on the choice of modes ($\mathscr{I}$.23), which in this case is encoded in a
pair of null coordinates $\\{u,v\\}$ in which these take the form of plane
waves. This dependence can be seen now through the fact that with a different
pair of null coordinates $\\{\tilde{u},\tilde{v}\\}$, the metric
($\mathscr{I}$.21) would take the form

$ds^{2}=-C[u(\tilde{u}),v(\tilde{v})]g(\tilde{u})h(\tilde{v})\,d\tilde{u}\,d\tilde{v},\qquad\text{with}\quad
g(\tilde{u})=\frac{du}{d\tilde{u}},\quad h(\tilde{v})=\frac{dv}{d\tilde{v}}.$
($\mathscr{I}$.29)

From here, all the above calculations follow in an analogous way, with the
different conformal factor $C\to\tilde{C}=Cgh$. The part of ($\mathscr{I}$.27)
which is written in terms of local curvature remains the same, but the non-
local part $\Theta_{\mu\nu}$ changes. In terms of components, the relation
between the RSETs in these two different vacua (written in the same coordinate
basis) is

$\displaystyle\expectationvalue{T_{uu}}_{\tilde{0}}$
$\displaystyle=\frac{1}{24\pi}\left(\frac{g^{\prime\prime}}{g^{3}}-\frac{3}{2}\frac{g^{\prime
2}}{g^{4}}\right)+\expectationvalue{T_{uu}}_{0},$ ($\mathscr{I}$.30a)
$\displaystyle\expectationvalue{T_{vv}}_{\tilde{0}}$
$\displaystyle=\frac{1}{24\pi}\left(\frac{h^{\prime\prime}}{h^{3}}-\frac{3}{2}\frac{h^{\prime
2}}{h^{4}}\right)+\expectationvalue{T_{vv}}_{0},$ ($\mathscr{I}$.30b)
$\displaystyle\expectationvalue{T_{uv}}_{\tilde{0}}$
$\displaystyle=\expectationvalue{T_{uv}}_{0},$ ($\mathscr{I}$.30c)

where $g^{\prime}\equiv\partial_{\tilde{u}}g(\tilde{u})$ and
$h^{\prime}\equiv\partial_{\tilde{v}}h(\tilde{v})$, and the subscripts $0$ and
$\tilde{0}$ refer to the quantisations with respect to the plane wave modes in
the $\\{u,v\\}$ and $\\{\tilde{u},\tilde{v}\\}$ coordinates respectively. We
can see that a change in the vacuum state translates into the addition of
right- and left-moving radiation flux terms (which we will identify with
ingoing and outgoing spherical fluxes through the Polyakov approximation in
spherically-symmetric spacetimes).

The fact that the state-dependent terms in the RSET are the ones which are
non-local in curvature brings to light an interesting observation: the
difference between RSETs in two vacuum states eliminates the terms which are
local in curvature, and thus can also remove the divergences present in the
operator expectation value before renormalisation. In other words, in
situations where the counter-terms for renormalisation have not been
constructed explicitly, one can still obtain information about vacuum energy
effects from differences between RSETs in different quantisations (see e.g.
[17, 75]).

Apart from the RSET, a useful tool for measuring energy content in BH
spacetimes is the effective temperature function (ETF) [76, 75], defined as

$\kappa_{\tilde{u}}^{u}\equiv-\left.\frac{d^{2}\tilde{u}}{du^{2}}\right/\frac{d\tilde{u}}{du}=\frac{g^{\prime}}{g^{2}},\quad\kappa_{\tilde{v}}^{v}\equiv-\left.\frac{d^{2}\tilde{v}}{dv^{2}}\right/\frac{d\tilde{v}}{dv}=\frac{h^{\prime}}{h^{2}}$
($\mathscr{I}$.31)

for the outgoing and ingoing (right- and left-moving) radiation sectors,
respectively. In the case of a spacetime representing the formation of a BH,
the usual Hawking effect is encoded in the constant value
$\kappa_{u_{in}}^{u_{out}}=1/2=2\pi T_{\rm H}$ between the “in” and “out”
vacuum states, where $T_{\rm H}$ is the Hawking temperature in natural units.
In more general terms, if $\kappa^{u}_{\tilde{u}}$ or $\kappa^{v}_{\tilde{v}}$
remains constant for a sufficiently long period of time (defined by an
adiabaticity condition), the vacuum state defined by the
$\\{\tilde{u},\tilde{v}\\}$ coordinates [through the modes in
($\mathscr{I}$.23)] will be seen by an observer with proper coordinates
proportional to $\\{u,v\\}$ as a state of outgoing or ingoing thermal
radiation respectively [76].

This function is also directly related to the outgoing and ingoing radiation
fluxes which appear in the RSET after a change of vacuum state [75].
Specifically, equations ($\mathscr{I}$.30) can be written as

$\displaystyle\expectationvalue{T_{uu}}_{\tilde{0}}$
$\displaystyle=\frac{1}{24\pi}\left(\frac{d\kappa_{\tilde{u}}^{u}}{du}+\frac{1}{2}(\kappa_{\tilde{u}}^{u})^{2}\right)+\expectationvalue{T_{uu}}_{0},$
($\mathscr{I}$.32a) $\displaystyle\expectationvalue{T_{vv}}_{\tilde{0}}$
$\displaystyle=\frac{1}{24\pi}\left(\frac{d\kappa_{\tilde{v}}^{v}}{dv}+\frac{1}{2}(\kappa_{\tilde{v}}^{v})^{2}\right)+\expectationvalue{T_{vv}}_{0},$
($\mathscr{I}$.32b) $\displaystyle\expectationvalue{T_{uv}}_{\tilde{0}}$
$\displaystyle=\expectationvalue{T_{uv}}_{0}.$ ($\mathscr{I}$.32c)

In other words, the information about the difference between the RSETs in two
different vacuum states in 1+1 dimensions is entirely contained in their
relative ETFs (and first derivatives thereof).

For the 3+1 dimensional spherically-symmetric spacetimes we will study, we are
interested in calculating these quantities for two special quantum vacuum
states: the “in” and the “out” states. The “in” (“out”) state is the one
defined by null coordinates proportional to the proper time of inertial
observers at past (future) null infinity in asymptotically flat regions. In
order to carry out calculations, we will want to extend these sets of
coordinates throughout the whole spacetime, if possible, and obtain the
relations between them. However, if there is a horizon present in the
geometry, one or both of these extensions may cover the spacetime only
partially. For example, in a collapse geometry which begins by being almost
flat and ends up forming a BH, the “in” state corresponds to the natural
Minkowski vacuum at the asymptotic past which then evolves according to the
dynamics of the system. On the other hand, the “out” state corresponds to the
Minkowski-like vacuum at future null infinity, the backwards extension of
which is ambiguous (as it requires additional data at either the horizon or
the singularity), but generally exhibits a singular behaviour at the horizon
[77]. This discrepancy leads to a variety of interesting horizon-related
semiclassical effects, which are the subject of the first half of this work.

#####  $\mathscr{I}$-2.3. Semiclassical Einstein equations

The semiclassical field equations we will work with are

$G_{\mu\nu}=8\pi(T_{\mu\nu}^{\rm class}+\expectationvalue{T_{\mu\nu}}).$
($\mathscr{I}$.33)

Aside from the RSET described above, $\expectationvalue{T_{\mu\nu}}$, we also
include an effectively classical matter source, $T_{\mu\nu}^{\rm class}$,
which takes into account the bulk of the macroscopic matter responsible for
the curvature of spacetime (planets, stars, etc.). It is standard to assume
that the two stress-energy tensors should be covariantly conserved
independently from each other. In other words, the only interaction between
vacuum energy and the effectively classical matter contemplated in this theory
is the one mediated by gravity itself.

