# On the Stability of a Wormhole in the Maximally-Extended Reissner-Nordström
Solution

Ross DeMott 111Please direct all correspondence to<EMAIL_ADDRESS>Sam
Major, and Alex Flournoy

(Department of Physics, Colorado School of Mines, Golden, CO 80401, USA.
)

###### Abstract

We consider the stability of the maximally-extended Reissner-Nordström
solution in a Minkowski, de Sitter, or anti-de Sitter background.222By
“background,” we mean the asymptotic behavior of the solution far from the
singularity.,333Throughout this article, we will refer to these solutions as
Reissner-Nordström or RN solutions, regardless of the background. In a broad
class of situations, prior work has shown that spherically symmetric
perturbations from a massless scalar field cause the inner horizon of an RN
black hole to become singular and collapse. Even if this is the case, it may
still be possible for an observer to travel through the inner horizon before
it fully collapses, thus violating strong cosmic censorship. In this work, we
show that the collapse of the inner horizon and the occurrence of a
singularity along the inner horizon are sufficient to prevent an observer from
accessing the white hole regions and the parallel universe regions of the
maximally extended RN space-time. Thus, if an observer passes through the
inner horizon, they will inevitably hit the central singularity. Throughout
this article, we use natural units where $c=G=4\pi\hskip
1.42262pt\epsilon_{0}=1$.

##  1 Introduction

By definition, a black hole is an object that has an event horizon. The
simplest black hole solution is the Schwarzschild solution, which possess an
event horizon and no other horizons. However, if a black hole has charge
and/or angular momentum, it exhibits an inner horizon as well as an event
horizon. Such a black hole may be described by the Kerr-Newman solution. When
the Kerr-Newman solution is maximally extended, the inner horizon forms the
entrance to a wormhole, leading to a parallel universe [1].
However, the Kerr-Newman solution is an idealization, as it contains no mass-
energy except at the central singularity. To determine whether the inner
horizon does actually form the entrance to a wormhole, we need to subject the
black hole to perturbations. Moreover, we would like to prove or disprove the
stability of the wormhole for a wide variety of perturbations. Eventually, we
would like to consider arbitrary perturbations that exhibit no symmetry. For
now, however, we will assume spherical symmetry to make the analysis more
manageable.
Any black hole with non-zero angular momentum lacks spherical symmetry, as the
rotation axis specifies a preferred direction in space. Thus, we must start
with a non-rotating black hole, and we must subject this black hole to
spherically symmetric perturbations. When the angular momentum is zero, the
Kerr-Newman solution reduces to the Reissner-Nordström solution.
A Reissner-Nordström (RN) black hole has both charge and mass, which are
concentrated in a point-like singularity at $r=0$. When the black hole is sub-
extremal, this solution describes a black hole with two horizons: an outer
event horizon and an inner Cauchy horizon. Between the event horizon and the
inner horizon, all mass-energy is inexorably drawn inwards. This is equivalent
to the statement that the radial coordinate $r$ is time-like in this region.
However, inside the inner horizon, $r$ again becomes space-like, so it is
possible for mass-energy to travel outwards from the singularity. The inner
horizon is also a Cauchy horizon. In other words, given generic boundary
conditions outside the inner horizon, it is generally impossible to find a
unique solution for the space-time inside the inner horizon [2, 3]. This
signals a breakdown of determinism.
In 1973, Simpson and Penrose demonstrated that the RN solution is unstable at
the inner horizon [4]. Therefore, Penrose proposed that perturbations to the
RN solution destroy the non-uniqueness of the unperturbed solution. This is
known as the strong cosmic censorship (SCC) conjecture [5]. More precisely,
the SCC conjecture states that the instability at the inner horizon produces a
singularity, preventing any observers from passing through it [6, 7]. If true,
this would imply that the region of space-time inside the inner horizon is
unphysical. Hence, the entire physical space-time manifold would be uniquely
specified by boundary conditions, and determinism would be restored.
Given boundary conditions outside the inner horizon, the inner horizon defines
the boundary of the region of space-time where a unique solution for the
metric can be found. Thus, it is possible to unambiguously describe the
evolution of the inner horizon. Previous research has analyzed the behavior of
the inner horizon in the presence of a spherically symmetric distribution of
mass-energy. Dafermos showed that the metric can be extended continuously
beyond the inner horizon, even in the presence of neutral scalar perturbations
[2, 5]. Later, Costa et. al. numerically demonstrated an analogous result for
near-extremal Reissner-Nordström (RN) black holes [6, 7, 8, 9, 10, 11].
In some scenarios, the in-falling mass-energy compresses near the inner
horizon, creating a null curvature singularity called the mass-inflation
singularity [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]. At first,
these results may appear to contradict those mentioned in the previous
paragraph. However, because the mass-inflation singularity is a weak
singularity, the metric is continuous at the Cauchy horizon [19, 22].
Altogether, the results in Refs. [2, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16,
17, 18, 19, 22] suggest that the region inside the inner horizon may be
accessible to observers.
Before we proceed, it is worth noting that the mass-inflation instability is
not generic to all relativistic theories of gravity. In some modified theories
of gravity, such as $f\left(R\right)$ gravity, the mass-inflation instability
is absent [25, 26, 27, 28, 29]. In these theories, models of stellar collapse
do not result in any singularities forming outside the physical singularity.
Additionally, there is some evidence that quantum effects may dampen or
eliminate the mass-inflation singularity [30]. We will leave the question of
wormhole stability in the absence of a mass-inflation singularity for future
work. In this paper, we assume that a mass-inflation instability occurs at the
inner horizon. We also assume that it is possible for an observer to pass
through the inner horizon. Under this assumption, we seek to determine the
ultimate fate of such an observer.
Given that the space-time inside the Cauchy horizon cannot be uniquely
specified, it is not possible to describe an observer’s trajectory inside the
Cauchy horizon with certainty. However, if we assume that space-time can still
be treated classically inside the inner horizon, it is possible to make
concrete statements about the observer’s ultimate fate. In the unperturbed,
maximally extended RN space-time, the region between the inner horizon and the
inner anti-horizon is a wormhole (see Figure 1), which leads to a white hole
and a parallel universe. (In fact, the maximally extended RN solution contains
an infinite number of parallel universes.) Using the Raychaudhuri equation
[31], we show that this wormhole collapses in the presence of spherically
symmetric perturbations satisfying the null energy condition. More precisely,
we show that, if a mass-inflation instability occurs on the Cauchy horizon,
any time-like or light-like observer who passes through the Cauchy horizon of
a perturbed RN black hole will inevitably hit the central singularity. Thus,
we conclude that the parallel universes of the unperturbed RN space-time are
unphysical.

##  2 Coordinate Systems for Reissner-Nordström Black Holes

Let us consider a black hole with mass $M$ and charge $Q$, which is subjected
to massless scalar perturbations. We assume that the metric and all fields are
spherically-symmetric. In spherical coordinates, a general spherically
symmetric metric takes the form

$ds^{2}=-f\left(r,t\right)\hskip 1.42262ptdt^{2}+f\left(r,t\right)^{-1}\hskip
1.42262ptdr^{2}+r^{2}\hskip 1.42262ptd\Omega^{2}.$ (1)

Let $\Lambda$ be the cosmological constant. In the absence of perturbations,
the RN metric has [15, 32]

$f\left(r\right)=1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}-\frac{1}{3}\hskip
1.42262pt\Lambda\hskip 1.42262ptr^{2}.$ (2)

Let $\mathcal{U}$ and $\mathcal{V}$ be general double null coordinates. In
such coordinates, we may rewrite Eqn. 1 as

$ds^{2}=-F\left(\mathcal{U},\mathcal{V}\right)\hskip
1.42262ptd\mathcal{U}\hskip
1.42262ptd\mathcal{V}+r^{2}\left(\mathcal{U},\mathcal{V}\right)\hskip
1.42262ptd\Omega^{2}.$ (3)

A sub-extremal RN black hole has two horizons: an event horizon and an inner
Cauchy horizon. In the absence of perturbations, such a space-time has the
following Penrose diagram:

Figure 1: This is a section of the maximally-extended Penrose diagram for an
unperturbed, sub-extremal RN black hole. This diagram uses Eddington-
Finkelstein (EF) coordinates $u$ and $v$, which we will define shortly.

In Figure 1, the bottom right square represents our universe, while the bottom
left square represents a parallel universe. Because the parallel universe is
inaccessible to any observer traveling from our universe, we may regard it as
unphysical. With this in mind, all perturbations must enter the black hole
through the event horizon in our universe.

###  2.1 Eddington-Finkelstein Double-Null Coordinates

Now, we define the Eddington-Finkelstein (EF) double-null coordinates $u$ and
$v$ for an unperturbed RN black hole [22]. Technically, there are two sets of
EF coordinates: one for the region inside the inner horizon, and one for the
region outside the inner horizon. Let $r_{-}$ denote the radius of the inner
horizon for an unperturbed RN black hole. Let $\kappa$ represent the
gravitational acceleration at the inner horizon. We may write $\kappa$ as [19,
22]

$\kappa=-\frac{1}{2}\hskip 1.42262pt\left.\frac{df}{dr}\right|_{r=r_{-}}.$ (4)

We define the tortoise coordinate $r^{*}$ to satisfy the relation

$\frac{dr}{dr^{*}}=f\left(r\right).$ (5)

Because $f\left(r\right)>0$ inside the inner horizon and $f\left(r\right)<0$
outside the inner horizon, $\kappa$ is positive. Close to $r=r_{-}$, we may
approximate $f\left(r\right)$ as

$f\left(r\right)\approx-2\kappa\hskip 1.42262pt\left(r-r_{-}\right).$ (6)

Let $A$ be an arbitrary real constant. Close to the inner horizon, we may
write the solution $r\left(r^{*}\right)$ as

$r\left(r^{*}\right)=r_{-}+A\hskip 1.42262pt\exp\left[-2\kappa\hskip
1.42262ptr^{*}\right].$ (7)

Now, we want to invert Eqn. 7 to obtain $r^{*}\left(r\right)$. To accomplish
this, we must consider the regions outside and inside the inner horizon
separately.
First, we consider the region outside the inner horizon ($r>r_{-}$). In order
for $r^{*}$ to be real, the constant $A$ must be positive. For simplicity, we
choose $A=1$. Thus, we may rewrite Eqn. 7 as

$\exp\left[-2\kappa\hskip 1.42262ptr^{*}\left(r\right)\right]=r-r_{-},$ (8)
$r^{*}\left(r\right)=-\frac{1}{2\kappa}\hskip
1.42262pt\ln\left(r-r_{-}\right).$ (9)

Next, we consider the region inside the inner horizon ($r<r_{-}$). In order
for $r^{*}$ to be real, the constant $A$ must be negative. For simplicity, we
choose $A=-1$. Thus, we may rewrite Eqn. 7 as

$\exp\left[-2\kappa\hskip 1.42262ptr^{*}\left(r\right)\right]=r_{-}-r,$ (10)
$r^{*}\left(r\right)=-\frac{1}{2\kappa}\hskip
1.42262pt\ln\left(r_{-}-r\right).$ (11)

Combining Equations 10 and 12, we may write $r^{*}$ as

$r^{*}=-\frac{1}{2\kappa}\hskip 1.42262pt\ln\left|r-r_{-}\right|.$ (12)

Finally, we introduce the Eddington-Finkelstein double-null coordinates $u$
and $v$, defined as

$u=t-r^{*},$ (13) $v=t+r^{*}.$ (14)

Just outside the inner horizon, we may approximate $F\left(u,v\right)$
(defined in Eqn. 3) as

$F\left(u,v\right)\approx-2\kappa\hskip 2.84526pte^{\kappa\hskip
1.42262pt\left(u-v\right)}.$ (15)

Just inside the inner horizon, we may approximate $F\left(u,v\right)$ as

$F\left(u,v\right)\approx 2\kappa\hskip 2.84526pte^{\kappa\hskip
1.42262pt\left(u-v\right)}.$ (16)

In the outer EF double-null coordinate system, the inner horizon lies at the
limits $v\to\infty$ and $u\to-\infty$. On the $u=-\infty$ section of the
Cauchy horizon, a severe singularity occurs [16, 18, 20, 22]. Because of this,
objects cannot pass through this section of the Cauchy horizon. Therefore, the
wormhole region beyond the $u=-\infty$ section of the Cauchy horizon is
unphysical. From here on, we shall primarily focus on the $v=+\infty$ section
of the Cauchy horizon.

