# Cylindrical black bounces and their field sources

Kirill A. Bronnikov<EMAIL_ADDRESS>Center of Gravitation and Fundamental
Metrology, VNIIMS, Ozyornaya ulitsa 46, Moscow 119361, Russia  Peoples’
Friendship University of Russia, ulitsa Miklukho-Maklaya 6, Moscow, 117198,
Russia  National Research Nuclear University “MEPhI”, Kashirskoe shosse 31,
Moscow 115409, Russia  Manuel E. Rodrigues<EMAIL_ADDRESS>Faculdade de
Ciências Exatas e Tecnologia, Universidade Federal do Pará
Campus Universitário de Abaetetuba, 68440-000, Abaetetuba, Pará, Brazil
Faculdade de Física, Programa de Pós-Graduação em Física, Universidade Federal
do Pará, 66075-110, Belém, Pará, Brazil  Marcos V. de S. Silva
<EMAIL_ADDRESS>Faculdade de Física, Universidade Federal do Pará, Campus
Universitário de Salinópolis, 68721-000, Salinópolis, Pará, Brazil

(today; LaTeX-ed August 28, 2024)

###### Abstract

We apply the Simpson-Visser phenomenological regularization method to a
cylindrically symmetric solution of the Einstein-Maxwell equations known as an
inverted black hole. In addition to analyzing some properties of thus
regularized space-time, including the Carter-Penrose diagrams, we show that
this solution can be obtained from the Einstein equations with a source
combining a phantom scalar field with a nonzero self-interaction potential and
a nonlinear magnetic field. A similar kind of source is obtained for the
cylindrical black bounce solution proposed by Lima et al. as a regularized
version of Lemos’s black string solution. Such sources are shown to be
possible for a certain class of cylindrically, planarly and toroidally
symmetric metrics that includes the regularized solutions under consideration.

###### pacs:

04.50.Kd,04.70.Bw

##  I Introduction

Space-time singularities are known to be an undesired though common feature of
general relativity (GR) and other classical theories of gravity. A natural
hope that all singularities will be inevitably suppressed by quantum gravity
effects is made somewhat uncertain by the fact that different models and
approaches of quantum gravity, being translated to the classical language,
generally lead to quite different results, see, e.g., QG1 ; QG2 ; QG3 ;
kelly20 ; achour20 ; alonso22 ; kunst20 ; ash20a ; bambi13 , see also a
discussion in we20 . One can find among them black hole-white hole transitions
QG2 ; QG3 ; kelly20 ; achour20 ; alonso22 , scenarios where the spherical
radius beyond a black hole horizon tends to a constant positive value kunst20
, configurations having no horizons at all bambi13 , etc. Such a diversity
evidently indicates a still unfinished situation in developing quantum gravity
by now.

It is therefore natural that attempts to simulate the possible effects of
quantum gravity in the framework of classical space-time, leaving aside the
details of quantization, cause rather large interest. The recent proposal of
Simpson and Visser (SV) simp18  can be regarded as one of such attempts: they
regularized the Schwarzschild metric by replacing the spherical radius $r$
with the nonzero expression $r(u)=\sqrt{u^{2}+b^{2}}$ ($b>0$ is a
regularization parameter), thus avoiding the Schwarzschild singularity at
$r=0$. With different values of $b$, the resulting geometry can represent a
wormhole (if $b>2m$, where $m$ is the Schwarzschild mass), a black hole with
two horizons if $b<2m$, and an intermediate case of an extremal black hole
with a single horizon if $b=2m$. In the black hole case $b<2m$, the sub-
horizon region is a Kantowski-Sachs type cosmology in which $u$ is a temporal
coordinate, and the hypersurface $u=0$, where the radius $r(u)$ has a regular
minimum, is actually a bounce of one of the two scale factors. This phenomenon
was termed a black bounce simp18 . One can also recall that such black bounces
are a common feature of one more class of space-time models, called black
universes, that is, black holes in which the internal region is a Kantowski-
Sachs cosmology with late-time isotropization bk-uni05 ; bk-uni06 ; bk-uni12 .
Such solutions have been obtained in GR with a phantom scalar field as well as
in some scalar-tensor theories clem09 ; azreg11 ; trap18 . In the case $b=2m$,
the hypersurface $u=0$ is null, being simultaneously a black hole horizon and
a throat, thus it may be called a black throat rahul22 .

Later on, a similar regularization was considered for the Reissner-Nordström
space-time franzin21 . Lobo et al. lobo20  constructed a large class of
regular black hole and wormhole space-times. The diversity and richness of
geometries obtained in this manner have attracted much attention, and their
rotating extensions and those with a NUT charge were also obtained and studied
mazza21 ; xu21 ; shaikh21 ; kam23 ; barr22 . A further analysis concerned
gravitational wave echoes at possible black hole/wormhole transitions,
quasinormal modes and gravitational lensing in such space-times churilova19 ;
yang21 ; guerrero21 ; tsukamoto21 ; islam21 ; cheng21 ; kb20 ; tsukamoto20 ;
lima21 ; nascimento20 ; franzin22 .

A question of interest emerging in this connection is whether these new
regular geometries can be presented as solutions of GR with some reasonable
sources, such as, for example, scalar and/or electromagnetic fields. As shown
in rahul22  and canate22 ; rodrigues23 , a large class of SV-like regular
metrics, including the regularized Schwarzschild and Reissner-Nordström
metrics, can be viewed as exact solutions to the Einstein equations with a sum
of two stress-energy tensors (SETs): that of a minimally coupled self-
interacting phantom scalar field and that of a nonlinear electromagnetic field
in the framework of nonlinear electrodynamics (NED) with a Lagrangian as a
function of the invariant $F=F_{\mu\nu}F^{\mu\nu}$. A scalar field or NED
taken separately cannot be such a source due to algebraic properties of their
SETs. For regularized Schwarzschild and Reissner-Nordström solutions, the
explicit forms of scalar and NED constructed were found as well as their
global structure diagrams for the cases of three and four horizons. Similar
regularizations for some cosmological models were considered in kam22 .

In kb22s , similar regularizations were obtained for two other families of
singular static, spherically symmetric solutions of GR: Fisher’s solution with
a massless scalar field fisher48  and a special subset of dilatonic black hole
solutions whose source consists of interacting scalar and electromagnetic
fields dil1 ; dil2 ; dil3 ; dil4 . In both cases, the SV trick
($x\mapsto\sqrt{u^{2}+b^{2}}$) is applied in the simplest way to a factor $x$
producing a space-time singularity at its zero value. Scalar-NED sources have
also been found in kb22s  for regularized versions of these space-times, and
it turned out that the necessary scalar fields cannot be only canonical
(possessing positive kinetic energy) or only phantom (having negative kinetic
energy), but change their nature from one region to another in a regular
manner. A situation where a scalar is phantom in a strong-field region and
canonical elsewhere has been termed a “trapped ghost” trap10 , and some
globally regular black hole and wormhole solutions with such fields were
obtained kroger04 ; trap10 ; don11 .

As a by-product, it has been shown in kb22s  that a sum of NED and minimally
coupled scalar SETs can provide a source for any static, spherically symmetric
metric, but in general the scalar can somewhere change its nature from
canonical to phantom. A way to describe an arbitrary static, spherically
symmetric metric using a single field source was found in kb23  in the
framework of the general (Bergmann-Wagoner-Nordtvedt) scalar-tensor theory of
gravity, but such a description turned out to be possible only piecewise. This
was demonstrated using as examples the Reissner-Nordström metric and the
regularized Schwarzschild metric according to simp18 .

