$v=\text{const.}$ (5.4)

For the outgoing ones we must solve the equation

$\frac{dr}{dv}=\frac{1}{2}f(r).$ (5.5)

For $v<v_{\rm f}$, $f(r)=1$ and the solution is simply

$r(v)=\frac{1}{2}(v-v_{0}),$ (5.6)

where $v_{0}$ is an integration constant, identified as the time at which the
light ray passes through the origin $r=0$. The “in” vacuum is constructed from
a pair of null coordinates $(v_{in},u_{in})$, which in this case are in fact
$v_{\rm in}=v$ and $u_{\rm in}=v_{0}$ (i.e. in the $u_{\rm in}=\text{const.}$
outgoing null trajectories the value of $u_{\rm in}$ is the value of $v$ when
said trajectories meet the origin). We have conveniently expressed the
integration constant in terms of $v_{0}$, and we need to do the same with all
solutions for outgoing light rays from here on, i.e. we need to trace them
back to the origin. For the region in which the BH has formed ($v>v_{\rm f}$),
we are only interested in analysing the vicinity of each horizon, where (5.3)
is sufficiently accurate. There, the solutions of (5.5) can be expressed in a
series around $d_{\rm f}=0$, a parameter corresponding to the distance of the
null ray from the horizon at $v_{\rm f}$,

$d_{\rm f}=r(v_{\rm f})-r_{\rm h}=\frac{v_{\rm f}}{2}-\frac{v_{0}}{2}-r_{\rm
h},$ (5.7)

the second equality coming from matching with (5.6) at said time. For the
purposes of the present calculation, it is sufficient to express the solution
of (5.5) up to order $d_{\rm f}^{3}$,

$\begin{split}r(v)\simeq r_{\rm h}&+e^{k_{1}(v-v_{\rm f})}d_{\rm
f}+\frac{k_{2}}{k_{1}}\left[-e^{k_{1}(v-v_{\rm f})}+e^{2k_{1}(v-v_{\rm
f})}\right]d_{\rm f}^{2}\\\
&+\left[\left(\frac{k_{2}^{2}}{k_{1}^{2}}-\frac{k_{3}}{2k_{1}}\right)e^{k_{1}(v-v_{\rm
f})}-2\frac{k_{2}^{2}}{k_{1}^{2}}e^{2k_{1}(v-v_{\rm
f})}+\left(\frac{k_{2}^{2}}{k_{1}^{2}}+\frac{k_{3}}{2k_{1}}\right)e^{3k_{1}(v-v_{\rm
f})}\right]d_{\rm f}^{3}.\end{split}$ (5.8)

At the outer horizon, where the surface gravity is positive, the coefficients
of this series increase exponentially, making it a bad approximation for any
finite $d_{\rm f}$ after sufficient time has passed. The reason for this can
be seen in fig. 5.2: outgoing light rays diverge away from the outer horizon,
thus away form where the expansion (5.3) is valid. The reverse happens at the
inner horizon, where the surface gravity is negative, as light rays converge
toward this horizon. However, this will translate inversely to the accuracy of
the RSET approximation in each region, as this quantity will depend on how the
rays evolve backwards in time.

Figure 5.2: Representation of the trajectories of outgoing radial light rays.
The BH geometry begins at $v>v_{\rm f}$. The shaded regions around each
horizon are a qualitative representation of the regions where our
approximation for the RSET is valid.

To switch from $(v,r)$ to $(u_{in},v_{in})\equiv(u,v)$ coordinates, we use the
relation

$dr=\left.\frac{\partial r}{\partial
v}\right|_{v_{0}=u}dv+\left.\frac{\partial r}{\partial
v_{0}}\right|_{v_{0}=u}du,$ (5.9)

and note from (4.25) that $\partial r/\partial v|_{v_{0}=u}=f(r)/2$, which
means that the term proportional to $dv^{2}$ in the metric cancels out. We are
then left with the line element for the BH region

$ds^{2}=2\left.\frac{\partial r}{\partial
v_{0}}\right|_{v_{0}=u}dudv+r^{2}(u,v)d\Omega^{2}=-C(u,v)dudv+r^{2}(u,v)d\Omega^{2},$
(5.10)

where $C=-2\partial r/\partial v_{0}|_{v_{0}=u}$ is the conformal factor of
the reduced 1+1 dimensional spacetime. From (5.8) and (5.7) we obtain the
expansion for this quantity up to order $d_{\rm f}^{2}$,

$\begin{split}C\simeq\;&e^{k_{1}(v-v_{\rm
f})}+2\frac{k_{2}}{k_{1}}\left[-e^{k_{1}(v-v_{\rm f})}+e^{2k_{1}(v-v_{\rm
f})}\right]d_{\rm f}\\\
&+3\left[\left(\frac{k_{2}^{2}}{k_{1}^{2}}-\frac{k_{3}}{2k_{1}}\right)e^{k_{1}(v-v_{\rm
f})}-2\frac{k_{2}^{2}}{k_{1}^{2}}e^{2k_{1}(v-v_{\rm
f})}+\left(\frac{k_{2}^{2}}{k_{1}^{2}}+\frac{k_{3}}{2k_{1}}\right)e^{3k_{1}(v-v_{\rm
f})}\right]d_{\rm f}^{2}.\end{split}$ (5.11)

With this we are ready to calculate the components of the RSET in the Polyakov
approximation, given in “in” coordinates by the standard expressions [see eq.
($\mathscr{I}$.27)]

$\displaystyle\expectationvalue{T_{uu}}$
$\displaystyle=\frac{1}{96\pi^{2}r^{2}}\left[\frac{\partial_{u}^{2}C}{C}-\frac{3}{2}\left(\frac{\partial_{u}C}{C}\right)^{2}\right],$
(5.12a) $\displaystyle\expectationvalue{T_{vv}}$
$\displaystyle=\frac{1}{96\pi^{2}r^{2}}\left[\frac{\partial_{v}^{2}C}{C}-\frac{3}{2}\left(\frac{\partial_{v}C}{C}\right)^{2}\right],$
(5.12b) $\displaystyle\expectationvalue{T_{uv}}$
$\displaystyle=\frac{1}{96\pi^{2}r^{2}}\left[\frac{\partial_{u}C\partial_{v}C}{C^{2}}-\frac{\partial_{u}\partial_{v}C}{C}\right].$
(5.12c)

Calculating these quantities and switching back to $(v,r)$ coordinates we
obtain the leading order of the local expressions around each horizon

$\displaystyle\expectationvalue{T_{vv}}$
$\displaystyle=-\frac{1}{96\pi^{2}r_{\rm
h}^{2}}\frac{k_{1}^{2}}{2}+\order{r-r_{\rm h}},$ (5.13a)
$\displaystyle\expectationvalue{T_{rv}}$
$\displaystyle=-\frac{1}{96\pi^{2}r_{\rm h}^{2}}2k_{2}+\order{r-r_{\rm h}},$
(5.13b) $\displaystyle\expectationvalue{T_{rr}}$
$\displaystyle=\frac{1}{96\pi^{2}r_{\rm
h}^{2}}3\frac{k_{3}}{k_{1}}\left[1-e^{-2k_{1}(v-v_{\rm
f})}\right]+\order{r-r_{\rm h}}.$ (5.13c)

There are several things to note here. The first one is that we can actually
obtain one additional order in the expansion of $\expectationvalue{T_{rv}}$,
and two in the expansion of $\expectationvalue{T_{vv}}$, from just the terms
in (5.11), as only deriving with respect to $u$ reduces its order. This will
be useful later in our calculation.

The second thing to note is the exponential in $\expectationvalue{T_{rr}}$.
Such time-dependent coefficients are also present in higher-order terms in the
remaining RSET components, and they affect the distance from $r_{\rm h}$ for
which the truncated series expansions are a good approximation. In particular,
due to the $C$ in the denominator of (5.12) and the subsequent change of
coordinates, they will be exponentials of positive multiples of $(-k_{1}v)$.
This means that for $k_{1}>0$ (outer horizon) the coefficients in the series
quickly tend to constants, while for $k_{1}<0$ (inner horizon) they grow
exponentially. This is the inverse effect on precision we alluded to earlier,
which can be understood easily by observing fig. 5.2. The light rays converge
toward the inner horizon $r_{\rm i}$ so that determining the RSET in a region
around it requires past information from larger and larger regions, where
(5.3) is no longer valid. On the other hand, the diverging light rays away
from the outer horizon imply that the approximation breaks down only when said
rays are sufficiently far away from this horizon for (5.3) to cease being
precise, and not due to incoming information. The latter case can be seen as a
manifestation of the universality of quantum effects around the external
horizon of a BH at late times, analysed in [76], while the former shows the
reverse being true for the inner horizon, making its long-term semiclassical
dynamics more difficult to pin down. Still, the fact that
$\expectationvalue{T_{rr}}$ evaluated at the inner horizon itself grows
exponentially makes it clear that semiclassical effects are not to be ignored
there, as they can overcome their Planck-scale suppression very quickly.

The third thing to note is the negative ingoing flux at both horizons, which
can be seen directly in the $\expectationvalue{T_{vv}}$ component. This term
is clearly non-local in curvature and is there solely due to the presence of
each horizon, being determined by their respective surface gravities. At the
outer horizon it is this negative flux which drives Hawking evaporation,
compensating the positive thermal flux at infinity, as discussed originally in
[72]. At the inner horizon one may therefore expect that this term would lead
to a similar, classically forbidden behaviour of the spacetime, such as
increasing the radial position of this horizon and reducing the size of the
trapped region from the inside. This indeed seems to occur, as we show below.

Finally, it is worth making some remarks regarding the approximation which we
use for the RSET (5.12). As discussed in the Introduction, one of its most
obvious drawbacks is the divergence it generally has at $r=0$. If either
horizon in our model were close to the origin (in units of the Planck length),
we would have to regularise this tensor (see e.g. [78, 27]) or use a different
approximation. However, we can simply restrict our geometric models to ones in
which $r_{\rm h}\gg l_{\rm p}$, which, for BHs, is also a requirement for the
semiclassical approximation itself to be valid.111This assumption is also
valid for the dynamical cases studied in the next section, perhaps with the
exception of the $n<1$ case of eq. (5.53) in its final stages, which however
occurs only after the bouncing effect we are interested in, and well outside
the range of validity of the approximation for the RSET that we use.

As stated above, the reason why we use the Polyakov approximation is twofold.
First, there is as yet no method to compute the exact 3+1 dimensional RSET for
the “in” vacuum with such generality; and second, this approximation seems to
be enough to capture horizon-related effects, that is, at least when it comes
to Hawking evaporation. However, it is easy to see that the terms local in
curvature would differ from the exact RSET by just looking at the trace
anomalies in 1+1 and 3+1 dimensions. The former is directly proportional to
the $\expectationvalue{T_{rv}}$ term (5.13b), which depends only on the
coefficient $k_{2}$ from (5.3), while the latter, calculated with the
expression given in e.g. [6], is at zeroth order in $(r-r_{\rm h})$ a
completely different function of the coefficients $k_{1}$, $k_{2}$ and
$k_{3}$. We therefore do not exclude the possibility that an exact 3+1
dimensional calculation of backreaction may lead to different dynamics. In
what follows it is shown, however, that the first perturbations on the
position of either horizon are driven by the non-local flux term in
$\expectationvalue{T_{vv}}$, making the resulting initial dynamics a robust
result whenever such a flux is present.

#####  5.1.2 Perturbed Einstein equations

In order to see the dynamical implications of backreaction near the two
horizons, we will perturb the metric (while maintaining spherical symmetry)
and source the first order perturbation with the RSET. Without loss of
generality, the perturbed metric can be written as

$ds^{2}=-\left[f(r)+\delta f(v,r)\right]dv^{2}+2\left[1+\delta
g(v,r)\right]dvdr+r^{2}d\Omega^{2}.$ (5.14)

Since we have expanded the RSET in powers of $(r-r_{\rm h})$, we must do the
same with these perturbations:

$\begin{split}\delta f(v,r)&=\delta f_{0}(v)+\delta f_{1}(v)(r-r_{\rm
h})+\cdots,\\\ \delta g(v,r)&=\delta g_{0}(v)+\delta g_{1}(v)(r-r_{\rm
h})+\cdots.\end{split}$ (5.15)

It is worth noting that since the RSET we are using is fixed entirely by the
background, the matter side of the Einstein equations may also require that a
perturbation of the classical matter (the stress-energy tensor of which
generates the background) be considered. However, with the series expansion
which we are using, we have found that at the order needed to determine
$\delta f_{0}$ and $\delta g_{0}$ the equations are compatible with this
additional perturbation being zero, i.e. the classical matter content
retaining its background functional form.

We therefore begin by equating the first order in $\delta f$ and $\delta g$
and zeroth order in $(r-r_{\rm h})$ of the Einstein tensor to the RSET (5.13)
times $8\pi l_{\rm p}^{2}$, where $l_{\rm p}$ is the Plank length. From the
$vv$ component we obtain

$(1-2k_{1}r_{\rm h})\delta f_{0}(v)-r_{\rm h}\delta
f^{\prime}_{0}(v)=-\frac{l_{\rm p}^{2}}{24\pi}k_{1}^{2}.$ (5.16)

Let us first analyse the implications of this equation in the case of the
external horizon, with $r_{\rm h}=r_{\rm e}$ and $k_{1}>0$. If we take the
familiar Schwarzschild case, where $2k_{1}r_{\rm e}=1$, with the initial
condition $\delta f(v_{\rm f},r)=0$ the equation simply integrates to

$\delta f_{0}=\frac{l_{\rm p}^{2}k_{1}^{2}}{24\pi r_{\rm e}}(v-v_{\rm
f})=\frac{l_{\rm p}^{2}k_{1}^{3}}{12\pi}(v-v_{\rm f}).$ (5.17)

Therefore, the redshift function tends to increase around the horizon, leading
to what can be identified as the first stages of Hawking evaporation of the
trapped region. This can be seen explicitly by noting from the metric (5.14)
that the modified equation for the trajectory of outgoing null radial
geodesics is

$\frac{dr}{dv}=\frac{1}{2}\frac{f(r)+\delta f(v,r)}{1+\delta g(v,r)}.$ (5.18)

Equating the right-hand side to zero and substituting the series (5.15), we
see that the first order change in the radial position of the external horizon
is

$r_{\rm e}\quad\to\quad r_{\rm e}-\frac{\delta f_{0}}{2k_{1}}=r_{\rm
e}-\frac{l_{\rm p}^{2}k_{1}^{2}}{24\pi}(v-v_{\rm f}).$ (5.19)

In other words, the Schwarzschild horizon has a tendency to shrink, albeit
slowly.

For a non-Schwarzschild horizon (e.g. Reissner-Nordström, Schwarzschild-dS or
-AdS, regular BH models, etc.), the general solution of (5.16) is

$\delta f_{0}=\frac{l_{\rm p}^{2}k_{1}^{2}}{24\pi(2k_{1}r_{\rm
e}-1)}\left[1-e^{-\frac{1}{r_{\rm e}}(2k_{1}r_{\rm e}-1)(v-v_{\rm
f})}\right].$ (5.20)

In the limit $2k_{1}r_{\rm e}\to 1$ we recover the above Schwarzschild case.
For other cases we can understand the initial tendencies by expanding this
expression around $v=v_{\rm f}$,

$\delta f_{0}=\frac{l_{\rm p}^{2}k_{1}^{2}}{24\pi r_{\rm e}}\left[(v-v_{\rm
f})-\frac{2k_{1}r_{\rm e}-1}{2r_{\rm e}}(v-v_{\rm f})^{2}+\cdots\right].$
(5.21)

For a horizon with a surface gravity greater than that of a Schwarzschild BH
of the same size, i.e. for $k_{1}>1/(2r_{\rm e})$, we see that at linear order
in $(v-v_{\rm f})$ the evaporation tends to be quicker while, when the
$(v-v_{\rm f})^{2}$ term becomes important, it slows down. The opposite is
true if $k_{1}<1/(2r_{\rm e})$, that is, the evaporation begins slower than in
Schwarzschild but then tends to quicken. Of course, no definite conclusions
can be drawn regarding the long-term evolution of the horizon due to the
various approximations involved. Particularly, even if the above exponentials
are good approximations initially, the fact that for an external horizon the
coefficient $2k_{1}r_{\rm e}-1$ can change sign as the surface gravity and
radius evolve makes it likely that the overall evolution has a kind of
intermediate behaviour, more akin to the Schwarzschild case.