###  2.2 Kruskal-Szekeres Transformation on the $v$ Coordinate for an
Unperturbed RN Black Hole

The Eddington-Finkelstein double-null coordinate system is the simplest double
null coordinate system to derive from standard spherical coordinates. From
here on, it will be convenient to use the Eddington-Finkelstein coordinate $u$
as one of our double null coordinates. However, we note that there are two
separate $v$ coordinates: one for the region outside the inner horizon and
another for the region inside the inner horizon. Fortunately, it is possible
to glue these two $v$ coordinates together by combining both of them into a
single Kruskal-Szekeres coordinate $V$. By convention, we choose the inner
horizon to be at $V=0$. We choose $V$ to be positive outside the inner horizon
and negative inside the inner horizon. Outside the inner horizon, we define
$V$ as

$V\left(v\right)=e^{-\kappa v}.$ (17)

Inside the inner horizon, we define $V$ as

$V\left(v\right)=-e^{-\kappa v}.$ (18)

Since $u$ and $V$ are both null coordinates, the metric still takes the form

$ds^{2}=-F\left(u,V\right)\hskip 1.42262ptdu\hskip
1.42262ptdV+r^{2}\left(u,V\right)\hskip 1.42262ptd\Omega^{2}.$ (19)

From Equations 15 and 16, we know how $F\left(u,v\right)$ behaves close to the
inner horizon in Eddington-Finkelstein coordinates. With basic calculus, we
may use the known expression for $F\left(u,v\right)$ to find
$F\left(u,V\right)$. Close to the inner horizon (both outside and inside), we
may approximate $F\left(u,V\right)$ as

$F\left(u,V\right)\approx 2\hskip 1.42262pte^{\kappa u}.$ (20)

We derived Eqn. 20 using an unperturbed RN black hole space-time. Thus, one
might be tempted to conclude that Eqn. 20 only applies to an unperturbed RN
black hole. However, in a perturbed space-time, we may assume that
$F\left(u,V\right)$ is positive, finite, and continuous everywhere along the
Cauchy horizon (except at the physical singularity) [19, 8, 9, 10]. Thus, even
for a perturbed black hole, we may perform a gauge transformation such that
$F\left(u,V\right)$ matches Eqn. 20 close to the inner horizon.

##  3 Space-Time Dynamics and the Raychaudhuri Equation in General Double Null
Coordinates

In this section, we describe the dynamics of an arbitrary spherically
symmetric metric coupled to a massless Klein-Gordon scalar field $\phi$. We
also calculate the Raychaudhuri scalar $\Theta$ and describe its dynamics.
These equations hold in arbitrary double null coordinates.

###  3.1 Einstein Field Equations and Dynamics of the Scalar Field

Let us consider a general spherically symmetric metric (Eqn. 3). Let $\phi$ be
a massless Klein-Gordon scalar field, and let $\Lambda$ be the cosmological
constant. We may write the Einstein field equations as

$r_{,uv}=-\frac{r_{,u}\hskip 1.42262ptr_{,v}}{r}-\frac{F}{4r}\hskip
1.42262pt\left(1-\frac{Q^{2}}{r^{2}}\right)+\frac{\Lambda}{4}\hskip
1.42262ptr\hskip 1.42262ptF,$ (21) $F_{,uv}=\frac{F_{,u}\hskip
1.42262ptF_{,v}}{F}+\frac{2F}{r^{2}}\hskip 1.42262ptr_{,u}\hskip
1.42262ptr_{,v}+\frac{F^{2}}{2r^{2}}\hskip
1.42262pt\left(1-\frac{2Q^{2}}{r^{2}}\right)-2F\hskip 1.42262pt\phi_{,u}\hskip
1.42262pt\phi_{,v}+\Lambda\hskip 1.42262ptF^{2},$ (22) $r_{,uu}-\left(\ln
F\right)_{,u}\hskip 1.42262ptr_{,u}+r\hskip
1.42262pt\left(\phi_{,u}\right)^{2}=0,$ (23) $r_{,vv}-\left(\ln
F\right)_{,v}\hskip 1.42262ptr_{,v}+r\hskip
1.42262pt\left(\phi_{,v}\right)^{2}=0.$ (24)

The derivation of Eqns. 21-24 may be found in Appendix B. The scalar field
$\phi$ satisfies the massless Klein-Gordon equation [19]:

$\phi_{,uv}+\frac{1}{r}\hskip 1.42262pt\left(r_{,u}\hskip
1.42262pt\phi_{,v}+r_{,v}\hskip 1.42262pt\phi_{,u}\right)=0.$ (25)

We assume that $\phi$ is a spherically symmetric function, so there is no
angular dependence. At the Cauchy horizon, infalling mass-energy creates a
singularity called the mass inflation instability. In effect, the mass-energy
“piles up” at the Cauchy horizon [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22].
Below, we have included a figure to illustrate this effect with massless
radiation.

Figure 2: A Penrose diagram illustrating the formation of the mass inflation
singularity at the Cauchy horizon. Light is emitted from the curve $AB$ in the
region U. As it gets closer to the Cauchy horizon, the light waves become
compressed and blue-shifted. This causes the energy of the light to increase
without bound, creating a singularity.

As discussed in the introduction, the mass inflation instability is a weak
singularity, which means that it is possible for an observer to pass through
the Cauchy horizon [19, 22].

###  3.2 Raychaudhuri Equation

For a more thorough derivation of all the statements in this subsection,
please see Appendix A. Let us define the vector fields $\ell^{\mu}$ and
$n^{\mu}$ as

$\ell^{\mu}=\begin{cases}F\left(u,v\right)^{-1},&\textrm{if }\mu=v\\\
0,&\textrm{otherwise}\end{cases},$ (26) $n^{\mu}=\begin{cases}2,&\textrm{if
}\mu=u\\\ 0,&\textrm{otherwise}\end{cases}.$ (27)

Let $\Theta$ be the Raychaudhuri expansion scalar in the $v$ direction. We may
write $\Theta$ as

$\Theta=2\hskip 1.42262ptF\left(u,v\right)^{-1}\hskip
1.42262ptr\left(u,v\right)^{-1}\hskip 1.42262pt\partial_{v}r\left(u,v\right).$
(28)

The scalar $\Theta$ obeys the Raychaudhuri equation

$\ell^{\mu}\hskip
1.42262pt\partial_{\mu}\Theta=-\frac{1}{2}\Theta^{2}-R_{\mu\nu}\hskip
1.42262pt\ell^{\mu}\ell^{\nu}.$ (29)

The total stress-energy tensor (Eqn. 152) satisfies the null energy condition

$T_{\mu\nu}\hskip 1.42262pt\ell^{\mu}\hskip 1.42262pt\ell^{\nu}\geq 0.$ (30)

Plugging Eqn. 30 into Eqn. 29, we find that

$\partial_{v}\Theta\leq 0.$ (31)

##  4 Sufficient Conditions for a Wormhole to be Unstable

In this section, we specify several conditions that are sufficient for a
spherically-symmetric wormhole, formed from a sub-extremal RN black hole
solution, to be unstable. We use the coordinate system from Subsection 2.2,
with $u$ an Eddington-Finkelstein coordinate and $V$ a Kruskal-Szekeres
coordinate. We assume that the black hole solution has a Cauchy horizon.

###  4.1 Regularity Conditions

We assume that the functions $F\left(u,V\right)$ and $r\left(u,V\right)$ are
finite, positive, and continuous for all $u$ and $V$, except at the physical
singularity [19]. At all points $u$, $V$ not at the physical singularity or on
the Cauchy horizon, we assume that $F\left(u,V\right)$ and $r\left(u,V\right)$
are twice continuously differentiable. We also assume that
$r_{,u}\left(u,V\right)$ is continuously differentiable for all $u$ and $V$
(including $V=0$), except at the physical singularity.
These regularity conditions are fairly generic, and we have not found any
situation (relevant to this paper) in which they have been shown to be
violated. Thus, we will assume that these conditions are true in all the
situations we consider in this paper.

###  4.2 Situation-Dependent Conditions

In addition to the regularity conditions, we have two conditions that cannot
be assumed to hold generally. Thus, we will have to determine whether these
conditions hold in each of the situations we consider.
We assume that there is at least one point on the Cauchy horizon, which we
call $u_{1}$, such that

$r_{,u}\left(u=u_{1},V=0\right)>\kappa\hskip
1.42262ptr\left(u=u_{1},V=0\right).$ (32)

We also assume that there exists another point $u_{2}$ on the Cauchy horizon,
which satisfies $u_{2}<u_{1}$, such that the following limit holds from both
sides:

$\lim_{V\to\hskip 1.42262pt0}r_{,V}\left(u=u_{2},V\right)=+\infty.$ (33)

###  4.3 Theorem 1: Collapse of the Cauchy Horizon to $r=0$

######  Theorem 1.

Let the assumptions described in Subsection 4.1 be true. Additionally, let
Eqn. 32 be true. Then, for all points $u<u_{1}$ on the Cauchy horizon (except
at the physical singularity),

$r_{,u}\left(u,V=0\right)>0.$ (34)

Additionally, there exists a finite value $u_{\textrm{min}}$ such that

$r\left(u=u_{\textrm{min}},V=0\right)=0.$ (35)

Proof. Using Eqn. 20, we may rewrite Eqn. 23 as

$r_{,uu}-\kappa\hskip 1.42262ptr_{,u}=-r\hskip
1.42262pt\left(\phi_{,u}\right)^{2}.$ (36)

Note that $\kappa$ is non-negative. Using the product rule, we may rewrite
Eqn. 36 as

$\frac{d}{du}\left(e^{-\kappa u}\hskip 1.42262ptr_{,u}\right)=-e^{-\kappa
u}\hskip 1.42262ptr\hskip 1.42262pt\left(\phi_{,u}\right)^{2}.$ (37)

From Eqn. 37, it is easy to see that

$\frac{d}{du}\left(e^{-\kappa u}\hskip 1.42262ptr_{,u}\right)\leq 0.$ (38)

For any value $u<u_{1}$, Eqn. 38 implies that

$e^{-\kappa u}\hskip 1.42262ptr_{,u}\left(u,V=0\right)\geq e^{-\kappa
u_{1}}\hskip 1.42262ptr_{,u}\left(u=u_{1},V=0\right).$ (39)