One more family of black hole space-times, those with cylindrical symmetry,
called black strings lemos95 ; cardoso01 , with the metric

$ds^{2}=A(r)dt^{2}-\frac{dr^{2}}{A(r)}-r^{2}(dz^{2}+d\varphi^{2}),\qquad
A(r)=\alpha^{2}r^{2}-\frac{b}{\alpha r},\qquad\alpha,b={\rm const}>0,$ (1)

singular at $r=0$, was also regularized in the manner of simp18 , by replacing
$r\to\sqrt{u^{2}+a^{2}}$, $a>0$, with, however, $dr\to du$ lima22 . (Note that
$r,u,a,\alpha^{-1}$ have here the dimension of length while $z$ and $b$ are
dimensionless.111We adopt the metric signature $(+,-,-,-)$ and use the
geometric units where $8\pi G=c=1$.) The resulting nonsingular metric has the
form

$ds^{2}=A(u)dt^{2}-\frac{du^{2}}{A(u)}-(u^{2}+a^{2})(dz^{2}+d\varphi^{2}),\qquad
A(u)=\alpha^{2}(u^{2}+a^{2})-\frac{b}{\alpha\sqrt{u^{2}+a^{2}}}.$ (2)

Both metrics (1) and (2) are asymptotically AdS at large $r$ or $u$, with the
cosmological constant $\Lambda=-\alpha^{2}/3$. A further study of the metric
(2) in lima22  concerned the energy conditions, the thermodynamic properties
of regular black strings, and possible stable and unstable circular orbits of
photons and massive particles.

In the present paper, we study a similar regularization of another static,
cylindrically symmetric metric of interest, called “inverted black holes” and
being a special solution to the Einstein-Maxwell equations kb79 ;
Bronnikov:2019clf . The metric has the form

$ds^{2}=A(x)dt^{2}-\frac{dx^{2}}{A(x)}-\frac{q^{2}x^{2}}{4k^{2}}(dz^{2}+d\varphi^{2}),\qquad
A(x)=\frac{16k^{4}(1-x)}{q^{2}x^{2}},$ (3)

where $q$ (characterizing the electric or magnetic charge density) and $k$ are
positive constants. The name “inverted black holes” was proposed in kb79
because, contrary to “normal” black holes, a static region $x<1$ in (3) with a
singularity at $x=0$ occurs at smaller values of the circular radius
$r(x)=qx/(2k)$, inside the horizon $x=1$, while the region $x>1$ where $A<0$
is nonstatic and represents a special kind of Bianchi-type I cosmology.

We regularize the metric (3) as before, replacing $x\to\sqrt{u^{2}+a^{2}}$, to
obtain

$ds^{2}=A(u)dt^{2}-\frac{du^{2}}{A(u)}-\frac{q^{2}(u^{2}+a^{2})}{4k^{2}}(dz^{2}+d\varphi^{2}),\qquad
A(u)=\frac{16k^{4}(1-\sqrt{u^{2}+a^{2}})}{q^{2}(u^{2}+a^{2})},$ (4)

and study its properties in a manner similar to lima22 .

In addition to such regularizations, we construct the Carter-Penrose global
structure diagrams and determine possible field sources for both regularized
metrics (2) and (4) in the framework of GR similarly to rahul22  in terms of a
combination of NED and a self-interacting scalar field.

It is necessary to mention that all the metrics (1)–(4) can be interpreted not
only as cylindrically symmetric ones (such that $z\in{\mathbb{R}}$ and
$\varphi\in[0,2\pi)$) but also in terms of two other kinds of symmetry:
planar, if both $z\in{\mathbb{R}}$ and $\varphi\in{\mathbb{R}}$, and toroidal,
if both $z$ and $\varphi$ range on finite segments with identified ends. All
local quantities discussed in this paper do not depend on these topological
assumptions. We will adhere to interpretations in terms of cylindrical
symmetry that seems to be of greater interest than the other two.

The paper is organized as follows. We begin the next Sec. II with giving some
general relations for cylindrically symmetric metrics and then consider some
properties of the regularized metrics (2) and (4). Section III is devoted to
finding and discussing field sources for these metrics, and Sec. IV is a
conclusion.

##  II Regularized metrics

###  II.1 General relations

In general, the line element that describes static space-times with
cylindrical symmetry is written as

$ds^{2}=A(x)dt^{2}-B(x)dx^{2}-C(x)dz^{2}-D(x)d\varphi^{2}.$ (5)

However, in this paper we are dealing with space-times possessing locally flat
orbits of the spatial isometry group, such that $C(x)=D(x)$, and using the
coordinate condition $B(x)=1/A(x)$, the metric can be written in the form222In
the whole paper, to avoid confusion, we denote by $r$ the quantity
$\sqrt{-g_{22}}=\sqrt{-g_{33}}$ that has a clear geometric meaning of a scale
factor of 2-surfaces (cylinders) parametrized by $x^{2}=z$ and $x^{3}=\varphi$
(similar to the spherical radius $r$ in the case of spherical symmetry). For
the radial coordinate $x^{1}$ we use other letters, $x$ or $u$.

$ds^{2}=A(x)dt^{2}-\frac{dx^{2}}{A(x)}-r^{2}(x)(dz^{2}+d\varphi^{2}),$ (6)

where we assume $z\in{\mathbb{R}}$ and $\varphi\in[0,2\pi)$ according to
cylindrical symmetry. The nonzero Riemann tensor components are

$\displaystyle K_{1}=R^{01}{}_{01}=-\frac{1}{2}A^{\prime\prime},\qquad\hskip
72.26999pt$ $\displaystyle
K_{2}=R^{02}{}_{02}=R^{03}{}_{03}=-\frac{A^{\prime}r^{\prime}}{2r},$
$\displaystyle{}K_{3}=R^{12}{}_{12}=R^{13}{}_{13}=-\frac{2Ar^{\prime\prime}+A^{\prime}r^{\prime}}{2r},\qquad$
$\displaystyle K_{4}=R^{23}{}_{23}=-\frac{Ar^{\prime 2}}{r^{2}},$ (7)

where the coordinates are numbered as $(t,x,z,\varphi)=(0,1,2,3)$, and primes
denote $d/dx$. Since the Kretschmann invariant
${\mathcal{K}}=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}=4K_{1}^{2}+8K_{2}^{2}+8K_{3}^{2}+4K_{4}^{2}$
is a sum of squares, it is clear that the finiteness of all $K_{i}$ from
(II.1) is a necessary and sufficient condition for finiteness of all algebraic
invariants of the Riemann tensor.

Also, in full similarity with the more familiar case of spherical symmetry,
regular zeros of $A(x)$ (provided $r\neq 0$) correspond to Killing horizons at
which the Killing vector $\partial_{t}$ that is timelike at $A>0$, becomes
null. At such horizons all $K_{i}$ are finite and well-behaved, thus
illustrating the well-known fact that Killing horizons are regular surfaces.

Let us also present the expressions for nonzero components of the Einstein
tensor $G_{\mu}^{\nu}=R_{\mu}^{\nu}-\frac{1}{2}\delta_{\mu}^{\nu}R$, to be
used in the Einstein equations $G_{\mu}^{\nu}=-T_{\mu}^{\nu}$, where
$T_{\mu}^{\nu}$ is the stress-energy tensor of matter:

$\displaystyle\
{}G^{0}_{0}=\frac{1}{r^{2}}\Big{[}A(2rr^{\prime\prime}+r^{\prime
2})+A^{\prime}rr^{\prime}\Big{]},$ $\displaystyle\
{}G^{1}_{1}=\frac{1}{r^{2}}\Big{[}Ar^{\prime 2}+A^{\prime}rr^{\prime}\Big{]},$
$\displaystyle\
{}G^{2}_{2}=G^{3}_{3}=\frac{1}{r}\Big{[}Ar^{\prime\prime}+A^{\prime}r^{\prime}+\frac{1}{2}A^{\prime\prime}r\Big{]}.$
(8)

###  II.2 The regularized black string

Let us begin with the metrics (1) and (2), already considered in lima22 . In
terms of (6), in the metric (1) we have

$r(x)=x,\qquad A(x)=\alpha^{2}x^{2}-b/(\alpha x).$ (9)

Outside the horizon, at $x>x_{h}=b^{1/3}/\alpha$, there is a static (R-)
region with an AdS-like asymptotic behavior at large $x$. Inside the horizon,
at $0<x<x_{h}$, there is a dynamic (T-) region with a special Bianchi-type I
geometry and a singularity at $x=0$.

The regularized metric (2), with $u\in{\mathbb{R}}$, has the same AdS-like
behavior at large $r$ as the original one, but now this happens at both limits
$u\to\pm\infty$ since $r=\sqrt{u^{2}+a^{2}}$.

