# Classical mass inflation vs semiclassical inner horizon inflation

Carlos Barceló<EMAIL_ADDRESS>Instituto de Astrofísica de Andalucía (IAA-
CSIC), Glorieta de la Astronomía, 18008 Granada, Spain  Valentin Boyanov
<EMAIL_ADDRESS>Departamento de Física Teórica and IPARCOS, Universidad
Complutense de Madrid, 28040 Madrid, Spain  Raúl Carballo-Rubio<EMAIL_ADDRESS>CP3-Origins, University of Southern Denmark, Campusvej 55, DK-5230 Odense M,
Denmark  Florida Space Institute, 12354 Research Parkway, Partnership 1,
Orlando, FL 32826-0650, USA  Luis J. Garay<EMAIL_ADDRESS>Departamento de
Física Teórica and IPARCOS, Universidad Complutense de Madrid, 28040 Madrid,
Spain  Instituto de Estructura de la Materia (IEM-CSIC), Serrano 121, 28006
Madrid, Spain

###### Abstract

We analyse the geometry of a spherically symmetric black hole with an inner
and outer apparent horizon which is perturbed by spherical null shells of
matter. On a classical level we observe that the mass inflation instability is
triggered, resulting in a growth of curvature and an inward displacement of
the inner horizon. We study in detail the inner structure of the mass-inflated
region and compare it with previous results obtained for the case in which the
perturbing matter content has a continuous distribution. We then perform an
approximate calculation of the renormalised stress-energy tensor of a quantum
field in the vicinity of the inner horizon, and analyse the semiclassical
backreaction on this region of the geometry. We find that the classical
tendency for this horizon to move inward due to mass inflation is challenged
and potentially overcome by a semiclassical tendency for it to inflate
outward.

##  I Introduction

A generic outcome of gravitational collapse is the formation of a trapped
region, and a generic characteristic of trapped regions is the presence of
both an outer and an inner apparent horizon. In classical General Relativity,
strict spherical symmetry, and no charge, any inner apparent horizon would
rapidly disappear into a singularity. However, the presence of the slightest
amount of angular momentum or electric charge, an effective regularized
central region, or even just a seemingly innocuous transient phenomenon, can
serve to breathe additional life into this horizon [1, 2, 3].

A long-lived inner horizon leads to the well-known mass inflation instability,
whereby small perturbations in the matter content of the geometry result in a
highly non-linear response in the increase of curvature [4]. For the charged
black hole and many other models, this increase in curvature can be related to
a growth of the Misner-Sharp mass [5], hence the name “mass inflation”. For an
initially static geometry with an inner horizon which is then perturbed, the
region where mass is “inflated” begins close to the initial position of this
horizon and extends below it. Marking the beginning of this large-curvature
region is a shockwave [6] located on a null hypersurface which remains in the
vicinity of the initial inner horizon position. Meanwhile, the inner horizon
moves inside this region along a timelike hypersurface, tending toward the
origin. If it reaches the origin before reaching the Cauchy horizon, a
spacelike singularity is formed, as observed in the numerical analysis in [7],
in addition to the null weak singularity at the Cauchy horizon itself [8, 9,
10, 11].

Classically, this entire process is hidden behind an event horizon and has no
effect on the outside universe. Only observers who enter the black hole may
worry about whether they will survive their altogether turbulent journey and
what, if anything, may await them beyond the Cauchy horizon. However, the
picture is quite different when quantum effects [12, 13] start to be taken
into account. To begin with, the energy content of the vacuum state of quantum
fields which reside on these spacetimes tends to a divergence at the Cauchy
horizon, and does so with enough strength to rule out the possibility of
extensions of spacetime beyond this horizon in a semiclassical regime [14, 15,
16, 17, 18, 19]. Additionally, for a dynamically formed black hole, if one
takes into account Hawking evaporation of the outer horizon [20, 21], then the
trapped region should disappear long before a Cauchy horizon or a null
singularity forms (assuming no stable extremal remnant is generated).

Furthermore, as shown in a previous work by the present authors [22], the
backreaction from the quantum vacuum around the inner apparent horizon (i.e.
the inner boundary of a dynamically formed trapped region) should also be
taken into account long before any consideration of the Cauchy horizon and its
divergences. Particularly, we used the renormalised vacuum expectation value
of the stress-energy tensor operator (which we will refer to as just the
“renormalised stress-energy tensor”, or RSET), constructed from a quantum
massless scalar field and calculated in the Polyakov approximation for
spherical symmetry, to analyse semiclassical backreaction perturbatively
around the inner horizon. What we found is a general tendency for the
perturbed inner horizon to move outward, reducing the size of the trapped
region. This movement is analogous to the Hawking evaporation from the other
side of the trapped region, as both are sourced by negative energy fluxes with
magnitudes related to the surface gravities of their respective horizons. The
main difference between the two is the fact that a quasi-stationary
approximation cannot be used for the inner horizon, as backreaction tends to
have a quicker accumulated effect there. In the same sense as the term
evaporation of the outer horizon conveys an idea of slowness, an adequate term
for its interior counterpart might be inflation of the inner horizon,
recalling its more explosive tendency.

It is the goal of this work to derive a similar perturbative semiclassical
backreaction scheme on top of classical backgrounds which capture more
realistically the evolution of the inner apparent horizon, i.e. backgrounds
which represent classical mass inflation. To this end, we construct a simple
geometric model with a spherically symmetric black hole perturbed by an
outgoing null shell travelling between its outer and inner horizons (akin to
Ori’s model [8]), and by a series of ingoing null shells with decreasing mass,
which represent the power-law decay of perturbations typically considered for
mass inflation [4]. Then, by deciding what mass and charges each shell carries
and how these affect the position of the inner horizon, we analyse the
possibilities for the resulting evolution of the trapped region and the global
causal structure of the spacetime.

On top of these spacetimes we construct a quantum “in” vacuum state and
calculate the RSET in the Polyakov approximation in the form of a series
expansion around the inner horizon. We once again find a tendency for the
semiclassical source to move the inner horizon outward, although in this case
it competes with the classical tendency for it to move inward due to mass
inflation. As the inner horizon moves inward, the classical term which drives
it to do so decreases, while the semiclassical term which pushes it outward
increases, eventually overcoming its Planck-scale suppression and dominating
over its classical counterpart when the surface gravity of this horizon
becomes Planckian.

This analysis of initial tendencies may seem to suggest that semiclassical
effects can only dominate when the spacetime curvature becomes Planckian
(which in itself is an important result), but a full semiclassically self-
consistent solution obtained from a family of particularly simple mass
inflation geometries reveals than this need not be the case. Terms sourced by
the RSET which drive the slow-down and reversal of the inner horizon tendency
toward the origin can in fact grow exponentially quickly, and lead to this
reversal while curvature is still far from being Planckian. This solidifies
the conclusion that horizon-related semiclassical effects play an important
role for the evolution of a generic black-hole trapped region.

This work is structured as follows. In section II we construct our shell-based
classical perturbation model which triggers mass inflation on an initially
static spherically symmetric black hole. In section III we analyse in detail
the inner regions of the resulting geometries, focusing in particular on the
evolution of the inner apparent horizon. We also show the global causal
structure which the purely classical evolution leads to, with its
corresponding weak and strong singularities. Then, in section IV, we calculate
the RSET in the Polyakov approximation for these spacetimes and tackle the
backreaction problem. In section V we conclude with an overview of our results
and we discuss the changes they may imply for the standard picture of black
hole formation and evaporation.