Rearranging Eqn. 39, we obtain

$r_{,u}\left(u,V=0\right)\geq e^{\kappa\hskip
1.42262pt\left(u-u_{1}\right)}\hskip 1.42262ptr_{,u}\left(u=u_{1},V=0\right).$
(40)

Eqn. 40 implies the first part of Theorem 1. For all points $u<u_{1}$ on the
Cauchy horizon (except at the physical singularity),

$r_{,u}\left(u,V=0\right)>0.$ (41)

Next, we integrate both sides of Eqn. 40, which yields

$\displaystyle r\left(u,V=0\right)$
$\displaystyle=r\left(u=u_{1},V=0\right)-\int_{u}^{u_{1}}r_{,u}\left(u^{\prime},V=0\right)\hskip
1.42262pt\mathrm{d}u^{\prime}$ (42) $\displaystyle\leq
r\left(u=u_{1},V=0\right)-r_{,u}\left(u=u_{1},V=0\right)\hskip
1.42262pt\int_{u}^{u_{1}}e^{\kappa\hskip
1.42262pt\left(u^{\prime}-u_{1}\right)}\hskip 1.42262pt\mathrm{d}u^{\prime}$
(43)
$\displaystyle=r\left(u=u_{1},V=0\right)-\frac{r_{,u}\left(u=u_{1},V=0\right)}{\kappa}+\frac{r_{,u}\left(u=u_{1},V=0\right)}{\kappa}\hskip
1.42262pte^{\kappa\hskip 1.42262pt\left(u-u_{1}\right)}.$ (44)

From the assumptions in Subsection 4.2, we know that

$r\left(u=u_{1},V=0\right)-\frac{r_{,u}\left(u=u_{1},V=0\right)}{\kappa}<0.$
(45)

As $u$ decreases, the third term in Eqn. 44 becomes arbitrarily small. Thus,
there exists a finite value $u_{\textrm{min}}$ such that

$r\left(u=u_{\textrm{min}},V=0\right)=0.$ (46)

###  4.4 Theorem 2: Divergence of $r_{,V}$ at the Cauchy Horizon

######  Theorem 2.

Let the assumptions described in Subsections 4.1 and 4.2 be true. Then, for
all $u\in\left(u_{\textrm{min}},u_{2}\right]$, the following statement is
true:

$\lim_{V\to\hskip 1.42262pt0}r_{,V}\left(u,V\right)=+\infty.$ (47)

Proof. We proceed via proof by contradiction. Let us assume that there is some
point $u_{\textrm{test}}\in\left(u_{\textrm{min}},u_{2}\right)$ such that

$\lim_{V\to 0}r_{,V}\left(u=u_{\textrm{test}},V\right)\neq+\infty.$ (48)

In any interval $V\in\left(-\epsilon,0\right)\cup\left(0,\epsilon\right)$,
there will be at least one line $V=V_{1}$ such that

$r_{,V}\left(u=u_{\textrm{test}},V=V_{1}\right)<r_{,V}\left(u=u_{2},V=V_{1}\right).$
(49)

Because $V_{1}\neq 0$, we know that $r_{,V}\left(u,V=V_{1}\right)$ is
continuous for all $u>u_{\textrm{min}}$. From Equation 21, we have the
following expression for $r_{,uV}$:

$r_{,uv}=-\frac{r_{,u}\hskip 1.42262ptr_{,v}}{r}-\frac{F}{4r}\hskip
1.42262pt\left(1-\frac{Q^{2}}{r^{2}}\right)+\frac{\Lambda}{4}\hskip
1.42262ptr\hskip 1.42262ptF.$ (50)

Since Equation 21 holds for any double null coordinate system, we have
replaced $v$ with the Kruskal-Szekeres coordinate $V$.
The functions $F\left(u=u_{2},V\right)$, $r\left(u=u_{2},V\right)$, and
$r_{,u}\left(u=u_{2},V\right)$ are finite, positive, and continuous. By
contrast, $r_{,V}\left(u=u_{2},V\right)$ grows without bound as $V\to 0$.
Therefore, if we choose $V_{1}$ to be sufficiently close to zero, we can make
$r_{,uV}\left(u=u_{2},V=V_{1}\right)$ negative.
Because $r_{,uV}$ is continuous along the line $V=V_{1}$, there must be some
interval around $u=u_{2}$ (meaning $u_{2}$ is not on the boundary) where
$r_{,uV}\left(u,V=V_{1}\right)$ is negative. Let $\mathcal{J}$ be the largest
interval, without any gaps, such that $u_{2}\in\mathcal{J}$ and such that, for
all $u\in\mathcal{J}$,

$r_{,uV}\left(u,V=V_{1}\right)<0.$ (51)

Let $u=u_{3}$ be the infimum of the set $\mathcal{J}$. If
$u_{3}=u_{\textrm{min}}$, then for all points
$u\in\left(u_{\textrm{min}},u_{2}\right)$,

$r_{,V}\left(u,V=V_{1}\right)>r_{,V}\left(u_{2},V=V_{1}\right).$ (52)

Clearly, Eqn. 52 contradicts our finding in Eqn. 49. Thus, if
$u_{3}=u_{\textrm{min}}$, Theorem 2 is proved.
Now, let us assume that $u_{3}>u_{\textrm{min}}$. Because
$r_{,uV}\left(u,V=V_{1}\right)$ is continuous, we know that

$r_{,uV}\left(u=u_{3},V=V_{1}\right)=0.$ (53)

For all $u\in\left(u_{3},u_{2}\right)$, we know that

$r_{,V}\left(u,V=V_{1}\right)>r_{,V}\left(u=u_{2},V=V_{1}\right).$ (54)

Because $r_{,V}\left(u,V=V_{1}\right)$ is continuous along the line $V=V_{1}$,
Eqn. 54 implies that

$r_{,V}\left(u=u_{3},V=V_{1}\right)>r_{,V}\left(u=u_{2},V=V_{1}\right).$ (55)

On the closed interval $\mathcal{I}=\left[u_{3},u_{2}\right]$, the functions
$F\left(u,V=V_{1}\right)$ and $r\left(u,V=V_{1}\right)$ are continuous and
positive. If we choose $V_{1}$ to be sufficiently close to $V=0$, then
$r_{,u}\left(u,V=V_{1}\right)$ will also be continuous and positive on
$\mathcal{I}$ (by Theorem 1).
According to the extreme value theorem [33], $F\left(u,V=V_{1}\right)$,
$r\left(u,V=V_{1}\right)$, and $r_{,u}\left(u,V=V_{1}\right)$ all have
positive lower and upper bounds on $\mathcal{I}$. Therefore, we may choose
$V_{1}$ such that, for all $u\in\mathcal{I}$:

$r_{,V}\left(u=u_{2},V=V_{1}\right)>-\frac{F\left(u,V=V_{1}\right)}{4r_{,u}\left(u,V=V_{1}\right)}+\frac{Q^{2}\hskip
1.42262ptF\left(u,V=V_{1}\right)}{4r\left(u,V=V_{1}\right)^{2}\hskip
1.42262ptr_{,u}\left(u,V=V_{1}\right)}.$ (56)

Using Equation 55, we may rewrite Eqn. 56 as

$r_{,V}\left(u=u_{3},V=V_{1}\right)>-\frac{F\left(u,V=V_{1}\right)}{4r_{,u}\left(u,V=V_{1}\right)}+\frac{r^{2}_{Q}\hskip
1.42262ptF\left(u,V=V_{1}\right)}{4r\left(u,V=V_{1}\right)^{2}\hskip
1.42262ptr_{,u}\left(u,V=V_{1}\right)}.$ (57)

Plugging Eqn. 57 into Eqn. 21, we find that

$r_{,uV}\left(u=u_{3},V=V_{1}\right)<0.$ (58)

However, Eqn. 58 conflicts with our earlier finding that

$r_{,uV}\left(u=u_{3},V=V_{1}\right)=0.$ (59)

Thus, if there is some point
$u_{\textrm{test}}\in\left(u_{\textrm{min}},u_{2}\right)$ such that

$\lim_{V\to 0}r_{,V}\left(u=u_{2},V\right)\neq\infty,$ (60)

we obtain a contradiction. Therefore, for all
$u\in\left(u_{\textrm{min}},u_{2}\right]$, we know that

$\lim_{V\to 0}r_{,V}\left(u,V\right)=+\infty.$ (61)

###  4.5 Theorem 3: Positivity of $r_{,V}$ Inside the Cauchy Horizon

######  Theorem 3.

Let the assumptions described in Subsections 4.1 and 4.2 be true. Then,

$r_{,V}\left(u,V\right)>0\textrm{ for all
}u,v\in\left(u_{\textrm{min}},u_{2}\right]\times\left(-\infty,0\right).$ (62)

Proof. Let us consider an arbitrary value
$u_{\textrm{test}}\in\left(u_{min},u_{2}\right]$. From Theorem 2, we have

$\lim_{V\to\hskip
1.42262pt0}r_{,V}\left(u=u_{\textrm{test}},V\right)=+\infty.$ (63)

Therefore, there exists some real number $\epsilon>0$ such that

$r_{,V}\left(u=u_{\textrm{test}},V\right)>0\textrm{ for all
}V\in\left(-\epsilon,0\right)\cup\left(0,\epsilon\right).$ (64)

Recall Equation 28 for the Raychaudhuri scalar $\Theta$:

$\Theta=2\hskip 1.42262ptF\left(u,v\right)^{-1}\hskip
1.42262ptr\left(u,v\right)^{-1}\hskip 1.42262pt\partial_{V}r\left(u,v\right).$
(65)

For any $u$ and $V$ in the physical space-time, if $r_{,V}\left(u,V\right)>0$,
then $\Theta\left(u,V\right)>0$. Therefore, there exists some real number
$\epsilon>0$ such that

$\Theta\left(u=u_{\textrm{test}},V\right)>0\textrm{ for all
}V\in\left(-\epsilon,0\right)\cup\left(0,\epsilon\right).$ (66)

From Subsection 3.2, we know that $\partial_{V}\Theta\leq 0$. Therefore, for
any $V<0$, we know that

$\Theta\left(u_{\textrm{test}},V\right)>0.$ (67)

This implies that, for all $V<0$,

$r_{,V}\left(u_{\textrm{test}},V\right)>0.$ (68)

Since $u_{\textrm{test}}$ is an arbitrary element of the interval
$\left(u_{min},u_{2}\right]$, we find that

$r_{,V}\left(u,V\right)>0\textrm{ for all
}u,V\in\left(u_{\textrm{min}},u_{2}\right]\times\left(-\infty,0\right).$ (69)

###  4.6 Application to Wormholes

Let us assume that the conditions in Subsections 4.1 and 4.2 are true.
Furthermore, let us assume that

$r_{,u}\left(u=u_{1},V=0\right)\geq\kappa\hskip
1.42262ptr\left(u=u_{1},V=0\right).$ (70)

Then, from Theorem 1, we know that $r\left(u,V=0\right)$ either reaches zero
at a finite value $u=u_{\textrm{min}}$, or

$\lim_{u\to-\infty}r\left(u,V=0\right)=0.$ (71)

According to Theorem 3,

$r_{,V}\left(u,V\right)>0\textrm{ for all
}u,V\in\left(u_{\textrm{min}},u_{1}\right]\times\left(-\infty,0\right).$ (72)

Therefore, for any $u_{0}\in\left(u_{\textrm{min}},u_{1}\right)$ and any
$V_{0}<0$,

$r\left(u=u_{0},V=V_{0}\right)<r\left(u=u_{0},V=0\right).$ (73)

Because

$\lim_{u\to u_{\textrm{min}}}r\left(u,V=0\right)\to 0,$ (74)

for any $V_{0}<0$, there exists some $u_{\textrm{zero}}\geq u_{\textrm{min}}$
such that

$\lim_{u\to u_{\textrm{zero}}}r\left(u,V=V_{0}\right)=0.$ (75)

Therefore, any object that passes through the Cauchy horizon will inevitably
hit the physical singularity $r=0$. Thus, the Penrose diagram for our space-
time looks similar to the one below.

