# Bubble universes and traversable wormholes

José P. S. Lemos<EMAIL_ADDRESS>Centro Multidisciplinar de Astrofísica
- CENTRA, Departamento de Física, Instituto Superior Técnico - IST,
Universidade de Lisboa - UL, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal
Paulo Luz<EMAIL_ADDRESS>Centro Multidisciplinar de Astrofísica -
CENTRA, Departamento de Física, Instituto Superior Técnico - IST, Universidade
de Lisboa - UL, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal  Departamento de
Matemática, ISCTE - Instituto Universitário de Lisboa, Portugal

###### Abstract

Bubble universes and traversable wormholes in general relativity can be
realized as two sides of the same concept. To exemplify it, we find, display,
and study in a unified manner a Minkowski-Minkowski closed universe and a
Minkowski-Minkowski traversable wormhole. By joining two 3-dimensional flat
balls along a thin shell two-sphere of matter, i.e., a spherical domain wall,
into a single spacetime one gets a Minkowski-Minkowski static closed universe,
i.e., a bubble universe. By joining two 3-dimensional complements of flat
balls along a thin shell two-sphere of matter, i.e., a spherical throat, into
a single spacetime one gets a Minkowski-Minkowski static open universe which
is a traversable wormhole. Thus, Minkowski-Minkowski bubble universes and
wormholes can be seen as complementary to each other. Is is also striking that
these two spacetimes, the Minkowski-Minkowski bubble universe and the
Minkowski-Minkowski traversable wormhole, have resemblances with two well-
known static universes of general relativity. The Minkowski-Minkowski static
closed universe, i.e., the Minkowski-Minkowski bubble universe, resembles in
many aspects the Einstein universe, i.e., a static closed spherical universe
homogeneously filled with dust matter and with a cosmological constant. The
Minkowski-Minkowski static open universe, i.e., the Minkowski-Minkowski
traversable wormhole, resembles the Friedmann static universe, i.e., a static
open hyperbolic universe homogeneously filled with negative energy density
dust and with a negative cosmological, which is a universe with two disjoint
branes, or branches, that can be considered a failed wormhole. In this light,
the Einstein static closed universe and the Friedmann static open universe
should also be seen as two sides of the same concept, i.e., they are
complementary to each other. The scheme is completed by performing a linear
stability analysis for the Minkowski-Minkowski bubble universe and the
Minkowski-Minkowski traversable wormhole and also by comparing it to the
stability of the Einstein static universe and the Friedmann static universe,
respectively. The complementarity between bubble universes and traversable
wormholes, that exists for these instances of static spacetimes, can be can
carried out for dynamical spacetimes, indicating that such a complementarity
is quite general. The overall study suggests that bubble universes and
traversable wormholes can be seen as coming out of the same concept, and thus,
if ones exist the others should also exist.

##  I Introduction

###  I.1 Minkowski-Minkowski bubble universe and Minkowski-Minkowski
traversable wormhole

General relativity is an excellent theory to study universe solutions and
wormhole solutions from which bubble universes and traversable wormholes can
arise as complementary to each other. To see this, one can attempt to find
within the theory, Minkowski-Minkowski bubble universes and Minkowski-
Minkowski traversable wormholes and study their properties. One picks up a
Minkowski spacetime and at constant time cuts a ball in it, to obtain two
spaces, namely, a 3-dimensional ball with a flat inside, and an infinite
extended 3-dimensional flat space with a hole, which is the complement of the
ball. Then one picks up another Minkowski spacetime and do the same, to get a
second ball and a second infinite extended flat space with a hole. If one
joins the two 3-dimensional balls along a 2-sphere, a shell containing matter,
one obtains a single 3-space that including time makes altogether a static
closed universe. If one joins the two complements, i.e., the two infinite
extended 3-dimensional flat spaces with a hole in each, along a 2-sphere, a
shell containing matter, one obtains a different single 3-space that including
time makes altogether another universe, which is a traversable wormhole. Thus,
one has a closed universe, which can be viewed as a bubble universe, and its
complement, an open universe, which is a traversable wormhole. To implement
the idea of a Minkowski-Minkowski closed universe, i.e., a bubble universe,
and a Minkowski-Minkowski open universe, i.e., a traversable wormhole, one
uses the equations of general relativity together with the appropriate thin
shell formalism Israel_1966 . When one has a thin shell in an ambient
spacetime one has the normal vector to the shell as an important quantity that
will allow to determine how the thin shell curves in that space, i.e., allows
to determine the extrinsic curvature of the shell, which besides the spacetime
metric itself, is one of the quantities that has to match at both sides of the
shell. Indeed, to find all possible shell solutions in an ambient spacetime
one has to understand the fact that the normal to a shell can have two
relative directions, such that, for static spherically symmetric spacetimes,
the normal to the shell may point towards or away from the center of the
coordinates. For intance, in an ambient Minkowski-Schwarzschild spacetime,
more precisely, for a shell with a Minkowski interior with a center and a
Schwarzschild exterior, usually called a fundamental shell, if the normal
points to increasing coordinate radius $r$ in the exterior, one has a star
shell, i.e., a shell that represents a star. In the same ambient Minkowski-
Schwarzschild spacetime, if the normal points to decreasing coordinate radius
$r$ one has a tension shell black hole, i.e., a shell supported by tension
that is in the other side of the Kruskal-Szekeres diagram as was noted by Katz
and Lynden-Bell Katz_Lynden-Bell_1991 . This can also be performed in an
ambient Minkowski-Reissner-Nordström spacetime, yielding, instead of two
fundamental electrically charged shell spacetimes, a bewildering variety of
fourteen fundamental electrically charged shell spacetimes with different
global spacetime structures lemosluz . Here, in place of using an ambient
Minkowski-Schwarzschild or an ambient Minkowski-Reissner-Nordström we use an
ambient Minkowski-Minkowski spacetime.

One possibility for a Minkowski-Minkowski spacetime is for a shell with a
Minkowski interior with a center, i.e., a fundamental shell, such that the
normal to the shell points towards decreasing radius $r$ in the Minkowski
exterior. One then finds the Minkowski-Minkowski closed universe, made of two
3-dimensional flat balls, or sheets, that are joined at some domain wall,
i.e., a 2-sphere shell with matter, to make a Minkowski-Minkowski bubble
universe. Note that for a shell with a Minkowski interior with a center such
that the normal to the shell points towards increasing radius $r$ in the
Minkowski exterior yields the trivial global Minkowski spacetime with a zero
shell.

There is yet another possibility for a Minkowski-Minkowski spacetime,
different from the fundamental shell. It comes from an exotic shell, i.e., a
shell attached to a Minkowski open interior, noting that interior is just a
name since it could as well be called exterior. For a shell with a Minkowski
open interior, when the normal to the shell points towards increasing $r$ in
the Minkowski exterior, one finds the Minkowski-Minkowski open universe, made
of two 3-dimensional flat open infinite sheets that are joined at some
2-sphere with matter, to make a Minkowski-Minkowski traversable wormhole. Note
that for a shell with a Minkowski open interior such that the normal to the
shell points towards decreasing radius $r$ in the Minkowski exterior yields
the trivial global Minkowski spacetime with a zero shell.

Universes and wormholes are usually envisaged as distinct objects. The two
Minkowski-Minkowski spacetimes demonstrate that they can be seen as
complementary to each other, i.e., they are two sides of the same concept. The
concept, i.e., a collection of two Minkowski spacetimes together, yields on
one side a closed universe, i.e., a bubble universe, and on the other side an
open universe which is a traversable wormhole. Surely, the collection of two
Minkowski spacetimes can also lead to two separate Minkowski spacetimes, but
this is the trivial case and needs not be considered.

###  I.2 Einstein static closed spherical universe and Friedmann static open
hyperbolic universe

There are two paradigmatic static homogeneous universes in general relativity.
There is the static closed spherical universe and there is the static open
hyperbolic universe, with two separated branes or branches. To implement the
idea of static uniform universes, one uses general relativity itself, i.e.,
Einstein equation modified to include a cosmological constant. From the
staticity condition one imposes that neither the geometry nor the matter
depend on time and from the homogeneous condition one imposes that the energy-
density is a constant in space.

This implementation, that the Universe, in particular a static universe, could
be described within general relativity, was put forward by Einstein. In
devising a way to realize Mach’s principle, a new interaction, namely, a
cosmological constant with repulsion features, was postulated. General
relativity with this new cosmological interaction is indeed the first modified
theory of gravitation. This repulsive cosmological term, that counterbalances
the self gravitational force due to the energy density of the matter supposed
pressureless, was then used to find a unique static solution for the Universe
which was also assumed to be closed, finite, and spheric einstein1917 . In the
limiting case that the universe would be spatially flat, the Einstein universe
disappeared in a Minkowski empty universe. The enforcing of Mach’s principle
in this way proved to be a dead end as exemplified by the de Sitter universe
with no matter and only a cosmological constant desitter , but the static
closed universe of Einstein was of great impact as it indeed started the
concept of universes. For instance, dynamic closed universes within general
relativity, like the Friedmann friedman1922  and Lemaître lemaitre  expanding
universes, came out of Einstein’s static one, which in turn, due to its
instability and propensity to grow, continued to be studied as a progenitor of
expanding closed universes edd ; bonnor1 ; harr2 ; gib1 ; barr ; barcelo .

Remarkably, Friedmann in his second paper on universes and cosmology proposed,
to start with, an open static hyperbolic universe as to exhaust the possible
static pressureless universes friedman1924 . In the process, a cosmological
constant, now with attraction features, was again introduced to counterbalance
the self gravitational repulsive force due to a matter energy density
necessarily negative. In the limiting case the universe would be spatially
flat, the Friedmann universe disappeared in a Minkowski empty universe. The
Friedmann static universe can be seen as an anti-Einstein universe and it
inaugurated the concept of open universes. Indeed, it was used by Friedmann in
friedman1924  to continue the analysis into dynamic open hyperbolic universes
and it was developed by Harrison harr  who, among many other universes, also
studied its stability. Now, the Friedmann static universe, being hyperbolic,
has two branches, or branes, which fail to communicate to each other by an
infinitesimal separation. This means that it can be considered a wormhole,
actually, a failed wormhole. It is not traversable but almost and it can be
thought as an embryo of a wormhole.

The Einstein static closed universe and the Friedmann static open universe can
be seen as complementary, i.e., they are two sides of the same concept. The
concept here is the constant spatial curvature of the spacetime, one side
gives positive spatial curvature, i.e., the Einstein universe, the other side
gives negative spatial curvature, i.e., the Friedmann static universe. The
trivial case here would also be two zero curvature spacetimes, i.e., two
separate Minkowski spacetimes, and needs not be considered.

###  I.3 Bubble universes and traversable wormholes

Bubble universes and traversable wormholes have been proposed as structures
that might arise if appropriate physical conditions are available. Indeed, the
Universe in its early phases, of which the inflationary period is an example,
filled with scalar and gauge fields, may have produced domain walls, cosmic
strings, and monopoles, which can still exist as frozen topological remains of
the symmetry breaking phase transition of that early era. In this connection,
a setting allowed by the prevailing physical conditions of that early
inflationary era or even of an epoch before it, is that bubble universes might
have unfolded within the Universe and also, conceivably, systems such as
traversable wormholes might have materialized to connect distant parts of the
Universe or distinct universes. In addition, a possibility also permitted by
the laws of physics, is that bubble universes and traversable wormholes might
be constructed if sufficient technology is available. General relativity is an
excellent theory to study universe solutions and wormhole solutions from which
bubble universes and traversable wormholes can emerge as complementary to each
other, and so they can be seen as duals of each other, leading to a better
understanding of both.

A bubble universe, a universe within the Universe, is a complete solution of
the Einstein’s equations. Bubble universes, together with baby universes, are
universes in themselves, somehow attached to our one. They made their
appearance in the physics of false vacuum decay within dynamic bubbles
colemandeluccia . Its interest and uses within the inflation theory was seen
in guth . The idea of bubble universes taking off out from our Universe was
developed in blau , general relativistic dynamic bubble universe solutions
with matter were proposed in ipser , several possible universe decays and
corresponding expanfing or contracting domain walls were thoroughly analyzed
in berezin , interesting scenarios with bubbles with different gravitational
constants were proposed in takamizumaeda , their intrinsic stability has not
been analyzed, see however gonzalez , and bubble universe astrophysical
connections to black holes and their formation were studied in dengvilenkin ;
deng .

A traversable wormhole, joining two otherwise distinct universes through two
mouths and a throat, is also a complete solution of the Einstein’s equations.
A wormhole is a concept with a history of its own that in a sense was
initiated by Einstein in the celebrated Einstein-Rosen bridge einsteinrosen .
The concept had further developments related to the quantization of the
spacetime geometry misnerwheeler , and it was essential to understand the
maximal extension of the Schwarzschild spacetime, now seen as a white hole
being converted to a black hole through a nontraversable wormhole connecting
two separated asymptotically flat spacetimes kruskal , which in turn gave rise
to the notion of multiply connected spacetimes fullerwheeler . Wormholes, in
particular traversable wormholes, abound in general relativity and in
gravitational theories. Traversable wormhole solutions were found first in
ellis ; bronnikov , were placed in a proper framework morristhorne , were
studied as inflating solutions in roman , received the status of being the
title of a book in visserbook , had their energy conditions analyzed in the
case of dynamic solutions in hochbergvisser , were built from vacuum stress-
energy tensors in krasnikov , were embedded in a cosmological constant setting
in llq , had the energy conditions at the wormhole’s throat generically
reassessed in zaslavskii , were analyzed in relation to their shadows and
quasinormal modes in bronkonpappas , had collisions between their own mouths
studied in flm , had a stability analysis performed in shinkaihayward ;
sushkovzhang ; Lemos_Lobo_2008 , see also visserbook , and had their possible
connection to astrophysics being proposed in kirillov ; ovgunetal ; piotrovich
.