##  II Mass inflation with thin shells

The model we will work with is a spherically symmetric geometry with a line
element

$ds^{2}=g_{tt}dt^{2}+g_{rr}dr^{2}+r^{2}d\Omega^{2},$ (1)

where $d\Omega^{2}$ is the line element of the unit sphere. The simplicity of
our construction lies in considering that this geometry is static, i.e. that
the metric components $g_{tt}$ and $g_{rr}$ are functions of $r$ only, in
patches separated by spherical null shells.

To begin with, consider that there are two null shells, one outgoing and one
ingoing, which intersect at a point $p_{1}=(t_{1},r_{1})$. Continuity of the
metric imposes the relation (see [23, 24, 25])

$\left|\frac{f_{2}}{f_{1}}\right|=\left|\frac{F_{2}}{F_{1}}\right|\quad\text{at}\quad
p_{1},$ (2)

where the functions $F$ and $f$ are the $g^{rr}$ component of the metric in
different patches, with upper (lower) case letters referring to the interior
(exterior) of the outgoing shell and the subscript “$1$” (“$2$”) referring to
the interior (exterior) of the ingoing shell, as shown schematically in fig.
1.

Figure 1: Static black hole with outer and inner horizons perturbed by an
outgoing shell and a series of ingoing shells with decreasing mass,
represented in advanced Eddington-Finkelstein coordinates. The trapped region
is shaded in grey. The ingoing shells are located at $v_{n}$, with
$n=1,2,\dots$; $m_{n}$ refers to the Bondi mass in the asymptotic region
before the corresponding ingoing shells (outside the outgoing one); $f_{n}$ is
the $g^{rr}$ metric component in each of these regions down to the outgoing
shell, while $F_{n}$ is the same metric component on the inside of the
outgoing shell. $r_{{\rm i},n}$ are the zeros of the $f_{n}$ functions
extended to the region below the outgoing shell (they are the radii which this
shell approaches exponentially in $v$ in each patch).

Now consider that this geometry has a trapped region (i.e. a region where
$g_{tt}$ becomes positive and $g_{rr}$ negative, which directly relates to the
expansion of outgoing null geodesics congruences becoming negative [26]). For
simplicity, consider also that $-g_{tt}=g^{rr}$ in the static coordinates of
each patch in the vicinity of the shells (as would be the case for e.g. the
charged black hole, and many regular black hole models [2, 3]). We will refer
to this metric component as the redshift function, this being the previously
defined $f$ (and $F$).111Throughout the text, it should be recalled that the
properties we require from this function are related to both $g_{tt}$ (which
often bears the name redshift function by itself) and $g^{rr}$; the former has
more to do with the trajectories of null geodesics, while the latter has to do
with the junction conditions which drive the dynamics of the spacetime. We
take the outgoing shell to be travelling inside the trapped region, between
the outer and inner horizons. Once we specify some aspects of the initial
conditions in the patches of $F_{1}$, $f_{1}$, and $f_{2}$, which amounts to
choosing the mass and charge carried by each shell, eq. (2) will tell us how
the redshift function in the future of the innermost region of the black hole,
$F_{2}$, behaves. We will then repeat the process by adding more ingoing
shells, as shown in fig. 1, representing an ingoing perturbation of decreasing
amplitude, and see when and how mass inflation is triggered.

In advanced Eddington-Finkelstein coordinates we can write the line element in
the $f_{1}$ patch as

$ds_{1}^{2}=-f_{1}(r)dv^{2}+2dvdr+r^{2}d\Omega^{2}.$ (3)

We note that the same coordinate $v$ can be used for all static
representations of the patches external to the outgoing shell (while for those
on the inside of this shell we will use a coordinate $V$). This outgoing shell
is travelling along a null geodesic for both the geometries in the $f_{1}$ and
$F_{1}$ patches. For now, we will describe its dynamics and its interactions
with the ingoing shell(s) in the coordinates of the external $f_{1}$ patch,
while in the next section we will go into more detail regarding its
description from the point of view of the interior patches. Being inside the
trapped region of (3), its movement is described by the equation

$\frac{dr_{\rm shell}}{dv}=\frac{1}{2}f_{1}.$ (4)

Since $f_{1}$ is negative inside the trapped region, the solution for its
radial position $r_{\rm shell}$ decreases in $v$, eventually tending toward
the inner gravitational radius of the $f_{1}$ patch, $r_{\rm i,1}$, where the
redshift function can be approximated by

$f_{1}(r)=-2\kappa_{\rm i}(r-r_{\rm i,1})+\mathcal{O}[(r-r_{\rm i,1})^{2}],$
(5)

with $\kappa_{1}$ being the absolute value of the surface gravity of this
inner radius (i.e. $-\frac{1}{2}\partial_{r}f_{1}|_{r=r_{\rm i,1}}$). To
clarify, $r_{\rm i,1}$ is the position the inner horizon would have if the
$f_{1}$ patch were continued to below the outgoing shell (i.e. the zero of the
$f_{1}(r)$ function when extended to below $r_{\rm shell}$). We note that
since below $r_{\rm shell}$ the geometry is described by the $F_{1}$ function,
the position of the actual inner apparent horizon of the geometry is in fact
the zero of $F_{1}$. This position will be denoted by $R_{\rm i,1}$ later on.

The solution which describes the tendency of the shell toward $r_{\rm i,1}$ at
sufficiently large values of $v$ can be obtained from the linear order in the
above expansion,

$r_{\rm shell}(v)=r_{\rm i,1}+(r_{0}-r_{\rm
i,1})e^{-\kappa_{1}v}+\order{e^{-2\kappa_{1}v}},$ (6)

where $r_{0}$ is a positive constant representing the position of the shell at
$v=0$. Substituting (6) into (5), the redshift function on the shell is then
approximated by

$f_{1}|_{\rm shell}=-2\kappa_{1}(r_{0}-r_{\rm
i,1})e^{-\kappa_{1}v}+\order{e^{-2\kappa_{1}v}}.$ (7)

Now consider that this outgoing shell is intersected by the ingoing one. If
the latter is assumed to have positive mass (more generally, satisfies the
null energy condition), the inner horizon can only be displaced inward, i.e.
$r_{\rm i,1}\geq r_{\rm i,2}$, following a timelike trajectory, in the same
way as the outer horizon can only be displaced in an outward (spacelike)
direction.222This is simply an extension of Hayward’s theorem for continuous
matter [26] to the case of shells. Intuitively, it can be seen from the fact
that the inner horizon acts as a surface of accumulation of geodesics, which
restricts its movement to causal directions. Let us also assume that this mass
which the shell carries to the black hole is small (compared to the total
black hole mass), as one may expect from perturbation tail analyses in an
astrophysical scenario. The redshift function after this shell can then be
approximated by

$f_{2}|_{\rm shell}=-2\kappa_{2}(r_{\rm shell}-r_{\rm
i,2})+\mathcal{O}[(r_{\rm shell}-r_{\rm i,2})^{2}].$ (8)

Particularly, the smallness of the mass carried by the shell is meant to
ensure that the order $(r_{\rm shell}-r_{\rm i,2})^{2}$ remains negligible (in
units of the characteristic length scale of the black hole, i.e. the initial
inner horizon radius), in the same way as the higher order terms in (5) are,
which amounts to a requirement that the that the displacement $(r_{\rm
i,1}-r_{\rm i,2})$ be small (which we will impose explicitly later on). At the
intersection point $p_{1}$, we can insert (6) for $r_{\rm shell}$ into eq. (8)
which, together with eq. (7), gives us the expression for one of the two
quotients from eq. (2):