Most of the spacetimes we will work with are spherically symmetric. Their line
element can be written as

$ds^{2}=-C(u,v)dudv+r(u,v)^{2}d\Omega^{2},$ ($\mathscr{I}$.34)

where $u$ and $v$ are radial null coordinates, $r$ is the area-radius of the
spherical slicing, and $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta\,d\varphi$ is
the line element of the unit sphere. The function $r$ is positive, and so is
$C$ if the null coordinates are regular throughout the spacetime. For the
RSET, we will use the operator expectation value derived from a massless
scalar field in the Polyakov approximation, by combining ($\mathscr{I}$.27)
and ($\mathscr{I}$.20). If we take $\\{u,v\\}$ to be the coordinates in which
the modes of the chosen 1+1 dimensional quantisation are written as plane
waves (or one with an equivalent vacuum state), then the components of the
semiclassical equations can be written explicitly as

$\displaystyle\frac{2C_{,u}r_{,u}}{Cr}-\frac{2r_{,uu}}{r}$ $\displaystyle=8\pi
T^{\rm class}_{uu}+\frac{l_{\rm p}^{2}}{12\pi
r^{2}}\left(\frac{C_{,uu}}{C}-\frac{3}{2}\frac{C_{,u}^{2}}{C^{2}}\right),$
($\mathscr{I}$.35a) $\displaystyle\frac{2C_{,v}r_{,v}}{Cr}-\frac{2r_{,vv}}{r}$
$\displaystyle=8\pi T_{vv}^{\rm class}+\frac{l_{\rm p}^{2}}{12\pi
r^{2}}\left(\frac{C_{,vv}}{C}-\frac{3}{2}\frac{C_{,v}^{2}}{C^{2}}\right),$
($\mathscr{I}$.35b)
$\displaystyle\frac{C}{2r^{2}}+\frac{2r_{,v}r_{,u}}{r^{2}}+\frac{2r_{,uv}}{r}$
$\displaystyle=8\pi T_{uv}^{\rm class}+\frac{l_{\rm p}^{2}}{12\pi
r^{2}}\left(\frac{C_{,u}C_{,v}}{C^{2}}-\frac{C_{,uv}}{C}\right),$
($\mathscr{I}$.35c)
$\displaystyle\frac{2C_{,v}C_{,u}r^{2}}{C^{3}}-\frac{2C_{,uv}r^{2}}{C^{2}}-\frac{4r_{,uv}r}{C}$
$\displaystyle=8\pi T_{\theta\theta}^{\rm class},$ ($\mathscr{I}$.35d)

where $l_{\rm p}$ is the Planck length, which we will write explicitly in
these equations in order to emphasise the suppression of the RSET with respect
to the classical terms. Note that, unlike standard classical stress-energy
tensors, the RSET contains second derivatives of the metric, changing the
principal part of the evolution equations (with whichever evolution parameter
one chooses). Though no explicit proof of the well-posedness of these
equations exists as of yet, they have been used in numerical calculations with
stable results [78, 79].

With exact calculations of the RSET in 3+1 dimensions (rather than the
Polyakov approximation), the situation gets even more complicated. Derivatives
of up to fourth order appear in the RSET, which, if taken at face value, can
lead to the semiclassical destabilisation of well-established classical
solutions, such as Minkowski spacetime, under arbitrary perturbations [80, 81,
82]. This issue has been subsequently formally remedied by considering the
semiclassical approximation as an expansion in $\hbar$, which allows an
effective reduction of the order of the derivatives through constraints
obtained form the zeroth order of this expansion (the classical limit) [83,
84, 85, 86].

In practice, the semiclassical analysis rarely gets far enough along for the
order reduction procedure to come into play. Indeed, due to the great
difficulty involved in obtaining the RSET in 3+1 dimensions, even in BH
spacetimes with angular symmetries, the calculation is usually only performed
on fixed backgrounds without complicated (or any) dynamics (see e.g. [14, 17,
68, 69]). The effects of backreaction are then only inferred in certain
regions of these spacetimes.

By contrast, the Polyakov approximation can be used for simple calculations of
the RSET in a large variety of dynamical spacetimes (albeit, restricted to
spherical symmetry), often allowing analytical perturbative analyses of
backreaction, and even full self-consistent solutions when numerics are
involved. In this thesis we mainly focus on analytical studies of the
magnitude of the RSET and backreaction, bringing to light the dynamical
scenarios in which semiclassical horizon-related corrections are relevant, and
thus paving the way for future numerical computations of self-consistent
solutions.

####  $\mathscr{I}$-3. Black holes in classical and semiclassical gravity

When one looks for a gravitational system in which deviations from
classicality are expected to occur, the most natural candidate is the densest
object in the observed universe: the astrophysical BH [36, 37]. BHs are a
generic outcome of the gravitational collapse of classical matter [87]. Their
classical description involves the formation a trapped region and, ultimately,
a singularity, which is a tell-tale sign of the breakdown of the classical
theory. These objects therefore provide a natural testing ground for theories
which model the interface between the gravitational and quantum realms [5, 88,
19, 18, 20, 89, 90, 91, 92, 93, 94, 95, 96]. We will now conclude this
introduction by presenting a brief overview of what the classical and
semiclassical theories have told us so far about BH formation and evolution,
comparing in particular what these two approaches (or partial admixtures of
the two) have argued the ultimate fate of these objects might be.

#####  $\mathscr{I}$-3.1. Classical BH formation and evolution

The first solution to the Einstein equations which described the formation of
a BH, obtained by Oppenheimer and Snyder [8], involved the collapse of a
perfectly spherical distribution of a homogeneous and pressureless ideal fluid
(or dust cloud). The end result of this collapse is (the future part of) a
Schwarzschild BH [97, 98], i.e. a spacelike curvature singularity enclosed by
a trapped region, as shown in fig. 1. For details on the Oppenheimer-Snyder
model, see Appendix B.

Figure 1: Left: Causal structure of the Oppenheimer-Snyder-type geometry. The
red curve represents the surface of the collapsing sphere of matter, and the
shaded part is the trapped region, delimited by the outer and inner horizons.
Right: The part of the Oppenheimer-Snyder geometry external to the matter
distribution is identified with a section of the Kruskal maximal extension of
Schwarzschild spacetime.

Initially it was not clear whether the BH singularity was a generic result, or
rather a consequence of the imposed idealisation of spherical symmetry [99].
The result was subsequently shown to indeed be generic with the singularity
theorems [87, 42, 100]: when a trapped region forms, if certain energy
positivity and causality conditions are satisfied [34], the collapse process
necessarily continues until a singularity (in the sense of inextendability of
incomplete geodesics) forms. However, these theorems do not give indications
as to the type of singularity, nor indeed the causal structure surrounding it.

As it turns out, the inner structure of a realistic BH is rather more
complicated than that of the Oppenheimer-Snyder solution. On the one hand, it
was shown that perturbations away from spherical symmetry can make spacelike
singularities develop in a significantly different way from their symmetric
counterparts [3, 101]. Particularly, it was shown that at the singular end of
a big-bang-like region of non-isotropic universes (akin to the interior of
BHs), a so-called Belinski-Khalatnikov-Lifshitz (BKL) singularity develops,
which has an unbounded oscillatory behaviour that is different in different
spatial directions, and which essentially evolves independently form the
matter content which generates it. A BKL singularity can also be found at the
endpoint of gravitational collapse [102, 103, 104].

On the other hand, it was discovered that the causal structure inside
realistic (spinning and/or electrically charged) BHs is altogether quite
different from that of the Schwarzschild solution, and that environmental
perturbations play a key role in its evolution. Particularly, the Kerr-Newman
solution [105, 106, 107] possesses not only an outer apparent horizon, but
also an inner one. Given that the trapped region does not go all the way down
to the singularity, the nature of this singularity becomes timelike.