The bubble universe and traversable wormhole solutions just mentioned are only
a few of the vast number of all the many interesting solutions found in the
literature, and it is not our intention to proceed with an analysis of so vast
a camp. Usually, bubble universes appear in dynamic contexts, whereas
traversable wormholes are generically realized in static backgrounds, but of
course they can both be static or dynamic. Bubble universes and traversable
wormholes can appear as duals of each other, a possibility being when quantum
gravity dominated or was still nonnegligible, bubble universes and traversable
wormholes would form alike out of the spacetime foam, and would stay stable or
metastable structures well into the classical regime. We stick to general
relativity and to the two static Minkowski-Minkowski spacetimes and the two
static homogeneous universes of Einstein and Friedmann. With these four
spacetimes it is possible to have two levels of comparison. On a first level
of comparison, on the one hand, one can compare the two Minkowski-Minkowski
spacetimes between themselves by investigating their similarities, and on the
other hand, one can also attempt to compare the two static, homogenewous,
presureless spacetimes of general relativity with cosmological constant
between themselves. On a second level of comparison, the two Minkowski-
Minkowski spacetimes are put face to face with the two static spacetimes of
Einstein and Friedmann. Let us be specific. When performing the first level
comparison between the two Minkowski-Minkowski spacetimes, i.e., the
Minkowski-Minkowski bubble universe and the Minkowski-Minkowski traversable
wormhole, one should take some steps, namely, one has to reveal in a unified
manner the two possible nontrivial cases in a Minkowski-Minkowski spacetime,
or more concretely, one has to find the fundamental shell spacetime, which is
a closed bubble universe, and find also the exotic shell spacetime, which is
an open traversable wormhole universe. In doing so, one classifies and
analyzes the possible junctions of Minkowski spacetimes through a static,
timelike, thin matter shell, which are the two nontrivial cases just
mentioned, the trivial one being the no shell pure Minkowski spacetime. A
linear stability study of these spacetimes completes the comparison. Through
this example, bubble universes and traversable wormholes can now be understood
in a unified light, in the sense that the Minkowski-Minkowski bubble universe
and the Minkowski-Minkowski traversable wormhole are two sides of the same
concept, in which instance, if one exists it makes a case to the existence of
the other. When performing the first level comparison between the two static,
homogenewous, presureless spacetimes with cosmological constant, i.e., the
Einstein closed universe and the Friedmann open universe, one should take some
steps, namely, one has to reveal in a unified manner these two possible
nontrivial cases, and display them in a new light. A linear stability study of
these spacetimes completes the analysis. In this new light, the Einstein and
Friedmann static universes can be compared, they are seen anew as a bubble
universe and a failed wormhole, respectively. On the second level of
comparison, the two Minkowski-Minkowski spacetimes are put face to face with
the two static homogeneous spacetimes, to find that the Minkowski-Minkowski
closed universe, a bubble universe, goes along with the Einstein closed
universe, which can then be seen then as a bubble universe, and the Minkowski-
Minkowski open universe, a traversable wormhole, goes along with the Friedmann
open universe, which is a failed wormhole. This comparison shows some striking
resemblances between those spacetimes on several grounds. In this sense, the
Einstein and Friedmann static universes are really seen anew as a bubble
universe and a failed wormhole, respectively, and also they are an example
that bubble universes and traversable wormholes can be perceived in a unified
light. The complementarity, or duality, between general relativistic bubble
universes and traversable wormholes exists for these examples of static
spacetimes. One can carry out this idea for dynamical spacetimes and show that
the complementarity, or duality, considered here is quite generic. Moreover,
following this rationale, if one finds inflating bubble universe solutions,
one should be able to find the corresponding inflating wormhole solutions, and
vice versa, so that, for instance, a given solution already found in one of
the sides could help in looking for the complementary solution in the other
side.

The paper is organized as follows. In Sec. II, we formalize in a unified way
the two possible junctions of two identical Minkowski spacetime regions and we
perform a linearized stability analysis of the Minkowski-Minkowski universes.
We then build in detail the Minkowski-Minkowski static closed universe, i.e.,
a bubble universe, and the Minkowski-Minkowski static open universe, i.e., the
Minkowski-Minkowski static traversable wormhole. In Sec. III, we formalize in
a unified way the two possible static homogeneous universes and we display a
linearized stability analysis of them. We then display in detail the Einstein
static spherical closed universe and we compare explicitly the Minkowski-
Minkowski static closed universe, i.e., the Minkowski-Minkowski bubble
universe, with the Einstein static spherical closed universe, and display the
Friedmann static hyperbolic open universe, i.e., the failed wormhole, and
compare explicitly the Minkowski-Minkowski static open universe, i.e., the
Minkowski-Minkowski static traversable wormhole, with the Friedmann failed
wormhole. In Sec. IV we conclude. Throughout the paper we work in geometrized
units system where the constant of gravitation $G$ and the speed of light $c$
are set to one, $G=1$ and $c=1$, and use the metric signature
$\left(-+++\right)$.

##  II Minkowski-Minkowski closed universe and Minkowski-Minkowski open
universe

###  II.1 Minkowski-Minkowski universes: Formal solutions and stability

####  II.1.1 Solutions

The Einstein field equations are

$G_{\alpha\beta}=8\pi T_{\alpha\beta}\,.$ (1)

where $G_{\alpha\beta}=R_{\alpha\beta}-\frac{1}{2}g_{\alpha\beta}\mathcal{R}$
is the Einstein tensor, $R_{\alpha\beta}$ and $\mathcal{R}$ are the Ricci
tensor and Ricci scalar, respectively, $g_{\alpha\beta}$ is the metric,
$T_{\alpha\beta}$ is the stress-energy tensor, and Greek indices run from o to
3 with 0 representing a time component and 1, 2, and 3 representing space
components. One wants to join consistently two solutions of Einstein field
equations and the Israel formalism provides the method needed to make the
junction between two different general relativistic spacetime regions
(Israel_1966, ). Consider then two spacetime manifolds with boundary, one is
$\mathcal{M}_{\mathrm{i}}$ with metric $g_{\mathrm{i}}$ and the other is
$\mathcal{M}_{\mathrm{e}}$ with metric $g_{\mathrm{e}}$. The spacetimes
$(\mathcal{M}_{\mathrm{i}},g_{\mathrm{i}})$ and
$(\mathcal{M}_{\mathrm{e}},g_{\mathrm{e}})$ are solutions of the theory of
general relativity and are to be glued together at a common boundary, forming
a new spacetime $\mathcal{M}$. In brief, $\mathcal{M}$ is partitioned by an
hypersurface $\mathcal{S}$ into two regions, the regions
$\mathcal{M}_{\mathrm{i}}$ and $\mathcal{M}_{\mathrm{e}}$. The formalism
applies directly to a hypersurface $\mathcal{S}$ that can be either timelike
or spacelike, the extension to the case of a null boundary hypersurface can
also be done with care.

We assume that it is possible to formally define a common coordinate system
$\left\\{x^{\alpha}\right\\}$ on both sides of the hypersurface $\mathcal{S}$.
We also admit the existence of a normal vector field $n$, well defined on both
sides of $\mathcal{S}$, which is orthogonal to the matching hypersurface at
each point. We choose $n$ to point from $\mathcal{M}_{\mathrm{i}}$ to
$\mathcal{M}_{\mathrm{e}}$ and, without loss of generality,
$n^{\alpha}n_{\alpha}=\varepsilon$, where $n^{\alpha}$ are the components of
$n$ in the coordinate system $\left\\{x^{\alpha}\right\\}$ and $\varepsilon$
is $+1$ or $-1$ depending on $n$ being spacelike or timelike, respectively.
The null case has $\varepsilon=0$ and it would have to be treated separately
which we do not do here. For a timelike normal vector field $n$ one has that
the corresponding hypersurface $\mathcal{S}$ is spacelike, and for a spacelike
$n$ one has that the corresponding hypersurface $\mathcal{S}$ is timelike, and
vice versa. Then, assuming $\left\\{y^{a}\right\\}$ to represent a local
coordinate system on $\mathcal{S}$, the normal vector field $n$ must be
orthogonal at each point to the tangent vectors to the hypersurface
$\mathcal{S}$, $e_{a}\equiv\frac{\partial\;\;}{\partial y^{a}}$, such that
$e_{a}^{\alpha}n_{\alpha}=0$, with $e_{a}^{\alpha}\equiv\frac{\partial
x^{\alpha}}{\partial y^{a}}$. The induced metric on $\mathcal{S}$ as seen from
each region $\mathcal{M}_{\mathrm{i}}$ and $\mathcal{M}_{\mathrm{e}}$, is
$h_{\mathrm{i}\,{ab}}=g_{\mathrm{i}\,\alpha\beta}e_{a}^{\alpha}e_{b}^{\beta}$,
$h_{\mathrm{e}\,ab}=g_{\mathrm{e}\,\alpha\beta}e_{a}^{\alpha}e_{b}^{\beta}$,
respectively, where $g_{\mathrm{i}\,\alpha\beta}$ and
$g_{\mathrm{e}\,\alpha\beta}$ are the components of the metrics
$g_{\mathrm{i}}$ and $g_{\mathrm{e}}$ in the coordinate system
$\left\\{x^{\alpha}\right\\}$. Notice that, in general, the induced metric on
$\mathcal{S}$ by each metric $g_{\mathrm{i}}$ and $g_{\mathrm{e}}$ may not
coincide, hence we use the notation $h_{\mathrm{i}\,ab}$ and
$h_{\mathrm{e}\,ab}$ to refer to the metric induced by the spacetime structure
of $\mathcal{M}_{\mathrm{i}}$ or $\mathcal{M}_{\mathrm{e}}$, respectively. The
extrinsic curvature $K_{\mathrm{i}\,ab}$ or $K_{\mathrm{e}\,ab}$ of the
hypersurface $\mathcal{S}$, as an embedded manifold in
$\mathcal{M}_{\mathrm{i}}$ or $\mathcal{M}_{\mathrm{e}}$, respectively, is
defined as
$K_{\mathrm{i}\,ab}=e_{a}^{\alpha}e_{b}^{\beta}\nabla_{\mathrm{i}\,\alpha}n_{\beta}$,
$K_{\mathrm{e}\,ab}=e_{a}^{\alpha}e_{b}^{\beta}\nabla_{\mathrm{e}\,\alpha}n_{\beta}$,
where $\nabla_{\mathrm{i}}$ and $\nabla_{\mathrm{e}}$ represent the covariant
derivatives with respect to $g_{\mathrm{i}}$ or $g_{\mathrm{e}}$. Their traces
are $K_{\mathrm{i}}=h_{\mathrm{i}}^{ab}K_{\mathrm{i\,ab}}$, and
$K_{\mathrm{e}}=h_{\mathrm{e}}^{ab}K_{\mathrm{i\,ab}}$, respectively.

Now, we need to give the conditions under which the matching of the two
spacetimes $\mathcal{M}_{\mathrm{i}}$ and $\mathcal{M}_{\mathrm{e}}$ form a
valid solution of the Einstein field equations, Eq. (1). Following the Israel
formalism, to join the two spacetimes $\mathcal{M}_{\mathrm{i}}$ and
$\mathcal{M}_{\mathrm{e}}$ at $\mathcal{S}$, such that the union of
$g_{\mathrm{i}}$ and $g_{\mathrm{e}}$ forms a valid solution to the Einstein
field equations (1), two junction conditions must be verified at the matching
surface $\mathcal{S}$: (i) The induced metric $h_{ab}$ as seen from each
region $\mathcal{M}_{\mathrm{i}}$ and $\mathcal{M}_{\mathrm{e}}$, must be the
same, i.e.,

$\left[h_{ab}\right]=0\,.$ (2)

(ii) If the extrinsic curvature $K_{ab}$ is not the same on both sides of the
boundary $\mathcal{S}$, then a thin shell is present at $\mathcal{S}$ with
stress-energy tensor $S_{ab}$ given by

$-\frac{\varepsilon}{8\pi}\left(\left[K_{ab}\right]-h_{ab}\left[K\right]\right)=S_{ab}\,,$
(3)

where $\left[K_{ab}\right]$ represents the difference of $K_{ab}$ as seen from
each sub-manifold at $\mathcal{S}$, i.e.,
$\left[K_{ab}\right]\equiv\left.{K_{\mathrm{e}\,ab}}\right|_{\mathcal{S}}-\left.{K_{\mathrm{i}\,ab}}\right|_{\mathcal{S}}$,
and similarly for $\left[K\right]$, and we use the notation
${K_{\mathrm{i}\,ab}}\equiv K_{ab}\left(\mathcal{M}_{\mathrm{i}}\right)$ and
${K_{\mathrm{e}\,ab}}\equiv K_{ab}\left(\mathcal{M}_{\mathrm{e}}\right)$ to
refer to $K_{ab}$ defined in $\mathcal{M}_{\mathrm{i}}$ or
$\mathcal{M}_{\mathrm{e}}$, respectively, and similarly for $K$.

The Minkowski spacetime with line element
$ds^{2}=-dt^{2}+dr^{2}+r^{2}d\Omega^{2}$, where $t$ and $r$ are the time and
radial coordinates, and $d\Omega^{2}\equiv d\theta^{2}+\sin^{2}\theta
d\varphi^{2}$, with $\theta$ and $\varphi$ being the angular coordinates, is a
solution of Einstein equations, see Eq. (1), in fact the simplest solution. We
assume now that the interior and exterior spacetime have Minkowski line
elements, and find and analyze all possible junctions of two Minkowski
spacetimes through a static, thin matter shell. Using the Israel formalism, we
consider two spacetimes, $\mathcal{M}_{\mathrm{i}}$ and
$\mathcal{M}_{\mathrm{e}}$, each endowed with the Minkowski metric tensor
field glued together at a common hypersurface, $\mathcal{S}$. To apply the
formalism, we will have to find the induced metric and extrinsic curvature
induced on an embedded hypersurface of each spacetime,
$\mathcal{M}_{\mathrm{i}}$ or $\mathcal{M}_{\mathrm{e}}$. We will start by
making the analysis in the interior spacetime, $\mathcal{M}_{\mathrm{i}}$, and
extend the results to the exterior spacetime, $\mathcal{M}_{\mathrm{e}}$.