$\left.\frac{f_{2}}{f_{1}}\right|_{r=r_{1}}=\frac{\kappa_{2}}{\kappa_{1}}+\frac{\kappa_{2}}{\kappa_{1}}\frac{r_{\rm
i,1}-r_{\rm i,2}}{r_{0}-r_{\rm
i,1}}e^{\kappa_{1}v_{1}}\left[1+\order{e^{-\kappa_{1}v_{1}}}\right],$ (9)

where $v=v_{1}$ corresponds to the position of the ingoing shell, and is thus
also the intersection time. The exponential in this expression, along with eq.
(2), already indicates that the value of redshift function $F_{2}$ can become
much greater than $F_{1}$, in a manner suggestive of mass inflation.
Particularly, one can easily see how this growth of $F_{2}$ relates to an
increase of mass in e.g. the Reissner-Nordström geometry (which we will use as
an illustrative example throughout this work), where
$F_{n}=1-2M_{n}/r+Q_{n}^{2}/r^{2}$. If the redshift function increases in
absolute value, and since it is negative ($r_{1}$ begin inside the trapped
region), it translates into an increase in the only negative term it contains:
the mass of the black hole. Assuming this term is already large from previous
shell crossings and dominates the behaviour of the redshift function $F_{1}$
close to the shell, we have the relation

$F_{1}|_{r=r_{1}}=-\frac{2M_{1}}{r_{1}}+\order{M_{1}^{0}},$ (10)

where $M_{1}$ is the mass on the inside of the outgoing shell before $v_{1}$.
With the same assumption for the region after the ingoing shell, i.e. that the
mass term dominates in the redshift function, we get the inflated mass of the
charged black hole

$M_{2}=M_{1}\left.\frac{f_{2}}{f_{1}}\right|_{r=r_{1}}+\order{M_{1}^{0}}.$
(11)

However, as the reader may have already noticed, the exponential in (9) has a
prefactor which must also be carefully analysed. For example, if the ingoing
shell is considered to have a particular charge and mass which make the
displacement of the inner gravitational radius $(r_{\rm i,1}-r_{\rm i,2})$
sufficiently small, the exponential growth could be cancelled. We will discuss
this in more detail in the following.

Let us make the perturbation by ingoing shells an iterative process: we
represent an ingoing, polynomially decreasing flux of radiation (which is
usually the source of mass inflation [4, 8] stemming from the decay of
perturbations on the geometry [27]) with a sequence of ingoing shells of
progressively smaller mass,

$\delta m_{n}=\frac{a}{v_{n}^{q}},$ (12)

where $\delta m_{n}$ refers to the change of the exterior mass (i.e. the Bondi
mass [28] related to past null infinity) produced by a particular ingoing
shell located at $v=v_{n}$, the power $q$ is positive,333$q$ has a lower bound
which depends on the spacing of the shells, needed to guarantee that the total
mass thrown into the black hole is finite. For example, for a linear
distribution of shells in $v$, $q>1$. If the shells become more spread-out,
then $q$ can be smaller, while if they become more concentrated it must be
larger. and $a$ is a positive constant with appropriate dimensions. We also
impose that there be infinitely many shells and that
$\lim_{n\to\infty}v_{n}=\infty$.

One of the key ingredients necessary for mass inflation is that this increase
in the mass (as seen from outside the object) is itself related polynomially
to the change in position of the inner gravitational radius $\delta r_{{\rm
i},n}$, i.e.

$\delta r_{{\rm i},n}=-\frac{\beta}{v_{n}^{p}},$ (13)

where the power $p$ is again positive (though it can be different from $q$),
and $\beta$ is again a positive constant (at least asymptotically in $v$). We
stress that eq. (13) is an assumption about how the geometry responds to the
ingoing perturbation (12). For example, in the Reissner-Nordström case, where
one has

$\delta r_{{\rm i},n}=\frac{1}{\sqrt{m^{2}-Q^{2}}}(-r_{{\rm i},n}\delta
m_{n}+Q\delta Q_{n}),$ (14)

the assumption (13) restricts the amount of (same sign) charge the ingoing
shells can carry. One can easily imagine a case in which the relation between
$\delta m_{n}$ and $\delta Q_{n}$ is such that e.g. $\delta r_{{\rm i},n}=0$,
which would lead to an absence of mass inflation. This is not limited to the
Reissner-Nordström case: (13) generally avoids the suppression of the
exponential in (9) by its prefactor, allowing mass inflation to take place.

This can be seen by looking at the evolution of the redshift function in the
interior of the outgoing shell after the $n$th ingoing shell has crossed it,
$F_{n}$. From eqs. (2), (9), and (13) we get the asymptotic relation at the
$n$th intersection point

$\begin{split}F_{n}&=F_{n-1}\frac{f_{n}}{f_{n-1}}\\\
&=F_{n-1}\frac{\kappa_{n}}{\kappa_{n-1}}\left(\frac{v_{n-1}}{v_{n}}\right)^{p}e^{\kappa_{n-1}\Delta
v_{n}}\left[1+\order{e^{-\kappa_{n-1}\Delta v_{n}}}\right]\\\
&=F_{n-1}e^{\kappa\Delta v_{n}}\left[1+\order{e^{-\kappa\Delta
v_{n}},\frac{\Delta v_{n}}{v_{n-1}}}\right],\end{split}$ (15)

where $\Delta v_{n}=v_{n}-v_{n-1}$ and in the last line we have directly
substituted the asymptotic value of the surface gravity of the $f_{n}$
geometries, $\kappa_{n}\to\kappa$. We have also used the fact that the
differences $v_{n}-v_{n-1}$ become negligible when compared to the absolute
value of $v_{n}$ asymptotically (for any distribution of infinite shells which
covers an infinite range in $v$), and that $e^{\kappa\Delta v_{n}}>1$, leaving
only the dominant contribution as leading order. If we use this relation
iteratively from an initial time $v=0$ when the interior redshift function was
$F_{0}<0$, we get the asymptotically exponential increase (of the absolute
value of) the redshift function at the shell

$F_{n}|_{\rm shell}\sim F_{0}e^{\kappa v_{n}}.$ (16)

The result is independent of the spacing between the ingoing shells: increased
spacing only leads to a higher jump in $F_{n}$ when a shell eventually falls.
This tendency continues for as long as more shells are thrown in. Returning to
the Reissner-Nordström case, we can once again easily associate this with a
proportional increase of the mass term through eq. (11). In more general
geometries, it can be related to an increase of the Misner-Sharp mass given by
[5, 26]

$M_{\rm MS}=\frac{1}{2}r(1-F),$ (17)

for this interior region. Let us recall that the Misner-Sharp mass provides a
quasi-local characterization of gravitational energy in spherical symmetry
[26].

Eq. (16) captures the main result of mass inflation: the exponential growth of
the redshift function (more generally, the $g^{rr}$ metric component) below a
certain radius, associated with a corresponding growth of the Misner-Sharp
mass. In our model it can be physically interpreted in terms of the exchange
of mass between the outgoing and ingoing shells. The outgoing shell, being
inside the trapped region, can be seen as having a negative asymptotic mass.
At the intersection points, the ingoing shells can therefore take away
positive mass by making the negative one of the outgoing shell increasingly
more so. This exchange is mediated by the dynamics of the gravitational field,
and in particular its exponential nature is only triggered if the outgoing
shell is taken from an initial proximity to the inner gravitational radius of
the $f_{n}$ geometry patches, and subsequently (after the interaction) ends up
deeper inside the trapped region due to the inward displacement of this inner
gravitational radius (13).