The global properties of space-time (2) depend on the value of the
regularization parameter $a$:

  * •

If $a<x_{h}$, it is a regular black hole, but now, unlike (1) (see diagram (a)
in Fig. 1), it has two horizons at $u=u_{h\pm}=\pm\sqrt{x_{h}^{2}-a^{2}}$
(diagram (b) in Fig. 1). At $u=0$ there is a minimum of $r(u)$ in a T-region,
in other words, a black bounce.

  * •

If $a=x_{h}$, it is a regular black hole with a single extremal horizon at
$u=0$, it separates two R-regions (diagram (c) in Fig. 1).

  * •

If $a>x_{h}$, it is a cylindrically symmetric wormhole with a throat at $u=0$.

Figure 1: Carter-Penrose diagrams for the black string metric (1) [panel (a)]
and its regularized version (2): panel (b) for $a<x_{h}$, and panel (c) for
$a=x_{h}$. All inner lines in the diagrams depict horizons, the letters R and
T mark static (R-) and dynamic (T-) regions, respectively. The dashed line in
panel (b) shows the black bounce instant $u=0$. Diagrams (b) and (c) are
infinitely continued upward and downward.

The physical properties of space-time (2) are discussed in detail in lima22 .

###  II.3 The regularized inverted black hole

The inverted black hole line element (3) contains the metric functions in
terms of (6)

$A(x)=\frac{16(1-x)k^{4}}{q^{2}x^{2}},\qquad
r^{2}(x)=\frac{q^{2}x^{2}}{4k^{2}}>0.$ (10)

This space-time contains a horizon at $x=1$ and a singularity at $x=0$, where
$r\to 0$ (a singular axis in cylindrically symmetric space-time that may be
interpreted as a charged thread repelling neutral particles since
$g_{00}=A(x)\to\infty$ there). The horizon at $x=1$ has a common feature with
a de Sitter horizon in that the metric has a cosmological, time-dependent
nature at larger values of $r$. The “far end”, $x\to\pm\infty$, corresponds to
an infinitely remote past or future in which the universe is highly
anisotropic ($x$ is a time coordinate there, the scale factor in the $t$
direction, now spatial, is $a_{t}\sim|x|^{-1/2}\to 0$, while the other two
scale factors are $a_{z}=a_{\varphi}\sim|x|\to\infty$). The corresponding
global causal structure diagram is presented in Fig. 2, panel (a).

In the regularized metric (4), we have, in terms of the new coordinate $u$,

$A(u)=\frac{16k^{4}\big{(}1-\sqrt{u^{2}+a^{2}}\big{)}}{q^{2}(u^{2}+a^{2})},\qquad
r^{2}(u)=\frac{q^{2}(u^{2}+a^{2})}{4k^{2}},\qquad u\in{\mathbb{R}}.$ (11)

As was the case with the black string metric, the global properties of this
space-time depend on the value of the parameter $a$:

  * •

If $a<1$, we obtain a cylindrically symmetric wormhole with a throat at $u=0$,
surrounded by two cosmological-type horizons at $u=u_{h\pm}=\pm\sqrt{1-a^{2}}$
(see diagram (b) in Fig. 2).

  * •

If $a=1$, it is a regular Bianchi-type I cosmological model with a single
extremal horizon at $u=0$ that separates two T-regions (diagram (c) in Fig.
2).

  * •

If $a>1$, it is a regular Bianchi-type I cosmological model with a bounce
(minimum of $r$) at $u=0$.

Figure 2: Carter-Penrose diagrams for the inverted black hole metric (3)
[panel (a)] and its regularized version (4): panel (b) for $a<1$, and panel
(c) for $a=1$. All inner solid lines in the diagrams depict horizons, the
letters R and T mark static (R-) and dynamic (T-) regions, respectively. The
dashed line in panel (b) shows the wormhole throat $u=0$. Diagrams (b) and (c)
are infinitely continued to the left and to the right.

By substituting (11) to the expressions (II.1) for $K_{i}$ it is
straightforward to verify that this metric is globally regular. Still we
present, for completeness, the corresponding expression for the Kretschmann
scalar:

$\mathcal{K}=\frac{256k^{8}\left[-3a^{4}(u^{2}-4)-40a^{2}u^{2}+9a^{6}+4u^{4}(3u^{2}+14)\right]}{q^{4}(a^{2}+u^{2})^{6}}-\frac{1024k^{8}(-11a^{2}u^{2}+5a^{4}+12u^{4})}{q^{4}(a^{2}+u^{2})^{11/2}},$
(12)

where it is evident, in particular, that ${\mathcal{K}}$ is finite at $u=0$
and decays as $u^{-6}$ at large $|u|$.

Substituting the quantities (11) to (II.1), we find the components of the
Einstein tensor:

$\displaystyle G^{0}_{0}$ $\displaystyle=$
$\displaystyle-\frac{16k^{4}\big{[}2a^{2}(\sqrt{a^{2}+u^{2}}-1)+u^{2}\big{]}}{q^{2}(a^{2}+u^{2})^{3}},$
(13) $\displaystyle G^{1}_{1}$ $\displaystyle=$
$\displaystyle-\frac{16k^{4}u^{2}}{q^{2}(a^{2}+u^{2})^{3}},$ (14)
$\displaystyle G^{2}_{2}$ $\displaystyle=$
$\displaystyle\frac{8k^{4}\big{(}2u^{2}-a^{2}\sqrt{a^{2}+u^{2}}\big{)}}{q^{2}(a^{2}+u^{2})^{3}}.$
(15)

Using the Einstein equations, we can find the effective stress-energy tensor
that can be a material source of the metric, and analyze the fulfillment or
violation of various energy conditions by a source of metric under
consideration.

It is, however, necessary to bear in mind that only in a static region ($A>0$)
we deal with the usual relations

$\rho=-G^{0}_{0},\qquad p_{r}=G^{1}_{1},\qquad p_{\bot}=G^{2}_{2},$ (16)

where $\rho$ is the density, $p_{r}$ the radial pressure, and $p_{\bot}$ the
tangential pressure. In T-regions, where $A<0$, we must identify

$\rho=-G^{1}_{1},\qquad p_{r}=G^{0}_{0},\qquad p_{\bot}=G^{2}_{2}.$ (17)

As we are working with solutions containing throats or bounces that require
exotic sources, it is of interest to analyze the fulfillment of the null
(NEC), weak (WEC), strong (SEC) and dominant (DEC) energy conditions: They can
be presented as follows:

$\displaystyle{\rm NEC}_{1,2}={\rm WEC}_{1,2}={\rm
SEC}_{1,2}\Longleftrightarrow\rho+p_{r,\bot}\geq 0,$ (18) $\displaystyle{\rm
SEC}_{3}\Longleftrightarrow\rho+p_{r}+2p_{\bot}\geq 0,$ (19)
$\displaystyle{\rm DEC}_{1,2}\Longrightarrow\rho-p_{r,\bot}\geq
0\quad\mbox{and}\quad\rho+p_{r,\bot}\geq 0,$ (20) $\displaystyle{\rm
DEC}_{3}={\rm WEC}_{3}\Longleftrightarrow\rho\geq 0.$ (21)

In essence, all energy conditions are violated once the NEC is violated, and
in our case the condition ${\rm NEC}_{1}$ is everywhere violated once
$r^{\prime\prime}/r>0$. The condition ${\rm DEC}_{1}$ is also violated since
it includes ${\rm NEC}_{1}$ as its component. In Fig. 3 we illustrate the
behavior of the other energy conditions. At small values of $a$, these
conditions are satisfied outside the horizon (i.e., where $A<0$).

In these and all subsequent figures we restrict ourselves to positive values
of $u$; for $u<0$ the plots are unnecessary due to the symmetry
$u\longleftrightarrow-u$.

Figure 3: Plots characterizing the fulfillment of energy conditions at $k=1$,
$q=0.5$, and different values of $a$. Solid lines present the functions inside
the horizon (at $A>0$), and dashed lines outside the horizon (at $A<0$).