Figure 3: Penrose diagram for a perturbed Reissner-Nordström black hole. The
values of the double-null coordinates u and v at the horizons are shown in
blue. Note that $u$ and $v$ are both Eddington-Finkelstein null coordinates.

##  5 Wormhole Stability for a Sub-Extremal RN Black Hole

In this section, we determine under what conditions a sub-extremal RN black
hole satisfies the assumptions in Subsection 4.2. We adopt a coordinate system
similar to that of Subsection 2.2, with $u$ an Eddington-Finkelstein
coordinate and $V$ a Kruskal-Szekeres coordinate.

###  5.1 Zero Cosmological Constant ($\Lambda=0$)

Let us consider a sub-extremal RN black hole in the absence of a cosmological
constant. The radius of the Cauchy horizon decreases monotonically as $u$
decreases [14, 16, 18, 19]. At a finite value $u=u_{\textrm{min}}$, the Cauchy
horizon intersects with the physical singularity at $r=0$ [18, 22].
According to Ref. [19], the derivative $r_{,V}\left(u,V\right)$ approaches
$+\infty$ as $V\to 0$ from the outside of the Cauchy horizon. Because
$r_{,V}\left(u,V\right)\to+\infty$ as $V\to 0$ outside the Cauchy horizon, it
is reasonable to assume that there is at least one line $u=u_{1}$ where the
same limit holds from both sides of the Cauchy horizon.
For $\Lambda=0$, all the conditions specified in Subsection 4.2 are satisfied.
Thus, a wormhole formed from a Reissner-Nordström black hole, in the absence
of a cosmological constant, is unstable. Note that this result holds for any
sub-extremal RN black hole, as long as $\Lambda=0$.

###  5.2 Positive Cosmological Constant ($\Lambda>0$)

Let us consider a black hole embedded in a background de Sitter space with
cosmological constant $\Lambda$. On the event horizon of the black hole, we
assume that the actual metric matches the RN metric and that the first
derivatives of the actual metric match the first derivatives of the RN metric
[8, 9, 10].
Let $r_{+}$ be the event horizon radius of an unperturbed RN black hole with
mass $M$ and charge $Q$. Let $r_{-}$ be the Cauchy horizon radius of such a
black hole. Now, we define the ratios $\sigma$ and $\Upsilon$ as [8, 9, 10]

$\sigma=\frac{r_{+}}{r_{-}},$ (76) $\Upsilon=\frac{\Lambda\hskip
1.42262ptr_{-}^{2}}{3}.$ (77)

In terms of $\sigma$ and $\Upsilon$, we may define the ratio $\rho$ as [8, 9,
10]

$\rho=\sigma^{2}\hskip 1.42262pt\frac{1-\Upsilon\hskip
1.42262pt\left(\sigma^{2}+2\sigma+3\right)}{1-\Upsilon\hskip
1.42262pt\left(3\sigma^{2}+2\sigma+1\right)}.$ (78)

In the limit of a Schwarzschild-de Sitter black hole ($r_{-}=0$), the ratio
$\rho$ approaches $+\infty$. In the limit of an extremal RN black hole
($r_{-}=r_{+}$), the ratio $\rho$ approaches unity [8, 9, 10]. Based on the
ratio $\rho$, we divide RN black holes into two classes. If $\rho>2$, we call
the black hole a near-Schwarzschild RN black hole. If $\rho\leq 2$, we call
the black hole a near-extremal RN black hole.

####  5.2.1 Near-Schwarzschild RN Black Hole

Let us assume that $\rho>2$, so that we are working with a near-Schwarzschild
RN black hole. Let $V_{0}$ be a finite number; we specify initial data for the
field $\phi$ on the surface $V=V_{0}$. For a broad variety of initial
conditions (more specifically, if $r_{,u}\left(u,V_{0}\right)$ fails to decay
sufficiently quickly close to the event horizon), a mass-inflation instability
will emerge on the Cauchy horizon of the black hole [8, 9, 10].
The radius of the Cauchy horizon satisifes [9, 10]

$\lim_{u\to+\infty}r\left(u,V=0\right)=r_{-}.$ (79)

Therefore, there exists some constant $U_{1}$ such that, for all $u>U_{1}$,

$r\left(u,V=0\right)>0.$ (80)

There exists a constant $U_{2}$ such that, for all $u>U_{2}$ [9, 10],

$r_{,u}\left(u,V=0\right)>0.$ (81)

On the event horizon, the first derivatives of the actual metric match the
first derivatives of the RN metric. Thus, the radius of the Cauchy horizon
should satisfy [9, 10]

$\lim_{u\to+\infty}r_{,u}\left(u,V=0\right)=0.$ (82)

Thus, there exists some value $U_{3}$ such that, for all $u>U_{3}$,

$r_{,u}\left(u,V=0\right)<+\infty.$ (83)

Let us define the renormalized Hawking mass $\omega\left(u,v\right)$ as [8, 9,
10]

$\omega\left(u,V\right)=\frac{e^{2}}{2}\hskip
1.42262ptr\left(u,V\right)^{-1}+\frac{1}{2}\hskip
1.42262ptr\left(u,V\right)-\frac{\Lambda}{6}\hskip
1.42262ptr\left(u,V\right)^{3}+2\hskip
1.42262ptF\left(u,V\right)^{-2}r\left(u,V\right)\hskip
1.42262ptr_{,u}\left(u,V\right)\hskip 1.42262ptr_{,V}\left(u,V\right).$ (84)

Note that $\omega\left(u,V\right)$ is a gauge invariant quantity. As discussed
above, a mass-inflation instability occurs for generic perturbations in this
space-time. Thus, there exists a constant $U_{4}$ such that, for all $u>U_{4}$
[10],

$\lim_{V\to 0}\omega\left(u,V\right)=+\infty.$ (85)

Let us select a constant $U$ that is larger than $U_{1}$, $U_{2}$, $U_{3}$,
and $U_{4}$. Combining Eqns. 80, 81, 83, 84, and 85, we find that, for all
$u>U$,

$\lim_{V\to 0}r_{,V}\left(u,V\right)=+\infty.$ (86)

All the conditions specified in Subsections 4.1 and 4.2 are satisfied. Thus,
under general initial conditions, a wormhole formed from a near-Schwarzschild
Reissner-Nordström black hole is unstable if $\Lambda>0$.

####  5.2.2 Near-Extremal RN Black Hole

Let us assume that $\rho<2$, so that we are working with a near-extremal RN
black hole. In general, it is not possible to guarantee that a mass-inflation
instability will occur on the Cauchy horizon of such a black hole. In fact,
there exist examples of near-extremal RN black holes that fail to develop a
mass-inflation instability under a broad range of initial conditions [8, 9,
10].
If no mass-inflation instability occurs, then the function
$\omega\left(u,v\right)$ remains finite at the Cauchy horizon. Thus, it is not
possible for all the conditions in Subsections 4.1 and 4.2 to be fulfilled.
This implies that a wormhole formed from a near-extremal RN black hole may be
stable if $\Lambda>0$.

###  5.3 Negative Cosmological Constant ($\Lambda<0$)

In the presence of a negative cosmological constant, the Cauchy horizon
collapses to $r=0$ [34, 35]. However, at all points on the Cauchy horizon
where $r\neq 0$, there is no singularity [34]. Therefore,
$r_{,V}\left(u,V=0\right)$ remains finite except at the physical singularity
($r=0$). This violates the conditions of Subsection 4.2. Therefore, the
wormhole that occurs inside an RN black hole may be stable if $\Lambda<0$.

##  6 Wormhole Stability for an Extremal RN Black Hole

An extremal RN black hole has only one horizon, which is simultaneously an
event horizon and a Cauchy horizon [36, 37, 38]. Let $r_{h}$ denote the radius
of this horizon. Let us define the tortoise coordinate $r^{*}$ as [38, 39]

$r^{*}\left(r\right)=r-\hskip 1.42262ptr_{h}+2r_{h}\hskip
1.42262pt\ln\left(\frac{\lvert
r-r_{h}\rvert}{r_{h}}\right)-\frac{r_{h}^{2}}{r-r_{h}}.$ (87)

We define the EF double null coordinates $u$ and $v$ as [38, 39]

$u=t-r^{*},$ (88) $v=t+r^{*}.$ (89)

In these EF coordinates, the horizon corresponds to the limit $u\to+\infty$
[38]. For a sub-extremal black hole, the Cauchy horizon corresponds to the
limit $v\to+\infty$. To rectify this disparity, we simply swap the definitions
of the coordinates $u$ and $v$.

$u=t+r^{*},$ (90) $v=t-r^{*}.$ (91)

In order to obtain meaningful results for physics at the horizon, we must
define a new coordinate $V$ that is regular at the horizon. We may do this via
the following Kruskal-Szekeres transformation [39, 38]:

$v=-2r^{*}\hskip 1.42262pt\left(r_{h}+V\right).$ (92)

In this new coordinate system, the horizon lies at $V=0$. The region outside
the horizon corresponds to $V>0$, while the region inside the horizon
corresponds to $V<0$. Close to the horizon, we have the following
approximation for $F\left(u,V\right)$ [39]:

$F\left(u,V\right)\approx 2.$ (93)

As in the sub-extremal case, we assume that the conditions in Subsections 4.1
and 4.2 are fulfilled. For an extremal black hole, the surface gravity
$\kappa$ is zero [39]. Thus, we may rewrite Eqn. 32 as

$r_{,u}\left(u=u_{1},V=0\right)>0\textrm{ for some }u_{1}\in\mathbb{R}.$ (94)

We may rewrite Eqn. 23 as

$r_{,uu}=-r\hskip 1.42262pt\left(\phi_{,u}\right)^{2}.$ (95)

Clearly, Eqn. 95 implies that $r_{,uu}$ is non-positive everywhere along the
Cauchy horizon. Thus, for all $u<u_{1}$,

$r_{,u}\left(u,V=0\right)>r_{,u}\left(u=u_{1},V=0\right).$ (96)

From Eqn. 96, we see that there exists a finite value $u_{\textrm{min}}$ such
that

$r\left(u=u_{\textrm{min}},V=0\right)=0.$ (97)

Having proved Theorem 1, one may prove Theorems 2 and 3 for an extremal RN
black hole in the same way as for a sub-extremal RN black hole. Thus, a
wormhole formed from an extremal RN black hole collapses. It is impossible for
an observer to access the parallel universe regions of the maximally-extended
extremal RN space-time.