To complete our picture of what occurs at the outer horizon, we can look at
the rest of the perturbed Einstein equations in search for a solution for
$\delta g_{0}$. Combining the $vr$ component at zero order in $(r-r_{\rm e})$
and the $vv$ component at first order [as we mentioned above, the first and
second orders of $\expectationvalue{T_{vv}}$ can be obtained easily from
(5.11)], we get

$\begin{split}(2k_{1}r_{\rm e}-1)\delta g^{\prime}_{0}(v)+\frac{1}{r_{\rm
e}}&(1-4k_{1}r_{\rm e})\delta g_{0}(v)=\\\ &\frac{1}{r_{\rm e}}(1+2k_{1}r_{\rm
e}+4k_{2}r_{\rm e}^{2})\delta f_{0}(v)+\frac{l_{\rm p}^{2}}{6\pi r_{\rm
e}}k_{2}(1-2k_{1}r_{\rm e})\end{split}$ (5.22)

In the Schwarzschild case, where $k_{1}=1/(2r_{\rm e})$ and $k_{2}=-1/(2r_{\rm
e}^{2})$, this reduces to

$\delta g_{0}=0.$ (5.23)

In more general scenarios, with different relations between $r_{\rm e}$ and
the coefficients $k_{i}$, $\delta g_{0}$ is a non vanishing function of time
which, looking at (5.18), causes a slightly faster or slower (depending on its
sign) divergence of null geodesics away from the horizon. From the metric
(5.14) its effect can also be interpreted physically as a contraction or
expansion of space in the radial direction, as discussed in chapter 3.

Having recovered the familiar evaporative tendency of the outer horizon, we
can now move on to the analysis of backreaction at the inner horizon. The
equations are exactly the same, except that for the sake of notation we switch
$r_{\rm e}\to r_{\rm i}$ and we have to keep in mind that $k_{1}$ is now
negative. To see how the position of the inner horizon changes we must look at
(5.20), which we can rewrite in a more convenient manner given the sign of
$k_{1}$ as

$\delta f_{0}=\frac{l_{\rm p}^{2}k_{1}^{2}}{24\pi(1-2k_{1}r_{\rm
i})}\left[e^{\frac{1}{r_{\rm i}}(1-2k_{1}r_{\rm i})(v-v_{\rm f})}-1\right].$
(5.24)

We see that, much like before, the redshift function tends to increase, and
this leads to a reduction of the size of the trapped region,

$r_{\rm i}\to r_{\rm i}+\frac{\delta f_{0}}{2|k_{1}|}.$ (5.25)

The series expansion around $v=v_{\rm f}$ is the same as (5.21), meaning that
if the absolute value of the surface gravity of the inner horizon is greater
than that of the outer horizon (which is usually the case, except in near-
extremal configurations), the initial tendency is for the trapped region to
begin evaporating more quickly from the inside than from the outside.
Additionally, we note that the coefficient multiplying $v$ in the exponential
in $\delta f_{0}$ is positive for any inner horizon, implying the possibility
that the exponential behaviour for the evaporation may be a more general
characteristic which is maintained beyond the initial tendency. We will
analyse this more closely for a particular family of dynamical solutions in
the following.

####  5.2 Dynamical horizons and self-consistent solutions

We have seen how the perturbations on the metric caused by the RSET behave
around the static inner and outer horizons of a BH. But either due to dynamics
in the classical sector, or to semiclassical backreaction itself, the position
of these horizons is generally not static. Calculating the RSET on a generic
dynamical background can be challenging even in the Polyakov approximation,
but we can work around this difficulty in a manner similar to what we employed
in the previous section. We will expand the redshift function $f(v,r)$ in a
series around a dynamical horizon $r_{\rm h}(v)$ and calculate the RSET in
terms of the time-dependent coefficients of this series. This will allow us to
again obtain the deviations in the metric $\delta f(v,r)$ and $\delta g(v,r)$
around each horizon. Doing so without completely specifying the dynamics of
the background will then allow us to arrive at approximate self-consistent
solutions in some particularly simple cases.

#####  5.2.1 RSET around dynamical horizons

To maintain the simplicity of the initial conditions for the quantum modes we
had in the previous section, we will once again consider a spacetime model
which is flat up to a time $v_{\rm f}$ and then transitions abruptly to a BH
geometry. The relation between the parameter $d_{\rm f}$ and the “in”
coordinate $u$, identified with $v_{0}$, is therefore still given by (5.7),
with $r_{\rm h}$ in this case being $r_{\rm h}(v_{\rm f})$.

The BH geometries which we will use for a background are of the form

$ds^{2}=-f(v,r)dv^{2}+2dvdr+r^{2}d\Omega^{2}.$ (5.26)

We note that considering both degrees of freedom of the spherically-symmetric
geometry yields an equally straightforward calculation for the RSET, the main
complication arising at the stage of resolution of the perturbed Einstein
equations. We will use the above form because the simplified cases in which we
will be able to obtain approximate self-consistent dynamics also retain it.
The expansion we will use for the redshift function is completely analogous to
the one in the static case,

$f(v,r)=2k_{1}(v)[r-r_{\rm h}(v)]+2k_{2}(v)[r-r_{\rm
h}(v)]^{2}+2k_{3}(v)[r-r_{\rm h}(v)]^{3}+\cdots,$ (5.27)

where now $k_{1},k_{2},\dots$ are functions of $v$. Assuming that the quantity
$r-r_{\rm h}(v)$ is small, the trajectories of outgoing null radial geodesics
can be expanded as

$r(v)=r_{\rm h}(v)+p_{1}(v)+d_{\rm f}e^{\tilde{k}_{1}(v)}+p_{2}(v,d_{\rm
f})+p_{3}(v,d_{\rm f})+\cdots,$ (5.28)

where

$\displaystyle\tilde{k}_{1}(v)$ $\displaystyle=\int_{v_{\rm
f}}^{v}k_{1}(\tilde{v})d\tilde{v},$ (5.29) $\displaystyle p_{1}(v)$
$\displaystyle=-e^{\tilde{k}_{1}(v)}\int_{v_{\rm
f}}^{v}e^{-\tilde{k}_{1}(\tilde{v})}r^{\prime}_{\rm h}(\tilde{v})d\tilde{v},$
(5.30) $\displaystyle p_{2}(v,d_{\rm f})$
$\displaystyle=e^{\tilde{k}_{1}(v)}\int_{v_{\rm
f}}^{v}e^{-\tilde{k}_{1}(\tilde{v})}k_{2}(\tilde{v})\left[p_{1}(\tilde{v})+d_{\rm
f}e^{\tilde{k}_{1}(\tilde{v})}\right]^{2}d\tilde{v},$ (5.31)
$\displaystyle\begin{split}p_{3}(v,d_{\rm
f})&=e^{\tilde{k}_{1}(v)}\int_{v_{\rm
f}}^{v}e^{-\tilde{k}_{1}(\tilde{v})}\left\\{2k_{2}(\tilde{v})p_{2}(\tilde{v})\left[p_{1}(\tilde{v})+d_{\rm
f}e^{\tilde{k}_{1}(\tilde{v})}\right]\right.\\\ &\hskip
102.43008pt\left.+k_{3}(\tilde{v})\left[p_{1}(\tilde{v})+d_{\rm
f}e^{\tilde{k}_{1}(\tilde{v})}\right]^{3}\right\\}d\tilde{v},\end{split}$
(5.32)

with $r^{\prime}_{\rm h}(v)=dr_{\rm h}/dv$. Unlike in the static case, this
expansion is performed around the first order solution for the separation from
the horizon, $r_{1}(v)\equiv\left[p_{1}(\tilde{v})+d_{\rm
f}e^{\tilde{k}_{1}(\tilde{v})}\right]$ (in units of the scale of each
horizon), making it progressively worse with time no matter how small the
initial parameter $d_{\rm f}$ is. In particular, looking at the expression for
$p_{1}$, the larger the rate of change of the horizon position
$r^{\prime}_{\rm h}(v)$ is, the quicker the approximation breaks down.
However, this can be delayed if the coefficients $k_{i}(v)$ with $i\geq 2$
remain small enough compared to powers of $k_{1}(v)$, as we will consider in
one of our simplified models below.

With these expressions we can obtain a generalisation of (5.13) for dynamical
backgrounds, valid for a small but finite time interval. This RSET at zero
order in $r_{1}(v)$ is

$\displaystyle\expectationvalue{T_{vv}}$
$\displaystyle\simeq-\frac{1}{96\pi^{2}r_{\rm
h}(v)^{2}}\left[\frac{1}{2}k_{1}(v)^{2}-k^{\prime}_{1}(v)\right],$ (5.33a)
$\displaystyle\expectationvalue{T_{rv}}$
$\displaystyle\simeq-\frac{1}{96\pi^{2}r_{\rm h}(v)^{2}}2k_{2}(v),$ (5.33b)
$\displaystyle\begin{split}\expectationvalue{T_{rr}}&\simeq\frac{1}{96\pi^{2}r_{\rm
h}(v)^{2}}e^{-2\tilde{k}_{1}(v)}\left\\{\int_{v_{\rm
f}}^{v}\left[8k_{2}(\tilde{v})e^{\tilde{k}_{1}(\tilde{v})}\int_{v_{\rm
f}}^{\tilde{v}}k_{2}(\bar{v})e^{\tilde{k}_{1}(\bar{v})}d\bar{v}+6k_{3}(\tilde{v})e^{2\tilde{k}_{1}(\tilde{v})}\right]d\tilde{v}\right.\\\
&\hskip 113.81102pt\left.-6\left[\int_{v_{\rm
f}}^{v}k_{2}(\tilde{v})e^{\tilde{k}_{1}(\tilde{v})}d\tilde{v}\right]^{2}\right\\},\end{split}$
(5.33c)

from which we can easily recover (5.13) in the static limit (paying attention
to the integration limits). As a side note, it is interesting to see that the
expression for $\expectationvalue{T_{vv}}$ has the same structure as the one
found in [75] and in ($\mathscr{I}$.32) for the difference between the values
of the RSET in two different vacuum states, when expressed in terms of the
effective temperature function (except for a sign difference between the ETF
and $k_{1}$). This suggests that the flux present in this term is of non-local
nature, stemming purely from the light-ray peeling or accumulation around a
horizon, and not from curvature.

#####  5.2.2 Perturbed equations and Hawking evaporation

With an expansion analogous to (5.15) for the metric perturbations, the
generalisation of eq. (5.16) is now

$\left[1-2r_{\rm h}(v)k_{1}(v)\right]\delta f_{0}(v)-r_{\rm h}(v)(\delta
f^{\prime})_{0}(v)-2k_{1}r_{\rm h}(v)r^{\prime}_{\rm h}(v)\delta
g_{0}(v)=-\frac{l_{\rm
p}^{2}}{24\pi}\left[k_{1}^{2}(v)-2k^{\prime}_{1}(v)\right],$ (5.34)

where $(\delta f^{\prime})_{0}=\delta f^{\prime}_{0}-\delta
f_{1}r^{\prime}_{\rm h}$ is the zeroth order of the derivative of $\delta f$
with respect to $v$ (while $\delta f^{\prime}_{0}$ is the derivative of the
zeroth order; the two only coincide for static backgrounds). We see that the
simple decoupling we had for $\delta f_{0}$ in the static case is not present
in general, unless the rate of change of the background is small enough for
$r^{\prime}_{\rm h}(v)\ll k_{1}(v)r_{\rm h}(v)$ to be satisfied. If the
dynamics are induced only by backreaction itself, then $r^{\prime}_{\rm h}$
and $k^{\prime}_{1}$ are initially of order $l_{\rm p}^{2}$ (in dimensions of
the horizon scale), and so are $\delta f$ and $\delta g$, implying that the
approximation

$\left[1-2r_{\rm h}(v)k_{1}(v)\right]\delta f_{0}(v)-r_{\rm h}(v)\delta
f^{\prime}_{0}(v)\simeq-\frac{l_{\rm p}^{2}}{24\pi}k_{1}^{2}(v),$ (5.35)

is reasonable. $r^{\prime}_{\rm h}$ being small also implies that the series
expansion itself is accurate for a longer period of time, as can be seem from
(5.30).

Let us once again start by looking at the evolution of an initially static
external Schwarzschild horizon. We have the initial condition $2r_{\rm
e}(v_{\rm f})k_{1}(v_{\rm f})=1$, and possible deviations from this equation
at later times $v$, once multiplied by $\delta f_{0}$, are of the same order
as the terms we have neglected above, allowing us to now neglect the first
term. We also have the relation $\delta f^{\prime}_{0}\simeq-r^{\prime}_{\rm
e}/r_{\rm e}$ for the displacement of the Schwarzschild horizon. Substituting
into this equation we get

$r^{\prime}_{\rm e}\simeq-\frac{l_{\rm p}^{2}}{96\pi}\frac{1}{r_{\rm e}^{2}}.$
(5.36)

The solution of this approximate self-consistent equation is simply

$r_{\rm e}(v)\simeq\left[r_{\rm e}(v_{\rm f})^{3}-\frac{l_{\rm
p}^{2}}{32\pi}(v-v_{\rm f})\right].$ (5.37)

To check that we are on the right track, we can see that expanding this
solution around $v_{\rm f}$ recovers the linear tendency form (5.19).
Additionally, we can extrapolate this solution to later times, assuming that
(5.37) remains approximately valid at the later stages of evaporation, and
obtain the Hawking evaporation time $v_{\rm H}$, defined by $r_{\rm e}(v_{\rm
H})=0$,

$v_{\rm H}\simeq 256\pi\frac{M^{3}}{l_{\rm
p}^{2}}\simeq\left(\frac{M}{M_{\odot}}\right)^{3}10^{73}\,\text{s},$ (5.38)

where $M=r_{\rm e}(v_{\rm f})/2$ is the initial BH mass and $M_{\odot}$ is the
solar mass. The approximations involved in obtaining this extrapolation are
equivalent to the quasi-stationary approximation used originally by Hawking to
estimate the lifetime of BHs [88]. The fact that the Polyakov approximation to
the RSET can capture this phenomenon bodes well for the validity of our
following calculations.