The interior Minkowski spacetime, $\mathcal{M}_{\mathrm{i}}$, is characterized
by the following line element, in spacetime spherical coordinates,

$ds_{\mathrm{i}}^{2}=-dt_{\mathrm{i}}^{2}+dr_{\mathrm{i}}^{2}+r_{\mathrm{i}}^{2}d\Omega^{2}\,,$
(4)

where $t_{\mathrm{i}}$ and $r_{\mathrm{i}}$ are the time and radial
coordinates, respectively, measured by a free-falling observer in
$\mathcal{M}_{\mathrm{i}}$, and again $d\Omega^{2}\equiv
d\theta^{2}+\sin^{2}\theta d\varphi^{2}$, with $\theta$ and $\varphi$ being
the angular coordinates. On one hand, for the solution itself we are
interested in studying the case where the hypersurface $\mathcal{S}$ is
timelike and static, i.e., static as seen from an observer free falling in the
interior Minkowski spacetime. On the other hand for the stability analysis
that we will take we have to allow for the hypersurface to be dynamical, an so
we compute the induced metric and extrinsic curvature of $\mathcal{S}$
allowing for a dynamical shell and when needed we take the static solution. In
the study of the properties of the matter shell at $\mathcal{S}$, we will then
restrict the setup to the static case. Since we assume the hypersurface to be
timelike, it is convenient to choose the coordinates on $\mathcal{S}$ to be
$\left\\{y^{a}\right\\}=\left(\tau,\theta,\varphi\right)$, where $\tau$ is the
proper time measured by an observer comoving with $\mathcal{S}$. In this
coordinate system, it follows that $e_{\tau}\equiv u$, where $u$ is the
4-velocity of an observer comoving with the shell. The hypersurface
$\mathcal{S}$, as seen from the interior $\mathcal{M}_{\mathrm{i}}$ spacetime,
is parameterized by $\tau$, such that the surface’s radial coordinate is
described by a function $R_{\mathrm{i}}=R_{\mathrm{i}}\left(\tau\right)$.
Then, the 4-velocity $u_{\mathrm{i}}$, where the subscript $\mathrm{i}$ is not
an index and as before denotes interior, as seen from the interior spacetime
is given by
$u_{\mathrm{i}}=\left(\frac{dt_{\mathrm{i}}}{d\tau},\dot{R}_{\mathrm{i}},0,0\right)$,
where overdot represents the derivative with respect to the proper time, i.e.,
$\dot{R}_{\mathrm{i}}\equiv\frac{dR_{\mathrm{i}}}{d\tau}$. Since $\mathcal{S}$
is a timelike hypersurface, it must verify
$u_{\mathrm{i}\,\alpha}u_{\mathrm{i}}^{\alpha}=-1$, therefore we find
$u_{\mathrm{i}}=\left(\sqrt{1+\dot{R}_{\mathrm{i}}^{2}},\dot{R}_{\mathrm{i}},0,0\right)$,
where we chose $\frac{dt_{\mathrm{i}}}{d\tau}>0$ as we assume $u_{\mathrm{i}}$
to point to the future. The expression for the 4-velocity of an observer
comoving with $\mathcal{S}$ and the condition $e_{a}^{\alpha}n_{\alpha}=0$
allow us to find the components of the normal vector field to $\mathcal{S}$,
$n$, as seen from the interior spacetime $\mathcal{M}_{\mathrm{i}}$,
$n_{\mathrm{i}}=\xi_{\mathrm{i}}\left(\dot{R}_{\mathrm{i}},\sqrt{1+\dot{R}_{\mathrm{i}}^{2}},0,0\right)$,
where $\xi_{\mathrm{i}}$ is a normalization factor. Using
$n^{\alpha}n_{\alpha}=\varepsilon$, Eq. (4), and the condition that the normal
vector field $n_{\mathrm{i}}$ is spacelike, yields $\xi_{\mathrm{i}}=\pm 1$.
Now, defining $\nabla_{\mathrm{i}}r_{\mathrm{i}}$ as the gradient of
$r_{\mathrm{i}}$, the choice $\xi_{\mathrm{i}}=+1$ or $\xi_{\mathrm{i}}=-1$
represents whether the inner product
$g_{\mathrm{i}}\left(n_{\mathrm{i}},\nabla_{\mathrm{i}}r_{\mathrm{i}}\right)>0$
or
$g_{\mathrm{i}}\left(n_{\mathrm{i}},\nabla_{\mathrm{i}}r_{\mathrm{i}}\right)<0$,
respectively. Under the Israel formalism both values for $\xi_{\mathrm{i}}$
are possible and we shall consider both cases. Using the induced metric
equation,
$h_{\mathrm{i}\,{ab}}=g_{\mathrm{i}\,\alpha\beta}e_{a}^{\alpha}e_{b}^{\beta}$,
and
$u_{\mathrm{i}}=\left(\sqrt{1+\dot{R}_{\mathrm{i}}^{2}},\dot{R}_{\mathrm{i}},0,0\right)$,
we find that the induced metric on $\mathcal{S}$ by the spacetime
$\mathcal{M}_{\mathrm{i}}$, is such that the line element can be written as
$\left.ds^{2}\right|_{\mathcal{S_{\mathrm{i}}}}=-d\tau^{2}+R_{\mathrm{i}}^{2}d\Omega^{2}$.
Gathering these results, we can compute the components of the extrinsic
curvature of $\mathcal{S}$ as seen from $\mathcal{M}_{\mathrm{i}}$,
$K^{\mathrm{i}}_{ab}$. In the case where the matching surface $\mathcal{S}$ is
timelike and spherically symmetric, the non-null components of the extrinsic
curvature are given by, dropping here the superscript i to not overcrowd the
notation, $K_{\tau\tau}=-a^{\alpha}n_{\alpha}$,
$K_{\theta\theta}=\nabla_{\theta}n_{\theta}$,
$K_{\varphi\varphi}=\nabla_{\varphi}n_{\varphi}$, where $a^{\alpha}\equiv
u^{\beta}\nabla_{\beta}u^{\alpha}$ represents the components of the
4-acceleration of an observer comoving with $\mathcal{S}$. Taking into account
Eq. (4) and
$u_{\mathrm{i}}=\left(\sqrt{1+\dot{R}_{\mathrm{i}}^{2}},\dot{R}_{\mathrm{i}},0,0\right)$,
we find that the non-trivial components of the exterior curvature as seen from
the interior Minkowski spacetime are given by
${{K_{\mathrm{i}}}^{\tau}}_{\tau}=\xi_{\mathrm{i}}\frac{\ddot{R}_{\mathrm{i}}}{\sqrt{1+\dot{R}_{\mathrm{i}}^{2}}}$,
and
${{K_{\mathrm{i}}}^{\theta}}_{\theta}={{K_{\mathrm{i}}}^{\varphi}}_{\varphi}=\xi_{\mathrm{i}}\frac{\sqrt{1+{\dot{R}}_{\mathrm{i}}^{2}}}{R_{\mathrm{i}}}$,
where the induced metric $h_{\mathrm{i}}^{ab}$ associated with the
hypersurface line element was used to raise the indices. Since we assume the
shell to be static, one has $\dot{R}_{\mathrm{i}}=0$: So in this static case,
in brief, one has, that the 4-velocity $u_{\mathrm{i}}$ and the normal
$n_{\mathrm{i}}$ are $u_{\mathrm{i}}=\left(1,0,0,0\right)$ and
$n_{\mathrm{i}}=\xi_{\mathrm{i}}\left(0,1,0,0\right)$, the line element on the
shell is

$\left.ds^{2}\right|_{\mathcal{S_{\mathrm{i}}}}=-d\tau^{2}+R_{\mathrm{i}}^{2}d\Omega^{2}\,,$
(5)

and the extrinsic curvature is given by

${{K_{\mathrm{i}}}^{\tau}}_{\tau}=0\,,\quad\quad{{K_{\mathrm{i}}}^{\theta}}_{\theta}={{K_{\mathrm{i}}}^{\varphi}}_{\varphi}=\frac{\xi_{\mathrm{i}}}{R_{\mathrm{i}}}\,.$
(6)

Under the Israel formalism both values for $\xi_{\mathrm{i}}$ are possible and
we shall consider both cases.

The exterior Minkowski spacetime, $\mathcal{M}_{\mathrm{e}}$, is characterized
by the following line element, in spacetime spherical coordinates,

$ds_{\mathrm{e}}^{2}=-dt_{\mathrm{e}}^{2}+dr_{\mathrm{e}}^{2}+r_{\mathrm{e}}^{2}d\Omega^{2}\,,$
(7)

where $t_{\mathrm{e}}$ and $r_{\mathrm{e}}$ are the time and radial
coordinates, respectively, measured by a free-falling observer in
$\mathcal{M}_{\mathrm{e}}$, and again $d\Omega^{2}\equiv
d\theta^{2}+\sin^{2}\theta d\varphi^{2}$, with $\theta$ and $\varphi$ being
the angular coordinates. Since the setup is the same as the one for the
interior we will sketch the calculations briefly in order to be complete. For
a timelike hypersurface it is convenient to choose the coordinates on
$\mathcal{S}$ to be $\left\\{y^{a}\right\\}=\left(\tau,\theta,\varphi\right)$,
where $\tau$ is the proper time measured by an observer comoving with
$\mathcal{S}$. In this coordinate system, it follows that the 4-velocity $u$
of an observer comoving with the shell is given by $e_{\tau}\equiv u$. The
hypersurface $\mathcal{S}$, as seen from the exterior
$\mathcal{M}_{\mathrm{e}}$ spacetime, is parameterized by $\tau$, such that
the surface’s radial coordinate is described by a function
$R_{\mathrm{e}}=R_{\mathrm{e}}\left(\tau\right)$. Then,
$u_{\mathrm{e}}=\left(\frac{dt_{\mathrm{e}}}{d\tau},\dot{R}_{\mathrm{e}},0,0\right)$,
where the subscript $\mathrm{e}$ is not an index and as before denotes
exterior, and $\dot{R}_{\mathrm{e}}\equiv\frac{dR_{\mathrm{e}}}{d\tau}$. Since
$\mathcal{S}$ is timelike, one has
$u_{\mathrm{e}\,\alpha}u_{\mathrm{e}}^{\alpha}=-1$, so
$u_{\mathrm{e}}=\left(\sqrt{1+\dot{R}_{\mathrm{e}}^{2}},\dot{R}_{\mathrm{e}},0,0\right)$,
where we chose $\frac{dt_{\mathrm{e}}}{d\tau}>0$ as we assume $u_{\mathrm{e}}$
to point to the future. From $e_{a}^{\alpha}n_{\alpha}=0$, one finds the
components of the normal vector field to $\mathcal{S}$, $n$, as seen from the
exterior spacetime, namely
$n_{\mathrm{e}}=\xi_{\mathrm{e}}\left(\dot{R}_{\mathrm{e}},\sqrt{1+\dot{R}_{\mathrm{e}}^{2}},0,0\right)$,
where $\xi_{\mathrm{e}}$ is a normalization factor. Using
$n^{\alpha}n_{\alpha}=1$ and Eq. (7), yields $\xi_{\mathrm{e}}=\pm 1$.
Defining $\nabla_{\mathrm{e}}r_{\mathrm{e}}$ as the gradient of
$r_{\mathrm{e}}$, the choice $\xi_{\mathrm{e}}=+1$ or $\xi_{\mathrm{e}}=-1$
represents whether the inner product
$g_{\mathrm{e}}\left(n_{\mathrm{e}},\nabla_{\mathrm{e}}r_{\mathrm{e}}\right)>0$
or
$g_{\mathrm{e}}\left(n_{\mathrm{e}},\nabla_{\mathrm{e}}r_{\mathrm{e}}\right)<0$,
respectively. Usingthe induced metric equation,
$h_{\mathrm{e}\,ab}=g_{\mathrm{e}\,\alpha\beta}e_{a}^{\alpha}e_{b}^{\beta}$,
and
$u_{\mathrm{e}}=\left(\sqrt{1+\dot{R}_{\mathrm{e}}^{2}},\dot{R}_{\mathrm{e}},0,0\right)$,
we find that the induced metric on $\mathcal{S}$ by the spacetime
$\mathcal{M}_{\mathrm{e}}$, is such that the line element can be written as
$\left.ds^{2}\right|_{\mathcal{S_{\mathrm{e}}}}=-d\tau^{2}+R_{\mathrm{e}}^{2}d\Omega^{2}$.
The non-null components of the extrinsic curvature are here given by, dropping
here the superscript e to not overcrowd the notation,
$K_{\tau\tau}=-a^{\alpha}n_{\alpha}$,
$K_{\theta\theta}=\nabla_{\theta}n_{\theta}$,
$K_{\varphi\varphi}=\nabla_{\varphi}n_{\varphi}$, where $a^{\alpha}\equiv
u^{\beta}\nabla_{\beta}u^{\alpha}$ represents the components of the
4-acceleration of an observer comoving with $\mathcal{S}$. Taking into account
Eq. (7) and
$u_{\mathrm{e}}=\left(\sqrt{1+\dot{R}_{\mathrm{e}}^{2}},\dot{R}_{\mathrm{e}},0,0\right)$
the non-trivial components of the exterior curvature as seen from the exterior
Minkowski spacetime are given by
${{K_{\mathrm{e}}}^{\tau}}_{\tau}=\xi_{\mathrm{e}}\frac{\ddot{R}_{\mathrm{e}}}{\sqrt{1+\dot{R}_{\mathrm{e}}^{2}}}$,
and
${{K_{\mathrm{e}}}^{\theta}}_{\theta}={{K_{\mathrm{e}}}^{\varphi}}_{\varphi}=\xi_{\mathrm{e}}\frac{\sqrt{1+{\dot{R}}_{\mathrm{e}}^{2}}}{R_{\mathrm{e}}}$,
where the induced metric $h_{\mathrm{e}}^{ab}$ associated with the
hypersurface line element was used to raise the indices. Since we assume the
shell to be static, one has $\dot{R}_{\mathrm{e}}=0$. So in this case one has
that the 4-velocity $u_{\mathrm{e}}$ and the normal $n_{\mathrm{e}}$ are
$u_{\mathrm{e}}=\left(1,0,0,0\right)$ and
$n_{\mathrm{e}}=\xi_{\mathrm{e}}\left(0,1,0,0\right)$, the line element on the
shell is

$\left.ds^{2}\right|_{\mathcal{S_{\mathrm{e}}}}=-d\tau^{2}+R_{\mathrm{e}}^{2}d\Omega^{2}\,,$
(8)

and the extrinsic curvature is given by

${{K_{\mathrm{e}}}^{\tau}}_{\tau}=0\,,\quad\quad{{K_{\mathrm{e}}}^{\theta}}_{\theta}={{K_{\mathrm{e}}}^{\varphi}}_{\varphi}=\frac{\xi_{\mathrm{e}}}{R_{\mathrm{e}}}\,.$
(9)

Both values for $\xi_{\mathrm{e}}$ are possible and we shall consider both
cases.