The interesting causal features do not end there. Observers which approach the
inner horizon in an outgoing manner actually have their proper time slowed
down with respect to the outside universe, to the point of making their
trajectories extendable beyond this universe through a Cauchy horizon [108,
1]. Even more strangely, the matter which forms the BH can itself cross the
Cauchy horizon before collapsing all the way to form a singularity [109, 1],
as shown in the right causal diagram of fig. 2 for a charged Reissner-
Nordström BH.

Figure 2: Causal diagrams for the formation of a charged BH, and their
analytical extensions. On the left diagram matter collapses to form a
singularity before crossing the Cauchy horizon. On the right, matter crosses
the Cauchy horizon before forming a singularity. The shaded regions are the
parts of the trapped regions external to the matter distribution in each case.
$r_{+}$ indicates the stationary part of the outer horizon and $r_{-}$ of the
inner and Cauchy horizons.

However, the picture changes yet again when taking into account the generic
decaying energy perturbations typically present in BH spacetimes [110]. The
so-called mass inflation instability is triggered [2, 111, 112], wherein the
small perturbation to the position of the inner horizon caused by the external
energy fluxes leads to an exponential growth of the energy contained in the
matter sector in the core of the BH, and consequently of the spacetime
curvature. Though this does not affect how the BH is perceived from the
outside (the core region of the BH being causally disconnected from the
outside universe), its inner structure does change significantly: the inner
horizon plummets to the origin (possibly forming a spacelike singularity at a
finite time [113]), and the Cauchy horizon is substituted by a (weak)
curvature singularity [32], which in the case of a rotating BH has an
oscillating structure [9] not unlike the BKL singularity. In part II of this
work, we will present a particular geometric model which captures the
essential characteristics of mass inflation, and we will discuss in more
detail the properties of the internal BH region.

All this said, for observers outside the BH outer horizon the Oppenheimer-
Snyder model, or indeed even the Schwarzschild solution itself, may be
considered sufficient for modelling the causal properties of a BH: a region of
spacetime from which nothing, not even light, can escape. One may then wonder
whether this more precise study of the inner structure of BHs is worthwhile,
given the lack of causal contact and the consequent impossibility of
observational corroboration (at least, for those of us not willing to leap
into BHs ourselves). However, as we will now see, this study becomes quite
relevant when semiclassical effects are considered.

#####  $\mathscr{I}$-3.2. Semiclassical BHs: the standard picture and beyond

###### Vacuum states on static BH backgrounds

One of the most frequently analysed problems in quantum field theory in curved
spacetimes is that of the vacuum energy present in BH geometries. If we take
the simplest BH solution, the Schwarzschild BH, the most direct way of
quantising a field in the region exterior to the horizon would be to take
advantage of the timelike Killing vector to separate the mode solutions into
positive and negative frequencies. The resulting quantum vacuum, commonly
referred to as the Boulware state [26], turns out to have non-regular
properties at both the past and future horizons (of the Kruskal maximal
extension, represented in fig. 1). This is directly related to the fact that
the time coordinate in the direction of symmetry becomes singular at the
horizons, these being Killing horizons beyond which the BH is no longer
static. The unbounded oscillatory behaviour of the modes (with respect to any
physical time parameter) as they approach the horizons translates to singular
energy expectation values for the field.

In fact, if one looks for a semiclassically self-consistent vacuum solution in
spherical symmetry compatible with staticity, one finds something which looks
like a Schwarzschild BH at large radii, where backreaction is small, but close
to the would-be horizon one finds instead a wormhole throat, which opens up
into a strange high-curvature region not hidden by horizons [27, 28]. The
large backreaction around the would-be horizon has in fact sparked a search
for static objects with classical (positive-energy) matter and semiclassical
vacuum energy, with the latter compensating the tendency for collapse of the
former, resulting in potentially stable ultra-compact configurations [21, 22].
It is also the inspiration for the work presented in part I of this thesis,
where effects related to how the dynamical “in” vacuum state can approach the
high-energy levels of the static Boulware state in the vicinity of would-be
horizons are studied.

In objects with actual trapped regions, the Boulware state is of course
considered non-physical. Eternal BHs are idealised objects, whereas realistic
BHs are expected to form dynamically from regular initial conditions, both
classically and semiclassically. The main argument against the physicality of
the Boulware state is the fact that if a quantisation is initially
renormalisable (i.e. divergences of quadratic operator expectation values are
of Hadamard type), then it continues as such throughout the Cauchy evolution
of the initial data [114]. Thus, the “in” state of gravitational collapse can
never evolve into the Boulware state if a closed trapped region is formed.

If one attempts to change the vacuum state to one which gives a regular RSET
at either the past or future horizons (or both), one finds an inevitable
introduction of energy fluxes at infinity [77]. In the case of regularising
only the future horizon, the resulting flux is none other than that of thermal
Hawking radiation. This vacuum is known as the Unruh state [115], and it
reproduces the general characteristics of the “in” state after BH formation,
independently of the details of the collapse. It is therefore useful for
studying the behaviour of quantum fields in evaporating BHs.

If we take a state which is regular at both past and future horizons, the
presence of fluxes at both past and future null infinity translates into a
time-invariant thermal bath in the bulk of the spacetime. This vacuum is known
as the Hartle-Hawking state [116, 117], and due to its time invariance it is
useful for a variety of semiclassical calculations (see e.g. [64, 118, 119]).

###### BH formation and evaporation

In order to study semiclassical backreaction on BHs, it is useful to work with
the “in” state of gravitational collapse. As mentioned earlier, this state is
defined from the Minkowski quantisation in the past region of asymptotically
flat spacetimes. When a BH forms, this state behaves much like the Unruh
state, giving a flux of outgoing thermal radiation at future null infinity,
and also a compensatory negative ingoing energy flux at the horizon [72].

When backreaction on dynamically formed BH was studied, it lead to one of the
most striking results of modern physics: the BH outer horizon, classically
seen as a one-way barrier from which nothing escapes, actually tends to
gradually reduce in size, and eventually lets out all that was trapped
inside—a process known as Hawking evaporation [5, 88]. Though it is not clear
what happens at the very end of this process, where the central singularity
comes into contact with the outside universe at the same moment in which it
disappears due to the depletion of its mass, the qualitative features of the
typically expected causal structure of an evaporating BH are represented in
fig. 3.

Figure 3: Causal diagram of BH formation and its subsequent Hawking
evaporation. Unlike in the classical case, the outer horizon area decreases,
until it reaches $r=0$. The point $p$ represents the so-called thunderbolt
event, where the singularity meets the outside universe, leading to an
ambiguity in the future evolution.

Remarkable though this result may be, it actually opens the door to a plethora
of problems for the consistency of BH solutions. For instance, it makes it
hard to sweep singularities under the rug, as is usually done in the classical
theory, formally through the well-known Cosmic Censorship Conjecture [120]. In
the semiclassical theory, the issue of the high-curvature central region has
to be dealt with eventually, but the question turns into how long this can be
avoided for.

One may believe that this evaporation picture is perfectly valid and free of
inconsistencies up until the outer horizon shrinks to a Planck size, from
where a quantum theory of gravity would resolve the issue. However, one
important observation bring this into question. The thermal nature of the
Hawking radiation flux, if continued throughout the whole evaporation process,
becomes incompatible with a unitary evolution of the field involved. However,
as mentioned above, this flux is necessary for the regularity of the RSET at
the outer horizon, and modifying it leads to a resurgence of an energetic
firewall at this horizon reminiscent of the Boulware state divergence [89].
While a transient Hawking evaporation phase can be compatible with unitarity,
if it is continued until the horizon reaches a region of Planckian curvature
(as the Hawking picture suggests), one would be left with an extremely small
region keeping the correlations which can purify the whole seemingly thermal
external region of the vacuum state. In other words, a Planck-sized region
would have to contain an amount of information regarding the field which is
well beyond what can be intuitively expected no matter what the quantum
gravity theory which describes this region might be. This issue is commonly
referred to as the information paradox of BHs, and it has been a central part
of semiclassical gravity research for several decades [121, 10, 122, 89, 123,
124, 11, 125].