In other scenarios where the quasi-stationary assumption is valid for a period
of time, we can still use eq. (5.35) to get a first glimpse at self-consistent
solutions. We can integrate for $\delta f_{0}$, obtaining

$\delta f_{0}(v)=e^{\int_{v_{\rm f}}^{v}[1/r_{\rm
h}(\tilde{v})-2k_{1}(\tilde{v})]d\tilde{v}}\int_{v_{\rm
f}}^{v}e^{-\int_{v_{\rm f}}^{\tilde{v}}[1/r_{\rm
h}(\bar{v})-2k_{1}(\bar{v})]d\bar{v}}\frac{l_{\rm
p}^{2}k_{1}(\tilde{v})^{2}}{24\pi r_{\rm h}(\tilde{v})}d\tilde{v}.$ (5.39)

Taking the right-hand side as a function of a slowly-evolving classical
background would just give a generalisation of (5.21). Modifying the right-
hand side with the backreaction due to $\delta f$ and $\delta g$ would give a
more accurate expression of backreaction in a quasi-stationary approximation,
the validity of which would have to be checked along the evolution in each
case. It is interesting to note that for an inner horizon, where $k_{1}(v)<0$,
unless $k_{1}(v)$ quickly tends to zero, we once again have the growing
exponential we had in (5.24), now somewhat hidden in the term given by the
lower bound of the middle integral (the one which is outside the
exponentials). To see how this exponential behaviour manifests itself in an
approximate self-consistent solution, we will now look at a specific set of
backgrounds for which we can solve this equation.

#####  5.2.3 Inner horizon expansion for a simple background

Let us consider for the BH region a redshift function which around the inner
horizon has the particular form

$f(v,r)=1-\frac{\lambda(v)}{2}r-\frac{\alpha(v)}{r},$ (5.40)

with $\lambda(v)$ a positive function and $\alpha(v)$ a function which
satisfies the initial condition $\alpha(v_{\rm f})=0$. While
$2\lambda\alpha<1$, the position and surface gravity of the inner horizon
within this geometry are

$\displaystyle r_{\rm i}$
$\displaystyle=\frac{1}{\lambda}(1+\sqrt{1-2\lambda\alpha})=\frac{2}{\lambda}-\alpha-\frac{1}{2}\alpha^{2}\lambda+\cdots,$
(5.41) $\displaystyle k_{1}$
$\displaystyle=\frac{1-2\lambda\alpha-\sqrt{1-2\lambda\alpha}}{4\alpha}=-\frac{1}{4}\lambda+\frac{1}{8}\alpha\lambda^{2}+\cdots,$
(5.42)

where the series expansions on the right-hand side are valid while
$|\alpha\lambda|\ll 1$.

For $\alpha=0$, the only non-zero term of the RSET on this background would be
$\expectationvalue{T_{vv}}$ from (5.33), and the local approximation for null
geodesics involved in obtaining it would actually be exact (i.e. higher order
terms in the expansion would be zero) in a finite region around the inner
horizon, akin to the left shaded region in fig. 5.2. There, we would thus have

$\expectationvalue{T_{vv}}=-\frac{1}{96\pi^{2}r^{2}}\left[\frac{1}{2}k_{1}(v)^{2}-k^{\prime}_{1}(v)\right].$
(5.43)

All this remains approximately true while $\alpha$ is small compared to
$\lambda$ in units of $r_{\rm i}$, and we will use this fact to simplify the
dynamical perturbation equations in order to obtain an approximate self-
consistent solution.

The Einstein tensor of this geometry is

$\displaystyle G_{vv}$
$\displaystyle=\frac{\lambda(v)}{r}-\lambda(v)^{2}+\frac{\lambda^{\prime}(v)}{2}-\lambda(v)\frac{\alpha(v)}{r^{2}}+\frac{\alpha^{\prime}(v)}{r^{2}},$
(5.44) $\displaystyle G_{vr}$ $\displaystyle=-\frac{\lambda(v)}{r},$ (5.45)
$\displaystyle G_{\theta\theta}$ $\displaystyle=\frac{G_{\varphi\varphi}}{{\rm
sin}^{2}\theta}=-\frac{r\lambda(v)}{2}.$ (5.46)

We see that $\alpha(v)$ appears only in two terms of the $vv$ component and is
divided by $r^{2}$. We can consider these two terms as the ones sourced by the
RSET (5.43), the rest corresponding to the classical background, which fixes
the function $\lambda(v)$ (keep in mind that the classical background is just
a toy model used to construct the causal structure we are interested in). Then
$\alpha(v)$ becomes the semiclassical perturbation satisfying the equation

$\alpha^{\prime}(v)-\lambda(v)\alpha(v)=-\frac{l_{\rm
p}^{2}}{24\pi}\left[k_{1}(v)^{2}-2k^{\prime}_{1}(v)\right].$ (5.47)

Integrating for $\alpha$ we obtain

$\alpha(v)=-e^{\int_{v_{\rm f}}^{v}\lambda(\tilde{v})d\tilde{v}}\int_{v_{\rm
f}}^{v}e^{-\int_{v_{\rm f}}^{\tilde{v}}\lambda(\bar{v})d\bar{v}}\frac{l_{\rm
p}^{2}}{24\pi}\left[k_{1}(\tilde{v})^{2}-2k^{\prime}_{1}(\tilde{v})\right]d\tilde{v}.$
(5.48)

If we take a background with $\lambda=\text{const.}$ and substitute $k_{1}$
for its zero-order value from the expansion (5.42), the integration yields

$\alpha(v)=-\frac{l_{\rm p}^{2}\lambda}{192\pi}\left[e^{\lambda(v-v_{\rm
f})}-1\right].$ (5.49)

This solution is valid until the initially zero $\alpha$ becomes comparable to
$\lambda$ (in units of $r_{\rm i}$), which, as (5.24) and (5.39) suggested,
does not take long due to the growing exponential. Introduced into the
geometry, this term behaves like a negative mass, tending to move the inner
horizon outward, as can be seen from (5.41). The exponential growth of this
negative mass and the displacement of the inner horizon can be seen as a
semiclassical manifestation of the inner horizon instability, with the
opposite effect to its classical counterpart.

Given this intriguing tendency to evaporate the trapped region from the
inside, it is only natural to ask oneself what may happen if the same
behaviour were to continue throughout the evolution of the inner horizon, up
until the disappearance of the trapped region. In other words, what would the
result be if the driving force of evaporation continued to be the local
$\expectationvalue{T_{vv}}$ term on the right-hand side of (5.47). Although
this assumption is less justifiable dynamically than the analogous one used
for Hawking evaporation in eq. (5.36), one may think of it as just an
extrapolation from the initial tendency. If nothing else, it serves as an
example of how the dynamics of the inner horizon can continue with an RSET
which continues violating the energy positivity conditions, as it seems likely
to do around a horizon, making the geometry evolve in a classically forbidden
manner.

To answer this question, we can take into account the change in surface
gravity due to the evolution of $\alpha$ on the right-hand side of (5.47)
through (5.42). Writing $\alpha$ in terms of $k_{1}$ as

$\alpha=\frac{1}{2}\frac{4k_{1}+\lambda}{(2k_{1}+\lambda)^{2}},$ (5.50)

equation (5.47) becomes

$\left[\frac{4k_{1}}{(\lambda+2k_{1})^{3}}+\frac{l_{\rm
p}^{2}}{12\pi}\right]k^{\prime}_{1}+\frac{\lambda^{\prime}}{2}\frac{\lambda+6k_{1}}{(\lambda+2k_{1})^{3}}+\frac{\lambda}{2}\frac{\lambda+4k_{1}}{(\lambda+2k_{1})^{2}}-\frac{l_{\rm
p}^{2}}{24\pi}k_{1}^{2}=0.$ (5.51)

This equation contains as solutions the initial behaviours in the backreaction
problem given by (5.48) and (5.49), along with their extensions. More
generally, it governs the evolution of a geometry whose dynamics is modified
by a (generally negative) ingoing flux of energy determined by the surface
gravity at its inner apparent horizon through (5.43).

Taking $\lambda$ as a positive constant, all solutions of (5.51) have the same
behaviour, shown in fig. 5.3: the decrease in $\alpha$ (also observed
perturbatively) initially increases the absolute value of $k_{1}$ but then
makes it tend to a constant (with a value $\lambda/2$). This surface gravity
then continues to feed the right-hand side of (5.47), extending the
exponential behaviour of $\alpha$ indefinitely. The radial position of the
inner horizon also continues to increase exponentially.

Figure 5.3: Plots of $|k_{1}(v)|$, $r_{\rm i}(v)$ and $\alpha(v)$, from left
to right, for the evaporation of the inner horizon. We have taken
$\lambda=0.7$ and $l_{\rm p}=10^{-8}$ (as a large difference in scales is
required but smaller values of $l_{\rm p}$ make numerical evaluation more
difficult and lead to no qualitative changes).

In summary, the geometry (5.40) fed by the flux (5.43) has an inner horizon
which moves outward exponentially quickly. This goes on indefinitely due to
the global structure of (5.40), which does not contain an outer bound for the
trapped region (being more akin to an inflationary universe than to a BH). In
a more realistic scenario, even if backreaction continues to be governed by a
term like (5.43), we expect such dynamics to end when the trapped region
disappears. The main conclusion we can extract from this is that incorporating
the modifications of the geometry due to backreaction on the right-hand side
of (5.47) does not tend to decrease the rate of the initially exponential
evaporation of the inner horizon.

If we assume that such a behaviour is the dominant factor in the elimination
of a trapped region, we can estimate a revised evaporation time for BHs with
an inner horizon. Considering a BH of mass $M$, with an outer horizon $r_{\rm
e}\sim M$ which evaporates slowly à la Hawking, and an inner horizon with an
initial position $r_{\rm i,0}$ and surface gravity $k_{1,0}$, the inner
horizon would meet the outer one after a time

$v_{\rm evap}\simeq\frac{1}{k_{1,0}+(2r_{\rm i,0})^{-1}}\log\frac{M}{l_{\rm
p}}\lesssim\frac{M}{M_{\odot}}\times 10^{-5}\,\text{s},$ (5.52)

where $M_{\odot}$ is the solar mass, and we have obtained the upper bound on
the right-hand side by assuming that the surface gravity at the inner horizon
is initially greater than that of the outer horizon, the latter of which we
take to be of the order $1/M$ (the logarithmic dependence has been omitted in
this bound as for no astrophysically reasonable object would it increase the
order of magnitude further). Needless to say, this process is much quicker
than the time it would take for a Schwarzschild BH to evaporate from the
outside, given by (5.38).

Therefore, using the word “evaporation” to describe the leading effects of
semiclassical backreaction on the inner horizon may not be adequate, as the
exponential behaviour of its outwards displacement may be better described as
an “inflation” process (akin to the classical mass inflation described in the
previous chapter). We therefore dub this phenomenon inner horizon inflation.

#####  5.2.4 Collapsing matter: singularity or bounce

So far our results in this section have been a direct generalisation of the
perturbation analysis in the previous one. But the treatment on dynamical
backgrounds and self-consistent extrapolation allow for a wider range of
solutions to be analysed, in particular ones in which the classical
backgrounds itself is dynamical. We expect the backreaction of a moving inner
horizon to have a similar effect as observed for the initially static
background: to push it outward and try to diminish the size of the trapped
region. Whether this tendency from backreaction can overcome its Planck-scale
suppression and dominate over the dynamics of the classical background is what
we will analyse here.

What we will look at is the backreaction problem around the dynamical inner
horizon of a gravitational collapse which would classically end in a
Schwarzschild-like BH. We construct a geometry around this horizon of the type
(5.40) with $\alpha=0$ (classically) and

$\lambda=\frac{\lambda_{0}^{1-n}}{(v_{\rm s}-v)^{n}},$ (5.53)

with $v\in(0,v_{\rm s})$, $n>0$ and $\lambda_{0}$ a constant (with the same
dimensions as $\lambda$) which defines the characteristic length scale of the
problem. Matching this with a Minkowski region through an ingoing null shell
at $v=0$, we get a picture of a collapse in which the inner horizon initially
travelled inward at light-speed but then slowed down before continuing to the
centre. This is once again a method of simplifying the initial conditions for
the quantum modes which enter the BH region by removing their dependence on
the details of the collapse in the far past, thus focusing only on the effects
caused by these modes entering the vicinity of the inner horizon at the final
stages of the collapse. This also goes hand in hand with our approximation
(5.47).

Introducing these backgrounds into eq. (5.49), we can analyse whether horizon-
related semiclassical effects can become relevant to the overall dynamics.
What we find is that there are two ways $\alpha$ can become large enough for
this to occur. First, if the integral of $\lambda$ diverges, which is the case
for $n\geq 1$, then $\alpha$ always diverges as the exponential of this
integral, making it clearly dominant over the classical background. Second,
regardless of whether the integral of $\lambda$ diverges or not, if the
interval $(0,v_{\rm s})$ is large enough, i.e. if the background dynamics is
slow enough for a long period of time, then an effect similar to what occurred
with a static background may dominate. Then the exponential of the integral of
$\lambda$ becomes large enough to overcome the Planck scale suppression, even
though it may not tend to a divergence.

Indeed, if we integrate (5.51), which contains these initial tendencies along
with their extrapolation to the regime in which semiclassical effects
dominate, we get two different types of solutions:

  1. 1.

For $n\geq 1$ the semiclassical backreaction always ends up overcoming the
contribution of the classical background, resulting in a bounce in the
position of the inner horizon, as shown in fig. 5.4. We note that the final
stages of the Oppenheimer-Snyder collapse correspond to a value $n=1$, as
shown in (B.10).

  2. 2.

For $n<1$, semiclassical backreaction can accumulate and lead to an initial
bounce for large enough values of $v_{\rm s}$, as can be seen in fig. 5.5.
Such a bounce indicates that the collapsing behaviour of the classical matter
has been temporarily counteracted, and may subsequently be inverted, making
the trapped region disappear completely. However, in our extrapolation, due to
the fact that the trapped region is not bounded from above, semiclassical
effects eventually lose out and a singularity forms.

Figure 5.4: Plots of $r_{\rm i}(v)$ for a background given by (5.53) with
$\lambda_{0}=1$ and $n=2$, with $v_{\rm s}=20$ on the left and with $v_{\rm
s}=30$ on the right (starting from zero perturbation at $v=0$). Units are once
again given by $l_{\rm p}=10^{-8}$, for the same computational and qualitative
reasons as above.

Figure 5.5: Plots of $r_{\rm i}(v)$ for a background given by (5.53) with
$\lambda_{0}=1$ and $n=1/2$, with $v_{\rm s}=20$ on the left and with $v_{\rm
s}=30$ on the right (starting from zero perturbation at $v=0$). Same units as
above.

We note that the time at which the bounce occurs depends on when this
exponential overcomes the Planck length suppression, which generally occurs
well before either the surface gravity or the radial position of the inner
horizon get close to the Planck scale, as can be checked by inspecting figures
5.4 and 5.5. On the one hand, this allows us to see that we have not obtained
an unnaturally large result due to using the Polyakov approximation for the
RSET [this already being obvious from the origin of the exponential in e.g.
(5.49), which grows even when this horizon is far away from $r=0$]. On the
other hand, it is an indication that such dynamics could be accurately
described with a semiclassical treatment.

In summary, horizon-related semiclassical effects during gravitational
collapse can only be avoided in this model if the classical trajectory of the
inner horizon goes to zero quickly (small $v_{\rm s}$, not giving the
exponential time to grow) and with a sharp peak at the end in $(v,r)$
coordinates ($n<1$, making the integral of $\lambda$ convergent). Otherwise,
in a regime where semiclassical effects are dominated by a term like (5.43),
the collapse will tend to a halt, followed by a quick extinction of the
trapped region from the inside, i.e. an inner horizon inflation process.

However, we remind the reader that the accuracy of our approximation for the
RSET (5.43) can break down when $\alpha$ becomes comparable to $\lambda$ at
the scale of $r_{\rm i}$. Furthermore, in these dynamical scenarios it may
become inaccurate even sooner if the inner horizon reaches a region
sufficiently close to the origin to cross paths with light rays which have
explored the core of the forming BH, i.e. when it steps out of the left shaded
region in fig. 5.2. Then the precise structure of this core must be specified
in order to calculate the RSET. Therefore, although this behaviour is the
natural extension of our approximation, we cannot claim with certainty that it
represents the complete semiclassical dynamics of gravitational collapse.
However, it is a very suggestive possibility.