##  7 Summary and Future Work

Under the assumptions of Subsections 4.1 and 4.2, we have shown that a
Reissner-Nordström wormhole collapses in the presence of spherically symmetric
perturbations from a massless scalar field. In this article, we have held
firmly to the assumption of spherical symmetry. In the future, it would be
interesting to investigate the stability of non-spherically symmetric
wormholes, such as those that arise from the Kerr-Newman solution.
Additionally, it would be interesting to incorporate non-spherically symmetric
distributions of mass-energy as perturbations.
As the wormhole collapses, it will eventually become sufficiently small that
quantum effects become important. At this scale, the classical analysis
performed in this paper is no longer valid. Hence, it is unknown whether the
wormhole will continue to collapse or not. Future work is needed to assess the
impact of quantum effects on the collapse of the wormhole.

## Appendix A: Computation of the Expansion Scalar and the Raychaudhuri
Equation in General Double Null Coordinates

Let $K$ be a scalar field, and let $\ell^{\mu}$ and $n^{\mu}$ be null vector
fields. Let $\nabla_{\nu}$ denote a covariant derivative. The null vector
fields must satisfy the following conditions:

$\ell^{\mu}\hskip 1.42262ptn_{\mu}=-1,$ (98) $\ell^{\nu}\hskip
1.42262pt\nabla_{\nu}\ell^{\mu}=K\hskip 1.42262pt\ell^{\mu}.$ (99)

We may define the expansion scalar $\Theta$ as

$\Theta=\nabla_{\mu}\ell^{\mu}-K.$ (100)

Next, let us define the transverse metric $h_{\mu\nu}$ as

$h_{\mu\nu}=g_{\mu\nu}+\ell_{\mu}\hskip 1.42262ptn_{\nu}+\ell_{\nu}\hskip
1.42262ptn_{\mu}.$ (101)

Finally, let us define the quantities $\sigma_{\mu\nu}$ and $\omega_{\mu\nu}$
as

$\sigma_{\mu\nu}=\frac{1}{2}\left(\nabla^{\sigma}\ell^{\rho}\right)\hskip
1.42262pt\left(h_{\mu\rho}\hskip 1.42262pth_{\nu\sigma}+h_{\nu\rho}\hskip
1.42262pth_{\mu\sigma}\right)-\frac{1}{2}h_{\mu\nu}\hskip 1.42262pt\Theta,$
(102)
$\omega_{\mu\nu}=\frac{1}{2}\left(\nabla^{\sigma}\ell^{\rho}\right)\hskip
1.42262pt\left(h_{\mu\rho}\hskip 1.42262pth_{\nu\sigma}-h_{\nu\rho}\hskip
1.42262pth_{\mu\sigma}\right).$ (103)

Let us make the definitions $\sigma^{2}:=\sigma_{\mu\nu}\hskip
1.42262pt\sigma^{\mu\nu}$ and $\omega^{2}:=\omega_{\mu\nu}\hskip
1.42262pt\omega^{\mu\nu}$). The Raychaudhuri equation takes the following form
[31]:

$\ell^{\mu}\hskip
1.42262pt\partial_{\mu}\Theta=-\frac{1}{2}\Theta^{2}-\sigma^{2}+\omega^{2}-R_{\mu\nu}\hskip
1.42262pt\ell^{\mu}\ell^{\nu}+K\Theta.$ (104)

We use the index $u$, not to be confused with $\mu$, to denote the component
of a vector in the positive $u$ direction. We use the index $v$, not to be
confused with $\nu$, to denote the component of a vector in the positive $v$
direction. We are free to choose the vector fields $\ell^{\mu}$ and $n^{\mu}$,
as long as they satisfy Equations 98 and 99. Therefore, we select the
following expressions for $\ell^{\mu}$ and $n^{\mu}$:

$\ell^{\mu}=\begin{cases}F\left(u,v\right)^{-1},&\textrm{if }\mu=v\\\
0,&\textrm{otherwise}\end{cases},$ (105) $n^{\mu}=\begin{cases}2,&\textrm{if
}\mu=u\\\ 0,&\textrm{otherwise}\end{cases}.$ (106)

To avoid confusion with the Latin indices $u$ and $v$, let us replace the
Greek indices $\mu$ and $\nu$ in Equation 99 with $\alpha$ and $\beta$. We may
expand $\ell^{\beta}\hskip 1.42262pt\nabla_{\beta}\ell^{\alpha}$ as follows

$\displaystyle\ell^{\beta}\hskip 1.42262pt\nabla_{\beta}\ell^{\alpha}$
$\displaystyle=\ell^{\beta}\hskip
1.42262pt\left(\partial_{\beta}\ell^{\alpha}+\Gamma^{\alpha}_{\beta\gamma}\hskip
1.42262pt\ell^{\gamma}\right)$ (107) $\displaystyle=\ell^{v}\hskip
1.42262pt\partial_{v}\ell^{\alpha}+\Gamma^{\alpha}_{vv}\hskip
1.42262pt\left(\ell^{v}\right)^{2}.$ (108)

We may write the Christoffel symbols $\Gamma^{\alpha}_{vv}$ as follows:

$\displaystyle\Gamma^{\alpha}_{vv}$ $\displaystyle=g^{\alpha\beta}\hskip
1.42262pt\partial_{v}g_{v\beta}$ (109)
$\displaystyle=\begin{cases}F\left(u,v\right)^{-1}\hskip
1.42262pt\partial_{v}F\left(u,v\right)&\textrm{if }\alpha=v\\\
0&\textrm{otherwise}\end{cases}.$ (110)

When $\alpha\neq v$, we find that $\ell^{\beta}\hskip
1.42262pt\nabla_{\beta}\ell^{\alpha}=0$. Therefore, the only non-trivial
component of Equation 99 occurs when $\alpha=v$. We may write this component
as

$\ell^{v}\hskip 1.42262pt\partial_{v}\ell^{v}+\Gamma^{v}_{vv}\hskip
1.42262pt\left(\ell^{v}\right)^{2}=K\hskip 1.42262pt\ell^{v}.$ (111)

Thus, we may write the scalar $K$ as

$\displaystyle K$ $\displaystyle=\partial_{v}\ell^{v}+\Gamma^{v}_{vv}\hskip
1.42262pt\ell^{v}$ (112)
$\displaystyle=-\left(F\left(u,v\right)\right)^{-2}\hskip
1.42262pt\partial_{v}F\left(u,v\right)+F\left(u,v\right)^{-2}\hskip
1.42262pt\partial_{v}F\left(u,v\right)$ (113) $\displaystyle=0.$ (114)

Using Equation 100, we may write $\Theta$ as

$\displaystyle\Theta$ $\displaystyle=\nabla_{\alpha}\ell^{\alpha}-K$ (115)
$\displaystyle=\partial_{\alpha}\ell^{\alpha}+\Gamma^{\alpha}_{\alpha\gamma}\ell^{\gamma}$
(116) $\displaystyle=\partial_{v}\ell^{v}+\Gamma^{\alpha}_{\alpha v}\ell^{v}.$
(117)

The index $\alpha$ is summed, since it does not denote a specific coordinate
direction. We may write the quantity $\Gamma^{\alpha}_{\alpha v}$ as

$\displaystyle\Gamma^{\alpha}_{\alpha v}$ $\displaystyle=\frac{1}{2}\hskip
1.42262ptg^{\alpha\beta}\hskip
1.42262pt\left(\partial_{v}g_{\alpha\beta}+\partial_{\alpha}g_{v\beta}-\partial_{\beta}g_{v\alpha}\right)$
(118) $\displaystyle=g^{uv}\hskip
1.42262pt\partial_{v}g_{uv}+\frac{1}{2}\hskip 1.42262ptg^{\theta\theta}\hskip
1.42262pt\partial_{v}g_{\theta\theta}+\frac{1}{2}\hskip
1.42262ptg^{\phi\phi}\hskip 1.42262pt\partial_{v}g_{\phi\phi}$ (119)
$\displaystyle=F\left(u,v\right)^{-1}\hskip
1.42262pt\partial_{v}F\left(u,v\right)+2\hskip
1.42262ptr\left(u,v\right)^{-1}\hskip 1.42262pt\partial_{v}r\left(u,v\right).$
(120)

Therefore, we may write $\Theta$ as

$\Theta=2\hskip 1.42262ptF\left(u,v\right)^{-1}\hskip
1.42262ptr\left(u,v\right)^{-1}\hskip 1.42262pt\partial_{v}r\left(u,v\right).$
(121)

Recall the expression for the transverse metric $h_{\mu\nu}$ ($\mu$ and $\nu$
are Greek indices):

$h_{\mu\nu}=g_{\mu\nu}+\ell_{\mu}\hskip 1.42262ptn_{\nu}+\ell_{\nu}\hskip
1.42262ptn_{\mu}.$ (122)

With $\ell^{\mu}$ and $n^{\mu}$ defined by Equations 105 and 106, it is easy
to show that $h_{\mu\nu}$ is non-zero if and only $\mu=\nu=\theta$ or
$\mu=\nu=\phi$. Hence, we have the following expression for $h_{\mu\nu}$:

$h_{\mu\nu}=\begin{cases}g_{\theta\theta}&\textrm{if }\mu=\nu=\theta\\\
g_{\phi\phi}&\textrm{if }\mu=\nu=\phi\\\ 0&\textrm{otherwise}\end{cases}.$
(123)

Now, we seek to prove that $\sigma^{2}=0$ and $\omega^{2}=0$. The tensors
$\sigma_{\mu\nu}$ and $\omega_{\mu\nu}$ are zero unless both $\mu$ and $\nu$
are elements of the set $\left\\{\theta,\phi\right\\}$. The tensor
$\sigma_{\mu\nu}$ is symmetric in $\mu$ and $\nu$, while the tensor
$\omega_{\mu\nu}$ is anti-symmetric in $\mu$ and $\nu$. Because
$\omega_{\mu\nu}$ is anti-symmetric in $\mu$ and $\nu$,
$\omega_{\theta\theta}$ and $\omega_{\phi\phi}$ must both be zero.
Below, we prove that $\omega_{\theta\phi}=0$:

$\displaystyle\omega_{\theta\phi}$ $\displaystyle=\frac{1}{2}\hskip
1.42262pt\left(\nabla^{\sigma}\ell^{\rho}\right)\hskip
1.42262pt\left(h_{\theta\rho}\hskip 1.42262pth_{\phi\sigma}-h_{\phi\rho}\hskip
1.42262pth_{\theta\sigma}\right)$ (124) $\displaystyle=\frac{1}{2}\hskip
1.42262pt\left(\nabla^{\phi}\ell^{\theta}\right)\hskip
1.42262pth_{\theta\theta}\hskip 1.42262pth_{\phi\phi}-\frac{1}{2}\hskip
1.42262pt\left(\nabla^{\theta}\ell^{\phi}\right)\hskip
1.42262pth_{\theta\theta}\hskip 1.42262pth_{\phi\phi}$ (125)
$\displaystyle=\frac{1}{2}\hskip 1.42262pth_{\theta\theta}\hskip
1.42262pth_{\phi\phi}\hskip
1.42262pt\left(\nabla^{\phi}\ell^{\theta}-\nabla^{\theta}\ell^{\phi}\right)$
(126) $\displaystyle=\frac{1}{2}\hskip 1.42262pth_{\theta\theta}\hskip
1.42262pth_{\phi\phi}\hskip 1.42262ptg^{\theta\theta}\hskip
1.42262ptg^{\phi\phi}\hskip
1.42262pt\left(\nabla_{\phi}\ell_{\theta}-\nabla_{\theta}\ell_{\phi}\right)$
(127) $\displaystyle=\frac{1}{2}\hskip
1.42262pt\left(\partial_{\phi}\ell_{\theta}-\partial_{\theta}\ell_{\phi}\right)$
(128) $\displaystyle=0.$ (129)