It is interesting to note that this process is independent of the
particularities of the infalling matter shells, only requiring that the
perturbations induce the polynomially decreasing response of the inner
gravitational radius (13). If the shift in its position were instead to
decrease, e.g., exponentially,

$\delta r_{{\rm i},n}=-\tilde{\beta}e^{-\sigma v},$ (18)

where $\tilde{\beta}$ and $\sigma$ are positive constants, then one can
observe from (9) and the corresponding equivalent of (15) that the outcome
would depend on the difference between the surface gravity $\kappa$ and the
coefficient $\sigma$. Particularly, mass inflation would only take place if
$\kappa>\sigma$, as has been shown in the case where $\sigma$ is the surface
gravity of a cosmological horizon in asymptotically de Sitter spacetimes,
governing the decaying tail of infalling radiation [29, 30].

It is worth noting that this shell-based model has some characteristics in
common with the case in which a continuous power-law flux of radiation is
present, but there are also some differences. For example, the exponential
which appears in the mass inflation of this model is directly related to the
exponentially decreasing separation between the outgoing shell and the inner
gravitational radius $r_{\rm i}$ (as seen from the outside) in each step
represented in fig. 1. Although the average of this distance taken over
several steps has the same inverse-polynomial decrease as is expected from a
continuous matter case (that is, if $r_{\rm i}$ had a position evolving as a
continuous version of (13)), taking the limit to a continuous ingoing
distribution of matter is not at all straightforward. Although the outgoing
shell seems to be a good model for the shockwave which generally appears in
this region even with continuous matter [6], the ingoing shells offer
qualitatively new features. This may also be inferred by the fact that a
generalisation of Ori’s model [31], in which the ingoing flux is continuous,
leads to a larger variety of asymptotic outcomes, unlike the single
exponential behaviour observed here. The question of whether the continuous or
the discrete model (or some combination of the two) fits best the behaviour of
small (possibly quantised) infalling perturbations may thus turn out to be an
important one for a better understanding of the singularity at the Cauchy
horizon, though we will not address it here.

##  III Geometry inside the mass-inflated region

This shell-based construction captures (albeit a simple variant of) the mass
inflation process and it can give us some additional insights into the
behaviour of the geometry inside the black hole at finite times. In
particular, to set up our subsequent semiclassical analysis, we want to see
how this model answers two questions about the depths of the mass-inflated
region. The first one is whether a stream of outgoing radiation below the
initial outgoing shockwave may also give rise to an effect similar to mass
inflation, which would further increase the rate of growth of mass in the
innermost region of the geometry, in the vicinity of the origin. The second
one is what particular path the inner apparent horizon actually follows in
this whole process, and whether it collapses to the origin to give rise to a
spacelike singularity at a finite time $v$, as suggested in the numerical
analysis of [7].

To answer these questions, we must begin by getting a better understanding of
the global structure of the geometry corresponding to the above construction
with just a single gravitating outgoing shell. The metrics on the outside and
inside of the outgoing shell respectively can be written as

$\displaystyle ds^{2}$ $\displaystyle=-f(v,r)dv^{2}+2dvdr+r^{2}d\Omega^{2},$
(19)
$\displaystyle\begin{split}dS^{2}&=-F(v,r)dV^{2}+2dVdr+r^{2}d\Omega^{2}\\\
&=A(v)\left[-A(v)F(v,r)dv^{2}+2dvdr\right]+r^{2}d\Omega^{2},\end{split}$ (20)

where lower and upper case letters are once again used for quantities in the
regions outside and inside the shell, respectively, and $A(v)=dV/dv=f/F|_{\rm
shell}$ is a positive function which allows us to switch between the
Eddington-Finkelstein coordinates of these two regions, as expressed on the
right-hand side of the latter equation.

For simplicity, we will take a smoothed-out average of the metric functions
from the previous section (particularly, their dominant behaviour at late
times). The redshift function on the shell evaluated on the inside (16) is
then

$F|_{\rm shell}=-|F_{0}|e^{\kappa v},$ (21)

and, from eq. (13) and the shell trajectory in each patch, the redshift
function on the outside satisfies

$f|_{\rm shell}=-\kappa\left[r_{\rm shell}-r_{\rm
i}(v)\right]=-\frac{b}{v^{p}},$ (22)

with $b$ a positive constant which depends on the average spacing between
ingoing shells (i.e. the average of the quantity $e^{\kappa\Delta v_{n}}$),
into which $\kappa$ (the outside region’s inner gravitational radius’s surface
gravity) and the constant $\beta$ from eq. (13) have also been absorbed. The
function which relates the outside and inside times, $v$ and $V$, then becomes

$A(v)=\frac{b}{|F_{0}|}\frac{e^{-\kappa v}}{v^{p}}.$ (23)

The results we have obtained thus far are valid for any redshift function, but
if we want to analyse the deeper regions of the geometry we have to be more
specific. Let us therefore first focus on the particular example we have used
previously, namely the Reissner-Nordström geometry. If we assume that the
ingoing and outgoing shells carry no electrical charge, we can take the
variation in $F$ to be solely due to a variation of the mass term. Then, from
eqs. (21) and (23), and with the Reissner-Nordström redshift function, we get
the geometry for the mass-inflated interior region (20) written in the $v$
coordinate

$\begin{split}dS^{2}=\frac{b}{|F_{0}|v^{p}}&e^{-\kappa
v}\left[-\frac{b}{v^{p}}e^{-\kappa v}\left(1-\frac{2M_{0}e^{\kappa
v}}{r}+\right.\right.\\\
&\left.\left.+\frac{Q^{2}}{r^{2}}\right)dv^{2}+2dvdr\right]+r^{2}d\Omega^{2},\end{split}$
(24)

where $M_{0}$ is a positive constant which represents the initial mass of the
black hole. Using this metric we see that the equation for outgoing null
geodesics (which we can later relate to trajectories of additional gravitating
outgoing shells) then takes the form

$\frac{dr}{dv}=\frac{b}{v^{p}}\left[-h_{0}(r)+h_{1}(r)e^{-\kappa v}\right],$
(25)

where $h_{0}(r)=M_{0}/r$ and $h_{1}(r)=(1+Q^{2}/r^{2})/2$ are positive
functions. Due to the exponential suppression of the $h_{1}$ term, it is clear
that the term with $h_{0}$ is dominant on the right-hand side of this
equation, except in a progressively smaller region around the origin, where
the inner apparent horizon is shrinking toward zero radius. In relation to
this, we will be able to distinguish between two types of outgoing geodesics
in the trapped region: ones whose dynamics is predominantly determined by the
$h_{0}$ term, and ones for which the two terms are comparable. The former
exist up to $v\to\infty$ only if $p>1$, which, given eq. (14), is directly
related to the mass thrown into the black hole from the outside being finite
(for our current example of Reissner-Nordström).444It is interesting to note
that it is quite easy to imagine a geometry in which e.g. due to a relation
$\delta r_{\rm i}\propto-(\delta m)^{1/2}$, $p$ ends up being 1 or smaller for
a finite accretion of mass. The absence of solutions of the type (26) would
then imply a lack of a Cauchy horizon below the outgoing shell, leaving just a
trapped region with a tendency toward the formation of a spacelike or null
singularity. If this condition is met and the $h_{0}$ term continues to
dominate, equation (25) generally integrates asymptotically to