Note that the assumptions underlying this discussion have been that the
classical BH formation scenario to be corrected is the one shown in fig. 1,
and that the RSET only produces notable backreaction at the outer horizon.
However, neither of these assumptions is actually justified. As discussed
above, the inner structure of a classical BH is notably more complex than
this, and backreaction from the RSET in this region has only been studied
partially.

###### Backreaction at the inner horizon

Studies of BHs are permeated by the idea that nothing can escape from these
objects. Though this is justified when working with classical matter, the
intuition has been extended even to semiclassical analyses. The Hawking effect
is indeed best seen as a consequence of effectively negative energy falling
into the BH, rather than something coming out, hence the information loss
problem. However, the modification of the causal structure of the BH produced
by this negative energy intake does imply that causal geodesics which were
once inside the trapped region can subsequently escape to the outside
universe. Although the amount of actual matter which could feasibly escape the
BH in this manner in a scenario like the one represented in fig. 3 is
miniscule (due to the time scale of Hawking evaporation), this need not be the
case in scenarios with a more complex inner structure. Even more crucially,
the idea that backreaction from the RSET can change the fate of classical
matter by altering the causal properties of the spacetime has far-reaching
implications. For instance, these modifications to the light-cone structure
may not be limited to the vicinity of the outer horizon.

Particularly, there is an obvious place where curvature may still be small
(making the semiclassical approximation valid), but where effects non-local in
curvature may lead to significant backreaction from the RSET: the inner
horizon. As discussed above, the inner horizon is classically believed to be
unstable, making the Cauchy horizon it generates weakly singular. As an add-on
to the study of this instability, semiclassical analyses have been performed
in the vicinity of BH Cauchy horizons [12, 13, 14, 15, 16, 17]. What is found
is that the RSET diverges in all initially renormalisable (Hadamard-type)
quantum states. This divergence is completely analogous to the one present in
the Boulware state at the outer horizon: the modes of the quantum field attain
a divergent behaviour with respect to the physical distance parameters at the
Cauchy horizon. The conclusions drawn from this behaviour are typically that
semiclassical effects would produce a stronger curvature singularity at the
Cauchy horizon and definitively prevent the extendability of the geometry into
another asymptotic region.

The fact that the RSET tends to be larger than the classical matter source in
the vicinity of the Cauchy horizon, which is the asymptotic limit of the inner
horizon, is indeed an interesting result. However, the conclusions drawn thus
far regarding backreaction in this region are, at the very least, incompatible
with the evaporation of the outer horizon, as a Cauchy horizon only tends to
form in these geometries if a trapped region continues to exist indefinitely
(as seen from the outside). Scenarios involving Cauchy horizons in the
framework of semiclassical gravity can only be consistent either if the BH is
extremal, or if Hawking evaporation ceases for some other reason (in which
case the RSET would also tend to a divergence at the outer horizon [77, 89]).

A fully consistent semiclassical analysis of the evolution of a generic
trapped region formed dynamically from gravitational collapse involves the
study of backreaction at both the outer and inner horizon at finite times. In
part II of this thesis we perform this analysis using the RSET in the Polyakov
approximation. We recover the result for Hawking evaporation of the outer
horizon, and we find that the inner horizon has a tendency to move outward,
reducing the size of the trapped region from the inside. Although our results
do not describe the evolution of the trapped region in its entirety, they are
highly suggestive of a process by which this region disappears from the
inside-out on a fairly short timescale, as will be discussed in detail.

##  Part I Semiclassical effects near the outer horizon

When quantising a field on a spherically-symmetric Schwarzschild spacetime,
there are three particular choices for a vacuum state which illustrate the
relation between the quantisation and the symmetries of the spacetime. If one
focuses on the staticity of the Kruskal maximal extension and quantises with
respect to the timelike Killing vector, one obtains the Boulware state [26].
This state is well behaved at infinity, having an energy content which goes to
zero there. However, its energy becomes divergent at the past and future
horizons, making it inconsistent with full semiclassical description of a BH.
If one attempts to regularise the state at both horizons while retaining a
stationary behaviour, one obtains the Hartle-Hawking state [116], in which
there are fluxes of Hawking radiation at both past and future null infinity,
resulting in an overall thermal bath in the bulk of the spacetime. However,
these eternal fluxes at infinity make this state incompatible with asymptotic
flatness when the semiclassical Einstein equations are considered. It thus
appears that eternal BHs are disallowed in semiclassical gravity.

A more physically reasonable scenario is that of an asymptotically flat
(ignoring cosmological backgrounds) spacetime with an initially dispersed
distribution of matter, which eventually collapses to form a BH. In it, we can
choose an asymptotically Minkowskian quantisation in the asymptotic past,
which can be extended to the whole spacetime (i.e. the “in” vacuum state).
This is in fact the state for which the Hawking radiation result was obtained
[5]. In it, observables are regular at the horizon, and the future asymptotic
structure is made consistent through the evaporation process, allowing for a
(nearly) complete semiclassically self-consistent description.

When using the “in” vacuum, the overall resulting picture is that any stellar-
mass object which _collapses rapidly toward the formation of a horizon_
generates extremely small RSETs. It is important to stress that “rapidly” here
corresponds precisely to the standard situation one would expect when working
in the framework of general relativity (defined by the Einstein field
equations coupled to matter satisfying the standard energy conditions [34])
and taking into account the forces that are known to play a role in stellar
evolution. In these situations, semiclassical effects appear so small that the
collapse can be expected to proceed in almost exactly the same manner as in
classical general relativity, forming a trapped surface and thus a BH (see
e.g. [72, 78] for the first treatments of this problem and [24, 126] for
modern retakes). The crucial hypothesis of “ _rapid approach toward the
formation of a horizon_ ” is, therefore, perfectly sensible in most scenarios,
even semiclassically. However, the possibility of modifications to the
geometry which begin inside BHs and propagate outwards (see e.g. [24, 127,
128, 93] and part II of this thesis) might lead to situations in which this
hypothesis is questionable. For instance, the divergent behaviour of the
Boulware vacuum may be taken as a hint of the possibility that even in a
physical vacuum, the surroundings of a black-hole outer horizon may be a
region where semiclassical corrections become large enough to be relevant to
the evolution of the system. Indeed, as we will show, the hypothetical
formation of ultracompact objects sustained very close to horizon formation
(an alternative to BHs [129, 130, 131, 132, 22]) appears to require at least a
semiclassical treatment. Generally, if the RSET contribution overcomes its
suppression by Planck’s constant and becomes comparable to the classical
stress-energy tensor, then a complete, non-perturbative semiclassical
treatment of the problem is in order.

In this first part of the thesis we study the values of the RSET for the “in”
vacuum of a free massless scalar field in spherically symmetric geometries
which approach the formation of a horizon in different ways. Previous works
with the same motivation have checked some of the semiclassical effects
produced by a collapse of matter which quickly decelerates just before
reaching the formation of a horizon [133, 134]. Our present goal is to more
generally identify the precise geometric characteristics of the dynamical
situations which would cause large backreaction close to horizon formation. We
will use the ETF introduced above, as well as the RSET, to measure these
effects.

For our geometric setup, we will mainly use a thin shell model with a
Schwarzschild exterior and a Minkowski interior, moving radially along some
timelike curve; in chapter 3, in the interest of analysing a larger variety of
causal structures and thermal effects, we will also generalise this setup to
arbitrary exterior and interior geometries. We will use trajectories for the
shell surface suitably chosen to represent scenarios in which the formation of
a horizon is approached in different ways. We note that the use of the term
“horizon” throughout this part of the thesis must be identified with the local
notion of an outer apparent/trapping horizon [135, 136], and not with the
global (and generally unobservable) event horizon [42, 137], unless explicitly
stated. Let us briefly make some comments on this.