####  5.3 Conclusions

In this chapter we produce a bare-bones picture of semiclassical backreaction
on black-hole spacetimes which have an inner horizon in addition to an outer
one. We construct a simple toy model of a spherically-symmetric geometry in
which a regular BH forms, and look at the perturbations caused by the RSET
around both the inner and outer apparent horizons. We find that treating these
perturbations locally yields analytical results, and we obtain a clear picture
of the initial tendencies of this double-horizon structure to evaporate. For
the RSET we use a massless scalar field and apply the Polyakov approximation.

At the external horizon, the RSET provides an ingoing flux of negative energy
(in accordance with the results of [72]). Backreaction from this flux
generates a small perturbation which tends to evaporate the horizon. For a
Schwarzschild geometry this perturbation initially grows proportionally to the
advanced Eddington-Finkelstein time coordinate $v$, with a suppression by a
Planck constant, i.e. a slow evaporation. For a background which is not a
vacuum spacetime (e.g. Schwarzschild-dS, Schwarzschild-AdS, regular BHs), the
modified relation between the radial position of the horizon and its surface
gravity results in a somewhat different behaviour for the perturbation: if the
surface gravity is larger than it would be for a Schwarzschild BH of the same
size, the evaporation is initially quicker than in Schwarzschild, but then has
a tendency to slow down, and vice versa if the surface gravity is smaller.

At the inner horizon, the RSET again gives us a negative ingoing flux. The
backreaction in this case again results in a reduction of the size of the
trapped region, i.e. the inner horizon moves outward. Most importantly, this
movement has an overall initial tendency to be much quicker that the
evaporation of the outer horizon. This calculation of first order
perturbations, if taken as indicative of the qualitative nature of the long-
term evolution, strongly suggests a revised picture for evaporation: instead
of the outer horizon slowly moving in and eventually revealing the core of the
BH, if an inner horizon is present, the trapped region may evaporate more
quickly from the inside out. For regular BHs, this coincides with the picture
described in [163, 164], which was motivated heuristically by the existence of
mass inflation due to classical perturbations [165], although without an
explicit discussion of the associated backreaction. Our results here show that
the backreaction from semiclassical effects contains the seeds that may lead
to a realization of this kind of picture.

In light of these results we extend our background geometries to include
dynamical horizons. On the one hand, we do so in order to obtain a better
approximation to the complete self-consistent semiclassical solutions which
start from a static background. On the other, we are also interested in the
backreaction around the dynamical inner horizon in models of spherical BH
formation (e.g. Oppenheimer-Snyder collapse [8]), where the trapped region
first appears close to the eventual outer horizon, and its inner bound quickly
moves inward, tending toward the origin (and the formation of a singularity).

Through analysing these additional geometries we indeed obtain approximations
for the self-consistent solutions in both static and dynamical backgrounds.
Though the range of validity of these approximations is limited, they at least
show us the initial tendency of the evolution quite clearly. We find that the
semiclassical tendency to inflate the inner horizon remains even in dynamical
backgrounds, though whether this can significantly affect the evolution of the
geometry varies on a case-by-case basis. Most notably, we find that in many
cases in which the background dynamics would make the inner horizon reach the
origin (Oppenheimer-Snyder-type collapse), there is a tendency for
semiclassical effects to become dominant before this occurs and bounce the
horizon back outward. Although the bounce itself occurs in most cases outside
the range of validity of the approximation we use for calculating the RSET,
the way this result depends on the divergent tendency of the surface gravity
is very suggestive of it being a generic property of geometries of this type.

Given that all astrophysical BHs are expected to have an inner horizon, our
results at the very least indicate that horizon-related semiclassical effects
should never be overlooked when analysing the formation and evolution of these
objects.

###  Chapter 6 Interlude: blueshift instabilities in the horizon structure of
warp drive spacetimes

Before we proceed with our analysis of BH evolution, we will briefly look at a
different type of geometry, one whose stability has been questioned on the
same grounds as that of a BH with an inner horizon: the Alcubierre warp drive
[166]. Let us first begin with some introductory remarks regarding this
spacetime.

Unlike BHs, the motivation behind its construction is purely geometrical,
curving the geometry and bending the lightcones locally in order to produce
what appears to be faster-than-light travel for faraway observers. While it
brings the idea of superluminal interstellar travel, usually reserved to
science fiction, to the realm of physics, its construction in practice
requires the creation and manipulation of large quantities of exotic matter
[167, 168]. In other words, the stress-energy tensor which generates the warp
drive solution of the Einstein equations violates every local energy
positivity condition [42]. In fact, this is a manifestation of an even more
general restriction: any asymptotically flat configuration which gives rise to
apparent superluminal travel seems to require exotic matter [169].

As of yet, there is no experimental evidence of the existence of exotic matter
capable of such spacetime distortions. In fact, attempts have been made to
provide a geometric interpretation for the absence of gravitating exotic
matter through the addition of an underlying causal structure which limits how
classical spacetimes can curve [170]. Alternatively, but in the same line of
reasoning, this underlying causal structure could be less rigid and allow
certain types of emergent warp-drive configurations, though never ones which
produce closed timelike curves [171]. Other arguments suggest that exotic
matter cannot be so easily dismissed, and can in fact be engineered at will.
One comes from interpreting the Casimir effect in terms of quantum vacuum
energy (for an alternative interpretation see [172]). Indeed, attempts have
been made to construct the required negative energy profile present in a warp
drive solution through manipulating the boundary conditions of quantum vacuum
modes in a Casimir-effect manner [173]. There are also other, more robust
proposals for methods in which effective exotic matter distributions can be
generated, such as the light-matter interaction and the protocol of quantum
energy teleportation [174].

Additionally, effective exotic matter appears naturally in quantum field
theory in curved spacetimes, as we have seen in previous chapters. However,
given that in this theory it is the curvature of spacetime itself which makes
quantum states react to potentially produce negative energies, it is not clear
whether such effects would work in favour or against building configurations
such as warp drives. The evaporative tendencies of horizons we have seen thus
far would in fact suggest the latter possibility, given that the warp drive
spacetime has causal aspects which are quite similar to a BH with both an
inner and outer horizon, as we will see below. Particularly, the inner-
horizon-like front end of the warp bubble can be argued to lead to a similarly
quick tendency toward destabilising backreaction.

Among the issues mentioned above, this semiclassical instability is perhaps
the most critical roadblock for the feasibility of warp-drive configurations
of any size. In [175] it was established that a 1+1 dimensional warp-drive
configuration, corresponding to the central axis of movement of higher-
dimensional drives, develops this instability. Then, in [176] this analysis
was generalised to warp drives formed dynamically from an initially flat
spacetime. Calculating the RSET of a quantum scalar field, it was shown that
the accumulation of geodesics, and correspondingly of modes of the quantum
field, at the front end of the drive leads to an exponential growth in the
vacuum energy density, analogous to that found at the inner horizon of BHs
[13, 14, 16].111It is interesting to note that in these works the
semiclassical inner horizon instability is quantified not through backreaction
analysis, but rather by considering the energy contained in the RSET as seen
by inertial observers which approach this horizon. Particularly, the finite
value of $\expectationvalue{T_{vv}}$ seen in chapter 5, combined with the
divergence of $v$ with respect to regular time coordinates at the Cauchy
horizon (such as the $V$ coordinate in chapter 4), leads to a divergent
physical energy. This can perhaps provide a more intuitive picture of why
backreaction seems to act so abruptly on such configurations. Furthermore, it
was shown that this instability survives even in the presence of a modified
dispersion relation at high energies [177]. Thus, this instability, being
caused by the very superluminal movement of the warp-drive with respect to the
quantum vacuum, does indeed appear unavoidable.

At least, this is the case in 1+1 dimensions, but whether and to what degree
this semiclassical instability is present in more realistic higher-dimensional
warp-drive spacetimes, including in the 3+1 dimensions of our universe, has so
far remained an open question. Given the lack of spherical symmetry in warp
drive geometries, we cannot directly use the Polyakov approximation, and
calculating the exact RSET in such spacetimes is both technically and
conceptually very challenging (see e.g. [17]). However, instead of performing
such a calculation, we can make use of some important intuitions gained from
the 1+1 example, as well as from the calculations in BH spacetimes, to give a
rough estimate of the vacuum energy in these warp drives of higher dimensions.
Particularly, the exponentially growing accumulation of energy can be related
to the presence of surfaces (or points) of infinite blueshift, which are known
to cause instabilities even on a classical level (cf. mass inflation
instability of BHs analysed in chapter 4). Therefore, a classical analysis of
geodesics can likely suffice to identify the regions which may cause such
instabilities.

In this chapter we analyse the geodesics of a warp-drive spacetime in 2+1
dimensions, focusing in particular on the vicinity of the walls of the warp
bubble, which posses horizon-like properties. Remarkably, we find that for
warp bubbles of finite spatial extension there is generally only a single
point where infinite blueshift can occur, suggesting that the semiclassical
singularity in 2+1 and higher dimensions is far weaker than its 1+1
dimensional counterpart. Particularly, by looking at the geodesics trapped in
an approach toward this point, as well as ones which get close to it but end
up deflected away, we estimate that the integrated semiclassical energy
density around this point should remain bounded in most cases. Furthermore, we
show that although changing the shape and trajectory of the warp bubble cannot
eliminate this point, it can serve to further disperse the geodesics in its
vicinity and, by extension, the semiclassical energy accumulation. Much like
how aircraft reduce their air resistance by having particular shapes and
adapting to air currents, warp drives must adapt their geometry and dynamics
in order for the quantum vacuum to offer as little resistance as possible to
their movement.

In section 6.1 we provide a brief introduction to the warp drive and its
semiclassical instability in 1+1 dimensions. In section 6.2 we analyse the
geodesics in a 2+1 dimensional drive which are relevant for determining its
causal structure and its potential instability-inducing points. We estimate
the semiclassical energy accumulation at and around these points, and we
analyse how this accumulation may be dispersed by changing the shape or
trajectory of the warp bubble. In section 6.3 we provide some concluding
remarks.

####  6.1 Warp drive and the semiclassical instability

The Alcubierre warp drive as an isolated system in an asymptotically flat
spacetime has the following metric:

$ds^{2}=-dt^{2}+[d{\bar{\bm{x}}}-{\bar{\bm{v}}}(t,{\bar{\bm{x}}})dt]^{2},$
(6.1)

where ${\bar{\bm{x}}}$ represents spatial coordinates and
${\bar{\bm{v}}}(t,{\bar{\bm{x}}})$ determines the velocity and shape of the
warp bubble (both these quantities are defined as Euclidean vectors with as
many components as spatial dimensions in the manifold). Flat spacetime is
recovered far away from the bubble by imposing that $|{\bar{\bm{v}}}|\to 0$ as
$|{\bar{\bm{x}}}|\to\infty$. We take $\bar{\bm{x}}_{\text{c}}(t)$ to be the
trajectory of the centre of the bubble in this asymptotically-Minkowskian
coordinate system, and for convenience we also define the comoving spatial
coordinates ${\bm{x}}={\bar{\bm{x}}}-\bar{\bm{x}}_{\text{c}}(t)$. We can then
write the usual definition
${\bar{\bm{v}}}(t,{\bm{x}})=\bar{f}({\bm{x}}){\bm{V}}(t)$, where
${\bm{V}}(t)=d\bar{\bm{x}}_{\text{c}}(t)/dt$ is the velocity of the bubble and
$\bar{f}({\bm{x}})$ determines its shape. At the centre of the bubble this
shape function must satisfy $\bar{f}(0)=1$, and as $|{\bm{x}}|\to\infty$ it
must tend to zero sufficiently quickly. In comoving coordinates the metric can
be written as

$ds^{2}=-dt^{2}+[d{\bm{x}}+{\bm{v}}(t,{\bm{x}})dt]^{2},$ (6.2)

where ${\bm{v}}(t,{\bm{x}})=f({\bm{x}}){\bm{V}}(t)$, with $f=1-\bar{f}$.

To understand this geometry better, let us start with a 1+1 dimensional
stationary (${\bm{V}}(t)=\text{const.}$) case. In this case, the line element
acquires the same form as that of the radial-temporal sector of a BH spacetime
written in Painlevé-Gullstrand coordinates [141]. Particularly, the front end
of the warp drive behaves like a white hole horizon (more precisely, a white
hole outer horizon or, equivalently, a BH inner horizon), and the rear end
like a BH horizon (time reverse of the former). In comoving coordinates, the
inside of the warp bubble appears located between two trapped regions, i.e.
between a white and a BH, as shown in fig. 6.1.

Figure 6.1: Causal structure of a stationary 1+1 dimensional warp drive
spacetime. The warp bubble is located between a white and a BH, separated by
the past and future horizons, denoted by $\mathscr{H}^{-}$ and
$\mathscr{H}^{+}$ respectively. $\mathscr{H}^{+}$ connects with a Cauchy
horizon to the past $\mathscr{C}^{-}$, and $\mathscr{H}^{-}$ with a Cauchy
horizon to the future $\mathscr{C}^{+}$. The curved lines correspond to lines
of $x=\text{const.}$, which are timelike inside the warp bubble, and spacelike
everywhere else.

In [176] it was shown that when such a configuration forms from an initially
flat region, the presence of any background quantum field, even in
vacuum,222The instability studied here is the same for any Hadamard state.
leads to an exponential growth of energy at the white hole horizon.
Particularly, the energy density obtained when contracting the RSET with the
velocity $u^{\mu}$ of a free-falling observer which approaches this horizon
grows as

$\rho=\expectationvalue{T_{\mu\nu}}u^{\mu}u^{\nu}\sim e^{2\kappa t},$ (6.3)

where $\kappa$ is the surface gravity of the horizon. This result was obtained
through a calculation much like the BH inner horizon analysis of the previous
chapter. Particularly, if we express the $\expectationvalue{T_{vv}}$ component
(5.13b) in a null coordinate which does not diverge at the Cauchy horizon, it
reveals the above exponential tendency. In BHs, the 1+1 dimensional
calculation in the radial-temporal sector generalises directly to higher
dimensions due to the isotropy of the light-trapping behaviour. In warp
drives, however, the generalisation of the horizon structure shown in fig. 6.1
to higher dimensions is not as straightforward. As it happens, for generic,
finite-sized warp bubbles, the dimension of the surface of infinite blueshift
does not grow with the dimension of the spacetime, but remains the same as in
1+1. We will now proceed to show this explicitly in 2+1 dimensions.

####  6.2 Geodesics and stability in 2+1 dimensions

Let us now turn our attention to the 2+1 dimensional warp drive. We will start
with a thorough analysis of a particularly simple, yet quite generic
configuration: a stationary warp bubble travelling in a straight line, with a
geometry which has a reflection symmetry with respect to a central axis
aligned with the direction of motion, as depicted in fig. 6.2. The comoving
spatial coordinates will be denoted by $\\{x,y\\}$, where $x$ is taken to be
aligned with the direction of motion and $y$ with the direction of symmetry.
The line element of the geometry is

$ds^{2}=-dt^{2}+[dx+v(x,y)dt]^{2}+dy^{2}.$ (6.4)

Taking $y=0$ as the position of the central axis, the function $v(x,y)$ has
even parity in $y$. The equations which determine the null geodesics of this
spacetime are

$\displaystyle(v^{2}-1)\dot{t}^{2}+2v\,\dot{t}\dot{x}+\dot{x}^{2}+\dot{y}^{2}$
$\displaystyle=0,$ (6.5) $\displaystyle(v^{2}-1)\,\dot{t}+v\,\dot{x}$
$\displaystyle=E,$ (6.6)
$\displaystyle\ddot{y}-\partial_{y}v(v\,\dot{t}^{2}+\dot{t}\dot{x})$
$\displaystyle=0,$ (6.7)

where $E$ is an integration constant and the dot indicates differentiation
with respect to the geodesic affine parameter $\sigma$.