##  III Field sources for the regularized metrics

###  III.1 General consideration

By analogy with rahul22 ; canate22 , let us find field sources for the
cylindrically symmetric metrics (2) and (4) in GR as NED plus a self-
interacting scalar field, with the total action

$S=\int\sqrt{-g}d^{4}x\left[R+2h(\phi)g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-2V(\phi)-L(F)\right],$
(22)

where $g$ is the determinant of the metric $g_{\mu\nu}$, $\phi$ is the scalar
field, $V(\phi)$ is its potential, $L(F)$ is the NED Lagrangian,
$F=F^{\mu\nu}F_{\mu\nu}$, and
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the
electromagnetic field tensor. The function $h(\phi)$ is included here not only
for generality (expressing the freedom of scalar field parametrization), but
also to include possible cases where $\phi$ changes its nature from canonical
($h>0$) to phantom ($h<0$), exhibiting a “trapped ghost” behavior trap10 ;
kb22s .

Varying the action (22) with respect to $\phi$, $A_{\mu}$, and $g^{\mu\nu}$,
we obtain the field equations

$\displaystyle\
{}\nabla_{\mu}\left[L_{F}F^{\mu\nu}\right]=\frac{1}{\sqrt{-g}}\partial_{\mu}\left[\sqrt{-g}L_{F}F^{\mu\nu}\right]=0,$
(23) $\displaystyle\
{}2h(\phi)\nabla_{\mu}\nabla^{\mu}\phi+\frac{dh}{d\phi}\partial^{\mu}\phi\partial_{\mu}\phi+\frac{dV(\phi)}{d\phi}=0,$
(24) $\displaystyle\ {}G_{\mu}^{\nu}=-T_{\mu}^{\nu}[\phi]-T_{\mu}^{\nu}[F],$
(25)

where $L_{F}=dL/dF$, $T_{\mu}^{\nu}[\phi]$ and $T_{\mu}^{\nu}[F]$ are the
stress-energy tensors of the scalar and electromagnetic fields, respectively:

$\displaystyle\
{}T_{\mu}^{\nu}[F]=\frac{1}{2}\delta^{\mu}_{\nu}L(F)-2L_{F}{F}^{\nu\alpha}F_{\mu\alpha},$
(26) $\displaystyle\
{}T_{\mu}^{\nu}[\phi]=2h(\phi)\partial^{\nu}\phi\partial_{\mu}\phi-\delta_{\mu}^{\nu}\big{(}h(\phi)\partial^{\alpha}\phi\partial_{\alpha}\phi-V(\phi)\big{)}.$
(27)

Let us show that any static, cylindrically symmetric metric of the form (6)
can be presented as a solution to Eqs. (23)–(25) with a proper choice of
$\phi$ and $F_{\mu\nu}$, quite similarly to the static, spherically symmetric
case studied in kb22s . The symmetry of the metric (6) makes us assume
$\phi=\phi(x)$ and single out among $F_{\mu\nu}$ either a “radial” electric
field with the only nonzero component $F_{01}=-F_{10}$ or a “radial” magnetic
field with $F_{23}=-F_{32}\neq 0$. For certainty, let us consider a magnetic
field, so that $F_{\theta\varphi}=-F_{\varphi\theta}=Q={\rm const}$, and $Q$
may be interpreted as a magnetic charge, while Eq. (23) is trivially
satisfied. Then the electromagnetic invariant $F$ is given by
$F=2Q^{2}/r^{4}$, and the stress-energy tensors (27) and (26) take the form

$\displaystyle\ {}T_{\mu}^{\nu}[\phi]=h(\phi)A(x)\phi^{\prime}{}^{2}\,{\rm
diag}\,(1,-1,1,1)+\delta_{\mu}^{\nu}V(\phi),$ (28) $\displaystyle\
{}T_{\mu}^{\nu}[F]=\frac{1}{2}\,{\rm diag}\,\Big{(}L,\ L,\
L-\frac{4Q^{2}}{r^{4}}L_{F},\ L-\frac{4Q^{2}}{r^{4}}L_{F}\Big{)}.$ (29)

For an arbitrary metric (6), all three components (II.1) of the Einstein
tensor are different; therefore, taken separately, a scalar or electromagnetic
field cannot solve the problem because of the equalities
$T^{0}_{0}[F]=T^{1}_{1}[F]$ for the electromagnetic field and
$T^{0}_{0}[\phi]=T^{2}_{2}[\phi]$ for the scalar field. So we will consider
them together.

Assuming that $A(x)$ and $r(x)$ are known and substituting them to Eqs. (25),
we find

$G^{2}_{2}-G^{0}_{0}=T^{0}_{0}-T^{2}_{2}=\frac{2Q^{2}}{r^{4}}L_{F},$ (30)

(the scalar field does not contribute to this combination of Eqs. (25)). Hence
we know $L_{F}$ as a function of $x$, and recalling that
$F(x)=2Q^{2}/r^{4}(x)$, is also known, we can write from (30)

$L^{\prime}=\frac{F^{\prime}}{F}(G^{2}_{2}-G^{0}_{0})=-\frac{4r^{\prime}}{r}(G^{2}_{2}-G^{0}_{0}),$
(31)

and integrate. We thus know $L(x)$, $F(x)$ and finally $L(F)$, at least in a
range where $r(x)$ is monotonic.

Furthermore, from (25) we have

$T^{0}_{0}-T^{1}_{1}=2h(\phi)A(x)\phi^{\prime}{}^{2}=G^{1}_{1}-G^{0}_{0}=-2A(x)\frac{r^{\prime\prime}}{r},$
(32)

which gives us $h(\phi)\phi^{\prime}{}^{2}$ as a function of $x$. It is easy
to notice that we have $h(\phi)\geq 0$ (the scalar field is canonical) as long
as $r^{\prime\prime}\leq 0$, and $h(\phi)\leq 0$ (the scalar field is phantom)
if $r^{\prime\prime}\geq 0$. In the general case, $r^{\prime\prime}(x)$ can
change its sign, and then the sign of $h(\phi)$ also varies. In all cases we
can freely choose the scalar field parametrization and conclude that we know
the functions $\phi(x)$ and $h(x)$. The only still unknown quantity, the
potential $V(\phi)$, can now be found from any components of Eqs. (25), for
example, $G^{t}_{t}=-T^{t}_{t}$. The scalar field equation (24) is known to
follow from the Einstein equations (25) (actually, as long as $\phi\neq{\rm
const}$, Eq. (24) directly follows from the “conservation” law
$\nabla_{\nu}T^{\nu}_{\mu}$, which is in turn a consequence of the Einstein
equations). So we can assert that the whole set of equations is fulfilled.

Another algorithm consists in using again (32) to determine $\phi$ and
$h(\phi)$ and then finding $V(\phi(x))$ from the scalar field equation (24).
With known $V(x)$, the function $L(F(x))$ is found from one more combination
of the Einstein equations,

$L(x)=-2V(x)-G^{0}_{0}-G^{1}_{1},$ (33)

and with known $F(x)$ it is then easy to find $L(F)$. Needless to say that
both algorithms must lead to the same result, up to the choice of integration
constants when finding $L(x)$ in the first algorithm and $V(x)$ in the second
one.

For our particular regular metrics (2) and (4), $r^{\prime\prime}>0$, hence we
will inevitably deal with a phantom scalar field and can safely put
$h(\phi)=-1$.

We can note that a representation of any metric (6) using an electric instead
of magnetic field is also possible but is slightly more complicated.

###  III.2 A field source for a regularized black string

According to the above, since $r^{\prime\prime}>0$ in the metric (2), we put
$h(\phi)=-1$. Let us try to model the solution with a magnetically charged
source and a phantom scalar field. The magnetic field and the invariant $F$
are given by

$F_{23}=Q,\qquad F(u)=\frac{2Q^{2}}{(u^{2}+a^{2})^{2}}.$ (34)

Let us use the second algorithm described above. From (32) we find

$\phi(u)=\arctan(u/a),$ (35)

and from the scalar equation (24) we determine, choosing the integration
constant so that $V$ vanishes at large $|u|$,

$V(u)=2a^{2}\left(\frac{b}{5\alpha(a^{2}+u^{2})^{5/2}}+\frac{\alpha^{2}}{a^{2}+u^{2}}\right).$
(36)

Next, Eq. (33) yields

$L(u)=\frac{6a^{2}b}{5\alpha(u^{2}+a^{2})^{5/2}}-6\alpha^{2},$ (37)

while the combination (30) leads to

$L_{F}(u)=\frac{3a^{2}b}{4\alpha Q^{2}\sqrt{u^{2}+a^{2}}},$ (38)

These quantities satisfy the condition

$L_{F}-\frac{dL}{du}\left(\frac{dF}{du}\right)^{-1}=0,$ (39)

confirming the correctness of our calculations. We can now write $L(F)$ and
$V(\phi)$ as

$\displaystyle L(F)$ $\displaystyle=$
$\displaystyle\frac{6a^{2}bF}{5\alpha}\bigg{(}\frac{F}{2Q^{2}}\bigg{)}^{5/4}-6\alpha^{2},$
(40) $\displaystyle V(\phi)$ $\displaystyle=$ $\displaystyle
2\alpha^{2}\cos^{2}\phi+\frac{2b}{5\alpha a^{3}}\cos^{5}\phi.$ (41)

Their behavior as functions of $u$ is illustrated in Figs. 4 and 5. The
magnitude of $V(u)$ and $L(u)$ decreases as $a$ and $\alpha$ increase, and
increases as $b$ increases.