$r=r_{\rm c}+\frac{h_{0}(r_{\rm
c})}{p-1}\frac{b}{v^{p-1}}+\mathcal{O}\left[\left(\frac{b}{v^{p-1}}\right)^{2}\right].$
(26)

The integration constant $r_{\rm c}$ represents the final radial position of
each light ray when it reaches the Cauchy horizon at $v\to\infty$, which is
different for each geodesic depending on initial conditions. In other words,
between the outgoing shell and the inner apparent horizon (which is rapidly
shrinking toward $r=0$) there are null geodesics which are trapped in a
tendency toward a finite radial position which is different from the
asymptotic position of this horizon. To see how this journey is perceived from
the point of view of an observer in the interior region itself, we can look at
the relation between the $v$ and $V$ coordinates, the latter of which, for
observers not tending toward the inner apparent horizon, is proportional to
the geodesic affine parameter. As we have seen, this relation is given by

$\frac{dv}{dV}=\left.\frac{F}{f}\right|_{\rm shell}=\frac{1}{A(v)},$ (27)

which asymptotically integrates to

$V=V_{\rm c}-\frac{b}{|F_{0}|\kappa}\frac{e^{-\kappa
v}}{v^{p}}+\order{\frac{e^{-\kappa v}}{v^{p+1}}}.$ (28)

These geodesics therefore reach the Cauchy horizon at a finite time parameter
$V=V_{\rm c}$, corresponding to the integration constant of the equation. This
gives us an interpretation for the behaviour seen in (26): outgoing null
geodesics are trapped in a tendency toward these different finite radial
positions $r_{\rm c}$ because the function $A$ acts to quickly freeze the
proper time, and consequently the movement, of observers in this region.
Because of this freezing function $A$, most of the outgoing radiation in the
trapped region would in fact reach the Cauchy horizon before getting close to
the inner apparent horizon. This tells us that interactions between the
ingoing shells with additional outgoing shells travelling along these
geodesics would not have time to produce any sort of amplification of the mass
inflation effect, as this would require proximity to the inner gravitational
radius $R_{\rm i}$ of this internal region (which in the absence of such
shells is in fact the inner apparent horizon position).

The region where eqs. (26) and (28) are valid begins on the inside of the
outgoing shell and increases in size toward the origin as the mass tends to
infinity due to its unbounded growth following eqs. (17) and (21). The radii
$r_{\rm c}$ at which null geodesics freeze can then vary continuously from
$r_{\rm i}$ (the inner gravitational radius of the external geometry) to zero,
as is observed in the analytical study of the Cauchy horizon in [10]. However,
this does not imply that all outgoing null geodesics are trapped in this way
and are unaffected by the shrinking inner apparent horizon. Due to the growing
mass, the radial position of this apparent horizon $R_{\rm i}$ can be
approximated by a series expansion of the lower of the two roots of the
Reissner-Nordström redshift function in $1/M(v)$, revealing its tendency to
zero

$R_{\rm
i}(v)=\frac{Q^{2}}{2M(v)}+\order{\frac{1}{M(v)}}^{3}=\frac{Q^{2}}{2M_{0}}e^{-\kappa
v}+\order{e^{-3\kappa v}},$ (29)

with a surface gravity (in absolute value)

$K_{\rm i}(v)=\frac{b}{v^{p}}\frac{2M_{0}^{3}}{Q^{4}}e^{2\kappa
v}+\order{\frac{1}{v^{p}}}.$ (30)

There are indeed many outgoing null geodesics which are sufficiently close to
this horizon to approach it asymptotically, i.e. to tend to zero from above
it. These are the second-type geodesics we mentioned before: the ones for
which the $h_{0}$ and $h_{1}$ terms of eq. (25) are comparable, with the
right-hand side of this equation being close to zero, i.e. $r$ being close to
$R_{\rm i}$. Their movement can be described by expanding the right-hand side
of this equation around $R_{\rm i}$,

$\frac{dr}{dv}\simeq-K_{\rm i}(v)[r-R_{\rm i}(v)].$ (31)

With the rapidly growing value of $K_{\rm i}$, it can be readily checked that
this equation has a family of solutions for which $r\to R_{\rm i}$. These
geodesics also reach $v=\infty$ with a finite affine parameter, where they
converge to $r=0$ along with $R_{\rm i}$, falling into a (strong) curvature
singularity.

If an additional mass inflation effect can take place, pushing the position of
the inner apparent horizon to below the one given by (29), it would be
triggered by placing an outgoing shell precisely on one of the geodesics
described by (31) which tend toward $R_{\rm i}$ from above. To keep the label
$R_{\rm i}$ for the zero of the $F$ function (29), i.e. the inner
gravitational radius of the $F$ patch, we will now call the actual inner
apparent horizon $\tilde{R}_{\rm i}$. Also, instead of working with the $v$
coordinate, here it will be more convenient to use $V$, with which we can
directly apply the relation (9) for the junction conditions on the
intersection points with the ingoing shells. The position of the inner
gravitational radius $R_{\rm i}$ and its surface gravity (taken as the
absolute value of $\partial_{r}g_{vv}/(2g_{vr})$ in each coordinate
system)555In other words, $K_{\rm i}^{V}$ is not just $K_{\rm i}$ with a
coordinate change, due to the $A(v)$ factor which multiplies the redshift
function after the coordinate change, as can be seen in (24). in the $V$
coordinate system $K_{\rm i}^{V}$ evolve as

$\displaystyle R_{\rm i}$ $\displaystyle=\xi_{1}(V_{\rm
c}-V)+\mathcal{O}[(V_{\rm c}-V)^{3}],$ (32) $\displaystyle K_{\rm i}^{V}$
$\displaystyle=\frac{\xi_{2}}{(V_{\rm c}-V)^{3}}+\order{\frac{1}{V_{\rm
c}-V}}$ (33)

where $\xi_{1}$ and $\xi_{2}$ are some positive constants which depend of the
asymptotic mass, the charge and the initial conditions. In the shell model we
once again consider that these functions actually have discrete jumps at a set
of points in $V$, corresponding to the positions of the infalling shells. If
we take these shells to be either equispaced in $v$, or at most distributed
with a density polynomially dependent in $v$, then in $V$ their spacing
decreases as

$\Delta V_{n}=V_{n}-V_{n-1}\sim V_{\rm c}-V_{n}$ (34)

as $V_{n}$ tends toward the Cauchy horizon $V_{\rm c}$. The jumps in the
position of the inner horizon between shells also decrease as

$\Delta R_{\rm i,n}\sim V_{\rm c}-V_{n}.$ (35)

Eq. (9) can be applied directly here because the outgoing shell has enough
time between each iteration to get exponentially closer to $R_{\rm i,n}$, as
can be seen by the fact that

$e^{-K^{V}_{\rm i,n}\Delta V_{n}}\sim e^{-1/(V_{\rm c}-V_{n})^{2}}\to
0,\quad\text{as}\quad V_{n}\to V_{\rm c}.$

The quotient of surface gravities in (9) once again tends to a constant in the
limit of interest, and so does the quotient of the differences between radial
positions in front of the exponential.666This can be seen explicitly by
solving the outgoing null geodesic equation between each infalling shell and
matching the solutions. The calculation is completely analogous to the one
performed in the vicinity of $r_{\rm i}$. This leaves the redshift function
below this new outgoing shell with an increase given by a multiplicative
factor

$e^{K^{V}_{\rm i,n}\Delta V_{n}}$

after each iteration, which diverges as $e^{\xi_{2}/(V_{\rm c}-V)^{2}}$ toward
the Cauchy horizon. In terms of $v$, this new mass inflation increases the
Misner-Sharp mass in the vicinity of the origin as an exponential of an
exponential. One can then imagine that each time we repeat this whole process
considering the new displacement of the inner horizon caused by this effect,
we would get an additional exponential in $v$ to the chain.