#####  6.2.1 Movement on the central axis

Figure 6.2: Warp bubble in 2+1 dimensions as seen in comoving coordinates.
Light cones show the permitted directions of movement for causal trajectories.
The point $p_{1}$ is the front end of the bubble, which produces a blueshift
instability toward the future. Likewise, $p_{2}$ produces an instability
toward the past. $p_{3}$ represents a generic point of the border of the
bubble which does not lie on the central axis.

The 1+1 dimensional example presented above corresponds to the movement of
geodesics along the central axis. We can recover the causal structure of fig.
6.1 by integrating the null geodesic equations with the initial conditions
$y_{0}=\dot{y}_{0}=0$ (which, through Eq. (6.7) and the fact that
$\partial_{y}v=0$ at $y=0$, implies $y(\sigma)=0$). It is worthwhile to do
this explicitly in the vicinity of the edges of the bubble, where $v$
approaches 1. Let $p_{1}=(x_{1},0)$ be the front end of the bubble, as shown
in fig. 6.2. Near this point we can consider the series expansion in the $x$
direction

$v(x,0)=1+\kappa(x-x_{1})+\cdots,$ (6.8)

where $\kappa$ is a positive constant (analogous to a horizon surface
gravity). At leading order, equations (6.5) and (6.6), with the expansion
(6.8), have a family of solutions

$x-x_{1}\simeq-E(\sigma-\sigma_{1}),\qquad
t\simeq-\frac{1}{\kappa}\log|\sigma-\sigma_{1}|,$ (6.9)

with $\sigma_{1}$ a constant. We see that these null geodesics reach the limit
$t\to\infty$ at a finite $\sigma$, corresponding to the future Cauchy horizon
in fig. 6.1. The past Cauchy horizon is obtained analogously by taking the
expansion (6.8) at the rear end of the drive ($p_{2}$ in fig. 6.2), making
$\kappa$ negative, and considering ${t\to-\infty}$.

We also note that, given that (6.5) is quadratic in the coordinate functions,
there are in fact two families of solutions of the geodesic equations around
$p_{1}$. Aside from the ones shown above, there are also solutions for which
$t$ does not diverge, which can be identified as the ones which cross the
black and white hole horizons depicted in fig. 6.1 (left-moving light rays).
Locally, these can be seen as backwards-directed trajectories relative to the
bubble.

The blueshift instability is triggered by the rays which take infinite $t$ to
reach $p_{1}$, as can be seen from the fact that the spatial separation
between any two distinct lightlike observers which approach this point tends
to zero, implying that the wavelength of individual perturbations also goes to
zero, i.e. they are infinitely blueshifted. Intuitively, one can then see that
backreaction from generic perturbations may well destabilise this
configuration even on a classical level. The semiclassical instability is also
a direct consequence of this blueshift, as the modes used to define a
quantisation become singular when approaching this point. In stationary,
eternal BH configurations, it has been shown that the semiclassical tendency
toward a singularity at the infinite blueshift surface is in fact stronger
than its classical counterpart [14, 17] (as we will see in the next chapter,
this appears to also be the case for BHs formed at a finite time).

#####  6.2.2 Other unstable points?

We have shown that the causal structure of the 1+1 case continues to be
present in higher dimensions, at least on one axis. However, it is not clear
whether other points of infinite blueshift besides $p_{1}$ (and $p_{2}$, if we
consider past instabilities) are present in other parts of the 2+1
configuration. In fact, for the simplest type of warp bubble, it turns out
that there are no other such points, as we will now show.

The points which one can expect to have special causal behaviour are the ones
which comprise the rest of the edge of the warp bubble, where $v=1$. Let
$p_{3}$ be one such point (see fig. 6.2). At this point, we can use the labels
$\kappa=\partial_{x}v|_{p_{3}}$ and $\zeta=\partial_{y}v|_{p_{3}}$. We define
the front and back end of the bubble ($p_{1}$ and $p_{2}$ in fig. 6.2) as the
points where $\zeta=0$, which for the symmetric bubble we are considering lie
on its intersection with the symmetry axis $y=0$. These are the points which
correspond to the future and past Cauchy horizons shown above. For a smooth
convex bubble, $\zeta\neq 0$ at all other points of its frontier.

We are interested in whether there are geodesics for which $t$ diverges at
finite $\sigma$ when approaching $p_{3}$. We can answer this by substituting
$v$ and its derivatives for their values on these points in the geodesic
equations, and checking whether $t$ can approach infinity while $\sigma$, $x$,
and $y$ remain bounded. In Eq. (6.6), the first term on the left-hand side,
$(v^{2}-1)\dot{t}$, can tend either to 0, a constant, or infinity. If it went
to infinity, then $\dot{x}$ would also diverge at $p_{3}$, which, given the
analiticity of the geometry, leads to no consistent solutions (one can check
this explicitly by taking an arbitrary inverse-polynomial or logarithmic
divergence for $\dot{x}$ and checking the requirements imposed on the other
derivatives in equations (6.5) and (6.7) at $p_{3}$, arriving at an
inconsistency). If this first term of (6.6) goes to a constant or to 0, then
equations (6.6) and (6.7) become

$\dot{x}|_{p_{3}}=\tilde{E}=\text{const.},\qquad\ddot{y}|_{p_{3}}=\zeta(\dot{t}^{2}+\tilde{E}\dot{t})|_{p_{3}}.$
(6.10)

Since the geometry is analytic, the divergence of $t$ implies the divergence
of its derivatives. Therefore, $\dot{t}$ and $\dot{t}^{2}$ would have
different rates of divergence, and $\ddot{y}$ would remain finite only if
$\zeta=0$. If $\zeta\neq 0$, then $\ddot{y}$ diverges, which does not occur
for any consistent solutions at the finite point $p_{3}$ (this can again be
seen by considering the analiticity of the geometry or checking explicitly for
such solutions in the geodesic equations).

In other words, the only points on the boundary of the bubble $v=1$ which have
the possibility of generating Cauchy horizons are the ones where
$\partial_{y}v=\zeta=0$, i.e. where the derivative of the shape function $v$
in the direction perpendicular to that of motion is zero. Stated as such, this
result can be seen to be independent of the particular shape or symmetry of
the bubble, as long as the configuration is stationary. For a smooth and
convex bubble, this implies that there are strictly only two points of
infinite blueshift akin to the ones present in 1+1 dimensions, and time
symmetry tells us that only one is unstable toward the future (and the other
toward the past).

If the bubble is not convex and additional points of $\partial_{y}v=0$ are
present, then the warp drive could be said to be less “aerodynamic” in its
motion within the quantum vacuum, as it would find further resistance to its
stability due to larger energy accumulation. However, as long as such points
are isolated from each other, the overall configuration could potentially be
stable, as the total amount of accumulated vacuum energy could be finite.

#####  6.2.3 Vicinity of the unstable points and vacuum energy divergence

To find out whether the single-point blueshift instabilities present in 2+1
(and higher) dimensions are actually detrimental to the stability of the whole
warp drive configuration, we must estimate the behaviour of the quantum vacuum
energy in a small vicinity around these points. An instability is present only
when the divergence at these points has a certain “width”, enough to produce a
singularity if backreaction is considered. Finding out whether this is the
case would generally involve calculating the RSET on this spacetime for a test
field in an appropriate vacuum state. However, due to the great technical
difficulty involved in such a calculation, we will resort to an estimation
based on an extension of the analogy between the movement of geodesics at and
around the central axis, and the 1+1 dimensional case.

To set this up, let us begin by considering a solution slightly away from the
central axis solutions (6.9), but still in the vicinity of $p_{1}$. We now
perform the expansion of $v$ around $p_{1}$ to leading order in both $x$ and
$y$,

$v(x,y)\simeq 1+\kappa(x-x_{1})+\xi y^{2n},$ (6.11)

where $\xi$ is a constant with appropriate inverse-length dimensions, positive
if the bubble is convex and negative if it has a concave peak (and zero if it
has a finite-sized flat peak), and $n$ a natural number. Larger values of $n$
make the peak more flat in the $y$ direction. We consider the deviation from
the solution (6.9) (where $y=0$),

$\displaystyle\delta x(t)$ $\displaystyle=x(t)-x_{1}-Ee^{-\kappa t},$ (6.12)
$\displaystyle\delta t(\sigma)$
$\displaystyle=t(\sigma)+\frac{1}{\kappa}\log|\sigma|,$ (6.13)

where $E$ is a constant. It is convenient to rewrite equations (6.5), (6.6)
and (6.7) at leading order in $\delta x$, $y$ and $\delta t$ (and their
derivatives) as

$\displaystyle y^{\prime\prime}+\kappa y^{\prime}-2n\xi y^{2n-1}$
$\displaystyle\simeq 0,$ (6.14) $\displaystyle\delta x^{\prime}+\kappa\delta
x+\xi y^{2n}+\frac{1}{2}(y^{\prime})^{2}$ $\displaystyle\simeq 0,$ (6.15)
$\displaystyle\delta\dot{t}+\frac{e^{2\kappa t}}{2\kappa^{2}E}(2\kappa\delta
x+2\xi y^{2n}+\delta x^{\prime})$ $\displaystyle\simeq 0,$ (6.16)

where the prime indicates a derivative with respect to $t$. Eq. (6.14) can
give us a description of the perturbation in $y$, from where we can use Eq.
(6.15) to obtain the perturbation $\delta x$, and Eq. (6.16) to find the
modification $\delta t$ to $t(\sigma)$. Particularly, Eq. (6.14) can be solved
directly, and the validity of the solution can be checked by making sure that
the approximations which lead to (6.14) are accurate, which can be done with
the solutions of (6.15) and (6.16).

Let us begin by looking at the case of a warp bubble edge with a finite-sized
region which is flat in $y$. This, as one might imagine, is not a very
“aerodynamic” shape, as it is not convex. The solutions of (6.14) with $n=0$
tending to this frontal region would be

$y\simeq c_{2}+c_{3}e^{-\kappa t},$ (6.17)

where $c_{2,3}$ are integration constants. The constant $c_{2}$ is indicative
of the fact that $p_{1}$ is no longer the only point which traps geodesics
into a tendency toward a Cauchy horizon. Eq. (6.15) with $n=0$ furthermore
shows us that for these solutions $x$ has the exact same behaviour at large
$t$ as it does on the central axis (i.e. $\delta x$ has the same solutions as
$x$ in (6.9) and can be absorbed in the integration constants of the latter);
and Eq. (6.16) shows the same for $t(\sigma)$, implying that the
approximations used to obtain (6.14) are accurate. Therefore, in this case
there would be a finite-sized region with a blueshift instability, and we can
expect that this configuration would be unstable under both classical and
semiclassical perturbations.

If the bubble has, say, a parabolic profile in $y$ (i.e. $n=1$), then the
solutions become

$y\simeq c_{2}e^{\eta_{-}t}+c_{3}e^{-\eta_{+}t},\qquad\delta x\simeq
c_{1}\,y^{2},$ (6.18)

with $c_{1}$, $c_{2}$ and $c_{3}$ constants, and

$\eta_{\pm}=\frac{\kappa}{2}\left(\sqrt{1+8\frac{\xi}{\kappa^{2}}}\pm
1\right).$ (6.19)

Let us first look at the case of a convex bubble, for which $\xi>0$ and
consequently $\eta_{\pm}>0$. In this case, only initial conditions which give
$c_{2}=0$ correspond to geodesics trapped in a tendency toward the tip of the
bubble from outside the axis, since $y\to 0$ as $t$ grows. These fine-tuned
geodesics (of measure zero within the total set of solutions) for each value
of $c_{3}$ represent a separatrix between solutions deflected away
(exponentially quickly, while the approximation is valid) to one side
($c_{2}>0$) and the other ($c_{2}<0$). As one might expect, the approximations
leading to Eq. (6.14) break down quickly when $c_{2}\neq 0$, and become
asymptotically exact when $c_{2}=0$.

Figure 6.3: Numerical integration in time of two bundles of null geodesics
launched from two different points in the interior of a stationary circular
warp bubble moving at twice the speed of light. $l$ is a characteristic length
scale of the bubble. The geodesics are launched in the forward direction (with
respect to the motion of the drive), with an initial angle dispersion of
$\pi/3$ between the first and last of each bundle of curves. The thicker lines
of each bundle represent the geodesics which get closest to the point $p_{1}$,
which in a vicinity of this point correspond to curves (6.18) with small
values of $c_{2}$.

A further check of this behaviour was performed numerically by directly
solving equations (6.5), (6.6), and (6.7) for geodesics launched from within a
moving convex warp bubble. The result is shown in fig. 6.3, where one observes
clearly the general behaviour of light within this geometry. The first thing
we note about the geodesics shown is that they follow the restrictions imposed
by the light cones represented in fig. 6.2: they can only move forward (toward
larger values of $x$) while inside the bubble, and when they approach its
edges they turn around. The thicker lines of each bundle of geodesics mark two
curves which get close to the central axis in the vicinity of the point of
runaway blueshift $p_{1}$. In these curves we see explicitly the behaviour
captured in Eq. (6.18) for solutions with small values of $c_{2}$. They
initially approach the axis, until the exponential growth of $e^{\eta_{-}t}$
overcomes the smallness of $c_{2}$ and pushes them away. As expected, for
geodesics launched from each point, the separatrix ($c_{2}=0$) between the
ones which end up on the left and on the right of $p_{1}$ turns out to be
impossible to capture numerically. This provides further evidence of the fact
that, although these solutions end up infinitely blueshifted, they are of
measure zero within the whole family of geodesics.

For a bubble with a more flat profile in $y$ at $p_{1}$ (i.e. $n>1$), there
are no analytical solutions to (6.14), but it can be seen that the source term
for the derivatives is smaller and deflection therefore has an initially
polynomial (rather than exponential) dependence on $t$. Aside from this, the
qualitative behaviour of the geodesics in such a bubble remains the same (this
has been checked numerically).

Returning to the $n=1$ parabolic profile, we can make an important observation
regarding the deflected geodesics. By taking the geodesic from (6.18) with
$c_{3}=0$ as representative of the generic qualitative behaviour of geodesics
with $c_{2}\neq 0$, we can write its solution in terms of the initial
condition $y(0)=y_{0}$ as

$y\simeq y_{0}e^{\eta_{-}t}.$ (6.20)

We can then define a deflection time $t_{\rm def}$ as the time it takes for
the solution to reach a fixed reference point $y_{\rm def}$,

$t_{\rm def}=\eta_{-}^{-1}\log(y_{\rm def}/y_{0}).$ (6.21)

The value of $y_{\rm def}$ is a characteristic length scale of the geometry
which can be defined e.g. as the separation for which the approximation which
lead to (6.14) fails. The important part is the dependence on $y_{0}$,
particularly, the logarithmic divergence as $y_{0}\to 0$.

In 1+1 dimensions, the presence of a Cauchy horizon and the corresponding
divergence of $t$ for finite $\sigma$ in (6.9) drives the exponential growth
of the energy density (6.3). In 2+1 and higher dimensions, one may then expect
the same type of growth only around points where the null geodesics behave the
same way (i.e. as if approaching a Cauchy horizon), which occurs only when
they approach the tip of the warp bubble. In other words, the exponential
growth of the energy density should only occur at a single point. As to what
happens in the vicinity of this point and how this energy accumulation falls
off away from it, the geodesics with $c_{2}\neq 0$ might provide a clue.