While in some of the geometries analysed below the position of
apparent/trapping horizon and event horizon are coincident, this should not be
taken as an indication that our results are tied in any way with the formation
of event horizons. In fact, it is always possible to deform these geometries
in a way that event horizons are removed completely, but the local geometric
conditions that eventually lead to their formation in the undeformed
geometries are maintained for arbitrarily long times (for geometries in which
apparent/trapping horizons are formed in finite time, this would imply that
they remain present for a large, but finite, amount of time), which would
yield the same results except for arbitrarily small deviations. One of the
shortcomings of these (quasi-)local definitions of the boundaries of BHs (with
respect to the notion of event horizon) is their non-uniqueness [138]. This
issue disappears in practice when dealing with spherically-symmetric
backgrounds, as one can focus on trapping horizons that are spherically-
symmetric as well, the location of which turn out to be determined by the
quasi-local Misner-Sharp mass [139] that measures the overall energy enclosed
in a given sphere [140]. When the external geometry to the shell is the
Schwarzschild geometry, the location of the horizon defined this way is simply
the Schwarzschild radius.

In this work, “close to horizon formation” will therefore mean that the shell
has trajectories exploring the surroundings of the Schwarzschild radius.
There, we expect to find interesting semiclassical effects, and we want to
understand their dependence on the precise dynamical properties of the
spacetime as it approaches this point. To this end, we have chosen three types
of shell trajectories, the study of which will be sufficient to provide a
general intuition for judging the magnitude of semiclassical effects in a much
larger variety of geometries. The first type of situation, studied in chapter
1, is that of a shell oscillating between two radii, outside but near the
Schwarzschild radius. This situation models in a simple way the effects of an
ultracompact horizonless object undergoing small pulsations. Varying the
characteristics of this oscillation will allow us to explore a wide range of
short-time dynamical behaviours.

For the second type of dynamical situation (chapter 2), we look at the case in
which a shell forms a horizon while moving at an arbitrarily low speed
(relative e.g. to stationary observers in its interior). Our analysis here,
which is an extension to [24], allows us to clearly see how the strength of
semiclassical effects depends crucially on the collapsing velocity at horizon
formation. This will provide a counterpoint to the already well-known results
for a shell collapsing at high velocities or even light speed (see e.g. [23,
126]).

The third type of situation (chapter 3) will explore the consequences of a
long-term monotonous dynamical behaviour, particularly one which we expect
(both a priori and based on results of chapter 1) to present interesting
semiclassical effects—a shell approaching the Schwarzschild radius
asymptotically in a regular time coordinate. In this study, we will even go
beyond the simple scenario of a thin shell in vacuum and attempt an exhaustive
analysis of all possible spherically symmetric geometries which present an
asymptotic tendency toward the trapping of outgoing light rays. This
asymptotic approach can be stopped at any time, so that these configurations
could model, for example, a relaxation phase towards an ultracompact object.
On the other hand, if the asymptotic process is continued indefinitely, the
spacetimes present the interesting feature of generating an event horizon
without having any trapped surfaces formed at finite time.

It is worth mentioning at this point that our analysis will be purely
geometrical, and thus goes beyond the Einstein equations. In other words, we
will be exploring the effects of a geometry on semiclassical quantities
without being concerned with how the geometry itself is generated. We will not
require that the evolution of the geometry be governed by the Einstein
equation with a stress-energy tensor which satisfies some energy conditions
(with the exception of chapter 3, where we consider energy conditions in the
generalisation of the thin shell model on either side of the object surface).
The goal of our geometry-based analysis is to point the way toward the
configurations which should be analysed in further detail in future works with
fully self-consistent semiclassical evolution in mind.

###  Chapter 1 Oscillating shell model and radiation emission

The first type of geometry we will look at is that of a spherical thin shell
of mass $M$ oscillating between two radii at high frequencies. The lower of
the radii will be close to (but above) the Schwarzschild radius $2M$, such
that the effect the geometry has on light rays and modes of quantum fields is
close to what a horizon would produce. Particularly, it is this closeness to a
horizon-like behaviour combined with the wide range of dynamics produced
during an oscillation that is worth a close examination through a
semiclassical lens. Aside from gaining a general intuition on effects produced
on backgrounds close to horizon formation, this model may also be useful for
analysing the (potentially observable) bursts of particle creation in
perturbations of semiclassically sustained horizonless BH mimickers [21, 22].

####  1.1 Spherical thin-shell geometries

The thin-shell geometries that we will analyse consist of an internal
Minkowskian region matched to an external Schwarzschild region of mass $M$
through a moving timelike shell. In the interior region one can write the
metric as

$ds_{-}^{2}=-du_{-}dv_{-}+r_{-}^{2}d\Omega^{2},$ (1.1)

where the subscript “$-$” refers to the interior region, and the radial null
coordinates are related to the Minkowski time $t_{-}$ and radius $r_{-}$
through

$u_{-}=t_{-}-r_{-},\quad v_{-}=t_{-}+r_{-}.$ (1.2)

Equivalently we can construct natural null coordinates in the Schwarzschild
region as

$ds_{+}^{2}=-|f(r_{+})|du_{+}dv_{+}+r_{+}^{2}d\Omega^{2},$ (1.3)

where $f(r)=1-2M/r$ is the redshift function, and in this case the null
coordinates are related to the Schwarzschild time $t_{+}$ and radius $r_{+}$
through

$u_{+}=\text{sign}\left[f(r_{+})\right](t_{+}-r_{+}^{*}),\quad
v_{+}=t_{+}+r_{+}^{*}.$ (1.4)

Here $r_{+}^{*}$ is the tortoise coordinate obtained by integrating
$dr_{+}^{*}=dr_{+}/f(r_{+})$. The $u_{+}$ coordinate goes from $-\infty$ to
$+\infty$, that is, between past null infinity and the Schwarzschild radius
(if the exterior region reaches that far in). Inside the Schwarzschild radius
(but outside the shell) we must define a different coordinate $u^{i}_{+}$,
given by the same relation to $t_{+}$ and $r_{+}^{*}$ as $u_{+}$ above, and
which goes from $-\infty$ at the horizon, until it reaches some point of the
spacelike singularity at some finite value. On the horizon itself, relations
with this variable can only be obtained as a limit from either side. The sign
of $f(r_{+})$ ensures that $u_{+}$ and $u^{i}_{+}$ advance in the same
direction as $u_{-}$, both outside and inside the horizon. Though the shells
we will use in this chapter will not cross the horizon, we define these more
general expressions to set up the formalism which will also be used in the
next chapter.

The two geometries are connected by a thin spherical shell of mass $M$. In
general, this matching is only possible if the shell’s radial position follows
a spacetime curve of the same causality type as seen from either side. In our
case, we will require that this be a timelike trajectory, parametrised by
$v_{-}=T_{-}(u_{-})$ from the inside and by $v_{+}=T_{+}(u_{+})$ from the
outside. Of course, given one of these curves the other is also fixed. For
convenience we will also define the velocity parameters

$\alpha_{-}\equiv\left.\frac{dv_{-}}{du_{-}}\right|_{\rm
shell},\qquad\alpha_{+}\equiv\left.\frac{dv_{+}}{du_{+}}\right|_{\rm shell}$
(1.5)

(which are simply the derivatives of $T_{\pm}$), both of which take values in
$(0,\infty)$ for a timelike trajectory. For an ingoing shell to approach the
speed of light would imply approaching the limit $\alpha_{\pm}\to 0$. On the
other hand, for an outgoing shell reaching the speed of light
$\alpha_{\pm}\to\infty$. A static shell has $\alpha_{\pm}=1$.