To complete the solution we have find the properties of the matter at the thin
shell. Indeed, having found previously the necessary expressions at the
hypersurface $\mathcal{S}$ as seen from the $\mathcal{M}_{\mathrm{i}}$ and
$\mathcal{M}_{\mathrm{e}}$ spacetimes, we can now use the Israel formalism to
glue together the two spacetimes. Imposing the first junction condition, Eq.
(2), and using Eqs. (5) and (8) for the induced metrics, we find that the
radial coordinate of $\mathcal{S}$ as seen from the interior and exterior
spacetimes, $R_{\mathrm{i}}$ and $R_{\mathrm{e}}$, respectively, must be the
same, $R_{\mathrm{i}}=R_{\mathrm{e}}$. We then denote by $R$ the value of the
radial coordinate of $\mathcal{S}$ as seen from both spacetimes,

$R\equiv R_{\mathrm{i}}=R_{\mathrm{e}}\,.$ (10)

We also assume that the stress-energy tensor $S_{ab}$ of the thin shell on
$\mathcal{S}$, can be cast in a perfect fluid form $S_{ab}=\sigma
u_{a}u_{b}+p\left(h_{ab}+u_{a}u_{b}\right)$, where $\sigma$ is the energy per
unit area, $p$ is the tangential pressure of the fluid, $h_{ab}$ is the
induced metric on $\mathcal{S}$, and $u_{a}$ is the fluid’s 3-velocity on
$\mathcal{S}$. Using the appropriate equations we find
$S^{\tau}{}_{\tau}=-\sigma$, and
$S^{\theta}{}_{\theta}=S^{\varphi}{}_{\varphi}=p$. Having found previously the
expressions for the extrinsic curvature of the hypersurface $\mathcal{S}$ as
seen from the $\mathcal{M}_{\mathrm{i}}$ and $\mathcal{M}_{\mathrm{e}}$
spacetimes, and knowing that the stress-energy tensor of the shell is that of
a perfect fluid, we are can now use the second junction condition, Eq. (3).
Applying it to our spherically symmetric problem gives that the only
nontrivial components of the stress-energy tensor $S^{a}{}_{b}$ are given by
$S^{\tau}{}_{\tau}=\frac{1}{4\pi}\left[K^{\theta}{}_{\theta}\right]$, and
$S^{\theta}{}_{\theta}=S^{\varphi}{}_{\varphi}=\frac{1}{8\pi}\left[K^{\tau}{}_{\tau}\right]+\frac{1}{4\pi}\left[K^{\theta}{}_{\theta}\right]$.
Using then for the interior,
${{K_{\mathrm{i}}}^{\tau}}_{\tau}=\xi_{\mathrm{i}}\frac{\ddot{R}}{\sqrt{1+\dot{R}^{2}}}$,
${{K_{\mathrm{i}}}^{\theta}}_{\theta}={{K_{\mathrm{i}}}^{\varphi}}_{\varphi}=\xi_{\mathrm{i}}\frac{\sqrt{1+{\dot{R}}^{2}}}{R}$,
and for the exterior,
${{K_{\mathrm{e}}}^{\tau}}_{\tau}=\xi_{\mathrm{e}}\frac{\ddot{R}}{\sqrt{1+\dot{R}^{2}}}$,
and
${{K_{\mathrm{e}}}^{\theta}}_{\theta}={{K_{\mathrm{e}}}^{\varphi}}_{\varphi}=\xi_{\mathrm{e}}\frac{\sqrt{1+{\dot{R}}^{2}}}{R}$,
we find
$\sigma=\left(\xi_{\mathrm{i}}-\xi_{\mathrm{e}}\right)\frac{\sqrt{1+\dot{R}^{2}}}{4\pi
R}$ and
$p=-\left(\xi_{\mathrm{i}}-\xi_{\mathrm{e}}\right)\frac{R\ddot{R}+\dot{R}^{2}+1}{8\pi
R\sqrt{1+\dot{R}^{2}}}$. From these two equations we derive the following
conservation law for the shell
$\dot{\sigma}+\frac{2\dot{R}}{R}\left(\sigma+p\right)=0$. For a static shell,
the time derivatives are zero and so, using directly if one wishes Eqs. (6)
and (9), one finds

$\displaystyle\sigma$
$\displaystyle=\quad\left(\xi_{\mathrm{i}}-\xi_{\mathrm{e}}\right)\frac{1}{4\pi
R}\,,$ (11) $\displaystyle p$
$\displaystyle=-\left(\xi_{\mathrm{i}}-\xi_{\mathrm{e}}\right)\frac{1}{8\pi
R}\,.$ (12)

From Eqs. (11) and (12) we derive

$\sigma+2p=0\,.$ (13)

The matter of the thin shell obeys necessarily this equation of state, namely,
$p=-\frac{1}{2}\sigma$, for a Minkowski-Minkowski static spacetime. From Eqs.
(11) and (12), we see that for $\xi_{\mathrm{i}}=\xi_{\mathrm{e}}$, and so
$\frac{1}{2}(\xi_{\rm i}-\xi_{\rm e})=0$ we get the trivial case, $\sigma=0$
and $p=0$. For $\xi_{\mathrm{i}}=1$ and $\xi_{\mathrm{e}}=-1$, and so
$\frac{1}{2}(\xi_{\rm i}-\xi_{\rm e})=1$, we get $\sigma=\frac{1}{2\pi R}$ and
$p=-\frac{1}{4\pi R}$. For $\xi_{\mathrm{i}}=-1$ and $\xi_{\mathrm{e}}=1$, and
so $\frac{1}{2}(\xi_{\rm i}-\xi_{\rm e})=-1$, we get $\sigma=-\frac{1}{2\pi
R}$ and $p=\frac{1}{4\pi R}$. All cases obey Eq. (13), i.e., the relation
between the surface energy density $\sigma$ and the surface pressure $p$ is
independent of $\xi_{\mathrm{i}}$ or $\xi_{\mathrm{e}}$. Besides the trivial
case, i.e., the Minkowski universe which has $\frac{1}{2}(\xi_{\rm i}-\xi_{\rm
e})=0$, there are two possible universes at this juncture, the Minkowski-
Minkowski static closed universe which has $\frac{1}{2}(\xi_{\rm i}-\xi_{\rm
e})=1$, and the Minkowski-Minkowski static open universe which has
$\frac{1}{2}(\xi_{\rm i}-\xi_{\rm e})=-1$.

####  II.1.2 Linearized stability analysis for Minkowski-Minkowski universes

An important question regarding the Minkowski-Minkowski static universe
solutions, i.e., the Minkowski-Minkowski closed and open universes, is if
these are stable under perturbations. Here we will discuss the linear
stability of the Minkowski-Minkowski solutions that we have found by analyzing
the equation of motion of the shell near the static configuration.

To study the linear stability of the Minkowski-Minkowski static universe
solution we have to find the evolution equation for the shell radius $R$ and
analyze the behavior of these solutions as we perturb the spacetime. The
analysis can be done in a unified way by making use of the parameter
$\frac{1}{2}(\xi_{\rm i}-\xi_{\rm e})$. The equation of motion of the thin
shell previously found, namely,
$\sigma=\left(\xi_{\mathrm{i}}-\xi_{\mathrm{e}}\right)\frac{\sqrt{1+\dot{R}^{2}}}{4\pi
R}$, can be inverted and put in the form

$\dot{R}^{2}+V(R)=0\,,$ (14)

where a dot means derivative with respect to time $t$, and the potential
$V(R)$ is given by

$V(R)=1-\left(\frac{4\pi
R\sigma}{\xi_{\mathrm{i}}-\xi_{\mathrm{e}}}\right)^{2}\,.$ (15)

A thin matter shell is stable if and only if the potential $V(R)$ at the
shell’s position is at a local minimum, i.e., if $V^{\prime}(R)=0$ and
$V^{\prime\prime}(R)\geq 0$, with the equality providing the marginal neutral
case and where a prime denotes the derivative with respect to $R$. Thus, we
have to calculate the matter properties and its derivatives. All these
properties are functions of the shell radius $R$, namely, $\sigma=\sigma(R)$,
$p=p(R)$, $\sigma^{\prime}=\sigma^{\prime}(R)$, and
$p^{\prime}=p^{\prime}(R)$. To find an expression for $\sigma^{\prime}(R)$, we
consider the conservation law for the shell already found, namely,
$\dot{\sigma}+\frac{2\dot{R}}{R}\left(\sigma+p\right)=0$, i.e.,
$\dot{\sigma}=-\frac{2\dot{R}}{R}\left(\sigma+p\right)$. Using the inverse
function theorem, we have $\sigma^{\prime}=\frac{\dot{\sigma}}{\dot{R}}$, and
so

$\sigma^{\prime}=-\frac{2\left(\sigma+p\right)}{R}\,.$ (16)

We have also to analyze the derivatives of the potential at the static
configuration. From Eq. (15) we get
$V^{\prime}(R)=-\frac{32\pi^{2}R\sigma}{\left(\xi_{\mathrm{i}}-\xi_{\mathrm{e}}\right)^{2}}\left(\sigma+R\sigma^{\prime}\right)$.
Taking another derivative we get
$V^{\prime\prime}(R)=\frac{32\pi^{2}}{\left(\xi_{\mathrm{i}}-\xi_{\mathrm{e}}\right)^{2}}\left[\left(\sigma+R\sigma^{\prime}\right)\left(\sigma+2p\right)+R\sigma\left(\sigma^{\prime}+2p^{\prime}\right)\right]$.
Now, if we introduce Eq. (16) into $V^{\prime}(R)$, we have
$V^{\prime}(R)=\frac{32\pi^{2}R\sigma}{\left(\xi_{\mathrm{i}}-\xi_{\mathrm{e}}\right)^{2}}\left(\sigma+p\right)$.
To analyze $V^{\prime\prime}(R)$ we have to find an expression for
$p^{\prime}(R)$. We assume that the thin matter shell is composed of cold
matter such that it verifies a generic equation of state of the form
$p=p\left(\sigma\right)$. Then, we can define the parameter
$\eta\left(\sigma\right)=\frac{\partial p}{\partial\sigma}$ such that,
$p^{\prime}=\eta\sigma^{\prime}$. Hence, using
$\sigma^{\prime}+2p^{\prime}=\sigma^{\prime}\left(1+2\eta\right)$ and Eq. (16)
we can write $V^{\prime\prime}(R)$ as
$V^{\prime\prime}(R)=-\frac{32\pi^{2}}{\left(\xi_{\mathrm{i}}-\xi_{\mathrm{e}}\right)^{2}}\left[2\sigma\left(\sigma+p\right)\left(1+2\eta\right)+\left(\sigma+2p\right)^{2}\right]$.
In brief, the derivatives of the potential $V(R)$ are

$V^{\prime}(R)=\frac{32\pi^{2}R\sigma}{\left(\xi_{\mathrm{i}}-\xi_{\mathrm{e}}\right)^{2}}\left(\sigma+p\right)\,,$
(17)

and

$V^{\prime\prime}(R)=-\frac{32\pi^{2}}{\left(\xi_{\mathrm{i}}-\xi_{\mathrm{e}}\right)^{2}}\left[2\sigma\left(\sigma+p\right)\left(1+2\eta\right)+\left(\sigma+2p\right)^{2}\right]\,.$
(18)

where $\eta\left(\sigma\right)=\frac{\partial p}{\partial\sigma}$.