We can now tackle answering the second question posed at the beginning of this
section: what is the actual path followed by the inner horizon, and what
causal structure does this movement give rise to. It is hardly surprising that
in the case of continuous matter the chain effect just described could produce
a spacelike singularity at finite $v$, as is observed numerically in [7] and
commented on in later works [10, 6]. Even if the exact analytical solution
were to only result in a very quick tendency toward the formation of this
singularity (i.e. a quick, but still asymptotic in $v$, approach of the inner
apparent horizon toward the origin), this approach would in fact be so quick
that it would likely be numerically indistinguishable from a singularity at a
finite time $v$; at any rate, the curvature would become Planckian very fast,
making a classical description of the geometry-matter interactions inadequate.
However, it is interesting to note that within the classical description, this
seemingly small difference between a singularity forming at strictly finite
$v$ and having just a tendency toward its formation, however quick, and only
forming it at $v\to\infty$, results in two different asymptotic structures,
represented in the two diagrams of fig. 2. In fact, the former case results in
the formation of a Schwarzschild-type spacelike singularity, while the latter
ends in a (strong) null singularity, both of which are at $r=0$ and are
attached to the weak null singularity which spans the Cauchy horizon, where
$r$ takes values up to $r_{\rm i}$. To clarify, we use the same criterion for
distinguishing strong and weak singularities as the one described in [8, 32]:
both have diverging curvature scalars, but the distortion suffered by an
observer of finite size remains bounded when crossing a weak singularity,
while it diverges when crossing a strong one. One can readily check that the
curvature blow-up of the geometry we use close to the Cauchy horizon [see eq.
(24)] is the same as in refs. [8, 7],
$R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\sim 48M_{0}^{2}r^{-6}e^{2\kappa v}$,
leading to the same type of weak singularity there.

Figure 2: Future part of the causal diagram of the mass inflation geometry.
The shaded part is the trapped region and the dashed line is the Cauchy
horizon (and a weak singularity). The null shell shown is the upper outgoing
one which tends to $r_{\rm i}$ and is responsible for the first mass inflation
effect. Left: the inner apparent horizon reaches $r=0$ at finite $v$ and forms
a Schwarzschild-type spacelike singularity. Right: the inner horizon only
tends to $r=0$ asymptotically in $v$, resulting in a strong null singularity
at $v\to\infty$ and $r=0$, at a finite affine distance for geodesics which
fall into it.

Up to here, all we have said regarding these questions has been based on the
Reissner-Nordström background. For other geometries besides Reissner-
Nordström, the inner structure of the mass-inflated region depends on how the
geometry reacts to the increase in mass provided by the infalling null shells,
particularly on the trajectory followed by the inner gravitational radius. The
relation from the charged black hole $\delta r_{\rm i}\propto(-\delta m)$ is
also satisfied in the case of a rotating black hole, but in regular black hole
spacetimes this tendency may be modified, depending on how the regularisation
of the origin is achieved in the first place. One may expect there to exist
trapped geodesics of the type (26), but the structure toward the origin may
differ, possibly avoiding the formation of a strong singularity altogether by
preventing the inner horizon from getting too close to $r=0$.

It is also interesting to note that in general, the explicit divergence of the
mass at $v\to\infty$ depends on the ever-smaller ingoing perturbations also
continuing up to infinity. Classically it is perfectly natural to consider
this to be the case, but one may imagine that a quantum description of the
interaction between the black hole and infalling matter may have a lower bound
on the energy which can actually affect the black hole. For example, we can
consider that this lower bound is given by the energy of a single photon with
a wavelength of the order of the black hole external mass $m$ (in geometric
units). Then, we can relate this energy to a mass and to a cutoff time $v_{\rm
cut}$ through (12), and for simplicity we can take the polynomial tail in (13)
to be the same as in (12), i.e. $p=q$. The result is that by $v_{\rm cut}$ the
mass in the interior region would have increased by a factor $e^{(M/l_{\rm
P})^{2/p}}$ from just the first exponential mass inflation effect around
$r_{\rm i}$, given by eqs. (16) and (17). Needless to say, the exponent in
this number is generally very large. For a solar mass black hole and a
polynomial decay with $p=12$ (as considered in [4]) the mass grows by a factor
larger than $e^{10^{6}}$, making the dynamics of the interior region enter a
full quantum gravity regime.

In conclusion, we can say that classically mass inflation would occur under
quite generic circumstances for charged black holes, and by extension for
rotating ones as well, where an analogous shell-based construction can be made
[25]. While in principle it is possible for the perturbations falling in to
carry enough charge or angular momentum to keep the inner horizon still enough
for the instability to not be triggered, this is unlikely to occur in an
astrophysical scenario over long periods of time. The only viable way to avoid
the instability seems to be to rely on a regularising mechanism which prevents
the inner horizon from getting too close to the origin. We will now explore
whether the quantum nature of matter can provide such a regularising tendency
in the semiclassical gravity regime.

##  IV Semiclassical backreaction

The main goal of this work is to see whether and how semiclassical physics can
have an influence on the evolution of the inner horizon inside a black hole
undergoing mass inflation. A recent work by the present authors [22] analyses
the backreaction around static and dynamical inner horizons in simple
geometries which do not incorporate mass inflation. The result was that the
inner horizon tends to be pushed outward due to semiclassical effects. The
initial tendency for its movement is exponential in time, and an extrapolation
of the process can thus be dubbed an _inflation of the inner horizon_ , which
leads to a quick extinguishing of the trapped region from the inside out. In
this section we will extend our semiclassical backreaction analysis to
geometries incorporating the mass inflation effect, the main new component
being the presence of the freezing function $A(v)$.

As in [22], we will use the RSET of a massless scalar field in the Polyakov
approximation as the source of backreaction. Although this approximation is
far from being the exact RSET of a 3+1 dimensional geometry, it is sufficient
to describe Hawking evaporation at the outer horizon, and is therefore a good
candidate to give us a first glimpse at horizon-related semiclassical effects
around the inner apparent horizon. We construct the “in” vacuum [33, 34] of
the 1+1 dimensional radial-temporal sector of the spacetime, used to calculate
the RSET in this approximation, by following the movement of light-like
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 black hole. As discussed in [22], 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 work 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 black hole with
an inner horizon, as shown in fig. 3. Effectively this is equivalent to the
black hole 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 3: Formation of a black hole 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 black hole is undergoing mass inflation and the
inner horizon is headed toward the origin.

In this section, 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 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 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.