Since all components of $\omega_{\mu\nu}$ are zero, the scalar $\omega^{2}$ is
also zero. Below, we prove that $\sigma_{\theta\theta}=0$:

$\displaystyle\sigma_{\theta\theta}$
$\displaystyle=\left(\nabla^{\theta}\ell^{\theta}\right)\hskip
1.42262pt\left(h_{\theta\theta}\right)^{2}-\frac{1}{2}\hskip
1.42262pth_{\theta\theta}\hskip 1.42262pt\Theta$ (130)
$\displaystyle=\nabla_{\theta}\ell_{\theta}-\frac{1}{2}\hskip
1.42262ptg_{\theta\theta}\hskip 1.42262pt\Theta$ (131)
$\displaystyle=-\Gamma_{\alpha\theta\theta}\hskip
1.42262pt\ell^{\alpha}-\frac{1}{2}\hskip 1.42262ptg_{\theta\theta}\hskip
1.42262pt\Theta$ (132) $\displaystyle=\frac{1}{2}\hskip
1.42262pt\ell^{v}\hskip
1.42262pt\partial_{v}g_{\theta\theta}-\frac{1}{2}\hskip
1.42262ptg_{\theta\theta}\hskip 1.42262pt\Theta$ (133)
$\displaystyle=F\left(u,v\right)^{-1}\hskip 1.42262ptr\left(u,v\right)\hskip
1.42262pt\partial_{v}r\left(u,v\right)-F\left(u,v\right)^{-1}\hskip
1.42262ptr\left(u,v\right)\hskip 1.42262pt\partial_{v}r\left(u,v\right)$ (134)
$\displaystyle=0.$ (135)

A similar calculation shows that $\sigma_{\phi\phi}=0$. Finally, we prove that
$\sigma_{\theta\phi}=0$:

$\displaystyle\sigma_{\theta\phi}$ $\displaystyle=\frac{1}{2}\hskip
1.42262pt\left(\nabla^{\phi}\ell^{\theta}\right)\hskip
1.42262pth_{\theta\theta}\hskip 1.42262pth_{\phi\phi}+\frac{1}{2}\hskip
1.42262pt\left(\nabla^{\theta}\ell^{\phi}\right)\hskip
1.42262pth_{\theta\theta}\hskip 1.42262pth_{\phi\phi}-\frac{1}{2}\hskip
1.42262pth_{\theta\phi}\hskip 1.42262pt\Theta$ (136)
$\displaystyle=\frac{1}{2}\hskip
1.42262pt\left(\nabla_{\theta}\ell_{\phi}+\nabla_{\phi}\ell_{\theta}\right)$
(137) $\displaystyle=-\Gamma_{\alpha\theta\phi}\hskip 1.42262pt\ell^{\alpha}$
(138) $\displaystyle=0.$ (139)

Because all components of $\sigma_{\mu\nu}$ are zero, we have proven that
$\sigma^{2}=0$. Thus, the Raychaudhuri scalar $\Theta$ obeys the equation

$\ell^{\mu}\hskip
1.42262pt\partial_{\mu}\Theta=-\frac{1}{2}\Theta^{2}-R_{\mu\nu}\hskip
1.42262pt\ell^{\mu}\ell^{\nu}.$ (140)

Let us define $T$ by $T=g^{ab}\hskip 1.42262ptT_{ab}$. Using the trace-
reversed Einstein field equations, we may write $R_{ab}$ as

$R_{ab}-\Lambda\hskip 1.42262ptg_{ab}=8\pi\hskip 1.42262ptT_{ab}-4\pi\hskip
1.42262ptT\hskip 1.42262ptg_{ab}.$ (141)

Next, we substitute the trace-reversed EFE into the Raychaudhuri equation.
Because $\ell^{\mu}$ is a null vector, the terms involving $T$ and $\Lambda$
disappear. Hence, we have the following equation:

$\ell^{\mu}\hskip
1.42262pt\partial_{\mu}\Theta=-\frac{1}{2}\Theta^{2}-8\pi\hskip
1.42262ptT_{\mu\nu}\hskip 1.42262pt\ell^{\mu}\ell^{\nu}.$ (142)

Now, we impose the null energy condition $T_{\mu\nu}\hskip
1.42262pt\ell^{\mu}\hskip 1.42262pt\ell^{\nu}\geq 0$. With this condition, it
is clear that $\ell^{\mu}\hskip 1.42262pt\partial_{\mu}\Theta$ is strictly
non-positive. In symbols,

$\ell^{\mu}\hskip 1.42262pt\partial_{\mu}\Theta\leq 0.$ (143)

Because the only non-zero component of $\ell^{\mu}$ is $\ell^{v}$, which is
positive, we know that $\partial_{v}\Theta\leq 0$.

## Appendix B: Spherically-Symmetric Einstein Field Equations with a
Cosmological Constant in Double Null Coordinates

With $\Lambda=0$, the Einstein field equations take the following form [19]:

$r_{,uv}=-\frac{r_{,u}\hskip 1.42262ptr_{,v}}{r}-\frac{F}{4r}\hskip
1.42262pt\left(1-\frac{Q^{2}}{r^{2}}\right),$ (144)
$F_{,uv}=\frac{F_{,u}\hskip 1.42262ptF_{,v}}{F}+\frac{2F}{r^{2}}\hskip
1.42262ptr_{,u}\hskip 1.42262ptr_{,v}+\frac{F^{2}}{2r^{2}}\hskip
1.42262pt\left(1-\frac{2Q^{2}}{r^{2}}\right)-2F\hskip 1.42262pt\phi_{,u}\hskip
1.42262pt\phi_{,v},$ (145) $r_{,uu}-\left(\ln F\right)_{,u}\hskip
1.42262ptr_{,u}+r\hskip 1.42262pt\left(\phi_{,u}\right)^{2}=0,$ (146)
$r_{,vv}-\left(\ln F\right)_{,v}\hskip 1.42262ptr_{,v}+r\hskip
1.42262pt\left(\phi_{,v}\right)^{2}=0.$ (147)

The scalar field $\phi$ has an associated stress-energy tensor
$T_{\alpha\beta}^{s}$ [19]:

$T_{\alpha\beta}^{s}=\frac{1}{8\pi}\hskip 1.42262pt\left(\phi_{,\alpha}\hskip
1.42262pt\phi_{,\beta}-\frac{1}{2}\hskip 1.42262ptg_{\alpha\beta}\hskip
1.42262ptg^{\gamma\delta}\hskip 1.42262pt\phi_{,\gamma}\hskip
1.42262pt\phi_{,\delta}\right).$ (148)

The charge $Q$ generates a Coulomb field around the black hole. This electric
field also has an associated stress-energy tensor
$T^{\textrm{EM}}_{\alpha\beta}$. We may write the non-zero components of
$T^{\textrm{EM}}_{\alpha\beta}$ as [19]

$T^{\textrm{EM}}_{uv}=T^{\textrm{EM}}_{vu}=\frac{Q^{2}}{16\pi}\hskip
1.42262ptF\left(u,v\right)\hskip 1.42262ptr\left(u,v\right)^{-4},$ (149)
$T^{\textrm{EM}}_{\theta\theta}=\frac{Q^{2}}{8\pi}\hskip
1.42262ptr\left(u,v\right)^{-2},$ (150)
$T^{\textrm{EM}}_{\phi\phi}=\frac{Q^{2}}{8\pi}\hskip
1.42262ptr\left(u,v\right)^{-2}\hskip 1.42262pt\sin^{2}\left(\theta\right).$
(151)

All other components of $T^{\textrm{EM}}_{\alpha\beta}$ are zero. The total
stress-energy tensor $T_{\alpha\beta}$ is the sum of the electromagnetic and
scalar field contributions.

$T_{\alpha\beta}=T^{\textrm{EM}}_{\alpha\beta}+T^{s}_{\alpha\beta}$ (152)

From Eqn. 105, we know that the only non-zero component of the null vector
field $\ell^{\alpha}$ is $\ell^{v}$. Thus, the null energy condition (Eqn. 30)
is trivially satisfied:

$\displaystyle T_{\alpha\beta}\hskip 1.42262pt\ell^{\alpha}\hskip
1.42262pt\ell^{\beta}$ $\displaystyle=T_{vv}\hskip
1.42262pt\left(\ell^{v}\right)^{2}$ (153) $\displaystyle=\frac{1}{8\pi}\hskip
1.42262pt\left(\phi_{,v}\right)^{2}\hskip 1.42262ptF\left(u,v\right)^{-2}$
(154) $\displaystyle\geq 0$ (155)

Using Eqns. 148-152, we may rewrite the Einstein field equations (Eqns.
144-147) as

$r_{,uv}+\frac{r_{,u}\hskip 1.42262ptr_{,v}}{r}+\frac{F}{4r}=4\pi\hskip
1.42262ptr\hskip 1.42262ptT_{uv},$ (156) $F_{,uv}-\frac{F_{,u}\hskip
1.42262ptF_{,v}}{F}-\frac{2F}{r^{2}}\hskip 1.42262ptr_{,u}\hskip
1.42262ptr_{,v}-\frac{F^{2}}{2r^{2}}=-\frac{8\pi\hskip
1.42262ptF^{2}}{r^{2}}\hskip 1.42262ptT_{\theta\theta},$ (157)
$r_{,uu}-\left(\ln F\right)_{,u}\hskip 1.42262ptr_{,u}=-4\pi\hskip
1.42262ptr\hskip 1.42262ptT_{uu},$ (158) $r_{,vv}-\left(\ln
F\right)_{,v}\hskip 1.42262ptr_{,v}=-4\pi\hskip 1.42262ptr\hskip
1.42262ptT_{vv}.$ (159)

Let us introduce the effective stress-energy tensor
$T^{\textrm{eff}}_{\mu\nu}$:

$T^{\textrm{eff}}_{\alpha\beta}=T_{\alpha\beta}-\frac{\Lambda}{8\pi}\hskip
1.42262ptg_{\alpha\beta}$ (160)

The tensor $T^{\textrm{eff}}_{\mu\nu}$ incorporates the effects of the
cosmological constant $\Lambda$. To account for $\Lambda$ in the Einstein
field equations, we should replace $T_{\alpha\beta}$ with
$T^{\textrm{eff}}_{\alpha\beta}$ in Eqns. 156-159. We may then write the
Einstein field equations as