In order to complete the definition of this geometry, we must require that the
metric be continuous at the shell. This will allow us to determine the
trajectory of the shell as seen from one side if it is defined on the other.
It will also allow us to extend the “$+$” coordinates into the “$-$” region
and vice versa.

From matching the null part of the line elements we obtain the functions,

$g=\frac{du_{+}}{du_{-}}=\left.\sqrt{\frac{\alpha_{-}}{|f|\alpha_{+}}}\right|_{\rm
shell},\qquad
h=\frac{dv_{+}}{dv_{-}}=\left.\sqrt{\frac{\alpha_{+}}{|f|\alpha_{-}}}\right|_{\rm
shell},$ (1.6)

which can be expressed in either the “$+$” or “$-$” variables. From matching
the radial parts we get the relation between the velocity parameters of the
shell from either side,

$\alpha_{+}=\text{sign}(f)+\frac{1}{2|f|}\frac{(1-\alpha_{-})^{2}}{\alpha_{-}}-\frac{1}{2|f|}\frac{1-\alpha_{-}}{\alpha_{-}}\sqrt{4\alpha_{-}f+(1-\alpha_{-})^{2}}.$
(1.7)

Thus if we define the trajectory in terms of $T_{-}$, we can obtain $T_{+}$ by
integrating $\alpha_{+}$ from the same initial radial position. We can also
obtain the relations $u_{+}(u_{-})$ and $v_{+}(v_{-})$ by integrating the
functions $g$ and $h$.

From the square root in (1.7) we deduce a condition for the continuous
matching of the geometries, namely that the $\alpha_{-}$ parameter which
defines the movement of their separation surface must be such that
$4\alpha_{-}f+(1-\alpha_{-})^{2}$ remains positive. In other words,
$\alpha_{-}$ must tend to zero (the infalling shell must approach light-speed)
if the shell goes below the Schwarzschild radius, in such a way as to
compensate the increasingly negative value of the redshift function. The
parameter $\alpha_{-}$ which satisfies

$4\alpha_{-}f|_{\rm shell}+(1-\alpha_{-})^{2}=0$ (1.8)

defines the slowest possible collapse below the Schwarzschild radius as seen
from the (rapidly disappearing) Minkowski region.

#####  1.1.1 Interpretation of the terms in $g$ and $h$

Let us focus on the function $g$ outside the Schwarzschild radius,

$g=\frac{du_{+}}{du_{-}}=\frac{1}{\sqrt{f}}\sqrt{\frac{\alpha_{-}}{\alpha_{+}}}.$
(1.9)

The presence of the term $1/\sqrt{f}$ is to be expected, as it represents the
redshift experienced by an outgoing light ray. This can be seen most clearly
in the case of a static shell (which, of course, would sit outside the
horizon), for which $\alpha_{\pm}=1$. There, this term is necessary for a
rescaling of the coordinates compatible with a matching of the angular parts
of the geometry.

The $\alpha_{-}/\alpha_{+}$ term has a purely dynamical origin. The velocity
of the shell seen by a static observer on one of its sides is different from
the one seen by a static observer on the other. In their respective null
coordinates this can be seen as a change in the slope of the line tangent to
the shell trajectory, namely $\alpha_{-}\to\alpha_{+}$ (see fig. 1.1). From
the perspective of the shell, which can use the appropriate coordinates for
each side, this looks something like a spacetime refraction phenomenon. If a
light ray incides with an angle $\theta$ with respect to the shell trajectory
from the inside, it exits with an angle $\theta^{\prime}$ related to the first
by

$\frac{\tan\theta^{\prime}}{\tan\theta}=\frac{\alpha_{-}}{\alpha_{+}}.$ (1.10)

For the angles formed by an ingoing ray, the relation is the inverse of the
above.

Figure 1.1: Change in angle with respect to the shell of on outgoing light
ray, as measured by static observers on either side.

Another way to interpret the $\alpha_{-}/\alpha_{+}$ term is as a kind of
Doppler effect. Even though technically there is no interaction between the
matter in the shell and the light ray crossing it which could cause absorption
and reemission, the similarity with the Doppler effect can be seen clearly
with the following. If we define

$R(t_{\pm})\equiv r|_{\rm
shell}(t_{\pm})\quad\text{and}\quad\dot{R}=dR/dt_{-},\quad
R^{\prime}=dR/dt_{+},$ (1.11)

then

$\alpha_{-}=\frac{1+\dot{R}}{1-\dot{R}},\quad\alpha_{+}=\frac{1+R^{\prime}/f(R)}{1-R^{\prime}/f(R)}.$
(1.12)

That is, the quotient $\alpha_{-}/\alpha_{+}$ represents the Doppler shift for
a ray that is “absorbed” at one side by a shell moving at a velocity $\dot{R}$
and “reemitted” on the other by a shell moving at a different velocity,
$R^{\prime}/f(R)$. If the geometry on both sides were the same, there would be
no net effect, as these velocities would be the same.

It is worth mentioning that there may be a difficulty in interpreting the
above expressions at the Schwarzschild radius, since the coordinate $t_{+}$
used for the derivative in the second equation in (1.12) is not regular there.
To see the behaviour of $\alpha_{+}$ more clearly we can switch to a regular
time coordinate, say the Painlevé-Gullstrand $\tau_{+}$ defined as the proper
time of a free-falling observer from infinity in the Schwarzschild region
[141], which satisfies

$d\tau_{+}=dt_{+}+\frac{\sqrt{1-f(r_{+})}}{f(r_{+})}dr_{+}.$ (1.13)

We can then define the radial velocity $R_{,\tau}\equiv dR/d\tau_{+}$, which
is regular at the horizon. Then the second equation in (1.12) becomes

$\alpha_{+}=\frac{1+R_{,\tau}/(1+\sqrt{2M/R})}{1-R_{,\tau}/(1-\sqrt{2M/R})},$
(1.14)

from which we can see that at $r=2M$, $\alpha_{+}=0$ and the function $g$ in
(1.9) diverges.

In light of these results, we will call the $1/\sqrt{|f|}$ terms in the
functions $g$ and $h$ the “redshift” terms, and the ones with a quotient of
$\alpha$’s the “Doppler” terms. Combining equations (1.6), (1.12) and (1.14)
we can write the functions $g$ and $h$ as

$\displaystyle
g(u)=\frac{1}{\sqrt{f}}\sqrt{\frac{1+\dot{R}}{1-\dot{R}}}\sqrt{\frac{1-R_{,\tau}/(1-\sqrt{2M/R})}{1+R_{,\tau}/(1+\sqrt{2M/R})}},$
(1.15a) $\displaystyle
h(v)=\frac{1}{\sqrt{f}}\sqrt{\frac{1-\dot{R}}{1+\dot{R}}}\sqrt{\frac{1+R_{,\tau}/(1+\sqrt{2M/R})}{1-R_{,\tau}/(1-\sqrt{2M/R})}},$
(1.15b)

in which all quantities are evaluated at the points where the lines $u=$
const. and $v=$ const. intersect the shell trajectory respectively. We could
work directly with these expressions instead of (1.6) by defining the
trajectory through the velocities $\dot{R}$ and $R_{,\tau}$, which must
satisfy a relation similar to (1.7). However, throughout this work we will
keep using the $\alpha_{\pm}$ parameters, as they are more natural and simple
when dealing with the relations between null coordinates needed for the
calculation of semiclassical quantities.

####  1.2 Oscillating shells and semiclassical effects

In this section we will study the behaviour of the functions $g$ and $h$,
which relate the “$+$” and “$-$” coordinates, when the shell gets near the
formation of a horizon, but does not reach it. We will then use equations
($\mathscr{I}$.30)-($\mathscr{I}$.32) and the properties of the static
Boulware state to directly obtain conclusions regarding the nature and
magnitude of semiclassical effects in geometries with this kind of light-
bending behaviour.