Figure 4: The potential $V(\phi(u))$, Eq. (36), as a function of the
coordinate $u$. In (a) we fixed $b=0.5$ and $\alpha=0.02$. In (b) we fixed
$a=0.5$ and $\alpha=0.02$. In (c) we fixed $a=0.5$ and $b=0.5$.

Figure 5: The function $L(F(u))$, Eq. (37), as a function of the coordinate
$u$. In (a) we fixed $b=0.5$ and $\alpha=0.02$. In (b) we fixed $a=0.5$ and
$\alpha=0.02$. In (c) we fixed $a=0.5$ and $b=0.5$. At large $|u|$, the
function $L(F(u))$ tends to $-6\alpha^{2}$ instead of zero, according to Eq.
(37), but this contribution is too small to be visible in the figure.

###  III.3 A field source for a regularized inverted black hole

Again, for a magnetically charged source, the magnetic field is given by

$F_{23}=Q,\qquad
F(u)=\frac{2Q^{2}}{r^{4}}=\frac{32Q^{2}k^{4}}{q^{4}(u^{2}+a^{2})^{2}}.$ (42)

Let us note that the magnetic charge $Q$ of our anticipated source has nothing
to do with the initial charge $q$ belonging to the “seed” solution with the
metric (3) and involved in the regularized metric (4). However, as a special
case, they can coincide.

As before, with $h(\phi)=-1$, for the scalar field we obtain from (32)

$\phi(u)=\arctan(u/a).$ (43)

Then the scalar field equation (24) yields

$V(u)=\frac{32a^{2}k^{4}}{15q^{2}(u^{2}+a^{2})^{3}}\left(-5+3\sqrt{u^{2}+a^{2}}\right).$
(44)

From the Einstein equations we then obtain

$\displaystyle L(u)$ $\displaystyle=$
$\displaystyle\frac{32k^{4}}{15q^{2}(u^{2}+a^{2})^{3}}\left[15(u^{2}+a^{2})+9a^{2}\sqrt{u^{2}+a^{2}}-20a^{2}\right],$
(45) $\displaystyle L_{F}(u)$ $\displaystyle=$
$\displaystyle\frac{q^{2}}{Q^{2}}\bigg{(}\frac{3a^{2}}{4\sqrt{a^{2}+u^{2}}}+\frac{u^{2}-a^{2}}{u^{2}+a^{2}}\bigg{)}.$
(46)

As in the previous case, the correctness of calculations is verified by Eq.
(39). With our magnetic source, the knowledge of $u(F)$ and $u(\phi)$ enables
us to write the Lagrangian $L(F)$ and the potential $V(\phi)$ as follows:

$\displaystyle L(F)$ $\displaystyle=$
$\displaystyle\frac{q^{2}F}{Q^{2}}\bigg{(}1+\frac{3a^{2}}{5x}-\frac{4a^{2}}{3x^{2}}\bigg{)},\qquad
x=\frac{2k}{q}\bigg{(}\frac{2Q^{2}}{F}\bigg{)}^{1/4},$ (47) $\displaystyle
V(\phi)$ $\displaystyle=$
$\displaystyle\frac{32k^{4}}{15q^{2}a^{4}}\cos^{5}\phi(3a-5\cos\phi).$ (48)

There is good reason to suppose the equality $Q=q$ since it provides $L\approx
F$ as $F\to 0$, the asymptotically Maxwell behavior of $L(F)$ at small
magnetic fields. So in what follows we put $Q=q$.

The behavior of $V(u)$ is shown in Fig. 6. With other values of the
parameters, the potential behaves qualitatively in the same way. As the charge
$q$ increases, the peaks of the potential become lower, while they grow at
increasing values of the parameter $a$.

Figure 6: The potential $V(\phi(u))$, Eq. (44), with $k=1$, as a function of
the coordinate $u$. In (a) we fixed $a=0.5$ and in (b) we fixed $q=0.5$.

Figure 7: The function $L(F(u))$, Eq. (45), with $k=1$, as a function of the
coordinate $u$. In (a) we fixed $a=0.5$ and in (b) we fixed $q=0.5$.

The function $L(u)$ is plotted in Fig. 7. The peaks in the electromagnetic
Lagrangian become lower as $a$ or $q$ increase.

The behavior of $L(F)$ and $V(\phi)$ is presented in Figs. 8 and 9. At small
$F$, the dominating term in the electromagnetic Lagrangian is $L(F)\approx F$.
The first nonlinear term that appears is of the order $F^{5/4}$. The potential
looks like a barrier as the scalar field changes. At zero scalar, the
potential tends to a constant, and the first correction term is proportional
to $\phi^{2}$, simulating a scalar field mass such that
$m_{\phi}^{2}=8(2-a)/(a^{2}q^{2})$.

Figure 8: The function $L(F)$, Eq. (47), with $k=1$. In (a) we fixed $a=0.5$,
and in (b) we fixed $q=0.5$.

Figure 9: The potential $V(\phi)$, Eq. (48), with $k=1$. In (a) we fixed
$a=0.5$ and in (b) we fixed $q=0.5$.

##  IV Conclusion

In this paper, we have considered SV-type regularizations for two
cylindrically symmetric space-times with Killing horizons: (1) the black
string proposed by Lemos lemos95  (previously analyzed by Lima et al. lima22 )
and (2) the so-called inverted black hole [51, 52]. For both families of
regularized space-times, we have studied the global causal structures,
constructing the appropriate Carter-Penrose diagrams, and found possible field
sources in the framework of GR, consisting of a phantom minimally coupled
scalar field with a nonzero potential and a nonlinear magnetic field. The
global regularity of these regularized metrics (2) and (4) is verified by
finiteness of the Riemann tensor components (II.1) or the Kretschmann
invariant.

Next, we proposed a way to find, in the framework of GR, field sources for a
class of cylindrically, planarly or toroidally symmetric metrics of the form
(6), by combining a minimally coupled scalar field $\phi$ with a self-
interaction potential $V(\phi)$ and a nonlinear magnetic field described by a
certain NED theory with a Lagrangian $L(F)$. As both regularized metrics (2)
and (4) belong to this class, we have obtained explicit forms of field sources
for both of them.

In particular, the regularized black string solution, which may be called a
black-bounce version of the black string proposed by Lemos lemos95 , has
different structures, depending on the value of the regularization parameter
$a$: at small $a$ we have a regular black hole with two horizons and a black
bounce at $u=0$, and at larger $a$ we obtain a regular black hole with a
single extremal horizon at $u=0)$ or a cylindrically symmetric wormhole with a
throat at $u=0$. (A similar study was recently carried out by Lima et al.
lima23  for the charged version of black strings lemos96 .)

In the regularized inverted black hole solution, at $a<1$ there is a throat at
$u=0$ surrounded by two cosmological-type horizons. At $a=1$, a static region
vanishes, and we have a regular Bianchi-type I cosmology with an extremal
horizon at $u=0$. At $a>1$, we have a regular Bianchi-type I cosmology with a
bounce at $u=0$.

In both regularized solutions, the null energy condition is violated
everywhere due to $r^{\prime\prime}/r>0$, and the scalar part of the source is
necessarily phantom; still some of the standard energy conditions are
respected at least in a certain part of space-time, as can be seen from Fig.
3.