We now linearize the equation of motion for the shell given by Eq. (14) around
a static solution. Defining $R_{0}$ as the circumferential radius of the
static thin shell and assuming the potential $V$ to be a differentiable
function at $R_{0}$, we can expand the potential given in Eq. (15) around
$R_{0}$ as

$V(R)=V(R_{0})+V^{\prime}(R_{0})\left(R-R_{0}\right)+\frac{1}{2}V^{\prime\prime}(R_{0})\left(R-R_{0}\right)^{2}\,,$
(19)

plus higher order terms of $\mathcal{O}\left[\left(R-R_{0}\right)^{3}\right]$.
A thin matter shell with radius $R_{0}$ is stable or neutrally stable if and
only if the potential $V(R)$ satisfies $V^{\prime}(R_{0})=0$ and
$V^{\prime\prime}(R_{0})\geq 0$. For a shell with radius $R_{0}$ the static
solutions found in the previous section are characterized generically by the
following expressions, $\sigma=\frac{\xi_{\mathrm{i}}-\xi_{\mathrm{e}}}{4\pi
R_{0}}$ and $p=-\frac{\xi_{\mathrm{i}}-\xi_{\mathrm{e}}}{8\pi R_{0}}$, see
Eqs. (11) and (12), where $\xi_{\mathrm{i}}\neq\xi_{\mathrm{e}}$ for the
nontrivial solutions. Substituting Eq. (11) into Eq. (15) we find
$V(R_{0})=0$, as expected. Substituting Eqs. (11) and (12) into Eq. (17) and
evaluating it at the static solution we find $V^{\prime}(R_{0})=0$. Evaluating
$V^{\prime\prime}(R)$ at the static solution, $R=R_{0}$, and using again Eqs.
(11) and (11), we find $V^{\prime\prime}(R_{0})=0$. In brief,

$V^{\prime}(R_{0})=0\,,$ (20)

and

$V^{\prime\prime}(R_{0})=0\,.$ (21)

Moreover, all higher order derivatives of the potential go to zero at $R_{0}$
for the static solutions.

Gathering these calculations, we conclude that, besides the trivial case,
i.e., the Minkowski universe which has $\frac{1}{2}(\xi_{\rm i}-\xi_{\rm
e})=0$ and is trivially neutraly stable, there is the Minkowski-Minkowski
static closed universe which has $\frac{1}{2}(\xi_{\rm i}-\xi_{\rm e})=1$ and
is nontrivially neutraly stable, and the Minkowski-Minkowski static open
universe which has $\frac{1}{2}(\xi_{\rm i}-\xi_{\rm e})=-1$ and is also
nontrivially neutraly stable. This neutral stability means that if we slightly
displace the thin shell, it will simply stay at the new radius. This confirms
our expectation, as the interior and exterior spacetimes are both described by
a Minkowski, i.e., flat, solution.

###  II.2 Minkowski-Minkowski universes: Geometry and physics

####  II.2.1  Minkowski-Minkowski static closed universe: A bubble universe

Here we display a Minkowski-Minkowski static closed universe as a solution of
general relativity. We rely on the results presented above. We assume that the
Minkowski line element is valid for a region, which we call interior
$\mathcal{M}_{\mathrm{i}}$, up to a radius $R$, i.e., $0\leq
r_{\mathrm{i}}\leq R$, where $r_{\mathrm{i}}$ denotes the interior radial
coordinate. We join this region to another region, which we call exterior
$\mathcal{M}_{\mathrm{e}}$, where the exterior radial coordinate is denoted by
$r_{\mathrm{e}}$. The junction is done at a common hypersurface $\mathcal{S}$
with circumferential radius $r_{\mathrm{i}}=r_{\mathrm{e}}=R$. Thus, the whole
spacetime is composed by the two regions plus the common hypersurface, which
is a domain wall, i.e., a thin shell. The common hypersurface $\mathcal{S}$ is
assumed to be static. Assuming the existence of a vector field $n$, normal, at
each point, to the common hypersurface $\mathcal{S}$, we have found that the
solution depends on the orientation of this normal field $n$. For each region,
$\mathcal{M}_{\mathrm{i}}$ and $\mathcal{M}_{\mathrm{e}}$, the orientation of
the normal is encoded in a single parameter, namely, one for the interior,
$\xi_{\mathrm{i}}$, and one for the exterior, $\xi_{\mathrm{e}}$. In both
cases, $\xi_{\mathrm{i}}$ and $\xi_{\mathrm{e}}$ can have values $+1$ or $-1$.
The value $+1$ indicates that the normal points in the direction of increasing
radial coordinate and the value $-1$ indicates that the normal points in the
direction of decreasing radial coordinate. The solutions with
$\xi_{\mathrm{i}}=\xi_{\mathrm{e}}$ are trivial as the resulting spacetime is
simply the full Minkowski flat universe. Here we consider the first non-
trivial solution, i.e., $\xi_{\mathrm{i}}=+1$ and $\xi_{\mathrm{e}}=-1$. This
static solution represents the case where the normal vector field $n$ points
in the direction of increasing radial coordinate as seen from the interior
spacetime $\mathcal{M}_{\mathrm{i}}$ and points in the direction of decreasing
radial coordinate as seen from the exterior spacetime
$\mathcal{M}_{\mathrm{e}}$. Since $n$ is assumed to point from
$\mathcal{M}_{\mathrm{i}}$ to $\mathcal{M}_{\mathrm{e}}$ this implies that in
the exterior region one also has $0\leq r_{\mathrm{e}}\leq R$. Thus, this
solution is composed by two spatially compact Minkowski spacetime regions
glued together at the common boundary $\mathcal{S}$. This solution then
represents a Riemann flat spacetime everywhere except at $\mathcal{S}$.
Overall it is a static closed Minkowski-Minkowski universe, for which the line
element can be written as, see also Eqs. (4) and (7),

$ds^{2}=-dt^{2}+dr_{\mathrm{i}}^{2}+r_{\mathrm{i}}^{2}d\Omega^{2},\quad 0\leq
r_{\mathrm{i}}\leq
R\,,\quad\quad\;\;\;ds^{2}=-dt^{2}+dr_{\mathrm{e}}^{2}+r_{\mathrm{e}}^{2}d\Omega^{2},\quad
0\leq r_{\mathrm{e}}\leq R\,.$ (22)

Now we turn to the properties of the matter at the domain wall, or shell, at
$\mathcal{S}$. In this case, putting $\xi_{\mathrm{i}}=+1$ and
$\xi_{\mathrm{e}}=-1$ into Eqs. (11) and (12), the energy density $\sigma$ and
the tangential pressure $p$ at the domain wall are given by

$\sigma=\;\;\frac{1}{2\pi R}\,,$ (23) $p=-\frac{1}{4\pi R}\,.$ (24)

This solution is then characterized by the presence of a surface layer in the
form of a domain wall at radius $R$, separating two Minkowski halves. The thin
domain wall is composed of a perfect fluid with positive energy density and is
supported by tension such that it obeys the equation of state $\sigma+2p=0$.
Moreover, from Eqs. (23) and (24), we find that the following inequalities are
verified: $\sigma\geq 0$, $\sigma+p\geq 0$, $\sigma+2p\geq 0$ and
$\sigma\geq\left|p\right|$, therefore, the matter at the domain wall verifies
the null, weak, strong, and dominant pointwise energy conditions. Since the
effective mass $m$ can be defined by the quantity $\sigma+2p$ and this latter
is zero, one has $m=0$. So the domain wall yields no total mass $m$ as it
should to have Minkowski spacetime on both sides of the domain wall. The
volume of this universe is $V=\frac{8\pi}{3}R^{3}$.

The spatial structure and the causal structure are also important to analyze.
A time slice $t={\rm constant}$ of the spacetime gives that the 3-space is a
highly squashed 3-sphere, i.e., it is made of two copies of two plane 3-balls
joined at a 2-sphere. To see this one makes an embedding. The embedding of
this 3-space can be easily done in 4-dimensional Euclidean space
$\mathbb{R}^{4}$. In Fig. 1 we show an embedding diagram of a
$\theta=\frac{\pi}{2}$ slice of the static Minkowski-Minkowski closed universe
in a 3-dimensional Euclidean space, displaying clearly the squashed character
of the 3-sphere. One can also make appropriate identifications between points
in the interior and exterior spherical pieces to turn the space into a
projective space. The causal structure of the resulting spacetime can be shown
in a Carter-Penrose diagram, as in Fig. 2. We use the hash symbol # to
represent the connected sum of the spacetime manifold in order to conserve the
conformal structure in the Carter-Penrose diagram of the total spacetime. We
see that it represents a universe in which the spatial sections are highly
squashed 3-spheres, i.e., two copies of two plane 3-balls joined at a
2-sphere, such that if we include time the total spacetime is a squashed
3-cylinder, the time line times the squashed 3-sphere. A timelike geodesic, or
a free-falling particle, initially moving along the radial coordinate towards
increasing values of it in one half of the spacetime, would reach the domain
wall at $r=R$ at some point, and then continue until it reaches the center of
coordinates at the other half where it would continue its trajectory into the
antipode point of the wall, and so on.

Figure 1:  Embedding diagram of a $t=\text{constant}$ and
$\theta=\frac{\pi}{2}$ slice of the Minkowski-Minkowski static closed universe
in 3-dimensional Euclidean space. The interior coordinate is in the range
$0\leq r_{\mathrm{i}}\leq R$, the exterior coordinate is in the range $0\leq
r_{\mathrm{i}}\leq R$, the radius of the domain wall, or shell, is $R$, and
the borders of the circumferences should be identified.  Figure 2:  Carter-
Penrose diagram of the Minkowski-Minkowski static closed universe. The hash
symbol # represents the connected sum of the spacetime manifold. The symbols
$i^{-}$ and $i^{+}$ represent past and future causal infinity, respectively.

The Minkowski-Minkowski static closed universe is marginally stable. Indeed,
one has from Eq. (21) that $V^{\prime\prime}(R_{0})=0$. The solution is in
neutral equilibrium, meaning that for a slight displacement the thin shell
stays at the new radius. This result confirms the expectations. as the
interior and exterior spacetimes are both described by a Minkowski, i.e.,
flat, solution.

This Minkowski-Minkowski static closed universe is a bubble universe which is
summarized in Eqs. (22)-(24) and in Figs. 1 and 2. Moreover, a fundamental
shell is defined as a shell with Minkowski interior with a center and one of
the three basic exterior spacetimes, Minkowski, Schwarzschild, and Reissner-
Nordström. Thus the Minkowski-Minkowski static closed universe we just found
completes the search of all the fundamental shells in the three basic ambient
spacetimes, namely, Minkowski-Minkowski, Minkowski-Schwarzschild, and
Minkowski-Reissner-Nordström, the latter two having been found previously.

The Minkowski-Minkowski static closed universe is a representative of the set
of closed universes. It can be compared with other such closed universes. This
will be done later.

####  II.2.2 Minkowski-Minkowski static open universe: A traversable wormhole

Here we display a Minkowski-Minkowski static open universe as a solution of
general relativity. We rely on the previous results. We assume that the
Minkowski line element is valid for a region, which we call interior
$\mathcal{M}_{\mathrm{i}}$, that goes from spatial infinity to a radius $R$,
i.e., $R\leq r_{\mathrm{i}}<\infty$, where $r_{\mathrm{i}}$ denotes the
interior radial coordinate. We join this region to another region, which we
call exterior $\mathcal{M}_{\mathrm{e}}$, where the exterior radial coordinate
is denoted by $r_{\mathrm{e}}$. The junction is done at a common hypersurface
$\mathcal{S}$ with circumferential radius $r_{\mathrm{i}}=r_{\mathrm{e}}=R$.
Thus, the whole spacetime is composed by the two regions plus the common
hypersurface, which is a thin shell. The common hypersurface $\mathcal{S}$ is
assumed to be static. Assuming the existence of a vector field $n$, normal, at
each point, to the common hypersurface $\mathcal{S}$, we have found that the
solution depends on the orientation of this normal field $n$. For each region,
$\mathcal{M}_{\mathrm{i}}$ and $\mathcal{M}_{\mathrm{e}}$, the orientation of
the normal is encoded in a single parameter, namely, one for the interior,
$\xi_{\mathrm{i}}$, and one for the exterior, $\xi_{\mathrm{e}}$. In both
cases, $\xi_{\mathrm{i}}$ and $\xi_{\mathrm{e}}$ can have values $+1$ or $-1$.
The value $+1$ indicates that the normal points in the direction of increasing
radial coordinate and the value $-1$ indicates that the normal points in the
direction of decreasing radial coordinate. The solutions with
$\xi_{\mathrm{i}}=\xi_{\mathrm{e}}$ are trivial as the resulting spacetime is
simply the full Minkowski flat universe. Here, we consider the second non-
trivial solution, i.e., $\xi_{\mathrm{i}}=-1$ and $\xi_{\mathrm{e}}=+1$. This
static solution represents the case where the normal vector field $n$ points
in the direction of decreasing radial coordinate as seen from the interior
spacetime $\mathcal{M}_{\mathrm{i}}$ and points in the direction of increasing
radial coordinate as seen from the exterior spacetime
$\mathcal{M}_{\mathrm{e}}$. Since $n$ is assumed to point from
$\mathcal{M}_{\mathrm{i}}$ to $\mathcal{M}_{\mathrm{e}}$ this implies that in
the exterior region one also has $R\leq r_{\mathrm{e}}<\infty$. Thus, this
solution is composed by two spatially open Minkowski spacetime regions glued
together at the common boundary $\mathcal{S}$. This solution then represents a
Riemann flat spacetime everywhere except at $\mathcal{S}$. Overall it is a
static closed Minkowski-Minkowski universe, for which the line element can be
written as, see also Eqs. (4) and (7),

$ds^{2}=-dt^{2}+dr_{\mathrm{i}}^{2}+r_{\mathrm{i}}^{2}d\Omega^{2},\quad R\leq
r_{\mathrm{i}}<\infty\,,\quad\quad\;\;\;ds^{2}=-dt^{2}+dr_{\mathrm{e}}^{2}+r_{\mathrm{e}}^{2}d\Omega^{2},\quad
R\leq r_{\mathrm{e}}<\infty\,.$ (25)

Now we turn to the properties of the matter shell at $\mathcal{S}$. In this
case, putting $\xi_{\mathrm{i}}=-1$ and $\xi_{\mathrm{e}}=+1$ into Eqs. (11)
and (12), the energy density $\sigma$ and the tangential pressure $p$ at the
thin shell are given by

$\sigma=-\frac{1}{2\pi R}\,,$ (26) $p=\;\;\frac{1}{4\pi R}\,.$ (27)

This solution is then characterized by the presence of a surface layer or thin
shell at radius $R$, separating two Minkowski open halves. The thin matter
shell is composed of a perfect fluid with negative energy density and is
supported by pressure such that it obeys the equation of state $\sigma+2p=0$.
Moreover, from Eqs. (26) and (27), we find that the following inequalities are
verified: $\sigma\leq 0$, $\sigma+p\leq 0$, $\sigma+2p\leq 0$ and
$\sigma\leq\left|p\right|$, therefore, the matter shell violates the null,
weak, strong and dominant pointwise energy conditions. Since the effective
mass $m$ can be defined by $\sigma+2p$ and this latter is zero, one has $m=0$.
So the shell yields no total mass $m$ as it should to have Minkowski spacetime
on both sides of the shell. The volume of this spacetime, in its simplest
personification, i.e., without making identifications for lare $r$, is
infinite.