Particularly, in the regime $t\ll t_{\rm def}$ these geodesics have a very
small deviation from the central solution (6.9), and one might expect that
they bring about a growth similar to (6.3), but instead of blowing up to
infinity as $t$ grows, tending to a finite cutoff value with a profile given
by the logarithm of $y_{\rm def}/y_{0}$ in (6.21), with $y_{0}$ representing
the separation from the central point.

Another argument in favour of this kind of asymptotic density profile can be
made by just considering the blueshift of light rays which could be randomly
launched in the general direction of the front of the warp bubble. The ones
which happen to tend exactly to the tip are the only ones which are trapped
and have a divergent tendency in their blueshift. In the rest of the bubble,
one may expect that a stationary situation is quickly reached if e.g. the rays
are launched at regular intervals. In a given time the same number of rays
would enter a given area as the ones which exit it, though the closer this
area is to the tip of the bubble, the longer their stay there and the larger
their transient blueshift, giving rise to the same logarithmic profile of
energy density.

In dimensions larger than 1+1, the “instability” for a convex warp bubble is
therefore just the growth at a single point, and integrating the energy in its
vicinity the result would not asymptotically tend to a divergence. Even if the
logarithmic profile we obtain for the asymptotic tendency for the
semiclassical energy density is not the correct description one would get from
calculating the RSET exactly in 2+1 or higher dimensions, at the very least
the fact that this profile is related to the accumulation of geodesics is
robust. Therefore, additional dispersion of these geodesics would translate
into further stabilisation of the semiclassical behaviour, as can be achieved
by decreasing $t_{\rm def}$ (e.g. by making the peak of the bubble sharper in
$y$, i.e. making $\xi$, and hence $\eta_{-}$, larger), or by making the
trajectory of the drive deviate from the straight line path we have considered
here, as we will show numerically below. On the other hand, making $t_{\rm
def}$ larger (e.g. by decreasing $\xi$ or increasing $n$) would have the
opposite effect and bring the drive closer to instability.

A convex shape with a very sharp peak, which offers the least resistance for
travel in the presence of a quantum field, even in vacuum, is reminiscent of
the shapes used for supersonic aircraft which minimise the frontal pressure
and drag that they experience. By extension of this analogy, one would
naturally expect that a warp bubble with a flat or concave peak would
experience much more resistance, i.e. a much stronger blueshift instability.
Indeed, in the case of a flat peak we found that the solutions which are
trapped in a tendency toward a Cauchy horizon are much more abundant (6.17).
For a concave peak with e.g. a locally parabolic profile, the solutions would
be the same as (6.18) but, $\xi$ being negative, $\eta_{-}$ would have a
negative real part, making both exponentials decreasing ones. This would be a
case in which a divergence of the order of that of a flat peak is concentrated
at a single point, making the instability even greater.

#####  6.2.4 Numerical analysis of non-stationary configurations: further
stabilising the warp drive

In light of these results, one may wonder how this behaviour generalises to
dynamical warp drive spacetimes, i.e. ones in which the warp bubble can change
its trajectory and velocity over time. Particularly, we want to see whether
the point of divergent blueshift $p_{1}$ remains, or whether some trajectories
for the bubble can “shake off” the potentially accumulated geodesics around
such a point at regular intervals.

There are two types of movement which have the potential to do this: a change
in direction, or a temporary reduction of the velocity to a subluminal one.
However, we have found through a numerical analysis that neither one of these
can fully eliminate the point of asymptotically infinite blueshift (and its
corresponding finite-time accumulation of vacuum energy). Nonetheless, they
can significantly disperse the geodesics in its vicinity, producing the same
effect as making the peak of a straight-line bubble sharper.

Figure 6.4: Numerical integration of null geodesics launched from the centre
of a circular warp bubble moving in a zig-zag pattern with subluminal and
superluminal intervals. The velocity of the drive in the $x$ direction changes
between 0.2$c$ and 10$c$ with frequency $2\pi/5$, while in the $y$ direction
it changes between 0.8$c$ and -0.8$c$ with frequency $\pi/5$. There are 6
geodesics emerging from the left side of the plot. The angle at which they are
emitted only varies by $10^{-4}$ between them, so they initially overlap. Each
time the drive becomes subluminal, the geodesics are released forward, only to
turn around once it becomes superluminal again. At the second release they
become dispersed enough to be visibly separate, while at the third release
only one of them remains (the rest having been dispersed to the sides).

The fact that the equivalent of $p_{1}$ cannot be eliminated can be deduced
from a simple consideration: if the warp bubble continues to exist
indefinitely (and has a well-defined average asymptotic direction), for
geodesics launched from each point in its interior there will always be a
separatrix between those which end up on one side or the other of its
asymptotic trajectory. What can be controlled, however, is the amount of
blueshift this separatrix and its adjacent geodesics experience at finite
times.

As an example, we present the result from a numerical analysis of a dynamical
warp drive configuration which combines the two modifications to the straight-
line scenario mentioned above: a change of direction, achieved through a
periodically varying velocity in the $y$ direction, and a change to a
subluminal velocity in $x$, also performed periodically. Figure 6.4 represents
the key features of the behaviour of light rays in such a geometry. Most null
geodesics launched from inside the bubble quickly escape to either side of the
averaged direction of motion, and never approach the bubble again, akin to
those plotted in fig. 6.3. The main difference is seen in the rays which are
launched approximately in the direction of motion. Particularly, those which
remain in the vicinity of the front end of the bubble long enough to catch one
of the changes in velocity have the chance to move in the forward direction
beyond the confines of the bubble while the drive is subluminal. Then, when
the drive becomes superluminal again, the bubble catches up to those rays once
again, and they are now deflected to either side of it from the outside. Those
which are close to the separatrix between the ones deflected to either side
can again remain in the vicinity of the edge of the bubble long enough to
catch the next change in velocity and move forward again. This process repeats
periodically, and there is once again a particular set of trajectories (which
are of measure zero within the total set of null geodesics, and which
asymptotically coincide) that define the separatrix between rays deflected to
either side of the drive.

Figure 6.4 represents 6 null geodesics which are launched in an approximately
forward direction, with an initial angle dispersion of the order of $10^{-4}$
(making them overlap initially, on the left side of the plot). We see that
each time the drive becomes subluminal, the geodesics are allowed to leave the
bubble in the forward direction (or, as seen from inside the drive, it is the
bubble that effectively expands to infinity). Then when it picks up
superluminal speed they again turn around. Due to the oscillatory nature of
the movement, we can expect that the separatrix also describes a periodic
movement in space. In fact, one of the geodesics in fig. 6.4 is very close to
such a behaviour: after the third time it is released in a forward direction
(i.e. the third time the bubble becomes subluminal) it nearly follows the same
trajectory as after the first time, though the small difference makes it so it
is deflected away (toward positive $y$) in the end.

While a convoluted trajectory for the warp drive would have a negative impact
on its initial purpose (i.e. shortening travel time), it can, on the other
hand, increase its semiclassical stability by reducing the accumulated
blueshift around its peak. Constructing an optimal warp bubble shape and
trajectory would therefore become a balancing act between having a short
travel time and minimising the (possibly already very small) accumulation of
unwanted vacuum energy and blueshifted classical perturbations. Of course,
this problem would likely have a secondary role when compared to the
inevitable engineering difficulties in constructing such configurations to
begin with.

####  6.3 Conclusions

We have analysed the semiclassical instability present in the Alcubierre warp-
drive spacetime through its relation to the behaviour of null geodesics. We
have argued that the strong instability found in 1+1 dimensional
configurations can actually be tamed in spacetimes of higher dimensions by
choosing appropriately the shape and the trajectory of the warp bubble.

First, the warp field should be chosen to have an “aerodynamic” shape, so as
to deflect null geodesics away from its unstable point in the shortest time
possible. Second, the trajectory of the drive can be chosen so as to further
facilitate this dispersion, particularly with slight changes in its direction
of movement (e.g. a small ziz-zag component to the motion), and with
alternating intervals of subluminal and superluminal warp field velocities.

Our findings are interesting even from a purely geometrical perspective. In
1+1 dimensions the front wall of a warp drive acts as a pure inner horizon.
However, in higher dimensions the warp drive does not have a closed inner
horizon (or indeed any closed trapped surfaces). Instead, the warp-drive
bubble can be interpreted as an interpolation between an inner horizon point
(the front end of the bubble) and an outer horizon point (the back end of the
bubble); then, in between we have a causal structure more similar to that of
an ergoregion, from which signals can in fact escape. This is the reason why
the geodesic accumulation, and the corresponding blueshift instability, is
limited to only single points, at least when the shape of the bubble is
smooth.

In contrast to BHs with an inner horizon, warp drives could indeed be stable
semiclassical configurations, if their shape and trajectory are appropriately
chosen, though the mechanism which generates their geometry in the first place
may lie outside the semiclassical theory. On the other hand, the less
“aerodynamic” configurations found, such as ones with a flat wall at the front
end of the bubble, are an interesting example of the fact that blueshift
instabilities (both classical and semiclassical) can appear without the
presence closed trapped surfaces.

###  Chapter 7 Classical mass inflation vs semiclassical inner horizon
inflation

The main goal of this second part of the thesis is to see whether and how
semiclassical physics can have an influence on the evolution of the inner
horizon of a BH. In chapter 4 we looked at an example of the purely classical
evolution of an inner horizon as it undergoes mass inflation. Then, in chapter
5 we analysed the semiclassical backreaction around static and dynamical inner
horizons in simple geometries which, however, do not incorporate mass
inflation. The result of this latter analysis was that the inner horizon tends
to be pushed outward due to backreaction from the RSET. The initial tendency
for this movement appears to be exponential in time, and extrapolating from it
(assuming that the RSET maintains its negative horizon-related flux) we
described a potential _inner horizon inflation_ process, through which the
trapped region is quickly extinguished from the inside out. In this chapter we
will extend our semiclassical backreaction analysis to background geometries
incorporating the causal properties of mass inflation.

As in previous chapters, we will use the RSET of a massless scalar field in
the Polyakov approximation as the source of backreaction. We construct the
“in” vacuum [23, 45] of the 1+1 dimensional radial-temporal sector of the
spacetime, used to calculate the RSET in this approximation, by following the
movement of lightlike geodesics from past null infinity up to the region of
interest, which in this case is the vicinity of the inner apparent horizon
inside the mass-inflated region of a dynamically formed BH. As discussed in
chapter 5, the accumulative effect that the inner horizon has on null
geodesics results in a sensitivity of the RSET to the past of a large part of
the collapse geometry, unlike what occurs for the outer horizon. We therefore
employ the same tactic as in that chapter to simplify the initial conditions
of the quantum modes in this region: we consider their propagation as being in
a flat spacetime up to an advanced time $v=v_{0}$, from where it continues
inside a BH with an inner horizon, as shown in fig. 7.1. Effectively, this is
equivalent to the BH being generated by the collapse of a null shell located
at the formation time $v=v_{0}$, but our motivation for the construction is
purely geometrical, to “clean up” the dependence on the details of the
collapse and leave only the part stemming from the quantum modes finding
themselves in the trapped region.

Figure 7.1: Formation of a BH by a ingoing null shell at $v=v_{0}$. The “in”
vacuum state is constructed by tracing light rays back to the flat region in
the past. Classically, the BH is undergoing mass inflation and the inner
horizon is headed toward the origin.

In this chapter, we will perform two distinct calculations to estimate
semiclassical backreaction in the vicinity of the inner horizon. The first
involves a series expansion of the RSET and the metric functions around the
point $\\{v_{0},R_{\rm i0}\\}$, where the (timelike part of the) inner horizon
$R_{\rm i}$ forms, which allows a simplified term-by-term calculation of the
semiclassical perturbations of the metric caused by the RSET. The second is a
full self-consistent solution for a classical background of the type (4.20)
with a particular family of functions $F(v,r)$, valid for a small but finite
time interval after $v_{0}$. From the latter calculation we find that the
initial tendencies seen in the series expansion can lead to very quick
accumulative effects which make semiclassical corrections relevant before the
spacetime curvature reaches Planckian scales.

####  7.1 Series expansion around horizon formation

Let us begin by considering the line element

$ds^{2}=A(v)\left[-B(v,r)dv^{2}+2dvdr\right]+r^{2}d\Omega^{2}.$ (7.1)

We will use this type of geometry to represent the inner region of a BH
undergoing mass inflation, particularly around its inner apparent horizon. In
relation to the construction of chapter 4, we are using the geometry obtained
with a single outgoing shell, resulting in a single exponential growth of
mass, which we expect to be the dominant effect in a transient period between
early and very late times, where we place our $v_{0}$. For the freezing
function $A$ we can use the dominant behaviour in $v$ of (4.23) and set
$A(v)=e^{-\kappa v}$, with $\kappa$ a positive constant. Neglecting the
$1/v^{p}$ part of (4.23) amounts to discarding corrections suppressed by an
additional $1/v$ factor in the backreaction calculations of this section,
which do not have a qualitative influence on our conclusions. For the function
$B$, which represents the product of $F$ and $A$ from (4.20), we only need to
impose that it has a zero at a radius $R_{\rm i}(v)$, corresponding to the
inner apparent horizon, with a negative slope in the $\partial_{r}$ direction.

To calculate the RSET for this spacetime, we first need to construct a
quantisation with a vacuum state which is physically adequate for the problem
at hand. As previously, we will use the “in” vacuum state, which is defined as
the Minkowski vacuum at the asymptotically flat region of past infinity and
its extension to the dynamical region through the propagation of the particle-
related modes. As the reader may recall, in the Polyakov approximation we only
need to do this in the 1+1 dimensional radial-temporal sector of our geometry,
which simplifies the problem greatly, given that 1+1 dimensional spacetimes
are conformally flat. Working with a pair of radial null coordinates
$\\{u,v\\}$, the line element (7.1) can generally be written as

$ds^{2}=-C(u,v)dudv+r^{2}d\Omega^{2}.$ (7.2)

At past null infinity, $C\sim 1$ and Poincaré invariance of the vacuum state
in flat spacetime amounts to supertranslation invariance of the “in” state
[5]. Following the construction of chapter 4, the $v$ coordinate used in (7.1)
is in fact one of these “in” coordinates, while $u$ must be obtained through
its relation to $r$.