The scalar field in both solutions is of kink type and varies between two
finite limits, while the potential $V(\phi)$, being everywhere finite, rapidly
tends to zero as $|u|\to\infty$, although these infinities are of drastically
different nature in the two solutions. Meanwhile, the NED sources of the
solutions are quite different in nature. In the case of a regularized black
string, $L(F)$ has no Maxwell weak field limit and has a constant contribution
$-6\alpha^{2}$ (see Eq. (37)) corresponding to the AdS asymptotic behavior of
the metric. Unlike that, in the regularized inverted black hole solution, both
$V(u)$ and $L(u)$ tend to zero as $|u|\to\infty$, and $L(F)$ has a Maxwell
weak field limit.

A common feature of these regular solutions is that the magnetic fields exist
without their own source (which could be imagined as some distributed current
or monopole charge density), their lines of force stretching from one infinity
to the other.

We can conclude that SV-type regularization of known solutions of GR with
horizons and singularities leads to a number of geometries of interest, and,
in turn, their possible field sources shed a certain new light on their
properties and can be useful, in particular, for studying their stability
under various kinds of perturbations.

It is clear that the SV regularization trick can be applied to many other
singular metrics, including general cylindrically symmetric ones, described by
Eq. (5). Since they contain one more degree of freedom as compared to (6),
they will require more general field sources than those described here, and
this can be a subject of a future study.

### Acknowledgements

M.E.R. thanks Conselho Nacional de Desenvolvimento Científico e Tecnológico -
CNPq, Brazil for partial financial support. K.B. acknowledges partial support
from the Ministry of Science and Higher Education of the Russian Federation,
Project “Fundamental properties of elementary particles and cosmology No.
0723-2020-0041 and from Project No. FSSF-2023-0003.

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0}\mathcal{E}_{\varepsilon}(\mu_{\varepsilon},\boldsymbol{\nu}_{\varepsilon})=\liminf_{\varepsilon\rightarrow
0}\mathcal{E}_{\varepsilon}(\mu_{\varepsilon},\boldsymbol{\nu}_{\varepsilon})<+\infty.$

Thus $\mathcal{F}_{\varepsilon}(\mu_{\varepsilon,1})$, as defined in equation
(2), must be bounded uniformly in $\varepsilon$, which shows that
$\lim_{\varepsilon\rightarrow 0}\|\mu_{\varepsilon,1}-m_{1}\|_{2}=0$. Since,
$\mu_{\varepsilon}\rightarrow\mu$ in
$C([0,1];{\mathcal{P}}_{1}({\mathbb{R}^{d}}))$, we have $\mu_{\varepsilon,1}$
converges to $\mu_{1}$ in ${\mathcal{P}}_{1}({\mathbb{R}^{d}})$. Thus, by
uniqueness of limits, $\mu_{1}=m_{1}$, so $\mathcal{F}(\mu_{1})=0$. Due to the
lower semicontinuity of the functional $\mathcal{G}$ from Lemma 3.1, we can
conclude that

$\mathcal{E}(\mu,\boldsymbol{\nu})=\mathcal{G}(\mu,\boldsymbol{\nu})\leq\liminf_{\varepsilon\rightarrow
0}\mathcal{E}_{\varepsilon}(\mu_{\varepsilon},\boldsymbol{\nu}_{\varepsilon}).$

Next, we show part (ii). We may assume without loss of generality that
$\mathcal{E}(\mu,\boldsymbol{\nu})<+\infty$, so $\mathcal{F}(\mu_{1})=0$ and
$\mu_{1}=m_{1}\in L^{2}({\mathbb{R}^{d}})$. The result then follows, since
$\mathcal{F}_{\varepsilon}(\mu_{1})\equiv 0$.

We now apply this $\Gamma$-convergence result to prove Proposition 1.1, which
shows that, as $\varepsilon\to 0$, minimizers of ($\rm MFC_{\varepsilon}$)
converge to a minimizer of (1.1), up to a subsequence.

######  Proof B.2 (Proof of Proposition 1.1).

First, note that by the feasibility of (1.1), there exist
$(\mu^{\prime},\mathbf{u}^{\prime})\in\mathcal{C}(m_{0})$ so that
$\mathcal{E}(\mu^{\prime},\mathbf{u}^{\prime})<+\infty$, so
$\mu^{\prime}_{1}=m_{1}$. Combining this with the fact that
$(\mu_{\varepsilon},\boldsymbol{\nu}_{\varepsilon})$ are minimizers of ($\rm
MFC_{\varepsilon}$), we have

$\mathcal{G}(\mu_{\varepsilon},\boldsymbol{\nu}_{\varepsilon})\leq\mathcal{E}_{\varepsilon}(\mu_{\varepsilon},\boldsymbol{\nu}_{\varepsilon})\leq\mathcal{E}_{\varepsilon}({\mu}^{\prime},{\mathbf{u}}^{\prime})=\mathcal{G}({\mu}^{\prime},{\mathbf{u}}^{\prime})+\mathcal{F}_{\varepsilon}({\mu}_{1}^{\prime})=\mathcal{G}({\mu}^{\prime},{\mathbf{u}}^{\prime})+0,\
\forall\varepsilon>0.$

Therefore,
$\sup_{\varepsilon}\mathcal{G}(\mu_{\varepsilon},\boldsymbol{\nu}_{\varepsilon})<+\infty$.
By Lemma 3.2, there exists $(\mu,\boldsymbol{\nu})\in\mathcal{C}(m_{0})$ so
that, up to a subsequence, (3) holds.

Furthermore, for any $(\mu^{\prime\prime},\boldsymbol{\nu}^{\prime\prime})\in
C(m_{0})$, Proposition 3.3 and the fact that
$(\mu_{\varepsilon},\boldsymbol{\nu}_{\varepsilon})$ are minimizers ensure
that

$\mathcal{E}({\mu}^{\prime\prime},{\boldsymbol{\nu}}^{\prime\prime})\geq\limsup_{\varepsilon\rightarrow
0}\mathcal{E}_{\varepsilon}({\mu}^{\prime\prime},{\boldsymbol{\nu}}^{\prime\prime})\geq\liminf_{\varepsilon\rightarrow
0}\mathcal{E}_{\varepsilon}(\mu_{\varepsilon},\boldsymbol{\nu}_{\varepsilon})\geq\mathcal{E}(\mu,\boldsymbol{\nu})$

Since
$({\mu}^{\prime\prime},{\boldsymbol{\nu}}^{\prime\prime})\in\mathcal{C}(m_{0})$
was arbitrary, this gives the result.

Next we prove Theorem 1.4, on the convergence of minimizers of ($\rm
MFC_{\varepsilon,\delta}$) to minimizers of ($\rm MFC_{\varepsilon}$) as
$\delta\to 0$.

######  Proof B.3 (Proof of Theorem 1.4).

First, note that, by the assumption that ($\rm MFC_{\varepsilon}$) is
feasible, there exist
$(\mu^{\prime},\boldsymbol{\nu}^{\prime})\in\mathcal{C}(m_{0})$ so that
$\mathcal{E}_{\varepsilon}(\mu^{\prime},\boldsymbol{\nu}^{\prime})<+\infty$.
Combining this with the fact that $(\mu_{\delta},\boldsymbol{\nu}_{\delta})$
are minimizers, we have

$\mathcal{G}(\mu_{\delta},\boldsymbol{\nu}_{\delta})\leq\mathcal{E}_{\varepsilon,\delta}(\mu_{\delta},\boldsymbol{\nu}_{\delta})\leq\mathcal{E}_{\varepsilon,\delta}({\mu}^{\prime},{\boldsymbol{\nu}}^{\prime})=\mathcal{G}({\mu}^{\prime},{\boldsymbol{\nu}}^{\prime})+\varepsilon^{-1}\|k_{\delta}*\mu_{1}^{\prime}-k_{\delta}*m_{1}\|_{L^{2}({\mathbb{R}^{d}})}^{2}.$

Since
$\mathcal{E}_{\varepsilon}(\mu^{\prime},\boldsymbol{\nu}^{\prime})<+\infty$,
$\mu_{1}^{\prime}\in L^{2}({\mathbb{R}^{d}})$, and the right hand side is
bounded uniformly in $\delta$. Thus, by Lemma 3.2, there exists
$(\mu,\boldsymbol{\nu})\in\mathcal{C}(m_{0})$ so that, up to a subsequence,
(6) holds.