###  IV.1 Series expansion around horizon formation

Let us begin by considering the line element

$ds^{2}=A(v)\left[-B(v,r)dv^{2}+2dvdr\right]+r^{2}d\Omega^{2}.$ (36)

We will use this type of geometry to represent the inner region of a black
hole undergoing mass inflation, particularly around its inner apparent
horizon. In relation to the previous section, 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 (23) and set
$A(v)=e^{-\kappa v}$, with $\kappa$ a positive constant.777Neglecting the
$1/v^{p}$ part of (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 (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. 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. 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 (36) can
generally be written as

$ds^{2}=-C(u,v)dudv+r^{2}d\Omega^{2}.$ (37)

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
[20]. The $v$ coordinate used in (36) 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

$C(u,v)=-2A(v)\frac{\partial r(u,v)}{\partial u}.$ (38)

The components of the RSET in the Polyakov approximation for the vacuum state
selected by the coordinate system $\\{u,v\\}$ are [33, 35]

$\displaystyle\expectationvalue{T_{uu}}$ $\displaystyle=\frac{l_{\rm
P}^{2}}{96\pi^{2}r^{2}}\left[\frac{\partial_{u}^{2}C}{C}-\frac{3}{2}\left(\frac{\partial_{u}C}{C}\right)^{2}\right],$
(39a) $\displaystyle\expectationvalue{T_{vv}}$ $\displaystyle=\frac{l_{\rm
P}^{2}}{96\pi^{2}r^{2}}\left[\frac{\partial_{v}^{2}C}{C}-\frac{3}{2}\left(\frac{\partial_{v}C}{C}\right)^{2}\right],$
(39b) $\displaystyle\expectationvalue{T_{uv}}$ $\displaystyle=\frac{l_{\rm
P}^{2}}{96\pi^{2}r^{2}}\left[\frac{\partial_{u}C\partial_{v}C}{C^{2}}-\frac{\partial_{u}\partial_{v}C}{C}\right],$
(39c)

where $l_{\rm P}$ is the Planck length. This tensor has a non-physical
divergence at the origin due to the process of dimensional reduction, making
it inadequate to use as such for the backreaction problem in the whole
spacetime. However, as discussed earlier, it is an adequate probe of quantum
effects in the vicinity of horizons, where it captures the non-local behaviour
which e.g. leads to the Hawking effect [36]. In this case, we will only use it
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).

To calculate the conformal factor (38), 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

$\frac{dr}{dv}=\frac{1}{2}B(v,r).$ (40)

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 subsection 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
$\\{v_{0},R_{\rm i0}\\}$,

$\displaystyle\begin{split}\frac{1}{2}B(v,r)&=k_{1}(v)(r-R_{\rm
i}(v))+k_{2}(v)(r-R_{\rm i}(v))^{2}+\\\ &\quad+k_{3}(v)(r-R_{\rm
i}(v))^{3}+\cdots,\end{split}$ (41) $\displaystyle R_{\rm i}(v)$
$\displaystyle=R_{\rm i0}+R_{\rm i1}v+R_{\rm i2}v^{2}+R_{\rm i3}v^{3}+\cdots,$
(42) $\displaystyle k_{n}(v)$
$\displaystyle=k_{n0}+k_{n1}v+k_{n2}v^{2}+k_{n3}v^{3}+\cdots,$ (43)

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 (40) we consider the series expansion

$r(v)=r_{0}+r_{1}v+r_{2}v^{2}+r_{3}v^{3}+\cdots$ (44)

Substituting this expression into (40), we obtain the coefficients of the null
trajectories (44) in terms of derivatives of $B$ [i.e. the coefficients of its
expansion (41)-(43)] 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. 3) 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 (38)
and then the RSET components (39) in the “in” coordinate system. Switching
them back to the Eddington-Finkelstein system, at zeroth order in the series
expansion, they are

$\displaystyle\begin{split}\expectationvalue{T_{vv}}&=\frac{l_{\rm
P}^{2}}{96\pi^{2}R_{\rm
i0}^{2}}\left(-\frac{1}{2}k_{10}^{2}+k_{11}-2k_{20}R_{\rm i1}-\right.\\\
&\qquad\qquad\qquad\left.-\frac{1}{2}\kappa^{2}+k_{10}\kappa\right)+\order{v,d},\end{split}$
(45a) $\displaystyle\expectationvalue{T_{rr}}$ $\displaystyle=\order{v,d},$
(45b) $\displaystyle\expectationvalue{T_{vr}}$ $\displaystyle=-\frac{2l_{\rm
P}^{2}k_{20}}{96\pi^{2}R_{\rm i0}^{2}}+\order{v,d},$ (45c)

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 $\order{v}$].

We now consider the semiclassical Einstein equations

$G_{\mu\nu}+\delta G_{\mu\nu}=T_{\mu\nu}^{\rm
class}+\expectationvalue{T_{\mu\nu}},$ (46)

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

$\delta G_{\mu\nu}=\expectationvalue{T_{\mu\nu}}+\order{v,d}.$ (47)

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

$\displaystyle G_{vv}$ $\displaystyle=\frac{2k_{10}R_{\rm i1}}{R_{\rm
i0}}+\order{v,d},$ (48a) $\displaystyle G_{rr}$ $\displaystyle=\order{v,d},$
(48b) $\displaystyle G_{vr}$ $\displaystyle=\frac{2k_{10}R_{\rm i0}-1}{R_{\rm
i0}^{2}}+\order{v,d}$ (48c) $\displaystyle G_{\theta\theta}$
$\displaystyle=2R_{\rm i0}(k_{10}+k_{20}R_{\rm i0}).$ (48d)

To construct $\delta G_{\mu\nu}$ we can consider perturbations to these
coefficients, e.g. $k_{10}\to k_{10}+\delta k_{10}$. Eq. (47) allows us to fix
one of the three coefficients present in the tensor components (48) 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 $k_{10}$
and the initial time derivative of the inner horizon trajectory $R_{\rm i1}$,
and using (47) we obtain two key relations,

$\displaystyle\delta k_{10}$ $\displaystyle=-\frac{l_{\rm
P}^{2}}{12\pi}\frac{k_{20}}{R_{\rm i0}},$ (49) $\displaystyle\delta R_{\rm
i1}$ $\displaystyle=\frac{l_{\rm P}^{2}}{48\pi}\left(-\frac{k_{10}}{R_{\rm
i0}}-\frac{\kappa^{2}}{R_{\rm i0}k_{10}}+\frac{1}{2}\frac{\kappa}{R_{\rm
i0}}+\frac{2k_{11}}{R_{\rm i0}k_{10}}\right).$ (50)

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. (30). Interestingly, the further along an evolution
of the type (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
(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 (50) 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 character. 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.

###  IV.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 (36) with

$B(v,r)=e^{-\kappa v}-\frac{1}{2}\lambda(v)r,$ (51)

where $\lambda(v)$ is a positive, but otherwise arbitrary function. The inner
horizon described by this geometry,

$R_{\rm i}(v)=2\frac{e^{-\kappa v}}{\lambda(v)},$ (52)

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 seen above (where
for $R_{\rm i}\sim e^{-\kappa v}$, the surface gravity has an increase with a
rate $e^{2\kappa v}$), 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

$\expectationvalue{T_{vv}}=\frac{l_{\rm
P}^{2}}{96\pi^{2}r^{2}}\left[-\frac{1}{4}\lambda^{\prime}-\frac{1}{32}\lambda^{2}-\frac{\kappa}{4}\lambda-\frac{\kappa^{2}}{2}\right],$
(53)

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, as we will see,
backreaction from the RSET around the inner horizon has the effect of changing
the $B$ function to

$B(v,r)=e^{-\kappa v}-\frac{1}{2}\lambda(v)r+\delta B(v,r),$ (54)

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 $r$,

$\delta B(v,r)=-\frac{\alpha(v)}{r}$ (55)

(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
(54), which has the non-zero components

$\displaystyle G_{vv}$
$\displaystyle=\frac{\lambda^{\prime}}{2}-\frac{\lambda^{2}}{2}+\frac{e^{-\kappa
v}\lambda}{r}+\frac{\kappa\lambda}{2}+\frac{1}{r^{2}}\left[\alpha^{\prime}-(\lambda-\kappa)\alpha\right],$
(56a) $\displaystyle G_{vr}$ $\displaystyle=-\frac{\lambda}{r},$ (56b)
$\displaystyle G_{\theta\theta}$
$\displaystyle=\frac{G_{\phi\phi}}{\sin^{2}\theta}=-\frac{r\lambda e^{\kappa
v}}{2},$ (56c)

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 (53), given that they have the same dependence in $r$
[the form of $\delta B$ in (55) 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, 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). For as long as this
perturbation of non-zero $\alpha$ is negligible for the calculation of the
RSET itself, and while (53) 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.