The conformal factor of the radial-temporal sector is given by

$C(u,v)=-2A(v)\frac{\partial r(u,v)}{\partial u}.$ (7.3)

The components of the RSET in the Polyakov approximation for the vacuum state
selected by the coordinate system $\\{u,v\\}$ are then given in terms of $C$
through eqs. (5.12). Like in chapter 5, we will only use this RSET in a small
vicinity around the inner horizon, and only while this horizon is considerably
farther away from the origin than a Planck length (i.e. early enough in the
evolution of the mass inflation background), so that the Polyakov
approximation can be qualitatively accurate. To calculate the conformal factor
(7.3), we need to obtain the function $r(u,v)$ from the solutions of radial
null geodesics. The ingoing geodesics are just $v=\text{const.}$, while the
outgoing ones are solutions to

$\frac{dr}{dv}=\frac{1}{2}B(v,r).$ (7.4)

At this point we must either specify the function $B$, or try to see what
general conclusions could be obtained from just the mere fact that there is an
inner horizon in this structure, i.e. that $B$ has a zero at some $R_{\rm
i}(v)$ with a negative slope. In the next section we will specify some
functions $B$ which can simplify our analysis while still reproducing the
causal properties of mass inflation, but for now we will maintain generality
and perform a perturbative analysis. Particularly, we will consider a generic
expansion of the function $B$ around the point at which this horizon forms
$\\{v_{0},R_{\rm i0}\\}$,

$\displaystyle\begin{split}\frac{1}{2}B(v,r)&=k_{1}(v)(r-R_{\rm
i}(v))+k_{2}(v)(r-R_{\rm i}(v))^{2}+\\\ &\quad+k_{3}(v)(r-R_{\rm
i}(v))^{3}+\cdots,\end{split}$ (7.5) $\displaystyle R_{\rm i}(v)$
$\displaystyle=R_{\rm i0}+R_{\rm i1}v+R_{\rm i2}v^{2}+R_{\rm i3}v^{3}+\cdots,$
(7.6) $\displaystyle k_{n}(v)$
$\displaystyle=k_{n0}+k_{n1}v+k_{n2}v^{2}+k_{n3}v^{3}+\cdots,$ (7.7)

with $n=1,2,\dots$, and where for simplicity we have set $v_{0}=0$. The only
conditions we impose on these series is that $R_{\rm i0}>0$ and $k_{10}<0$
(this being the inner horizon). The smallness of the terms in the expansions
of quantities with (inverse) length dimensions, here and throughout this
section, can be measured in terms of their respective initial values at $v=0$,
or in units of the characteristic initial scale $R_{\rm i0}$.

For the solution of (7.4) we consider the series expansion

$r(v)=r_{0}+r_{1}v+r_{2}v^{2}+r_{3}v^{3}+\cdots$ (7.8)

Substituting this expression into (7.4), we obtain the coefficients of the
null trajectories (7.8) in terms of derivatives of $B$ [i.e. the coefficients
of its expansion (7.5)-(7.7)] and a free parameter fixed by an initial
condition. We will use $d_{0}=r_{0}-R_{\rm i0}$ as this parameter. Tracing
back the null trajectories through the Minkowski region $v<0$ (see fig. 7.1)
we find that our missing “in” coordinate is $u=-2d_{0}$ (up to a constant
which fixes the origin of $u$, taken as zero).

Constructing $r(u,v)$ in this manner, we calculate the conformal factor (7.3)
and then the RSET components (5.12) in the “in” coordinate system. Switching
them back to the Eddington-Finkelstein system, at zeroth order in the series
expansion, they are

$\displaystyle\begin{split}\expectationvalue{T_{vv}}&=\frac{l_{\rm
P}^{2}}{96\pi^{2}R_{\rm
i0}^{2}}\left(-\frac{1}{2}k_{10}^{2}+k_{11}-2k_{20}R_{\rm i1}-\right.\\\
&\qquad\qquad\qquad\left.-\frac{1}{2}\kappa^{2}+k_{10}\kappa\right)+\order{v,d},\end{split}$
(7.9a) $\displaystyle\expectationvalue{T_{rr}}$ $\displaystyle=\order{v,d},$
(7.9b) $\displaystyle\expectationvalue{T_{vr}}$ $\displaystyle=-\frac{2l_{\rm
P}^{2}k_{20}}{96\pi^{2}R_{\rm i0}^{2}}+\order{v,d},$ (7.9c)

where $d=r-R_{\rm i0}$ [here $\order{d}$ can be though of as $\order{d_{0}}$
or $\order{u}$, as the difference between them is $\order{v}$]. Note that if
we take (5.33) and incorporate the expansions in the $v$ direction, we recover
the above expressions with $\kappa=0$. The introduction of a non-trivial
freezing function $A$ is what makes the calculation in this chapter more
involved, requiring the additional series expansion in time (or the
specification of a particularly simple function $B$, as in the next section)
to obtain a meaningful result.

We now consider the semiclassical Einstein equations

$G_{\mu\nu}+\delta G_{\mu\nu}=T_{\mu\nu}^{\rm
class}+\expectationvalue{T_{\mu\nu}},$ (7.10)

where $\delta G_{\mu\nu}$ is a perturbation to the background Einstein tensor
and $T_{\mu\nu}^{\rm class}$ is the classical matter content sourcing the
zeroth order background. In general, the perturbation caused by the RSET would
also affect $T_{\mu\nu}^{\rm class}$ through its dependence on the metric.
However, since we want to remain as agnostic as possible about this classical
matter, we consider that $\delta T_{\mu\nu}^{\rm class}=0$ at zeroth order in
the series expansion (in its functional form in Eddington-Finkelstein
coordinates), i.e. that

$\delta G_{\mu\nu}=\expectationvalue{T_{\mu\nu}}+\order{v,d}.$ (7.11)

This simplifying assumption serves two purposes: on the one hand, it allows us
to continue to work in purely geometric terms, with as few ingredients in the
dynamics as possible. On the other hand, it exemplifies well what backreaction
from the RSET can look like in its purest form, where the potentially
negative-energy terms in this tensor directly source $\delta G_{\mu\nu}$.
Technically, considering $\delta T_{\mu\nu}^{\rm class}=0$ would over-
determine the system of equations, but it turns out to work consistently up to
second order in our series expansion, allowing us this first geometric glimpse
into backreaction.

Using the expansion of the function $B$ and its coefficients, the Einstein
tensor of our generic background is

$\displaystyle G_{vv}$ $\displaystyle=\frac{2k_{10}R_{\rm i1}}{R_{\rm
i0}}+\order{v,d},$ (7.12a) $\displaystyle G_{rr}$ $\displaystyle=\order{v,d},$
(7.12b) $\displaystyle G_{vr}$ $\displaystyle=\frac{2k_{10}R_{\rm
i0}-1}{R_{\rm i0}^{2}}+\order{v,d}$ (7.12c) $\displaystyle G_{\theta\theta}$
$\displaystyle=2R_{\rm i0}(k_{10}+k_{20}R_{\rm i0}).$ (7.12d)

with $G_{\phi\phi}=\sin^{2}(\theta)G_{\theta\theta}$. To construct $\delta
G_{\mu\nu}$ we can consider perturbations to the coefficients in (7.12), e.g.
$k_{10}\to k_{10}+\delta k_{10}$. Eq. (7.11) allows us to fix one of the three
coefficients present in the tensor components (7.12) to its classical value as
an initial condition, and we choose the initial position of the inner horizon
$R_{\rm i0}$. Then, perturbing the surface gravity $k_{10}$, the initial time
derivative of the inner horizon trajectory $R_{\rm i1}$, and the second
spatial derivative of the redshift function $k_{20}$, we obtain the leading
order of the perturbed tensor

$\displaystyle\delta G_{vv}$ $\displaystyle=2\frac{\delta k_{10}R_{\rm
i1}+k_{10}\delta R_{\rm i1}}{R_{\rm i0}}+\order{v,d},$ (7.13a)
$\displaystyle\delta G_{rr}$ $\displaystyle=\order{v,d},$ (7.13b)
$\displaystyle\delta G_{vr}$ $\displaystyle=\frac{2\delta k_{10}R_{\rm
i0}-1}{R_{\rm i0}^{2}}+\order{v,d}$ (7.13c) $\displaystyle\delta
G_{\theta\theta}$ $\displaystyle=2R_{\rm i0}(\delta k_{10}+\delta k_{20}R_{\rm
i0}).$ (7.13d)

As per eq. (7.11), we equate these components to the RSET generated by the
background (7.9), keeping in mind that
$\expectationvalue{T_{\theta\theta}}=\expectationvalue{T_{\phi\phi}}=0$ in the
Polyakov approximation, thus implying $\delta G_{\theta\theta}=0$. From the
other two equations we obtain two key relations,

$\displaystyle\delta k_{10}$ $\displaystyle=-\frac{l_{\rm
P}^{2}}{12\pi}\frac{k_{20}}{R_{\rm i0}},$ (7.14) $\displaystyle\delta R_{\rm
i1}$ $\displaystyle=\frac{l_{\rm P}^{2}}{48\pi}\left(-\frac{k_{10}}{R_{\rm
i0}}-\frac{\kappa^{2}}{R_{\rm i0}k_{10}}+\frac{1}{2}\frac{\kappa}{R_{\rm
i0}}+\frac{2k_{11}}{R_{\rm i0}k_{10}}\right).$ (7.15)

On the one hand, we can see that the semiclassical contribution to the
modification of the surface gravity can be either positive or negative,
depending on the sign of the background coefficient $k_{20}$. This initial
semiclassical contribution very much depends on the details of the initial
background geometry. On the other hand, the modification of the derivative of
the inner horizon trajectory is almost always positive, implying a decrease in
the rate at which it moves inward. This is a first indication of the
regularising tendency which semiclassical corrections can add to the inner
horizon dynamics in these spacetimes. Particularly, it can be seen from the
fact that $k_{10}<0$, $\kappa>0$, $R_{\rm i0}>0$, and the assumption that
$k_{11}<0$, i.e. that the background surface gravity tends to become
increasingly more negative, which is certainly the case in mass inflation, as
can be seen from e.g. eq. (4.30). Interestingly, the further along an
evolution of the type (4.29) we set our initial conditions for semiclassical
backreaction, the smaller the background value of $R_{\rm i1}$ would be
(approaching zero as $v\to\infty$) and the larger $\delta R_{\rm i1}$ would be
in comparison. If the background surface gravity $k_{10}$ has a divergent
behaviour akin to (4.30), then the growth of $\delta R_{\rm i1}$ as we take
our initial radial position $R_{\rm i0}$ to 0 cannot be said to be a
consequence of the unphysical divergent $1/r^{2}$ factor present in the RSET
approximation. More generally, if $k_{10}$ diverges at least as strongly as
$1/R_{\rm i0}$, then a regularised version of this perturbation $R_{\rm
i0}^{2}\delta R_{\rm i1}$ remains at least finite, as opposed to the
background $R_{\rm i1}$ which is expected to approach zero unless a spacelike
singularity forms at finite $v$.

Using these initial tendencies as an estimate of the magnitude of this effect
later on in the evolution, one is led to the hypothesis that semiclassical
backreaction will always become dominant at some point before a singularity is
formed, perhaps leading instead to a non-singular future. The only caveat
might appear when interpreting the semiclassical effects as only the first set
of corrections towards a quantum gravity theory. Then, if the corrections
occurred only when curvature becomes Planckian, i.e. when the background
values in (7.15) overcome the suppression by the $l_{\rm P}^{2}$ factor, one
could argue that these semiclassical effects would have been already
superseded by other effects of unknown nature. However, as we will see in the
following example, a full time evolution can lead to a very different result,
in which semiclassical corrections have a much quicker accumulative effect.

####  7.2 Time-integrable example

In order to see what the semiclassical evolution of the inner horizon could
look like beyond the initial tendencies calculated above, we can use a
particular family of geometries for the classical background which simplify
our semiclassical analysis greatly. Particularly, we will use geometries
which, in a vicinity around the inner horizon, take the form (7.1) with

$B(v,r)=e^{-\kappa v}-\frac{1}{2}\lambda(v)r,$ (7.16)

where $\lambda(v)$ is a positive, but otherwise arbitrary function. This type
of geometry, along with the below perturbation (7.19), is a straightforward
generalisation of (5.40) from chapter 5. The inner horizon described by this
geometry,

$R_{\rm i}(v)=2\frac{e^{-\kappa v}}{\lambda(v)},$ (7.17)

moves toward the origin as long as $\lambda$ does not decrease faster than
$e^{-\kappa v}$. The relation between this position and its surface gravity is
not quite the same as in e.g. the Reissner-Nordström case (where for $R_{\rm
i}\sim e^{-\kappa v}$, the surface gravity has an increase with a rate
$e^{2\kappa v}$, as seen in chapter 4), but a growing surface gravity can
still be replicated by choosing a $\lambda$ which increases in time.

The RSET in the Polyakov approximation corresponding to this classical
background geometry has a single non-zero component: the ingoing flux

$\expectationvalue{T_{vv}}=\frac{l_{\rm
P}^{2}}{96\pi^{2}r^{2}}\left[-\frac{1}{4}\lambda^{\prime}-\frac{1}{32}\lambda^{2}-\frac{\kappa}{4}\lambda-\frac{\kappa^{2}}{2}\right],$
(7.18)

which is negative as long as $\lambda^{\prime}\geq 0$ (this being a reasonable
requirement for a mass inflation background, i.e. that the surface gravity of
the inner horizon does not decrease).

The main motivation for using these geometries is that, much like the
dynamical examples of chapter 5, backreaction from the RSET around the inner
horizon has the effect of changing the $B$ function to

$B(v,r)=e^{-\kappa v}-\frac{1}{2}\lambda(v)r+\delta B(v,r),$ (7.19)

where the first two terms are just its background form, and the perturbation
$\delta B$, obtained from the semiclassical Einstein equations, will have a
particularly simple form in terms of its dependence in $r$,

$\delta B(v,r)=-\frac{\alpha(v)}{r}$ (7.20)

(the minus sign serves to make an analogy with a mass term, as discussed
below). This can be readily checked by calculating the Einstein tensor with
(7.19), which has the non-zero components

$\displaystyle G_{vv}$
$\displaystyle=\frac{\lambda^{\prime}}{2}-\frac{\lambda^{2}}{2}+\frac{e^{-\kappa
v}\lambda}{r}+\frac{\kappa\lambda}{2}+\frac{1}{r^{2}}\left[\alpha^{\prime}-(\lambda-\kappa)\alpha\right],$
(7.21a) $\displaystyle G_{vr}$ $\displaystyle=-\frac{\lambda}{r},$ (7.21b)
$\displaystyle G_{\theta\theta}$
$\displaystyle=\frac{G_{\phi\phi}}{\sin^{2}\theta}=-\frac{r\lambda e^{\kappa
v}}{2},$ (7.21c)

where primes denote derivatives with respect to $v$. We see that $\alpha$
appears only in $G_{vv}$, and that the terms which contain it can be directly
equated to the RSET flux (7.18), given that they have the same dependence in
$r$ [the form of $\delta B$ in (7.20) was obtained by requiring this]. Here we
are once again assuming that the classical part of the equations, i.e. the
rest of the terms of $G_{\mu\nu}$ and their source $T_{\mu\nu}^{\rm class}$,
remain unchanged. Analogously to the series expansion calculation above (and
the one in chapter 5), our motivation for doing this is to introduce as little
information about the classical matter content as possible, while also getting
a cleaner backreaction problem for the Einstein tensor sourced solely by the
RSET (in this case it is the particular form of the background which allows us
to do this without over-determining the system of equations). As long as this
perturbation of non-zero $\alpha$ is negligible for the calculation of the
RSET itself, and while (7.18) is accurate (which is the case for a finite time
interval after $v_{0}$, which is smaller for faster dynamics of the
background), this gives an approximate self-consistent solution to the
semiclassical Einstein equations. In the language of eq. (7.11), we are once
again considering a $\delta G_{\mu\nu}$ which is equated to the Polyakov RSET.
The only non-zero component of this perturbation tensor is thus $\delta
G_{vv}$, sourced by (7.18), corresponding to the terms involving $\alpha$ in
(7.21a), while all other $\delta G_{\mu\nu}$ are zero.

Note that the function $\alpha$ in $B$ is analogous to the $M_{0}$ term in
(4.24), which is constant in the product of $A$ and $F$ used to construct $B$
(up to the decaying inverse polynomial $1/v^{p}$ terms, which we have
neglected in our construction here), and represents the classically
exponentially growing mass. Thus, if $\alpha$ grows, toward either positive of
negative values, it could be taken as an indication of the semiclassical
backreaction tending to become the dominant source of dynamics. We will now
see that $\alpha$ in fact becomes negative, and in many cases its absolute
value tends to grow exponentially quickly.