The spatial structure and the causal structure are also important to analyze.
A time slice $t={\rm constant}$ of the spacetime gives that the 3-space is a
universe in which the spatial sections are two copies of the complements of
two plane 3-balls joined at a 2-sphere, the throat, yielding a non-simply-
connected open universe, more precisely, a traversable wormhole. To see this
one makes an embedding. The embedding of this 3-space can be easily done in
4-dimensional Euclidean space $\mathbb{R}^{4}$. In Fig. 3 we show an embedding
diagram of a constant $\theta=\frac{\pi}{2}$ slice of the static Minkowski-
Minkowski open universe, or traversable wormhole. One can also make
appropriate identifications of the two open sheets, and turn the space into,
e.g., a flat 3-torus, in which case the space is closed. The causal structure
of the resulting spacetime can be shown in a Carter-Penrose diagram, as in
Fig. 4. We use the hash symbol # to represent the connected sum of the
spacetime manifolds in order to conserve the conformal structure in the
Carter-Penrose diagram of the total spacetime. We see that it represents a
universe in which the spatial sections are two copies of the complements of
two plane 3-balls joined at a 2-sphere, the throat, yielding a traversable
wormhole, such that if we include time, the total spacetime has the topology
$\mathbb{R}\times\Sigma$, where $\Sigma$ is a 3-manifold with nontrivial
topology, whose boundary $\partial\Sigma\sim S^{2}$. A causal geodesic, or a
free-falling particle, initially moving in the direction of decreasing radial
coordinate in one half of the spacetime, would reach the shell at $r=R$ and
then continue until it reaches infinity at the other sheet of the wormhole.

Figure 3:  Embedding diagram of a $t=\text{constant}$ and
$\theta=\frac{\pi}{2}$ slice of the Minkowski-Minkowski static open universe,
or traversable wormhole, in 3-dimensional Euclidean space. The interior
coordinate is $R\leq r_{\mathrm{i}}<\infty$, the exterior coordinate is $R\leq
r_{\mathrm{e}}<\infty$, the radius of the shell is $R$, and the borders of the
circumferences should be identified.  Figure 4:  Carter-Penrose diagram of the
Minkowski-Minkowski static open universe, or traversable wormhole. The hash
symbol # represents the connected sum of the spacetime manifold. The symbols
$i^{-}$, $i^{0}$, and $i^{+}$ represent past timelike infinity, spatial
infinity, and future timelike infinity, respectively, and the symbols
$\mathscr{I}^{-}$ and $\mathscr{I}^{+}$, represent past and future null
infinity, respectively.

The Minkowski-Minkowski static open universe, i.e., the Minkowski-Minkowski
traversable wormhole, is marginally stable. Indeed, one has from Eq. (21) that
$V^{\prime\prime}(R_{0})=0$. The solution is in neutral equilibrium, meaning
that for a slight displacement the thin shell stays at the new radius. This
result confirms the expectations. as the interior and exterior spacetimes are
both described by a Minkowski, i.e., flat, solution.

This Minkowski-Minkowski static open universe is a traversable wormhole which
is summarized in Eqs. (25)-(27) and in Figs. 3 and 4. This open universe is an
exotic shell spacetime rather than a fundamental shell spacetime, as its
interior does not contain a center or origin, instead the interior opens up to
infinity. This further possibility for a shell spacetime, i.e., that its
interior opens up to infinity, implies that the spacetime is a traversable
wormhole spacetime, and thus the matter properties of the shell must be exotic
since they necessarily violate the energy conditions. The study of the
Minkowski-Minkowski open spacetime, or traversable wormhole, introduces the
prospect of analyzing all possible exotic shells, i.e., shells for which the
Minkowski interior has no center, in the other two basic ambient spacetimes,
namely, Schwarzschild and Reissner-Nordström spacetimes.

The Minkowski-Minkowski static open universe is a representative of the set of
traversable wormholes. It can be compared with other such open universes and
traversable wormholes. This will be done later.

###  II.3 Minkowski-Minkowski universes: One concept with two sides

The two, close and open, Minkowski-Minkowski spacetimes demonstrate that they
can be seen as complementary to each other, i.e., they are two sides of the
same concept. The concept, i.e., a collection of two Minkowski spacetimes
together, that when cut into spherical regions yield on one side a closed
universe, a bubble universe, and on the other side an open universe which is a
traversable wormhole. The formalism presented in analyzing the two Minkowski-
Minkowski universes is well suited to show this point. Indeed, from an
algebraic point of view, one side is given by $\frac{1}{2}(\xi_{\rm
i}-\xi_{\rm e})=1$, the other side is given by $\frac{1}{2}(\xi_{\rm
i}-\xi_{\rm e})=-1$, where $\xi_{\rm i}$ and $\xi_{\rm e}$ are the
characteristics of the interior and exterior normals to the shell,
respectively. This algebraic side appears clearly in the evaluation of the
matter properties as displayed in Eqs. (11) and (12). More formally, to
implement the idea of a Minkowski-Minkowski closed universe, i.e., a bubble
universe, and a Minkowski-Minkowski open universe, i.e., a traversable
wormhole, one uses the equations of general relativity together with the
appropriate thin shell formalism. For a shell with a Minkowski interior with a
center, when the normal to the shell in the exterior region points towards
decreasing $r$, i.e., $\frac{1}{2}(\xi_{\rm i}-\xi_{\rm e})=1$, one finds the
Minkowski-Minkowski closed universe, made of two 3-dimensional flat balls, or
sheets, that are joined at some domain wall, i.e., a 2-sphere shell with
matter, to make a Minkowski-Minkowski bubble universe. For a shell with a
Minkowski open interior, when the normal to the shell points towards
increasing $r$ in the Minkowski exterior, i.e., $\frac{1}{2}(\xi_{\rm
i}-\xi_{\rm e})=-1$, one finds the Minkowski-Minkowski open universe, made of
two 3-dimensional flat open infinite sheets that are joined at some throat, to
make a Minkowski-Minkowski traversable wormhole. From a matter point of view
the two universes show a form of complementarity, as for $\frac{1}{2}(\xi_{\rm
i}-\xi_{\rm e})=1$ the matter obeys the energy conditions while for
$\frac{1}{2}(\xi_{\rm i}-\xi_{\rm e})=-1$ the matter violates the energy
conditions. From a geometrical point of view, the two sides of the concept
appear when one picks up a Minkowski spacetime and at constant time cuts a
ball in it, to obtain two spaces, namely, a 3-dimensional ball with a flat
inside, and an infinite extended 3-dimensional flat space with a hole, which
is the complement of the ball. Then one picks up another Minkowski spacetime
and do the same, to get a second ball and a second infinite extended flat
space with a hole. One side is given if one joins the two 3-dimensional balls
along a 2-sphere, a shell containing matter, to obtain a single 3-space that
including time makes altogether a static closed Minkowski-Minkowski universe,
a bubble universe. The other side is given if one joins the two complements,
i.e., the two infinite extended 3-dimensional flat spaces with a hole in each,
along a 2-sphere, a shell containing matter, to obtain a different single
3-space that including time makes altogether another Minkowski-Minkowski
universe, which is a traversable wormhole. Comparison of Fig. 1 with Fig. 3
for a spatial geometrical representation of the bubble universe and the
traversable wormhole, respectively, displays the complementarity of the two
spaces clearly, which can be further strengthened with the comparison of the
spacetime drawings in the form of Carter-Penrose diagrams, given in Fig. 2 and
in Fig. 4, respectively. From a stability point of view it is also interesting
that both spacetimes are marginally stablee, showing thus here some form of
neutral complementarity.

So, that the two Minkowski-Minkowski spacetimes demonstrate that they can be
seen as complementary to each other, i.e., they are two sides of the same
concept, is clear. It can be raised the point that the bubble universe has
matter that obeys the energy conditions, whereas the traversable wormhole has
matter that does not obey those conditions. This is obviously true, but there
is no real problem with it. In an early era of the Universe, when quantum
gravity dominates, there is really no obeyance to the energy conditions and
the closed and open universes, created as bubble universes with domain walls
and traversable wormholes with throats out of the spacetime foam must coexist
together. Some kind of inflation would grow these objects to macroscopic
dimensions turning them into new structures inhabiting the Universe itself,
showing that bubble universes and traversable wormholes are distinct but
connected objects, some obeying the energy conditions and others not.

##  III Einstein static closed universe and Friedmann static hyperbolic open
universe

###  III.1 Einstein and Friedmann static universes: Formal solutions and
stability

####  III.1.1 Solutions

Two paradigmatic solutions of the theory of general relativity for static
universes are the Einstein and the hyperbolic Friedmann spacetimes. These two
solutions have various resemblances with the open and closed Minkowski-
Minkowski universes studied in the previous section, and we will present them
in a form suited for comparing their properties. Consider then the Einstein
field equations with a non-vanishing cosmological constant $\Lambda$,

$G_{\alpha\beta}+\Lambda g_{\alpha\beta}=8\pi T_{\alpha\beta}\,,$ (28)

where
$G_{\alpha\beta}=\mathcal{R}_{\alpha\beta}-\frac{1}{2}g_{\alpha\beta}\mathcal{R}$
is the Einstein tensor, $\mathcal{R}_{\alpha\beta}$ and $\mathcal{R}$ are the
Ricci tensor and Ricci scalar, respectively, $g_{\alpha\beta}$ is the
spacetime metric, and $T_{\alpha\beta}$ is the stress-energy tensor.

One assumes a static, homogeneous and isotropic spacetime and, in addition,
one supposes that $T_{\alpha\beta}$ corresponds to a perfect fluid which has
energy density $\rho$ and vanishing pressure $p$, i.e., a dust-like fluid.
With these assumptions, the solution of the field equations (28) in spacetime
spherical coordinates $\left(t,r,\theta,\varphi\right)$ is given by the line
element

$ds^{2}=-dt^{2}+dr^{2}+R^{2}\left[\frac{1}{\sqrt{k}}\sin\left(\sqrt{k}\frac{r}{R}\right)\right]^{2}d\Omega^{2}\,,$
(29)

where $t$ is the time coordinate, $r$ is the radial coordinate,
$d\Omega^{2}\equiv d\theta^{2}+\sin^{2}\theta d\varphi^{2}$, with $\theta$ and
$\varphi$ being the spherical angular coordinates, $0\leq\theta\leq\pi$ and
$0\leq\varphi\leq 2\pi$, $R$ is a positive constant scale factor representing
a characteristic radius of the universe, and $k$ is related with the Ricci
curvature scalar by $\mathcal{R}=\frac{6k}{R^{2}}$ and may take the values
$k=1,0,-1$. Furthermore, the field equations (28) also give expressions for
the energy density of the fluid and for the cosmological constant, namely,

$\displaystyle\rho$ $\displaystyle=\frac{k}{4\pi R^{2}}\,,$ (30)
$\displaystyle\Lambda$ $\displaystyle=\frac{k}{R^{2}}\,,$ (31)

and $p=0$. From Eqs. (30) and (31) one finds that

$\rho-\frac{\Lambda}{4\pi}=0\,,$ (32)

and so $\Lambda$ counteracts the gravitational pulling effects of $\rho$.
Noticing that the term $\Lambda g_{\alpha\beta}$ can be thought as a perfect
fluid contribution to the stress-energy tensor with energy density
$\bar{\rho}=\frac{\Lambda}{8\pi}$ and pressure
$\bar{p}=-\frac{\Lambda}{8\pi}$, this static homogeneous universe solution can
then be seen as a solution of a two fluid system, one fluid with energy
density $\rho=\frac{\Lambda}{4\pi}$ and pressure $p=0$, and the other fluid, a
vacuum fluid, with energy density $\bar{\rho}=\frac{\Lambda}{8\pi}$ and
pressure $\bar{p}=-\frac{\Lambda}{8\pi}$, such that
$\rho+\bar{\rho}+3\bar{p}=0$. Besides the trivial case, i.e., the Minkowski
universe which has $k=0$, there are two possible universes at this juncture,
the Einstein static closed universe which has $k=1$, and the Friedmann static
open universe which has $k=-1$.

####  III.1.2 Linearized stability analysis for Einstein and Friedmann static
universes

An important question regarding the static homogeneous universes, i.e., the
Einstein and hyperbolic Friedmann static universes, is if these are stable
under perturbations. Here we will discuss the linear stability of these
cosmological solutions by analyzing the equation of motion of the universe
near the static configuration. The result for the Einstein universe is well
known, whereas the stability of the static hyperbolic Friedmann universe is
less known.