The shell trajectories we will look at cover a wide range of dynamical
configurations, in which both the redshift and Doppler effects have
significant contributions to the values of these functions and their
derivatives. Namely, we will consider a high-speed radial oscillation about a
point just above the surface with radius $r_{\rm s}=2M$ (in the following, we
will always take $r_{\rm s}=1$ for numerical evaluations). We will use three
parameters to describe this movement: the distance $d$ of the centre of
oscillation to the gravitational radius $r_{\rm s}$, the amplitude $A$ and the
frequency $\omega$. Then, the radius at which the shell is located will follow
the spacetime curve (see fig. 1.2)

$R(t_{-})=r_{\rm s}+d+A\sin(\omega t_{-}).$ (1.16)

In order to avoid the formation of a horizon and maintain a timelike
trajectory, the parameters must satisfy the relations

$A<d\quad\text{and}\quad A\omega<1.$ (1.17)

We stress once again that the purpose of this study is to gain a better
understanding of the relation between dynamical regimes close to horizon
formation and the magnitude of semiclassical effects, and not to provide a
self-consistent solution with a classical matter content which satisfies some
energy conditions. Thus we only impose that the shell be causal, with no
further restrictions to its trajectory.

Figure 1.2: Oscillatory radial trajectory of the shell (with parameters
$d=0.1$, $A=0.099$ and $\omega=10$). The dashed line represents the $r=2M$
$(=1)$ surface, and the diagonal lines represent a light ray entering and
exiting the interior region. Although not perceived in the figure, the thick
oscillatory curve does not touch the $r=2M$ line.

Since the trajectory is described in terms of the interior coordinate system,
we can obtain the simple expression for the interior velocity parameter

$\alpha_{-}=\frac{r_{\rm s}+A\omega\cos(\omega t_{-})}{r_{\rm
s}-A\omega\cos(\omega t_{-})},$ (1.18)

while for $\alpha_{+}$ we must use eq. (1.7). To evaluate these quantities on
the points where the shell trajectory intersects the lines of constant $u$ or
$v$ we must solve a transcendental equation, which we will do numerically.
First we will obtain the individual values of the functions $g$ and $h$, which
represent the change in the coordinate description of outgoing and ingoing
radial light rays respectively. Then we will calculate the quotient $g/h$ with
$h$ evaluated at a point of entry $v_{-}$ of a light ray into the Minkowski
region and $g$ evaluated at the point of exit $u_{-}$, which carries
information of how light rays suffer a temporal dispersion by passing through
this region. With the convention fixed in eq. (1.2) we can see that an ingoing
ray $v_{-}$ connects with an outgoing ray through $u_{-}=v_{-}$, so the
quotient we are looking for is $g(v_{-})/h(v_{-})$. This quantity will also
describe the evolution of the “in” quantum vacuum state, defined at the
asymptotically flat region at past null infinity, and its comparison with the
“out” vacuum state, defined at future null infinity.

Figure 1.3: Functions $g$ and $h$ for an oscillation with parameters $d=0.1$,
$A=0.099$ and three different frequencies: $\omega=10$, $\omega=0.5$ and
$\omega=0.1$. The peaks are produced when the shell is nearly at the closest
point to the horizon, as will be discussed below. We observe that at low
frequencies the functions practically coincide since the light rays enter and
leave the interior region in a time much smaller than $\omega^{-1}$, so the
in-crossing and out-crossing dispersion effects would almost cancel out (i.e.
$g(u_{-})/h(v_{-}=u_{-})\simeq 1$). At somewhat larger frequencies the light
rays enter and exit at appreciably different points of the oscillation and the
functions attain a relative displacement. Finally, at frequencies which make
the shell move at nearly light-speed the displacement is greater still, and
the peaks become somewhat tilted to one side for each function, due to the
fact that the peaks of the sine function in $t_{-}$ become tilted when seen in
the $u_{-}$ and $v_{-}$ coordinates (in opposite directions).

In figure 1.3 we observe the values of the functions $g$ and $h$ evaluated at
$u_{-}$ and $v_{-}=u_{-}$, representing the dispersion of a light ray when it
is exiting and entering the interior region respectively. The net effect,
given by $g/h$, reduces to nearly unity when $g(u_{-})\simeq h(u_{-})$, which
occurs when the shell is oscillating very slowly (at low $\omega$) compared to
the time it takes for light to cross it (in the static limit, $g/h=1$). At
higher frequencies the light rays enter and exit at completely different
points of the oscillation, as in the case represented in fig. 1.2, and the net
effect becomes appreciable. It is easy to notice that there are some special
cases for this net effect corresponding to different resonances between the
oscillation frequency and the crossing time of the light ray: say, when it
enters crossing a maximum and also exits crossing one, or crossing a minimum,
and a few other such situations. These will be studied in more detail in the
following subsection.

#####  1.2.1 Resonance between in-crossing and out-crossing effects

From equations (1.6) we can obtain the expression for the total temporal
dispersion suffered by a light ray entering the shell at a point “in” and
exiting at a point “out”,

$\frac{du_{+,{\rm out}}}{dv_{+,in}}=\frac{g|_{\rm out}}{h|_{\rm
in}}=\frac{\sqrt{f}|_{\rm in}}{\sqrt{f}|_{\rm
out}}\left.\sqrt{\frac{\alpha_{-}}{\alpha_{+}}}\right|_{\rm
out}\left.\sqrt{\frac{\alpha_{-}}{\alpha_{+}}}\right|_{\rm in},$ (1.19)

where in the first step we have made use of the fact that for rays reflecting
at the origin $dv_{-}|_{\rm in}/du_{-}|_{\rm out}=1$, as can be seen from eq.
(1.2). We can see again that for a static shell, for which the surface
redshift function would be constant and $\alpha_{\pm}=1$, this quotient
reduces to unity. For a moving shell the effects can cancel out again only in
one special case, which occurs when not only the $in$ and $out$ redshift
functions are the same, but also when $\alpha_{-}=1$ (and therefore
$\alpha_{+}=1$ as well, as can be seen from eq. (1.7)) at both points. For the
case of an oscillating shell this can occur only when a light ray exists such
that it both enters and exits at a minimum or at a maximum of $R(t_{-})$. Then
the effects cancel out locally, but they continue being non-trivial for the
rest of the light rays. These local resonances are possible only when the
frequency, amplitude and distance from the horizon satisfy the relations

$\omega=\frac{n\pi}{r_{\rm s}+d-A},\quad\text{with }n\text{ integer less than
}\quad\frac{r_{\rm s}+d-A}{a\pi},$ (1.20)

for a ray entering and exiting at a minimum, and likewise

$\omega=\frac{n\pi}{r_{\rm s}+d+A},\quad\text{with }n\text{ integer less than
}\quad\frac{r_{\rm s}+d+A}{a\pi},$ (1.21)

for a maximum. These expressions are obtained simply by comparing the ray
crossing time and the oscillation periods in the coordinate $t_{-}$. The upper
bound on the values of $n$ comes from the causal restriction $A\omega<1$. For
a shell following an arbitrary (known) radial motion such cases can be found
just as easily.

On the other hand, if we want to see when a maximisation of $du_{+,{\rm
out}}/dv_{-,{\rm in}}$ in eq. (1.19) takes place (which will be related to the
largest bursts of radiation in the RSET, as we will see), a more detailed
analysis is necessary. First, we may notice that when a light ray enters at a
maximum of $R(t_{-})$ and exits at a minimum, the total redshift effect is
maximised. For such a ray to exist, the relation between the parameters must
be

$\omega=\frac{\pi}{2}\frac{2n+1}{r_{\rm s}+d},\quad\text{with }n\text{ integer
less than }\quad\frac{r_{\rm s}+d}{A\pi}-\frac{1}{2}.$ (1.22)

In fig. 1.4 we observe the three terms of the right-hand side of eq. (1.19)
plotted (without the square roots) for this case. The peaks of the redshift
term, which correspond to precisely the light rays described, reach their
highest possible values for the parameters $A$ and $d$ used.