Equating (7.18) to the terms containing $\alpha$ in (7.21a), leaving the
remaining terms as fixed by the background, the evolution of this
semiclassically sourced $\alpha$ is given by the equation

$\alpha^{\prime}(v)-\eta_{1}(v)\alpha(v)=\frac{l_{\rm
P}^{2}}{48\pi}\eta_{2}(v)+\order{l_{\rm P}^{4}},$ (7.22)

where

$\eta_{1}=\lambda-\kappa,\quad\eta_{2}=-\lambda^{\prime}-\frac{1}{8}\lambda^{2}-\kappa\lambda-2\kappa^{2}$
(7.23)

are two functions determined by the choice of background. The general solution
of this equation is

$\alpha(v)=\frac{l_{\rm
P}^{2}}{48\pi}e^{\int^{v}\eta_{1}(\tilde{v})d\tilde{v}}\left[c_{1}+\int^{v}e^{-\int^{\tilde{v}}\eta_{1}(\bar{v})d\bar{v}}\eta_{2}(\tilde{v})d\tilde{v}\right],$
(7.24)

where $c_{1}$ is the integration constant which can be fixed by initial
conditions.

Let us first look at the simple case in which $\lambda$ is constant, which
represents an inner horizon shrinking in radius proportionally to $e^{-\kappa
v}$ while maintaining a constant surface gravity. Here we already see the main
difference from the situations studied in chapter 5. Depending on whether
$\kappa$ or $\lambda$ is larger, $\eta_{1}$ can be either a positive or
negative constant, while $\eta_{2}$ is always a negative constant. With the
condition of the semiclassical perturbation being initially zero,
$\alpha(0)=0$, we get the solution

$\alpha(v)=\frac{l_{\rm
P}^{2}}{48\pi}\frac{\eta_{2}}{\eta_{1}}[e^{\eta_{1}v}-1].$ (7.25)

Whether $\eta_{1}$ is positive or negative, $\alpha$ evolves toward negative
values. In the former case its absolute value grows exponentially quickly,
while in the latter it tends to a constant. There is also the particular case
in which $\lambda=\kappa$, for which, since $\eta_{1}=0$, the solution is

$\alpha(v)=\frac{l_{\rm P}^{2}}{48\pi}\eta_{2}v,$ (7.26)

with $\eta_{2}$ being once again negative. In all these cases, the
(increasingly) negative values of $\alpha$ tend to push the inner horizon
outward. Expanding the radial position of this horizon around $\alpha=0$,

$R_{\rm i}=2\frac{e^{-\kappa v}}{\lambda}-\alpha e^{\kappa v}+\cdots,$ (7.27)

we see that once $\alpha$ acquires a non-vanishing value, even in the case
where it tends to a constant, this radius quickly acquires non-perturbative
corrections. The inner horizon thus begins to move outward, which, although in
apparent violation of causal evolution, is hardly surprising considering that
the source given by the RSET (7.18) is an ingoing flux of negative energy.

For more general backgrounds given by different functions $\lambda(v)$, we can
see from equations (7.22) and (7.23) that starting from $\alpha(0)=0$,
$\alpha(v)$ will tend to decrease and the inner horizon will tend to move
outward unless $\lambda(v)$ decreases sufficiently quickly for $\eta_{2}$ to
become positive. For example, if $\lambda$ tends to zero asymptotically in $v$
and its tendency is quicker than $1/v$ (but slower than $e^{-\kappa v}$, so
the inner horizon does not move outward classically), and if $\kappa$ is
initially negligible in (7.23), there can be a period of time in which
$\alpha$ increases toward positive values. However, except in these specific
scenarios, $\eta_{2}$ will generally be negative and $\alpha$ will acquire
negative values, making the semiclassical movement of the inner horizon an
outward one.

Therefore, the results obtained in chapter 5 appear to still hold in most of
these mass inflation geometries. In other words, while the flux (7.18)
dominates the RSET, backreaction tends to push the inner horizon outward.
However, it is worth reminding the reader that the conditions for which we
have been able to show that this flux is dominant only hold true for a short
period of time after the formation of the BH, given by the time it takes for
outgoing null geodesics which come from outside the region where (7.19) is
accurate (with $\delta B$ sufficiently small) to intersect the inner horizon.
The result for the movement of the inner horizon is therefore only accurate as
an initial tendency. Still, the fact that the RSET is likely to keep violating
energy positivity conditions even in the later parts of the evolution makes
the possibility that the trapped region continue to evaporate from the inside
a likely one.

Figure 7.2: Trajectory of the inner apparent horizon in an extrapolated
semiclassical solution, with a classical background which simulates the
behaviour of the interior of a mass-inflated charged BH. Length units are
taken in terms of the exterior mass $M_{0}$, the charge is taken to be
$Q=M_{0}/2$ and the Planck length is set as $10^{-5}M_{0}$ (as a large
difference in orders of magnitude is necessary, but a smaller value only
increases the computational difficulty while giving no qualitative
difference). Note that the bounce occurs at $r\simeq 10^{-3}M_{0}$, which
illustrates that semiclassical effects can become dominant without the radius
of the inner horizon becoming Planckian

.

Assuming that a term like (7.18) continues to dominate the RSET even after
backreaction has become significant, we can extrapolate the movement of the
inner horizon further along the evaporation process. For example, for a
classical background in which the surface gravity increases exponentially as
in the Reissner-Nordström case, the extrapolated self-consistent solution can
be observed in fig. 7.2. The inner horizon initially moves inward while the
classical background still dominates, but when the semiclassical perturbation
has enough time to accumulate it produces an outward bounce and a rapid
inflationary extinction of the trapped region from the inside.

####  7.3 Conclusions and discussion

In this chapter we have extended our analysis of backreaction on BH inner
horizons from chapter 5 to background geometries which incorporate the causal
features of a classical mass inflation process, as analysed in chapter 4. The
initial formation of the BH, which defines the “in” vacuum state, is modelled
the same way as in chapter 5: an ingoing radial null surface separates a past
Minkowski region from a future BH. The BH region is taken to initially be in
the early stages of mass inflation, where the inner horizon is approaching the
origin as a decaying exponential, with a timescale $\kappa$ which is typically
related to the initial surface gravity of this horizon before mass inflation
began, as seen in chapter 4.

First, using series expansions in the radial and temporal direction, we have
looked at the initial perturbative effects the RSET has on the background
geometry. We have found that the tendency for the negative ingoing flux of the
RSET to push the inner horizon outward is still present. Due to the Planck
scale suppression, this tendency is initially very small (for BHs at
astrophysical scales). However, this does not preclude the possibility that
the exponential accumulation observed over time in the geometries of chapter 5
be present once again. Indeed, following the steps in that chapter, we found a
particularly simple family of geometries in which backreaction at finite times
can be analysed.

The result we found is that semiclassical backreaction once again has a
tendency to source a negative mass term, which now turns out to grow more
quickly than the positive mass in classical mass inflation. We are thus lead
to the conclusion that an inner horizon may well undergo an outward inflation,
even in the presence of mass-inflation-inducing classical perturbations, as an
extrapolation of our results shows in figure 7.2.

With this we conclude our analytical analysis of semiclassical inner horizon
evolution. The next steps in this study will require numerical computations of
self-consistent solutions with particular classical matter sources, which we
leave for future work.

### Conclusions: the evolution of black holes in semiclassical gravity

> Lying on his back, he looked up into the high, cloudless sky . “Do I not
> know,” thought he, “that that is infinity of space, and not a vault of blue
> stretching above me? But, however I strain my sight, I can see only a
> vaulted dome; and, in spite of my knowledge of infinite space, I have more
> satisfaction in looking at it as a blue, vaulted dome, than when I try to
> look beyond.”
>
> \- Lev Tolstoy, Ana Karenina (1878).

Modern physics lies at the interface between two spheres of reality. The first
is the material world, which we can observe and manipulate, and which we aim
to describe. The second is the mathematical and logical constructs of our
minds, which allow us to extrapolate, generalise, and ultimately to
understand. From observation and experiment, we find logical patterns in the
behaviour of matter, which we then aim to distil into mathematically concise
theories. The full descriptive capability of these theories then typically
goes beyond the observations and intuitions which resulted in their
conception, leading to predictions of entirely new and as yet unobserved
aspects of the physical world.

In their inception, black holes (BHs) were one such prediction. They resulted
from an extrapolation of a very particular solution of the Einstein field
equations with a high degree of symmetry, which was not expected to be
realisable in genuine physical scenarios, at least initially [97, 99].
However, subsequent theoretical developments showed that classical matter does
indeed form trapped regions and singularities under generic circumstances [87,
42]. This then brought the spotlight to the question of where exactly the
limits of applicability of the classical theory lie. Singularities themselves
are certainly considered a step too far, but what then is to be thought of the
long-lived trapped regions which bend causality in a singularity-directed
manner?

If we take the standard approach and assume that corrections to the classical
gravitational picture only occur when curvature reaches the Planck scale, then
long-lived BH trapped regions may well be allowed to exist in our universe.
However, quantum field theory in curved spacetimes and semiclassical gravity
show that not all quantum corrections to the geometry need to be triggered by
local curvature-related effects. Indeed, non-local effects seem to also be
present, and they turn out to be of particular importance precisely when
causality is bent to the point of forming trapped regions. Most importantly,
the backreaction effect they have on the dynamics of the geometry seems to
always head in the direction of undoing these causal knots and returning
spacetime to normal. Aside from being highly suggestive that quantum
regularising tendencies would affect not only singularities, but trapped
regions themselves, these semiclassical results already suffice to challenge
the classical BH paradigm.

The first result in this direction was that of Hawking evaporation [5]: the
peeling of light rays off of the outer horizon of a dynamically formed BH has
an effect on quantum fields (in the appropriate “in” quantisation) which
causes an outgoing flux of positive energy to appear far away from the BH, and
a compensatory ingoing flux of negative energy to appear near the BH itself
[72], encoded in the renormalised stress-energy tensor (RSET) of the fields.
The BH thus tends to lose mass and its horizon area tends to shrink.

The effect on the outer horizon is even more drastic for static BH spacetimes.
If one quantises a field on top of a static BH in a manner which respects the
time-reversal symmetry and asymptotic flatness, then one ends up with the
Boulware state [26, 23], in which the RSET has a divergence at the horizon. A
non-perturbative extrapolation of backreaction from this RSET (in the absence
of other sources of gravity) tends to eliminate the trapped region altogether,
modifying the geometry toward a wormhole configuration [28, 27].

While eternal BHs are not expected to exist in our universe (even less so
given this result), this state of quantum matter does have potential physical
significance in the case of static horizonless stellar configurations,
particularly in the limit in which their compactness tends to that of a BH.
Such configurations, even when formed dynamically, have a vacuum which relaxes
to a Boulware-like state in their exterior, making the RSET around (and
usually also below) their surface have extremely large values. And once again,
backreaction from the RSET in these stellar objects seems to pull the
configuration away from trapped region formation, allowing objects to remain
in equilibrium with a compactness arbitrarily close to that of a BH [21, 22].

The study of these objects is of paramount importance due to the fact that, if
they turn out to be stable, they could potentially mimic astrophysical BHs in
observations of both electromagnetic and gravitational nature. However, the
precise mechanism behind their hypothetical formation is as yet unknown. One
essential ingredient to understand this mechanism is the relation between the
“in” vacuum of gravitational collapse, which when nearing the compactness of a
BH is usually expected to have negligible energy contributions [72, 78, 24,
126], and the final static vacuum, which has an extremely large semiclassical
energy content. In part I of this thesis we explored this relation with a
series of ad hoc dynamical, spherically-symmetric geometric constructions
which connect the collapse and static scenarios in various ways, focusing in
particular on the magnitude of the semiclassical effects they produce, as
encoded in the RSET and effective temperature function.

In chapter 1 we begun with a simple model which captures a large variety of
dynamical behaviours: an oscillating spherical shell of matter which
periodically approaches the formation of a horizon but bounces back just
before it is formed. At each bounce, where matter slows down its motion, we
indeed found large values of the RSET. We analysed this by looking at the
bursts of radiation at infinity, obtained from the effective temperature
function. These bursts in turn also guarantee the presence of large values of
the RSET close to the matter surface. Curiously, between each bounce we also
found a period of emission of nearly thermal Hawking-like radiation. Aside
from having large values of the RSET and thus significant semiclassical
effects in its motion, an object with this type of dynamics (e.g. a perturbed
ultracompact horizonless object) would also produce a non-negligible emission
of radiation which would likely act to reduce its overall mass and size, much
like is what expected to occur for BHs.

In chapter 2 we again used a thin-shell model, and analysed the case of a
trajectory which forms a horizon in finite time. The parameter we were
interested in was the speed at which the shell crosses its Schwarzschild
radius. As a function of this parameter, we calculated the values of the RSET
at the horizon, and the corresponding energy density, flux and pressure
perceived by free-falling observers. We saw that these physical quantities can
become arbitrarily large (and also stay large for longer) the lower this speed
parameter is, approaching a divergence in the static limit.

To further explore the low-velocity regime in gravitational collapse, in
chapter 3 we analysed surface trajectories which approach the horizon radius
monotonously, but reach it only asymptotically. In this case we observed that
semiclassical effects are very sensitive to the structure of the geometry
close to the surface, and for completeness we went beyond the thin-shell
approximation and used an arbitrary spherically-symmetric geometry near
horizon formation (in the sense of light-ray trapping). With minimal
assumptions, we showed that this type of dynamics results in the emission of
thermal Hawking radiation, with values of the effective temperature function
and RSET at late times depending only on a few characteristics of the geometry
through one of its degrees of freedom, which we called the generalised
redshift function: particularly, the speed at which its minimum approaches
zero (i.e. the speed at which the formation of an apparent horizon is
approached) and its spatial derivatives on both sides of this minimum (a
generalisation of the notion of surface gravity). Depending on these
quantities, the dynamical “in” vacuum can behave as in the usual case of
black-hole formation in finite time, or it can become similar to the static
Boulware vacuum (generally at lower speeds of approach). In the latter case,
while thermal emission is still possible, it is at a temperature below that of
Hawking evaporation (related to the generalised surface gravity), and the RSET
thus acquires very large values around the horizon radius, tending to a
divergence asymptotically in time. As a side note, we also found that this
asymptotic tendency toward the formation of an apparent horizon, if maintained
indefinitely, actually leads to the formation of an event horizon, despite the
absence of trapped surfaces. The causal structure of these spacetimes and the
energy conditions they can satisfy were also analysed.

A clear-cut result from all the above situations is that semiclassical
backreaction on the geometry (through the RSET) is indeed a necessary
ingredient in analysing any geometry in which matter happens to be contracting
or expanding at very low velocities (much lower than the speed of light) when
it is so compact that it approaches crossing its own gravitational radius and
forming a trapped surface. As a purely kinematic exercise, our analysis shows
the richness of the situations around the threshold of horizon formation.
Beyond that, if a scenario in which matter actually enters such low-velocity
regimes, it is possible that backreaction may tend to further halt the
collapse, and one may expect that such dynamics would connect with the static
semiclassical stellar ultracompact equilibrium solutions [21, 22] as a final
state.

The remaining question is how these low-velocity initial conditions could be
achieved in a region where the gravitational pull is so strong that a trapped
region is nearly formed. In standard astrophysical collapse, matter is
expected to be moving at high speeds when it reaches this level of
compactness, making it likely that semiclassical effects are insufficient to
prevent the initial formation of a trapped region [72, 78, 24, 126]. Since it
is generally believed that once a trapped region forms, a region of Planckian
curvature is also inevitable, one may decide to relegate the issue of the