Note that the function $\alpha$ in $B$ is analogous to the $M_{0}$ term in
(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. Indeed, we
will see that $\alpha$ becomes negative and in many cases its absolute value
tends to grow exponentially quickly.

Equating (53) to the terms containing $\alpha$ in (56a), leaving the remaining
terms as fixed by the background, the evolution of this semiclassically
sourced $\alpha$ is given by the equation

$\alpha^{\prime}(v)-\eta_{1}(v)\alpha(v)=\frac{l_{\rm
P}^{2}}{48\pi}\eta_{2}(v)+\order{l_{\rm P}^{4}},$ (57)

where

$\eta_{1}=\lambda-\kappa,\quad\eta_{2}=-\lambda^{\prime}-\frac{1}{8}\lambda^{2}-\kappa\lambda-2\kappa^{2}$
(58)

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

$\alpha(v)=\frac{l_{\rm
P}^{2}}{48\pi}e^{\int^{v}\eta_{1}(\tilde{v})d\tilde{v}}\left[c_{1}+\int^{v}e^{-\int^{\tilde{v}}\eta_{1}(\bar{v})d\bar{v}}\eta_{2}(\tilde{v})d\tilde{v}\right],$
(59)

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 [22]. 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

$\alpha(v)=\frac{l_{\rm
P}^{2}}{48\pi}\frac{\eta_{2}}{\eta_{1}}[e^{\eta_{1}v}-1].$ (60)

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

$\alpha(v)=\frac{l_{\rm P}^{2}}{48\pi}\eta_{2}v,$ (61)

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 $\alpha=0$,

$R_{\rm i}=2\frac{e^{-\kappa v}}{\lambda}-\alpha e^{\kappa v}+\cdots,$ (62)

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 causality, is hardly surprising considering that the
source given by the RSET (53) is an ingoing flux of negative energy.

For more general backgrounds given by different functions $\lambda(v)$, we can
see from equations (57) and (58) 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
(58), 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 [22] appear to still hold in most of these
mass inflation geometries. In other words, while the flux (53) 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 black hole, given by the time it takes for outgoing
null geodesics which come from outside the region where (54) is accurate 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.

Assuming that a term like (53) 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. 4. 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.

Figure 4: 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 black hole. 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

.

##  V Conclusions and discussion

The inner horizon and the process of mass inflation are components of black
hole physics which are indispensable when constructing a global picture of the
spacetimes of these objects, from their formation to their ultimate fate.
However, semiclassical analyses of this full picture have thus far been
incomplete. Past works have mostly focused on calculating the RSET in
background geometries with a Cauchy horizon in order to see how generic its
divergent behaviour is on this horizon. This observation is then typically
used to suggest that strong cosmic censorship could be saved (from its
classical problem of extensions past the weak null singularity) through
semiclassical backreaction, as generic initial conditions would seemingly
change the Cauchy horizon into a strong singularity. Less often, it has been
indeed appreciated [15, 16, 19] that semiclassical backreaction might lead to
“defocusing” or “expansion” of the Cauchy horizon to large radii due to the
addition of negative mass (essentially coinciding with our result). However,
the effect this could have on a dynamically formed trapped region at finite
times has not been addressed. Crucially, the idea that the trapped region
might have a finite lifetime has not been incorporated into these analyses.

In this work, we have provided a first glimpse into what a complete picture of
black hole formation and evolution may look like in semiclassical physics. To
this end, we have first used a simple shell-based construction to understand
better the classical dynamics of the trapped region in a realistic black hole
formation scenario. That has lead to interesting results in its own right: on
the one had, the mass inflation instability being triggered depends strongly
on the mass to charge (and, by extension, angular momentum) ratio of the
infalling matter. In more general black hole constructions with an inner
horizon (e.g. singularity-free black holes) the necessary condition for this
instability appears to be even more strongly model-dependent, as it comes down
to how the infalling perturbations affect the inner horizon position in time.

In the semiclassical picture, for a static (or nearly static) inner horizon
not undergoing classical mass inflation, our previous work [22] has shown that
backreaction from the RSET has a tendency to induce an inflationary
instability, wherein this horizon is displaced in an outward direction in an
exponential manner, reminiscent of (a negative version of) the mass inflation
effect itself. Extrapolating from this tendency, the trapped region can be
assumed to have a very short lifetime, acting rather as a transient between
collapse and the formation of a final horizonless or extremal configuration.

Here we have extended this analysis to black holes which, in the absence of
other regularising mechanisms, are undergoing classical mass inflation. We
first performed a perturbative analysis, which by itself already suggested
that semiclassical effects always become dominant at some point in the
evolution, as they depend on the surface gravity of the inner horizon which
grows exponentially during mass inflation. We then obtained a simple set of
self-consistent solutions which, in the absence of the RSET source, can
reproduce the structure of a mass inflation geometry, but in a complete
semiclassical treatment reveal a very different behaviour. They present a
tendency for the semiclassically induced outward push on the inner horizon to
accumulate exponentially quickly, leading to an evolution similar to the case
of a classically static inner horizon.

Figure 5: Qualitative picture of the extrapolated evolution of a trapped
region in semiclassical physics.

Within the framework established here, the semiclassical evolution of black
holes could have one of three outcomes. First, we must admit that the standard
picture remains a possibility. Classical mass inflation may continue to
dominate the dynamics around the inner horizon in later parts of the
evolution, where the approximations we have used to estimate semiclassical
backreaction cease to be accurate. In this case Hawking evaporation of the
outer horizon would dominate the first part of the semiclassical evolution, up
to the point at which the mass-inflated region (the upper part of which need
not be close to the origin) is revealed to the external universe, where a more
detailed analysis would be necessary. The physics of the inner horizon plays a
secondary role in this picture until very late times.

However, our present results point to two alternative possibilities which
could be realized in a fully self-consistent semiclassical evolution. The
first one is that the inner horizon moves outward due to backreaction from the
RSET, but only up to the point at which it meets the outer one, leaving an
extremal black hole remnant. The second alternative, represented in fig. 5, is
that the trapped region disappears completely. The dissipation from this
transient process (or several iterations thereof) could bring matter to the
initial conditions necessary for the formation of semiclassically self-
sustained horizonless black hole mimickers [37, 38, 39, 40]. In both these
possibilities, the dynamical evolution of the inner horizon plays an essential
role. Obviously, further work is necessary to fully probe the viability of
these scenarios. However, our analyses already highlight that these not much
contemplated possibilities deserve full attention.

## Acknowledgements

Financial support was provided by the Spanish Government through the projects
PID2020-118159GB-C43 and PID-2020-118159GB-C44, and by the Junta de Andalucía
through the project FQM219. VB is funded by the Spanish Government fellowship
FPU17/04471. CB acknowledges financial support from the State Agency for
Research of the Spanish MCIU through the “Center of Excellence Severo Ochoa”
award to the Instituto de Astrofísica de Andalucía (SEV-2017-0709). RCR
acknowledges financial support through a research grant (29405) from VILLUM
fonden.

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