To study the linear stability of the static Einstein and Friedmann universes
we have to find the evolution equation for the scalar factor $R$ and analyze
the behavior of these solutions as we perturb the spacetime. The analysis can
be done in a unified way by making use of the parameter $k$. From the general
relativity field equations we find the Friedmann equation, namely,

$\dot{R}^{2}+V(R)=0\,,$ (33)

where we have introduced the potential

$V(R)=k-\frac{8\pi\rho+\Lambda}{3}R^{2}\,,$ (34)

and the scale factor $R$ is now a function of the time coordinate $t$,
$R=R(t)$, and a dot represents derivative with respect to it, and again $k$
represents the sectional curvature of constant time slices such that, $k=+1$
for the closed universe, $k=0$ for the flat universe, and $k=-1$ for the open
universe. Note anew that a universe is stable if, and only if, the potential
$V(R)$ is at a local minimum, i.e., if $V^{\prime}(R)=0$ and
$V^{\prime\prime}(R)\geq 0$, with the equality providing the marginal neutral
case, where a prime denotes the derivative with respect to $R$. Thus, we have
to calculate the matter properties and its derivatives. All these properties
are functions of the shell radius $R$, namely, $\rho=\rho(R)$, $p=p(R)$,
$\rho^{\prime}=\rho^{\prime}(R)$, $p^{\prime}=p^{\prime}(R)$, and the radius
itself is a function of time $R(t)$. The conservation equation, which can be
taken from the field equations, is $\dot{\rho}+\frac{3\dot{R}}{R}\rho=0$, so
that the equation for $\rho^{\prime}$, where prime denotes the derivative with
respect to $R$, is

$\rho^{\prime}=-3\frac{\rho}{R}\,.$ (35)

We have now to analyze the derivatives of the potential $V(R)$ given in Eq.
(34) at the static configuration. From Eq. (34) we get
$V^{\prime}(R)=-\frac{1}{3}\left(8\pi\rho^{\prime}\right)R^{2}-\frac{2}{3}\left(8\pi\rho+\Lambda\right)\,R$,
where we assume that $\Lambda$ is a constant. Taking the derivative of it we
get
$V^{\prime\prime}(R)=-\frac{1}{3}\left(8\pi\rho^{\prime\prime}\right)R^{2}-\frac{4}{3}\left(8\pi\rho^{\prime}\right)R-\frac{2}{3}\left(8\pi\rho+\Lambda\right)$.
Now, if we introduce Eq. (35) into $V^{\prime}(R)$ we obtain
$V^{\prime}(R)=\frac{2}{3}\left(4\pi\rho-\Lambda\right)R$. Simplifying also
$V^{\prime\prime}(R)$ we obtain
$V^{\prime\prime}(R)=-\frac{2}{3}\left(8\pi\rho+\Lambda\right)$. In brief, the
derivatives of the potential $V(R)$ are

$V^{\prime}(R)=\frac{2}{3}\left(4\pi\rho-\Lambda\right)R\,,$ (36)

and

$V^{\prime\prime}(R)=-\frac{2}{3}\left(8\pi\rho+\Lambda\right)\,,$ (37)

where we assume that $p=0$ throughout, i.e., the cold generic equation
$p=p(\rho)$ is the trivial one, so here
$\eta\left(\rho\right)\equiv\frac{\partial p}{\partial\rho}$ is zero,
$\eta\left(\rho\right)=0$.

Following the usual reasoning, we linearize Eq. (33) around a static solution.
Defining $R_{0}$ as the radius of the static universe and assuming the
potential $V$ to be a differentiable function at $R_{0}$, we can expand the
potential (34) around $R_{0}$ as

$V(R)=V(R_{0})+V^{\prime}(R_{0})\left(R-R_{0}\right)+\frac{1}{2}V^{\prime\prime}(R_{0})\left(R-R_{0}\right)^{2}\,,$
(38)

plus higher order terms of $\mathcal{O}\left[\left(R-R_{0}\right)^{3}\right]$.
Now, a universe with radius $R_{0}$ is stable if, and only if, the potential,
$V(R)$, is at a local minimum, i.e., if $V^{\prime}(R_{0})=0$ and
$V^{\prime\prime}(R_{0})\geq 0$, with the equality providing the marginal
neutral case. The static solutions found in the previous sections are
characterized by the following expressions for the energy density and the
cosmological constant of the fluid, $\rho=\frac{k}{4\pi R_{0}^{2}}$ and
$\Lambda=\frac{k}{R_{0}^{2}}$, see Eqs. (30) and (31), with the fluid pressure
$p$ being zero, $p=0$. Then putting Eq. (30) into Eq. (34) we find
$V(R_{0})=0$, as expected. Substituting it into Eq. (36) and evaluating at the
static solution we find $V^{\prime}(R_{0})=0$. Evaluating
$V^{\prime\prime}(R)$ at the static solution, $R=R_{0}$, we find
$V^{\prime\prime}(R_{0})=-\frac{2k}{R_{0}^{2}}$. In brief, defining $R_{0}$ as
the value of the scale factor of the static Einstein and hyperbolic Friedmann
universes, expanding the potential $V(R)$ around $R_{0}$, Eq. (38), we find
that

$V^{\prime}(R_{0})=0\,,$ (39)

and

$V^{\prime\prime}(R_{0})=-\frac{2k}{R_{0}^{2}}\,.$ (40)

Gathering these calculations, we conclude that, besides the trivial case,
i.e., the Minkowski universe which has $k=0$ and is trivially neutraly stable,
there is the Einstein static closed universe which has $k=1$ and so is
unstable, and the Friedmann static open universe which has $k=-1$ and so is
stable. The instability of the $k=1$ Einstein static closed universe means
that if we slightly displace the scale radius $R$ towards larger or smaller
values, the universe will expand in the former displacement or collapse in the
latter displacement, and the stability of the $k=-1$ Friedmann static closed
universe means that if we slightly displace the scale radius $R$ towards
larger or smaller values, the universe will get back to the initial value $R$.

###  III.2  Einstein and Friedmann static universes: Geometry and physics

####  III.2.1 Einstein static closed universe

The Einstein universe is a solution of the general theory of relativity for a
dust source with energy density $\rho$, pressure $p$ equal to zero, a positive
cosmological constant $\Lambda$, and positive curvature, $k=1$. In spacetime
spherical coordinates $(t,r,\theta,\varphi)$ it is characterized by the line
element given in Eq. (29) with $k=1$, i.e.,

$ds^{2}=-dt^{2}+d{r}^{2}+R^{2}\sin^{2}\left(\frac{r}{R}\right)\,d\Omega^{2}\,,$
(41)

where $t$ is the time coordinate, $r$ is the radial coordinate with $0\leq
r\leq\pi\,R$, and $d\Omega^{2}\equiv d\theta^{2}+\sin^{2}\theta d\varphi^{2}$,
with $\theta$ and $\varphi$ being the spherical angular coordinates,
$0\leq\theta\leq\pi$ and $0\leq\varphi\leq 2\pi$. In addition, $R$ is a
positive scale factor which here is a constant, and which gives the
characteristic radius of the universe. The Ricci scalar for the Einstein
universe is given by $\mathcal{R}=\frac{6}{R^{2}}$. This solution then
represents a static spacetime, a 3-dimensional sphere, and so is a closed
universe. Now we turn to the properties of the matter in the Einstein
universe. Assuming a perfect fluid made of dust, i.e., the matter has energy
density $\rho$ and pressure $p=0$, the Einstein field equations with
cosmological constant $\Lambda$ for the line element given in Eq. (41) yield

$\rho=\frac{1}{4\pi R^{2}}\,,$ (42) $\Lambda=\frac{1}{R^{2}}\,,$ (43)

see Eqs. (30) and (31) with $k=1$. Note that $\rho-\frac{\Lambda}{4\pi}=0$,
see Eq. (32), and so $\Lambda$ being positive is repulsive everywhere and thus
assumes the function of a pressure that acts against the gravitational pull of
the matter specified by $\rho$. This system can be seen as a two fluid system,
one fluid with energy density $\rho$, the other fluid, a vacuum fluid, with
energy density ${\bar{\rho}}=\frac{\Lambda}{8\pi}$ and pressure
${\bar{p}}=-\frac{\Lambda}{8\pi}$, such that $\rho+\bar{\rho}+3\bar{p}=0$. All
the matter energy conditions are satisfied. The volume of this universe is
$V=2\pi^{2}R^{3}$ and its mass is is $m=2\pi^{2}R^{3}\rho$.

The spatial and causal structure of the spacetime can also be presented.
Considering a slice of constant time of the spacetime, $t=\rm{constant}$, we
find that the 3-space is diffeomorphic to a 3-sphere. To show this, we can
embed the 3-space in 4-dimensional Euclidean space, $\mathbb{R}^{4}$. Defining
the Euclidean spatial coordinates $(w,x,y,z)$ as $w=R\cos\frac{r}{R}$,
$x=R\sin\frac{r}{R}\sin\theta\cos\phi$,
$y=R\sin\frac{r}{R}\sin\theta\sin\phi$, and $z=R\sin\frac{r}{R}\cos\theta$,
the line element of the embedded surface is given by
$ds^{2}=dw^{2}+dx^{2}+dy^{2}+dz^{2}$, and the surface verifies the equation
$w^{2}+x^{2}+y^{2}+z^{2}=R^{2}$, showing that indeed it can be regarded as a
3-sphere in $\mathbb{R}^{4}$. To visualize the embedding one makes a
$\theta=\frac{\pi}{2}$ slice, i.e., $z=0$ in the Euclidean coordinates. In
Fig. 5 we show such an embedding for the static spherical Einstein universe.
By making appropriate identifications between points in the two hemispheres,
the spherical space turns into a projective spherical space also called an
elliptical space. In Fig. 6 we show the causal structure of the resulting
spacetime in a Carter-Penrose diagram. The Einstein universe, a static
spacetime, models a universe with spherical spatial sections such that if we
include time the total spacetime is a 3-cylinder, $\mathbb{R}\times S^{3}$. A
timelike geodesic, or a free-falling particle, initially moving from $r=0$ in
the direction of increasing radial coordinate would reach the other pole at
$r=\pi R$ and then continue until it reaches back the center of coordinates
and so forth.

Figure 5:  Embedding diagram of a $t=\text{constant}$ and
$\theta=\frac{\pi}{2}$ slice of the Einstein static closed universe in
3-dimensional Euclidean space. The radial coordinate $\bar{r}$ related to the
area defined by it, namely, ${\bar{r}}=R\sin\left(\frac{r}{R}\right)$, is the
radial coordinate used in the diagram. This coordinate runs from 0 at one
pole, to $R$ at the equatior, and then back to 0 at the other pole, with $R$
being the characteristic radius of the Einstein universe.  Figure 6:  Carter-
Penrose diagram of the Einstein static closed universe. The vertical lines
represent the two poles of the sphere. The radial coordinate $\bar{r}$ related
to the area defined by it, namely, ${\bar{r}}=R\sin\left(\frac{r}{R}\right)$,
is the radial coordinate used in the diagram. One pole is situated at the
origin with ${\bar{r}}=0$. The other pole has also ${\bar{r}}=0$. The two
lines denoted by $R$ constitute the equatior. The hash symbol # represents the
connected sum of the spacetime manifold. The symbols $i^{-}$ and $i^{+}$
represent past and future causal infinity, respectively.

The Einstein static closed universe is unstable. Indeed, from Eq. (40) one has
that for $k=1$, $V^{\prime\prime}(R_{0})<0$, recovering the well known result
that the Einstein static closed universe is unstable under perturbations. This
result confirms the expectations. For the static Einstein universe a small
increase in the radius of the universe means less gravitational field due to
matter and more cosmological repulsion field from $\Lambda$, so it is a
runaway expanding unstable solution, and reversing the argument for a small
decrease in the radius one finds a runaway contracting unstable solution. So,
although it obeys the energy conditions and is a priori not problematic, it is
unstable, giving rise to an expanding bubble universe.

This Einstein static closed universe is well known. It was extremely important
in initiating the science of cosmology. The requirement that the boundary
conditions on the gravitational field should be finite and consistent led to a
closed universe, which in turn was also relevant to make the point that
general relativity could be Machian, i.e., that geometry and inertia would
arise solely from matter. The requirement that the universe should be static,
as was thought at the time, yielded a new constant to physics, the
cosmological constant. The corresponding cosmological term added to the
original general theory of relativity provided in turn the first modified
gravitational theory.

We can now make a comparison between the Minkowski-Minkowski static closed
universe and the Einstein universe. Although the two universes are, of course,
totally distinct solutions of the general relativistic field equations, there
are differences and also striking similarities between them. In relation to
the matter properties, the Minkowski-Minkowski closed universe is highly
nonuniform, it is vacuum everywhere except at a thin shell with
circumferential radius $R$, made of a perfect fluid with a positive energy
density $\sigma$ and a positive, repulsive, pressure $p$ to hold it static
against gravitational collapse or expansion. The Einstein universe, with
characteristic radius $R$, is uniform, permeated by a fluid with a positive
energy density $\rho$ and a repulsive cosmological constant to hold it static
against gravitational collapse or expansion. Thus, both universes obey the
energy conditions, they have positive densities and have some form of
pressure, negative tangential shell pressure in one case and positive
cosmological constant pressure in the other case, to hold them static. In
relation to the geometric and causal properties, one can compare the figures
drawn, namely, a $t=\text{constant}$ and $\theta=\frac{\pi}{2}$ slice of the
Minkowski-Minkowski closed universe and the corresponding Carter-Penrose
diagram shown in Figs. 1 and 2, respectively, and a $t=\text{constant}$ and
$\theta=\frac{\pi}{2}$ slice of the Einstein closed universe and the
corresponding Carter-Penrose diagram shown in Figs. 5 and 6. The comparison
leads to the conclusion that the two universes have an evident similar
geometrical structure. The Minkowski-Minkowski closed universe models a
universe with squashed spherical spatial sections such that the total
spacetime is a squashed 4-cylinder with $\mathbb{R}\times S^{3}$, with the
possibility of further identifications in $S^{3}$. The Einstein closed
universe models a universe with spherical spatial sections such that the total
spacetime is a 4-cylinder $\mathbb{R}\times S^{3}$, with the possibility of
further identifications in $S^{3}$. Both universes have thus spherical spatial
topology and similar causal structures. The Minkowski-Minkowski closed
universe is the result of compressing, in a sense, the evenly distributed
matter of the Einstein universe into a thin shell leaving the rest of the
spacetime empty. The Minkowski-Minkowski closed universe is stable,
marginally, and the Einstein closed universe is unstable, so, since there are
no topological obstructions, a possible endpoint of the Einstein closed
universe, if perturbed at constant universe radius, could be the Minkowski-
Minkowski closed universe.