$r_{,uv}+\frac{r_{,u}\hskip 1.42262ptr_{,v}}{r}+\frac{F}{4r}=4\pi\hskip
1.42262ptr\hskip 1.42262ptT_{uv}-\frac{\Lambda\hskip 1.42262ptr}{2}\hskip
1.42262ptg_{uv},$ (161) $F_{,uv}-\frac{F_{,u}\hskip
1.42262ptF_{,v}}{F}-\frac{2F}{r^{2}}\hskip 1.42262ptr_{,u}\hskip
1.42262ptr_{,v}-\frac{F^{2}}{2r^{2}}=-\frac{8\pi\hskip
1.42262ptF^{2}}{r^{2}}\hskip 1.42262ptT_{\theta\theta}+\frac{\Lambda\hskip
1.42262ptF^{2}}{r^{2}}\hskip 1.42262ptg_{\theta\theta},$ (162)
$r_{,uu}-\left(\ln F\right)_{,u}\hskip 1.42262ptr_{,u}=-4\pi\hskip
1.42262ptr\hskip 1.42262ptT_{uu},$ (163) $r_{,vv}-\left(\ln
F\right)_{,v}\hskip 1.42262ptr_{,v}=-4\pi\hskip 1.42262ptr\hskip
1.42262ptT_{vv}.$ (164)

Because $g_{uu}$ and $g_{vv}$ are both zero, there is no cosmological constant
term in Eqns. 163 and 164. From Eqns. 148-152, we know all the components of
the stress-energy tensor $T_{\alpha\beta}$, which does not include the effects
of $\Lambda$. After plugging these expressions for $T_{\alpha\beta}$ into
Eqns. 161-164, we obtain

$r_{,uv}=-\frac{r_{,u}\hskip 1.42262ptr_{,v}}{r}-\frac{F}{4r}\hskip
1.42262pt\left(1-\frac{Q^{2}}{r^{2}}\right)+\frac{\Lambda}{4}\hskip
1.42262ptr\hskip 1.42262ptF,$ (165) $F_{,uv}=\frac{F_{,u}\hskip
1.42262ptF_{,v}}{F}+\frac{2F}{r^{2}}\hskip 1.42262ptr_{,u}\hskip
1.42262ptr_{,v}+\frac{F^{2}}{2r^{2}}\hskip
1.42262pt\left(1-\frac{2Q^{2}}{r^{2}}\right)-2F\hskip 1.42262pt\phi_{,u}\hskip
1.42262pt\phi_{,v}+\Lambda\hskip 1.42262ptF^{2},$ (166) $r_{,uu}-\left(\ln
F\right)_{,u}\hskip 1.42262ptr_{,u}+r\hskip
1.42262pt\left(\phi_{,u}\right)^{2}=0,$ (167) $r_{,vv}-\left(\ln
F\right)_{,v}\hskip 1.42262ptr_{,v}+r\hskip
1.42262pt\left(\phi_{,v}\right)^{2}=0.$ (168)

## Appendix C: Nonlinear Electrodynamics and Wormhole Collapse

In this section, we generalize the analysis of the previous sections to
theories of modified gravity. Of course, it is not practical to address all
theories of modified gravity in this section. Thus, we focus on models that
couple general relativity to non-linear electrodynamics. As before, we assume
spherical symmetry and we use double null coordinates. Thus, in general double
null coordinates $\mathcal{U}$ and $\mathcal{V}$, the metric takes the form
given in Eqn. 3.

### Wormhole Collapse in a Bardeen Black Hole

The Reissner-Nordström solution arises from general relativity coupled to a
standard, Maxwellian electromagnetic field. However, it is possible to couple
general relativity to non-linear theories of electrodynamics. We focus on a
well-known black hole solution in general relativity coupled to non-linear
electrodynamics: the Bardeen black hole.
Let $F_{\mu\nu}$ be the electromagnetic field strength tensor. We define the
scalar EM field strength as

$F=\frac{1}{4}\hskip 1.42262ptF_{\mu\nu}\hskip 1.42262ptF^{\mu\nu}.$ (169)

For the Bardeen model, we have the following expression for $F$ [40]:

$F=\frac{g^{2}}{2r^{4}}.$ (170)

In the Bardeen model, the center of the black hole has mass $m$ and a magnetic
charge $g$. The Lagrangian $\mathcal{L}_{EM}$ for the electromagnetic field
depends on these parameters. We have the following expression for
$\mathcal{L}_{EM}$ [40]:

$\mathcal{L}_{EM}=\frac{3m}{\lvert g\rvert^{3}}\hskip
1.42262pt\left(\frac{\sqrt{2g^{2}\hskip 1.42262ptF}}{1+\sqrt{2g^{2}\hskip
1.42262ptF}}\right)^{\frac{5}{2}}.$ (171)

We define the function $\mathcal{L}_{EM}^{\prime}$ as

$\displaystyle\mathcal{L}_{EM}^{\prime}$
$\displaystyle=\frac{\partial\mathcal{L}_{EM}}{\partial F}$ (172)
$\displaystyle=\frac{15m}{2\lvert g\rvert}\hskip
1.42262pt\left(\frac{\sqrt{2g^{2}\hskip 1.42262ptF}}{1+\sqrt{2g^{2}\hskip
1.42262ptF}}\right)^{\frac{3}{2}}\hskip 1.42262pt\frac{1}{\sqrt{2g^{2}\hskip
1.42262ptF}\hskip 1.42262pt\left(1+\sqrt{2g^{2}\hskip
1.42262ptF}\right)^{2}}.$ (173)

From Eqns. 170 and 173, we know that $\mathcal{L}_{EM}^{\prime}$ is non-
negative for the Bardeen solution. We may write the electromagnetic stress-
energy tensor as [40]

$T^{EM}_{\mu\nu}=2\mathcal{L}_{EM}^{\prime}\hskip
1.42262ptF_{\mu\lambda}\hskip 1.42262ptF_{\nu}^{\lambda}-2g_{\mu\nu}\hskip
1.42262pt\mathcal{L}_{EM}.$ (174)

Let the null vector field $\ell^{\mu}$ be defined by Eqn. 26. It is easy to
verify that Eqn. 174 satisfies the null energy condition for all possible
configurations of the electromagnetic field. In symbols,

$\displaystyle T^{EM}_{\mu\nu}\hskip 1.42262pt\ell^{\mu}\hskip
1.42262pt\ell^{\nu}$ $\displaystyle=2\mathcal{L}_{EM}^{\prime}\hskip
1.42262ptF_{\mu\lambda}\ell^{\mu}\hskip
1.42262ptF_{\nu}^{\lambda}\ell^{\nu}-2g_{\mu\nu}\hskip
1.42262pt\ell^{\mu}\ell^{\nu}\hskip 1.42262pt\mathcal{L}_{EM}$ (175)
$\displaystyle=2\mathcal{L}_{EM}^{\prime}\hskip 1.42262pt\hskip
1.42262pt\left(F_{\mu\lambda}\ell^{\mu}\right)^{2}$ (176) $\displaystyle\geq
0.$ (177)

Let us define the function $f_{B}\left(r\right)$ as

$f_{B}\left(r\right)=1-\frac{2m\hskip
1.42262ptr^{2}}{\left(r^{2}+g^{2}\right)^{\frac{3}{2}}}.$ (178)

In spherical coordinates, the metric for a Bardeen black hole takes the form
[40]

$ds^{2}=-f_{B}\left(r\right)\hskip 1.42262ptdt^{2}+f\left(r\right)^{-1}\hskip
1.42262ptdr^{2}+r^{2}\hskip 1.42262ptd\Omega^{2}.$ (179)

Just like RN black holes, Bardeen black holes may be classified as sub-
extremal or extremal. (Neither the super-extremal RN space-time nor the super-
extremal Bardeen space-time contain any horizons.) For simplicity, we focus on
sub-extremal Bardeen black holes, which satisfy the condition [40]

$g^{2}<\frac{16}{27}\hskip 1.42262ptm^{2}.$ (180)

If Eqn. 180 is satisfied, the Bardeen solution possesses an outer event
horizon and an inner Cauchy horizon. After maximal extension (in the absence
of perturbations), the sub-extremal Bardeen solution exhibits a wormhole
similar to the wormhole that appears in the maximally-extended RN space-time
[40]. To describe the Cauchy horizon and the space-time on either side of it,
we adopt the coordinate system defined in Subsection 2.2.
Let us define the Schwarzschild mass function as [41]

$M\left(u,V\right)=\frac{1}{2}r\left(u,V\right)\hskip
1.42262pt\left(1+4F\left(u,V\right)^{-1}\hskip
1.42262ptr_{,u}\left(u,V\right)\hskip 1.42262ptr_{,V}\left(u,V\right)\right).$
(181)

Everywhere along the Cauchy horizon, the mass function $M\left(u,V\right)$
blows up [41]. In symbols,

$\lim_{V\to 0}M\left(u,V\right)=+\infty.$ (182)

Let us assume the conditions in Subsection 4.1 are satisfied, along with Eqn.
32. Then, according to Theorem 1,

$r_{,u}\left(u,V=0\right)>0\textrm{ for all }u<u_{1}.$ (183)

In Subsection 4.1, we assumed that $r_{,u}\left(u,V\right)$ is continuously
differentiable (and thus finite) at the Cauchy horizon. Thus, in order for
Eqns. 182 and 183 to simultaneously be true, the following must be true:

$\lim_{V\to 0}r_{,V}\left(u,V\right)=+\infty.$ (184)

Thus, all the preconditions for Theorem 3 are satisfied. This implies that the
wormhole in the maximally-extended Bardeen solution collapses, and it is
impossible for an observer to pass through it.

### Generalization to Other Models of Nonlinear Electrodynamics

Of course, Eqn. 171 is not the only possible expression for
$\mathcal{L}_{EM}$. Indeed, it is possible to couple general relativity to
electrodynamics regardless of the expression for any $\mathcal{L}_{EM}$. Some
of these models exhibit solutions similar to the Bardeen black hole [42, 43,
44]. In the sub-extremal case, each of these space-times contains an event
horizon and an inner Cauchy horizon. After maximal extension, these space-
times contain wormholes and parallel universe regions.
Let $F$ be defined as in Eqn. 169. For a general model of non-linear
electrodynamics, we may express the electromagnetic Lagrangian
$\mathcal{L}_{EM}$ as a function of the scalar $F$. In symbols,

$\mathcal{L}_{EM}=\mathcal{L}_{EM}\left(F\right).$ (185)

Let us consider general relativity coupled to a model of non-linear
electrodynamics with Lagrangian $\mathcal{L}_{EM}\left(F\right)$. Furthermore,
let us assume that this model has a spherically-symmetric black hole solution
with an event horizon, a Cauchy horizon, and a wormhole region. If the
conditions in Subsections 4.1 and 4.2 are true, we may complete the proof of
Theorem 3 only if the null energy condition holds.
In Eqns. 154-155, we showed that a massless Klein-Gordon scalar field
satisfies the null energy condition. Thus, we may assume the null energy
condition holds for the total stress-energy tensor if it holds for the EM
stress-energy tensor. The electromagnetic stress-energy tensor is still

$T^{EM}_{\mu\nu}=2\mathcal{L}_{EM}^{\prime}\hskip
1.42262ptF_{\mu\lambda}\hskip 1.42262ptF_{\nu}^{\lambda}-2g_{\mu\nu}\hskip
1.42262pt\mathcal{L}_{EM}.$ (186)

Let $\mathcal{L}_{EM}^{\prime}\left(F\right)$ be defined as in Eqn. 172. As
long as $\mathcal{L}_{EM}^{\prime}\geq 0$, the EM stress-energy tensor
satisfies the null energy condition. In symbols,

$T^{EM}_{\mu\nu}\hskip 1.42262pt\ell^{\mu}\hskip 1.42262pt\ell^{\nu}\geq 0.$
(187)

Thus, as long as $\mathcal{L}_{EM}^{\prime}\geq 0$, all the sufficient
conditions for Theorem 3 are met. Thus, the wormhole inside the Cauchy horizon
collapses, and it is impossible for an observer to reach the parallel universe
regions of the unperturbed maximally-extended space-time.