Furthermore, for any $(\mu^{\prime\prime},\boldsymbol{\nu}^{\prime\prime})\in
C(m_{0})$, Proposition 3.4 and the fact that
$(\mu_{\delta},\boldsymbol{\nu}_{\delta})$ are minimizers ensure that

$\mathcal{E}_{\varepsilon}({\mu}^{\prime\prime},{\boldsymbol{\nu}}^{\prime\prime})\geq\limsup_{\delta\rightarrow
0}\mathcal{E}_{\varepsilon,\delta}({\mu}^{\prime\prime},{\boldsymbol{\nu}}^{\prime\prime})\geq\liminf_{\delta\rightarrow
0}\mathcal{E}_{\varepsilon,\delta}(\mu_{\delta},\boldsymbol{\nu}_{\delta})\geq\mathcal{E}(\mu,\boldsymbol{\nu})$

Since
$({\mu}^{\prime\prime},{\boldsymbol{\nu}}^{\prime\prime})\in\mathcal{C}(m_{0})$
was arbitrary, this gives the result.

Now, we prove Proposition 1.6, which develops sufficient conditions to ensure
that a solution of ($\rm MFC_{\varepsilon,\delta,N}$) exists. Our proof is a
mild adaptation of [30, Proposition 4.2], extending to the case when $L_{N}$
satisfies Assumption 1.5(ib).

######  Proof B.4 (Proof of Proposition 1.6).

First, we consider feasibility of ($\rm MFC_{\varepsilon,\delta,N}$). As
observed in the paragraph following [30, equation (3.8)], classical results on
existence of solutions to ordinary differential equations ensure that there
exists $(\mathbf{y},0)\in\mathcal{A}_{N}(\mathbf{y}_{0})$ is nonempty. If
$L_{N}$ satisfies Assumption 1.5(ia), then for
$(\mathbf{y},0)\in\mathcal{A}_{N}(\mathbf{y}_{0})$,
$\mathcal{E}_{\varepsilon,\delta,N}(\mathbf{y},\mathbf{u})<+\infty$.

On the other hand, suppose $L_{N}$ satisfies Assumption 1.5(ib), the control
is unconstrained $U={\mathbb{R}^{d}}$, and the initial particle locations are
contained in $\\{L_{N}<+\infty\\}$. If we consider the curve $y_{i}(t)\equiv
y_{i,0}$ and the velocity
$u_{i}(t)=-\mathbf{F}_{N}(y_{i}(t),\mathbf{y}(t))=-\mathbf{F}_{N}(y_{i,0},\mathbf{y}_{0})$,
then $(\mathbf{y},\mathbf{u})\in\mathcal{A}_{N}(\mathbf{y}_{0})$, and since
$y_{i}(t)\in\\{L_{N}<+\infty\\}$ for all $t\in[0,1]$, we have
$\mathcal{E}_{\varepsilon,\delta,N}(\mathbf{y},\mathbf{u})<+\infty$.

Now, suppose ($\rm MFC_{\varepsilon,\delta,N}$) is feasible, and we will show
that a minimizer exists. Take a minimizing sequence
$(\mathbf{y}_{k},\mathbf{u}_{k})\in\mathcal{A}_{N}(\mathbf{y}_{0})$. Since

$\displaystyle\sup_{k}\frac{1}{N}\sum_{i=1}^{N}\int_{0}^{1}\psi(u_{i,k}(t))dt<+\infty,$

as in [30, Proposition 4.2], we obtain that there exist
$(\mathbf{y},\mathbf{u})\in\mathcal{A}_{N}(\mathbf{y}_{0})$ so that, up to a
subsequence, $\mathbf{y}_{k}\to\mathbf{y}$ in
$C([0,1];({\mathbb{R}^{d}})^{N})$ and $\mathbf{u}_{k}\to\mathbf{u}$ in
$L^{1}([0,1];U^{N})$. Furthermore, the proof of [30, Proposition 4.2] shows
that

$\displaystyle\liminf_{k\to+\infty}\frac{1}{N}\sum_{i=1}^{N}\int_{0}^{1}\psi(u_{i,k}(t))dt+\frac{1}{N}\sum_{i=1}^{N}\int_{0}^{1}L_{N}(y_{i,k}(t),\mathbf{y}_{k}(t))dt$
$\displaystyle\qquad\qquad\geq\frac{1}{N}\sum_{i=1}^{N}\int_{0}^{1}\psi(u_{i}(t))dt+\frac{1}{N}\sum_{i=1}^{N}\int_{0}^{1}L_{N}(y_{i}(t),\mathbf{y}(t))dt.$

By continuity of $K_{\delta}$ and $K_{\delta}*m_{1}$, we may likewise pass to
the limit in $k$ in the third term in $\mathcal{E}_{\varepsilon,\delta,N}$.
This shows
$\liminf_{k\to+\infty}\mathcal{E}_{\varepsilon,\delta,N}(\mathbf{y}_{k},\mathbf{u}_{k})\geq\mathcal{E}_{\varepsilon,\delta,N}(\mathbf{y},\mathbf{u}).$
Since $(\mathbf{y}_{k},\mathbf{u}_{k})$ was a minimizing sequence, this shows
that $(\mathbf{y},\mathbf{u})$ is a minimizer of ($\rm
MFC_{\varepsilon,\delta,N}$).

Now, we prove Theorem 3.10, which shows that, for fixed
$\varepsilon,\delta>0$, as $N\to+\infty$, minimizers of the spatially discrete
problem ($\rm MFC_{\varepsilon,\delta,N}$) converge to a solution of ($\rm
MFC_{\varepsilon,\delta}$), up to a subsequence.

######  Proof B.5 (Proof of Theorem 3.10).

First, we will show that there exists such a sequence $\mathbf{y}_{N,0}$. Let

(81) $\displaystyle
e_{m_{0}}:=\inf_{(\mu,\boldsymbol{\nu})\in\mathcal{C}(m_{0})}\mathcal{E}_{\varepsilon,\delta}(\mu,\boldsymbol{\nu}),\quad\mathcal{E}_{\varepsilon,\delta}(\mu,\boldsymbol{\nu}):=\mathcal{G}(\mu,\boldsymbol{\nu})+\mathcal{F}_{\varepsilon,\delta}(\mu_{1}).$

By our hypothesis that ($\rm MFC_{\varepsilon,\delta}$) is feasible, we have
that $e_{m_{0}}<+\infty$. Thus, for all $N\in\mathbb{N}$, there exists
$(\mu_{N},\boldsymbol{\nu}_{N})\in C(m_{0})$ so that
$\mathcal{E}_{\varepsilon,\delta}(\mu_{N},\boldsymbol{\nu}_{N})<e_{m_{0}}+1/N$.
By Proposition 3.8(ii), there exists $\mathbf{y}_{N,0}$ so that (37-36) hold
and there exist
$(\tilde{\mathbf{y}}_{N},\tilde{\mathbf{u}}_{N})\in\mathcal{A}_{N}(\mathbf{y}_{N,0})$
so that
$\mathcal{E}_{\varepsilon,\delta,N}(\tilde{\mathbf{y}}_{N},\tilde{\mathbf{u}}_{N})\leq
e_{m_{0}}+2/N$. Now, suppose
$(\mathbf{y}_{N},\mathbf{u}_{N})\in\mathcal{A}_{N}(\mathbf{y}_{N,0})$ is an
optimizer of ($\rm MFC_{\varepsilon,\delta,N}$). Then, we have

(82)
$\displaystyle\limsup_{N\to+\infty}\mathcal{E}_{\varepsilon,\delta,N}(\mathbf{y}_{N},\mathbf{u}_{N})\leq\limsup_{N\to+\infty}\mathcal{E}_{\varepsilon,\delta,N}(\tilde{\mathbf{y}}_{N},\tilde{\mathbf{u}}_{N})\leq
e_{m_{0}}$

and

$\displaystyle\sup_{N}\frac{1}{N}\sum_{i=1}^{N}\int_{0}^{1}\psi(u_{i,N}(t))dt\leq\sup_{N}\mathcal{E}_{\varepsilon,\delta,N}(\mathbf{y}_{N},\mathbf{u}_{N})\leq\sup_{N}\mathcal{E}_{\varepsilon,\delta,N}(\tilde{\mathbf{y}}_{N},\tilde{\mathbf{u}}_{N})\leq
e_{m_{0}}+1<+\infty.$

Then [30, Theorem 3.1] ensures that there exists
$(\mu,\boldsymbol{\nu})\in\mathcal{C}(m_{0})$ so that (39)-(40) hold.
Furthermore, for any such limit point $(\mu,\boldsymbol{\nu})$, by Proposition
3.8(i),

(83) $\displaystyle
e_{m_{0}}\leq\mathcal{E}_{\varepsilon,\delta}(\mu,\boldsymbol{\nu})\leq\liminf_{N\to+\infty}\mathcal{E}_{\varepsilon,\delta,N}(\mathbf{y}_{N},\mathbf{u}_{N}).$

Combining (82) and (83), we obtain (38) and that $(\mu,\boldsymbol{\nu})$ is a
minimizer of ($\rm MFC_{\varepsilon,\delta}$).