Figure 1.4: Left: total redshift and Doppler terms plotted separately (the
square root of their product gives $du_{\rm out}/dv_{\rm in}$) for on
oscillation with parameters $d=0.1$, $A=0.099$ and $\omega\simeq 9.996$, given
by (1.22) with $n=3$. We see that even though the shell reaches 99% of the
speed of light (as seen in the Minkowski coordinates) and the Doppler terms
become quite large, the redshift term clearly gives the dominant contribution
around its maxima. Right: in-crossing and out-crossing Doppler terms plotted
separately. We observe that the peaks and valleys are completely out of phase
between the two.

Also in fig. 1.4, we observe that the individual Doppler terms have distinct
maxima. For the in-crossing term the maximum is produced for a ray which
enters slightly after the one which maximises redshift (which enters at a
maximum of $R$), during the in-fall of the shell. For the out-crossing term it
is produced for a ray which exits slightly before redshift maximising one
(which exits at a minimum of $R$), so again during an in-fall of the shell.
Guided by this result, we can look for the conditions which maximise the
individual Doppler terms, and also see whether there is a frequency for the
shell at which the two peaks coincide to make a maximum net effect. In fig.
1.5 we can directly see the values which $\alpha_{-}/\alpha_{+}$ takes at
different redshifts $f$ and velocity parameters $\alpha_{-}$.

Figure 1.5: Values of the Doppler term $\alpha_{-}/\alpha_{+}$ as a function
of $f$ and $\alpha_{-}$. The axes of $\alpha_{-}$ and $\alpha_{-}/\alpha_{+}$
have been rescaled with a function $\text{at}(x)=\frac{2}{\pi}\tan[-1](x)$ to
scale down their whole range into $(0,1)$. The curve drawn on top of the
surface on the left represents the values taken during a period of oscillation
with parameters $d=0.1$, $A=0.099$ and $\omega=10$, and the curve on the right
during an oscillation with parameters $d=10$, $A=9.99$ and $\omega=0.1$. The
region with a sharp gradient close to the horizon ($f=0$) is produced around
$\alpha_{-}=1$ ($1/2$ in the graphic), corresponding to the transition from
falling inward (during which time $\alpha_{-}<1$) to going outward (during
which $\alpha_{-}>1$). At $f\to 1$ the value of the Doppler term tends to 1
smoothly, as the interior and exterior geometries become the same.

At the very minimum of the oscillation of the shell (the closest point to
$r=r_{\rm s}$), $\alpha_{-}/\alpha_{+}=1$ and there is no Doppler effect for
any redshift $f$. When the shell is moving outward ($\alpha_{-}>1$) but is
still close to the horizon ($f\ll 1$), from eq. (1.7) we get

$\frac{\alpha_{-}}{\alpha_{+}}\simeq\frac{\alpha_{-}^{2}}{(1-\alpha_{-})^{2}}f,$
(1.23)

that is, at a constant velocity the Doppler term has a linear dependence on
the redshift function, with a slope which grows rapidly as $\alpha_{-}\to
1^{+}$ and which tends to 1 as $\alpha_{-}\to\infty$. On the other hand, when
the shell is falling in, the function close to the horizon can be expressed as

$\frac{\alpha_{-}}{\alpha_{+}}\simeq\frac{(1-\alpha_{-})^{2}}{f},$ (1.24)

which grows parabolically as $\alpha_{-}\to 0$ (as the in-fall speed
increases) and hyperbolically as $f\to 0$ (as the formation of the horizon is
approached).

With the above equations and fig. 1.5 we can see that the point of the shell
trajectory where the Doppler effect reaches a maximum appears in the
$\alpha_{-}<1$ region, and that its precise position is influenced by two
factors: at a constant $f$ it is maximum at the highest velocity (lowest
$\alpha_{-}$), increasing parabolically as $\alpha_{-}$ decreases, while at a
constant velocity it is maximum at the lowest $f$, with a hyperbolic
divergence at $f=0$. If $\alpha_{-}$ approaches 1 while $f$ approaches 0, that
is, if the shell tends to a full stop just before the formation of the
horizon, then, when $f$ is sufficiently small, the hyperbolic divergence
dominates over the parabolic tendency to zero and the maximum is reached at a
point very close to the minimum value of $f$, just before the region of very
large gradient observed in fig. 1.5 is entered. If, on the other hand, the
shell oscillations are produced far away from the Schwarzschild radius $r_{\rm
s}$, the maximum Doppler effect is reached closer to the point of maximum in-
fall velocity (minimum $\alpha_{-}$).

As an example, in fig. 1.6 we can see the almost-coincidence of the two
Doppler peaks (it looks exact in the figure) for an oscillation which bounces
at $d-A=10^{-3}r_{\rm s}$, with a frequency $\omega$ which allows rays which
enter at a minimum of $R$ to also exit at a minimum. The rays which maximise
the in-crossing and out-crossing Doppler effects almost coincide with the ones
which cancel out the redshift effect, and even more so with each other. Even
when the peaks do not exactly coincide, due to their widths the net Doppler
effect given by their product can be very close to its maximum possible value.

Figure 1.6: Left: total redshift and Doppler terms plotted separately (the
square root of their product gives $du_{\rm out}/dv_{\rm in}$) for an
oscillation with parameters $d=0.1$, $A=0.099$ and $\omega\simeq 9.42$, given
by eq. (1.20) with $n=3$. In this case we see the Doppler term clearly
dominates. Right: in-crossing and out-crossing Doppler terms plotted
separately. We observe the almost-coincidence of the two Doppler peaks,
produced for two very close rays passing through the shell slightly before the
one which enters and exits at a minimum of the oscillation. This near-
coincidence results in the dominance of the net Doppler term in the left
graph.

To conclude, these resonant cases have allowed us to understand the behaviour
of the quotient $g/h$ around its highest values, and relate it to specific
dynamical regimes of the shell. As we will see, the observed regions of rapid
increase or decrease will have significant influence on the behaviour of
semiclassical effects.

#####  1.2.2 Semiclassical effects

So far we have studied the dispersion of light rays (or analogously, of modes
of the massless scalar field) which cross the oscillating shell and pass
through the interior Minkowski region. From these results we can directly
calculate the semiclassical quantities discussed earlier, namely the ETF and
the RSET. The behaviour of these quantities will be similar to that of the
dispersion functions described above, as the former are constructed simply
from derivatives of the latter. The structure of peaks and plateaus for each
period of the oscillation will merely become more exaggerated for these new
functions. For reference, the structure of the ETF due to a single interval of
deceleration during collapse has been previously studied with some detail in
[134]. Our study is based on quite different dynamics, but the results are
qualitatively similar.

In fig. 1.7 we can see the ETF $\kappa_{u_{\rm in}}^{u_{\rm out}}$, which
contains information of the flux of particles seen in the “in” vacuum state by
an inertial observer at future null infinity, calculated with the relation
between the “in” and “out” coordinates given by the product of the functions
plotted in fig. 1.4 through eq. (1.19). As can be guessed by observing the
curves in fig. 1.4, the smaller peaks in $\kappa_{u_{\rm in}}^{u_{\rm out}}$
are produced around the maxima of the Doppler effect contributions. On the
other hand, the largest negative and positive peaks are produced in the
regions of large gradient on either side of the maximum of the redshift
contribution (keep in mind that the horizontal axes of the two plots are
rescaled versions of each other). Between each set of peaks there is a region