## References

  * [1] Brandon Carter  “Global Structure of the Kerr Family of Gravitational Fields”  In _Phys. Rev._ 174 American Physical Society, 1968, pp. 1559–1571  DOI: 10.1103/PhysRev.174.1559
  * [2] M. Dafermos  “The interior of charged black holes and the problem of uniqueness in general relativity”  In _Communications on Pure and Applied Mathematics_ 58.4, 2003, pp. 445–504
  * [3] G.. Jeffery  “The Field of an Electron on Einstein’s Theory of Gravitation”  In _Proc. R. Soc. Lond. A_ 99.697, 1921, pp. 123–134
  * [4] M. Simpson and R. Penrose  “Internal instability in a Reissner-Nordström black hole”  In _Internatl. J. Theor. Phys._ 7, 1973, pp. 183–197
  * [5] Mihalis Dafermos  “Stability and instability of the Cauchy horizon for the spherically symmetric Einstein-Maxwell-scalar field equations”  In _Annals of Mathematics_ 158, 2003, pp. 875–928
  * [6] Vitor Cardoso et al.  “Quasinormal Modes and Strong Cosmic Censorship”  In _Phys. Rev. Lett._ 120 American Physical Society, 2018, pp. 031103  DOI: 10.1103/PhysRevLett.120.031103
  * [7] Vitor Cardoso et al.  “Strong cosmic censorship in charged black-hole spacetimes: Still subtle”  In _Phys. Rev. D_ 98 American Physical Society, 2018, pp. 104007  DOI: 10.1103/PhysRevD.98.104007
  * [8] João Costa, Pedro Girão, Jose Natario and Jorge Silva  “On the global uniqueness for the Einstein–Maxwell-scalar field system with a cosmological constant: I. Well posedness and breakdown criterion”  In _Classical and Quantum Gravity_ 32, 2015  DOI: 10.1088/0264-9381/32/1/015017
  * [9] João Costa, Pedro Girão, Jose Natario and Jorge Silva  “On the Global Uniqueness for the Einstein–Maxwell-Scalar Field System with a Cosmological Constant: Part 2. Structure of the Solutions and Stability of the Cauchy Horizon”  In _Communications in Mathematical Physics_ 339, 2015, pp. 903–947  DOI: 10.1007/s00220-015-2433-6
  * [10] João Lopes Costa, Pedro M. Girāo, José Natário and Jorge Drumond Silva  “On the Global Uniqueness for the Einstein–Maxwell-Scalar Field System with a Cosmological Constant: Part 3. Mass Inflation and Extendibility of the Solutions”  In _Annals of PDE_ 3, 2014, pp. 1–55
  * [11] João L. Costa, Pedro M. Girāo, José Natário and Jorge Drumond Silva  “On the Occurrence of Mass Inflation for the Einstein–Maxwell-Scalar Field System with a Cosmological Constant and an Exponential Price Law”  In _Communications in Mathematical Physics_ 361, 2018, pp. 289–341
  * [12] William A. Hiscock  “Evolution of the interior of a charged black hole”  In _Physics Letters A_ 83.3, 1981, pp. 110–112  DOI: https://doi.org/10.1016/0375-9601(81)90508-9
  * [13] S. Chandrasekhar and J.. Hartle  “On Crossing the Cauchy Horizon of a Reissner-Nordström Black-Hole”  In _Proc. R. Soc. Lond. A_ 384.1787, 1982, pp. 301–315
  * [14] Eric Poisson and Werner Israel  “Internal Structure of black holes”  In _Physical Review D_ 41.6, 1990, pp. 1796–1809
  * [15] Patrick R. Brady and Eric Poisson  “Cauchy horizon instability for Reissner-Nordstrom black holes in de Sitter space”  In _Class. Quantum Grav._ 9, 1992, pp. 121–125
  * [16] P. Brady and J. Smith  “Black Hole Singularities: A Numerical Approach”  In _Phys. Rev. Letters_ 75.7, 1995, pp. 1256–1259
  * [17] Amos Ori and Éanna É. Flanagan  “How generic are null spacetime singularities?”  In _Phys. Rev. D_ 53.4, 1996
  * [18] Lior M. Burko  “Structure of the Black Hole’s Cauchy-Horizon Singularity”  In _Phys. Rev. Lett._ 79 American Physical Society, 1997, pp. 4958–4961  DOI: 10.1103/PhysRevLett.79.4958
  * [19] Lior M. Burko and Amos Ori  “Analytic study of the null singularity inside spherical charged black holes”  In _Phys. Rev. D_ 57.12, 1998, pp. R7084–R7088
  * [20] Lior M. Burko  “Singularity deep inside the spherical charged black hole core”  In _Phys. Rev. D_ 59 American Physical Society, 1998, pp. 024011  DOI: 10.1103/PhysRevD.59.024011
  * [21] H. Maeda, T. Trorli and T. Harada  “Novel Cauchy-horizon instability”  In _Phys. Rev. D_ 71.064015, 2005
  * [22] D. Marolf and A. Ori  “Outgoing gravitational shock-wave at the inner horizon: The late-time limit of black hole interiors”  In _Phys. Rev. D_ 86.12, 2012, pp. 124026
  * [23] Yekta Gürsel, Vernon D. Sandberg, Igor D. Novikov and A.. Starobinsky  “Evolution of scalar perturbations near the Cauchy horizon of a charged black hole”  In _Phys. Rev. D_ 19 American Physical Society, 1979, pp. 413–420  DOI: 10.1103/PhysRevD.19.413
  * [24] Amos Ori  “Evolution of perturbations inside a charged black hole: Linear scalar field”  In _Phys. Rev. D_ 55 American Physical Society, 1997, pp. 4860–4871  DOI: 10.1103/PhysRevD.55.4860
  * [25] Adnan Malik, Asifa Ashraf, Uzma Naqvi and Zhiyue Zhang  “Anisotropic spheres via embedding approach in f(R) gravity”  In _International Journal of Geometric Methods in Modern Physics_ 19.05, 2022, pp. 2250073  DOI: 10.1142/S0219887822500736
  * [26] Adnan Malik, Iftikhar Ahmad and Kiran  “A study of anisotropic compact stars in f(R,$\phi$,X) theory of gravity”  In _International Journal of Geometric Methods in Modern Physics_ 19.02, 2022, pp. 2250028  DOI: 10.1142/S0219887822500281
  * [27] M. Shamir and Adnan Malik  “Bardeen compact stars in modified f(R) gravity”  In _Chinese Journal of Physics_ 69, 2021, pp. 312–321  DOI: https://doi.org/10.1016/j.cjph.2020.12.009
  * [28] Ammara Usman and M. Shamir  “Collapsing stellar structures in f(R) gravity using Karmarkar condition”  In _New Astronomy_ 91, 2022, pp. 101691  DOI: https://doi.org/10.1016/j.newast.2021.101691
  * [29] Mushtaq Ahmad, G. Mustafa and M. Shamir  “A comparative analysis of self-consistent charged anisotropic spheres”  In _International Journal of Modern Physics A_ 36.27, 2021, pp. 2150203  DOI: 10.1142/S0217751X21502031
  * [30] Hao Tang, Bin Wu, Rui-Hong Yue and Cheng-Yi Sun  “Entropy of Vaidya Black Hole on Event Horizon with Generalized Uncertainty Principle Revisited”  In _Communications in Theoretical Physics_ 71.1 IOP Publishing, 2019, pp. 075  DOI: 10.1088/0253-6102/71/1/75
  * [31] Simone Speziale  “Raychaudhuri and optical equations for null geodesic congruences with torsion”  In _Phys. Rev. D_ 98 American Physical Society, 2018, pp. 084029  DOI: 10.1103/PhysRevD.98.084029
  * [32] Chris M. Chambers  “The Cauchy horizon in black hole de sitter space-times”  Israel Phys. Soc., 1997, pp. 33–84
  * [33] William F. Trench  “Introduction to real analysis”  Pearson College Div, 2003
  * [34] T.. Helliwell and D.. Konkowski  “Stability of the Cauchy horizon in anti–de Sitter spacetime”  In _Phys. Rev. D_ 51 American Physical Society, 1995, pp. 5517–5521  DOI: 10.1103/PhysRevD.51.5517
  * [35] D.. Konkowski and T.. Helliwell  “Improved Cauchy horizon stability conjecture”  In _Phys. Rev. D_ 54 American Physical Society, 1996, pp. 7898–7901  DOI: 10.1103/PhysRevD.54.7898
  * [36] Stefanos Aretakis  “Stability and Instability of Extreme Reissner-Nordström Black Hole Spacetimes for Linear Scalar Perturbations I”  In _Commun. Math. Phys._ 307.17, 2011  URL: https://doi.org/10.1007/s00220-011-1254-5
  * [37] Stefanos Aretakis  “Stability and Instability of Extreme Reissner-Nordström Black Hole Spacetimes for Linear Scalar Perturbations II”  In _Ann. Henri Poincaré_ 12, 2011, pp. 1491–1538  URL: https://doi.org/10.1007/s00023-011-0110-7
  * [38] James Lucietti, Keiju Murata, Harvey S. Reall and Norihiro Tanahashi  “On the horizon instability of an extreme Reissner-Nordström black hole”  In _Ann. Henri Poincaré_ 12, 2011, pp. 1491–1538  URL: https://doi.org/10.1007/s00023-011-0110-7
  * [39] Keiju Murata, Harvey S Reall and Norihiro Tanahashi  “What happens at the horizon(s) of an extreme black hole?”  In _Classical and Quantum Gravity_ 30.23 IOP Publishing, 2013, pp. 235007  DOI: 10.1088/0264-9381/30/23/235007
  * [40] Eloy Ayon-Beato and Alberto Garcia  “The Bardeen model as a nonlinear magnetic monopole”  In _Physics Letters B_ 493.1, 2000, pp. 149–152  DOI: https://doi.org/10.1016/S0370-2693(00)01125-4
  * [41] Alfio Bonanno, Amir-Pouyan Khosravi and Frank Saueressig  “Regular black holes with stable cores”  In _Phys. Rev. D_ 103 American Physical Society, 2021, pp. 124027  DOI: 10.1103/PhysRevD.103.124027
  * [42] Eloy Ayon-Beato and Alberto Garcia  “Regular Black Hole in General Relativity Coupled to Nonlinear Electrodynamics”  In _Phys. Rev. Lett._ 80 American Physical Society, 1998, pp. 5056–5059  DOI: 10.1103/PhysRevLett.80.5056
  * [43] Eloy Ayon-Beato and Alberto Garcia  “New regular black hole solution from nonlinear electrodynamics”  In _Physics Letters B_ 464.1, 1999, pp. 25–29  DOI: https://doi.org/10.1016/S0370-2693(99)01038-2
  * [44] Manuel E. Rodrigues and Marcos V. S.  “Bardeen regular black hole with an electric source”  In _Journal of Cosmology and Astroparticle Physics_ 2018.06 IOP Publishing, 2018, pp. 025–025  DOI: 10.1088/1475-7516/2018/06/025