Now, we turn to the proof of Corollary 1.7, which provides sufficient
conditions to ensure existence of minimizers to the continuum optimization
problems we consider.

######  Proof B.6 (Proof of Corollary 1.7).

Since (1.1) is feasible, we have that ($\rm MFC_{\varepsilon}$) and ($\rm
MFC_{\varepsilon,\delta}$) are feasible for all $\varepsilon,\delta>0$.
Likewise, Proposition 1.6 ensures that, for any
$\mathbf{y}_{0}\in({\mathbb{R}^{d}})^{N}$ with

(84) $\displaystyle y_{i,0}\in\\{L_{N}<+\infty\\}\text{ for all }i=1,\dots,N,$

minimizers of ($\rm MFC_{\varepsilon,\delta,N}$) exist.

By Theorem 3.10, for all $N\in\mathbb{N}$, there exists
$\mathbf{y}_{N,0}\in({\mathbb{R}^{d}})^{N}$ satisfying (84) so that, for any
sequence of minimizers
$(\mathbf{y}_{N},\mathbf{u}_{N})\in\mathcal{A}(\mathbf{y}_{N,0})$ of ($\rm
MFC_{\varepsilon,\delta,N}$), up to a subsequence,
$(\mathbf{y}_{N},\mathbf{u}_{N})$ converges to a minimizer of ($\rm
MFC_{\varepsilon,\delta}$). Thus, minimizers of ($\rm
MFC_{\varepsilon,\delta}$) exist.

By Theorem 1.4, any sequence of minimizers of ($\rm MFC_{\varepsilon,\delta}$)
converges, up to a subsequence, to a minimizer of ($\rm MFC_{\varepsilon}$),
so minimizers of ($\rm MFC_{\varepsilon}$) exist.

Finally, by Proposition 1.1, any sequence of minimizers of ($\rm
MFC_{\varepsilon}$) converges, up to a subsequence, to a minimizer of (1.1),
so minimizers of (1.1) exist.

We conclude with the proof of our main theorem, Theorem 1.10, on the
convergence of minimizers of ($\rm MFC_{\varepsilon,\delta,N}$) to a minimizer
of (1.1) as $\varepsilon\to 0,\delta\to 0$, and $N\to+\infty$.

######  Proof B.7 (Proof of Theorem 1.10).

Recall that, as a consequence of our assumption that (1.1) is feasible, we
also have that ($\rm MFC_{\varepsilon}$) and ($\rm MFC_{\varepsilon,\delta}$)
are feasible.

By Theorem 1.8, for any $\varepsilon,\delta>0$, if
$(\mathbf{y}_{N}^{\varepsilon,\delta},\mathbf{u}_{N}^{\varepsilon,\delta})\in\mathcal{A}_{N}(\mathbf{y}_{N,0})$
minimizes ($\rm MFC_{\varepsilon,\delta,N}$), there is a subsequence
$N_{l}^{\varepsilon,\delta}$ of $N$ and
$(\mu_{\varepsilon,\delta},\boldsymbol{\nu}_{\varepsilon,\delta})\in\mathcal{C}(m_{0})$
that minimizes ($\rm MFC_{\varepsilon,\delta}$) for which

(85)
$\displaystyle\frac{1}{N_{l}^{\varepsilon,\delta}}\sum_{i=1}^{N_{l}^{\varepsilon,\delta}}\delta_{y^{\varepsilon,\delta}_{i,N_{l}^{\varepsilon,\delta}}}\xrightarrow{l\to+\infty}\mu_{\varepsilon,\delta}\text{
in }C([0,1];{\mathcal{P}}_{1}({\mathbb{R}^{d}}))$ (86)
$\displaystyle\frac{1}{N_{l}^{\varepsilon,\delta}}\sum_{i=1}^{N_{l}^{\varepsilon,\delta}}u^{\varepsilon,\delta}_{i,N_{l}^{\varepsilon,\delta}}(t)\delta_{y^{\varepsilon,\delta}_{i,N_{l}^{\varepsilon,\delta}}(t)}dt\xrightarrow{l\to+\infty}\boldsymbol{\nu}_{\varepsilon,\delta}\text{
in }{\mathcal{M}}({\mathbb{R}^{d}}\times[0,1];U).$

Likewise, for any $\varepsilon$, Theorem 1.4 ensures that there exists a
subsequence $\delta_{k}^{\varepsilon}$ of $\delta$ so that
$(\mu_{\varepsilon,\delta_{k}^{\varepsilon}},\boldsymbol{\nu}_{\varepsilon,\delta_{k}^{\varepsilon}})$
converges to some
$(\mu_{\varepsilon},\boldsymbol{\nu}_{\varepsilon})\in\mathcal{C}(m_{0})$,
where $(\mu_{\varepsilon},\boldsymbol{\nu}_{\varepsilon})$ minimizes ($\rm
MFC_{\varepsilon}$). Finally, by Theorem 1.1, up to a subsequence,
$(\mu_{\varepsilon},\boldsymbol{\nu}_{\varepsilon})$ converges to some
$(\mu,\boldsymbol{\nu})$, where $(\mu,\boldsymbol{\nu})$ is a minimizer of
(1.1).

To obtain the result, we now apply a standard diagonal argument. For
simplicity of notation, recall that convergence in the topologies of equations
(14-15) is metrizable, and let $d$ denote such a metric. For $m\in\mathbb{N}$,
we may choose a subsequence
$(\mu_{\varepsilon_{m}},\boldsymbol{\nu}_{\varepsilon_{m}})$ so that

$d\left((\mu_{\varepsilon_{m}},\boldsymbol{\nu}_{\varepsilon_{m}}),(\mu,\boldsymbol{\nu})\right)<\frac{1}{3m}.$

Likewise, we may choose a subsequence
$\left(\mu_{\varepsilon_{m},\delta_{m}^{\varepsilon_{m}}},\boldsymbol{\nu}_{\varepsilon_{m},\delta_{m}^{\varepsilon_{m}}}\right)$
so that

$d\left(\left(\mu_{\varepsilon_{m},\delta_{m}^{\varepsilon_{m}}},\boldsymbol{\nu}_{\varepsilon_{m},\delta_{m}^{\varepsilon_{m}}}\right),(\mu_{\varepsilon_{m}},\boldsymbol{\nu}_{\varepsilon_{m}})\right)<\frac{1}{3m}.$

Finally, we may choose a subsequence
$\left(\mathbf{y}^{\varepsilon_{m},\delta_{m}}_{N_{m}^{j_{m},k_{m}}},\mathbf{u}^{\varepsilon_{m},\delta_{m}}_{N_{m}^{j_{m},k_{m}}}\right)$
so that, defining $\mu_{m}$ and $\boldsymbol{\nu}_{m}$ as on the left hand
side of (12-13),

$d\left((\mu_{m},\boldsymbol{\nu}_{m}),\left(\mu_{\varepsilon_{m},\delta_{m}^{\varepsilon_{m}}},\boldsymbol{\nu}_{\varepsilon_{m},\delta_{m}^{\varepsilon_{m}}}\right)\right)<\frac{1}{3m}.$

The result then follows by the triangle inequality.

Acknowledgements: K. Craig would like to thank Amir Sagiv for an interesting
discussion on the case of optimal transport around obstacles and the
connection to manifolds with holes.

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