####  III.2.2 Friedmann static hyperbolic open universe: A failed wormhole

The Friedmann static universe is a solution of the general theory of
relativity for a dust source with negative energy density $\rho$, pressure $p$
equal to zero, a negative cosmological constant $\Lambda$, and negative
curvature $k=-1$. In spacetime hyperspherical coordinates
$(t,r,\theta,\varphi)$ it is characterized by the line element given in Eq.
(29) with $k=-1$, i.e.,

$ds^{2}=-dt^{2}+d{r}^{2}+R^{2}\sinh^{2}\left(\frac{r}{R}\right)\,d\Omega^{2}\,,$
(44)

where $t$ is the time coordinate, $r$ is the radial coordinate with $0\leq
r<\infty$, and $d\Omega^{2}\equiv d\theta^{2}+\sin^{2}\theta d\varphi^{2}$,
with $\theta$ and $\varphi$ being the spherical angular coordinates,
$0\leq\theta\leq\pi$ and $0\leq\varphi\leq 2\pi$. In addition, $R$ is a
positive scale factor which here is a constant, and which gives the
characteristic radius of the universe. The Ricci scalar for the Friedmann
universe is given by $\mathcal{R}=-\frac{6}{R^{2}}$, so the Friedmann universe
is a negative constant curvature spacetime. Clearly it is a static hyperbolic
spacetime, and so an open universe. Now we turn to the properties of the
matter in the static Friedmann universe. Assuming a perfect fluid made of
dust, i.e., the matter has energy density $\rho$ and pressure $p=0$, the
Einstein field equations with negative cosmological constant $\Lambda$ for the
line element given in Eq. (44) yield

$\rho=-\frac{1}{4\pi R^{2}}\,,$ (45) $\Lambda=-\frac{1}{R^{2}}\,,$ (46)

see Eqs. (30) and (31) with $k=-1$. Note that $\rho-\frac{\Lambda}{4\pi}=0$,
see Eq. (32), and so $\Lambda$ being negative is attractive everywhere, and
thus assumes the function of a tension that acts against the gravitational
push of the matter specified by a negative energy density $\rho$. This system
can be seen as a two fluid system, one fluid with negative energy
density$\rho$, the other fluid, a vacuum fluid, with negative energy density
${\bar{\rho}}=\frac{\Lambda}{8\pi}$ and tension
${\bar{p}}=-\frac{\Lambda}{8\pi}$ such that $\rho+\bar{\rho}+3\bar{p}=0$. The
matter energy conditions are violated. The volume of this hyperbolic universe,
in its open form, is $V=\infty$ and its mass is also infinite.

Figure 7:  Embedding diagram of a $t=\text{constant}$ and
$\theta=\frac{\pi}{2}$ slice of the Friedmann static open universe in
3-dimensional Minkowski space. The radial coordinate $\bar{r}$ related to the
area defined by it, namely, ${\bar{r}}=R\sinh\left(\frac{r}{R}\right)$, is the
radial coordinate used in the diagram. This coordinate runs from 0 at one
pole, to infinity, passing through $R$ a some point, and the same at the other
pole, it runs from 0, to infinity, passing through $R$ a some point, with $R$
being the characteristic radius of the Friedmann static universe. The poles
are at the origin which is common to both branches, sheets, or branes of the
spacetime, although this cannot be seen explicit in the embedding. The
branches are separated, and the solution is a failed wormhole. Indeed, there
are two disconnected spacetimes almost connected at ${\bar{r}}=0$.

The spatial and causal structure of the spacetime can also be presented.
Considering a time slice $t={\rm constant}$ of the spacetime gives two copies
of the hyperbolic 3-space. To see this, one makes an embedding. The hyperbolic
3-space cannot be embedded in the 4-dimensional Euclidean space, but it can be
embedded to an open region of the 4 dimensional Minkowski spacetime. Defining
the Minkowski coordinates $(w,x,y,z)$, where $w$ is a time coordinate and
$(x,y,z)$ are spatial coordinates, as $w=R\cosh\frac{r}{R}$,
$x=R\sinh\frac{r}{R}\sin\theta\cos\phi$,
$y=R\sinh\frac{r}{R}\sin\theta\sin\phi$, and $z=R\sinh\frac{r}{R}\cos\theta$,
the line element of the embedded surface is given by
$ds^{2}=-dw^{2}+dx^{2}+dy^{2}+dz^{2}$, and the surface verifies the equation
$w^{2}-x^{2}-y^{2}-z^{2}=R^{2}$. So the surface, which represents a time slice
of the Friedmann static universe, is in fact given by two copies of a
3-dimensional hyperboloid. To visualize the embedding, one makes a
$\theta=\frac{\pi}{2}$ slice, i.e., $z=0$ in the Minkowski coordinates. In
Fig. 7 we show an embedding for the static hyperbolic open Friedmann universe.
Clearly there are two sheets, i.e., the universe has two branes, the two
copies of the Friedmann static universe. Notice that we opted to map the two
asymptotic flat regions to open sets of the future and past light cones to
show both regions, although one should bear in mind that, first, this has no
physical relevance as both regions are equivalent and, second, this has no
relation with time reversal. Moreover, admitting it might not be clear from
the embedding diagram, the vertices of the hyperbolas are identified as the
same point hence, the static hyperbolic Friedmann universe can also be seen as
a model of a failed wormhole where two asymptotic flat regions have a common
point with circumferential radius $R\sinh\frac{r}{R}=0$, i.e., $r=0$, so that
the wormhole’s throat is a point, a zero measure set. Since the two
hyperboloid branes are independent there is indeed no wormhole, it is a failed
wormhole. By making appropriate identifications each of the two open infinite
sheets turns into some closed 3-space, in which case the volume of such a
universe would be finite. In Fig. 8 we show the causal structure of the
resulting spacetime in a Carter-Penrose diagram. We use the hash symbol # to
represent the connected sum of the spacetime manifolds in order to conserve
the conformal structure in the Carter-Penrose diagram of the total spacetime.
We see that it represents a universe in which the spatial sections are two
copies of a 3-hyperboloid. The Friedmann hyperbolic universe is a spacetime
composed of time times hyperbolic 3-space, actually two copies of it. A
geodesic, or a free-falling particle, initially moving in the upper brane in
the direction of decreasing radial coordinate would reach $r=0$ and would
continue until it reaches infinity, without interacting with a mirror
geodesic, or a mirror free-falling particle, initially moving in the lower
brane in the direction of decreasing radial coordinate reaching the same $r=0$
and continuing until it reaches the infinity of its own brane.

Figure 8:  Carter-Penrose diagram of the Friedmann static open universe. The
vertical lines represent the poles of each hyperboloid branch, ${\bar{r}}=0$,
where $\bar{r}$ is the radial coordinate related to the area defined by it,
namely, ${\bar{r}}=R\sinh\left(\frac{r}{R}\right)$. There is no hash symbol #
here because the two spacetimes are disconnected, no geodesic can pass from
one spacetime to the other. The symbols $i^{-}$, $i^{0}$, and $i^{+}$
represent past timelike infinity, spatial infinity, and future timelike
infinity, respectively, and the symbols $\mathscr{I}^{-}$ and
$\mathscr{I}^{+}$, represent past and future null infinity, respectively. The
timelike line $R$ is drawn to call attention that the Friedmann static open
universe has a characteristic intrinsic radius.

The Friedmann static open universe is stable. Indeed, from Eq. (40) one has
that for $k=-1$, $V^{\prime\prime}(R_{0})>0$. This result confirms the
expectations. For the static Friedmann universe, a small increase in the
radius of the universe means less gravitational field due to matter with
negative energy density, so less repulsion, and more cosmological tension
field from $\Lambda$, so the universe oscillates around the original radius in
a stable situation.

This open static universe proposed by Friedmann came after a suggestion by
Fock, and was worked out by Friedmannbefore introducing, in the same paper,
the new expanding time-dependent hyperbolic solutions. Friedmann’s main
motivation for presenting it was that the solution represented the other side
of Einstein’s static universe, the two solutions, Einstein’s and Friedmann’s,
are indeed complementary to each other. It is a much forgotten universe. Since
this static solution has a negative energy density and a negative cosmological
constant, and violates all energy conditions, it seemed a physically
inadmissible strange universe that could hardly captured any attention. This
prejudice against solutions that violate the energy conditions came to an end
when traversable wormholes, systems that violate several energy conditions,
jumped into the limelight. We see here that the Friedmann static universe is a
wormhole, albeit a failed one. Moreover, although the solution does not obey
the energy conditions, Friedmann’s static universe is interestingly stable.
Thus, Friedmann had a prescient foresight in contemplating working out in
detail the mathematics of this static solution.

We can now make a comparison between the Minkowski-Minkowski static open
universe, or traversable wormhole, and the Friedmann static hyperbolic
universe, or failed wormhole. Although the two universes are, of course,
totally distinct solutions of the general relativistic field equations, there
are both differences and similarities between them, although the smiliarites
here are not so compelling. In relation to the matter properties, the
Minkowski-Minkowski open universe, i.e., the traversable wormhole universe, is
highly nonuniform, it is vacuum everywhere except at a thin shell throat with
circumferential radius $R$, made of a perfect fluid with a negative energy
density $\sigma$ and a positive pressure $p$ to hold it static. The Friedmann
static open universe with characteristic radius $R$ is uniform, permeated by a
fluid with a repulsive negative energy density $\rho$ and a negative,
attractive, cosmological constant $\Lambda$ to hold it static. Thus, both
spacetimes violate the energy condition, they have negative energy densities
and have some form of pressure, positive tangential shell pressure in one case
and negative cosmological constant pressure in the other case, to hold them
static. In relation to the geometric and causal properties, one can compare
the figures drawn, namely, a $t=\text{constant}$ and $\theta=\frac{\pi}{2}$
slice of the Minkowski-Minkowski open universe, or traversable wormhole, and
the corresponding Carter-Penrose diagram, shown in Figs. 3 and 4,
respectively, and a $t=\text{constant}$ and $\theta=\frac{\pi}{2}$ slice of
the Friedmann open universe, or failed wormhole, and the corresponding Carter-
Penrose diagram shown in Figs. 7 and 8. The comparison leads to the conclusion
that the two universes have some similarities. Both universes for large radii
have two distinct open sheets, although the circumferential radius in the
Minkowski-Minkowski open universe is finite not zero, and so composes a
traversable wormhole, whereas in the Friedmann static open universe the
circumferential radius goes to zero, and the wormhole fails to happen. The
bare geometrical structure of the two universes is different, the Minkowski-
Minkowski open universe has geometry $\mathbb{R}\times\Sigma$, where $\Sigma$
is a 3-space with nontrivial topology, and the Friedmann static open universe
has geometry $\mathbb{R}\times\mathbb{H}^{3}$, with negative curvature in the
two copies of the spatial sections. The Minkowski-Minkowski open universe
could be thought of as being the result of compressing, in a sense, the evenly
distributed matter of the Friedmann static universe into a thin shell at some
throat radius leaving the rest of the spacetime empty. The Minkowski-Minkowski
open universe is stable, marginally, and the Friedmann open universe is
stable. But here there are topological obstructions, one universe is
connected, although not simply connected, the other universe is disconected,
it has two separate branches, and so one cannot pass from one universe to the
other without changing the topology.

###  III.3 Einstein and Friedmann static universes: One concept with two sides

The Einstein and Friedmann static universes can be seen as complementary to
each other, i.e., they are two sides of the same concept. The concept, i.e., a
collection of static constant curvature homogeneous universes, yields on one
side of the concept a closed universe, a bubble universe, and on the other
side of the concept an open universe which is a failed wormhole. The formalism
presented in analyzing the two universes is well suited to show this point.
From an algebraic point of view, one side of the concept is given by $k=1$,
the other side is given by $k=-1$, where $k$ is a characteristic that gives
how space curves, positively in one case, negatively in the other,
respectively. This algebraic side appears clearly in the evaluation of the
matter properties as displayed in Eqs. (30) and (31). More formally, to
implement the idea of a closed universe, i.e., a bubble universe, and an open
universe, i.e., a failed wormhole, one uses the equations of general
relativity. For one universe one picks up $k=1$, a 3-dimensional sphere. For
the other universe one picks up $k=-1$, a 3-dimensional hyperboloyd. From a
matter point of view the two universes show a form of complementarity, as for
$k=1$ the matter obeys the energy conditions while for $k=-1$ the matter
violates the energy conditions. From a geometrical point of view, the two
sides of the concept appear when one picks up a manifold spacetime and at
constant time imposes a space with constant curvature. One side is for
positive curvature, a bubble universe, the other side for negative curvature,
a failed wormhole. Comparison of Fig. 5 with Fig. 7 for a spatial geometrical
representation of the bubble universe and the traversable wormhole,
respectively, displays some complementarity of the two spaces, which can be
further strengthened with the comparison of the spacetime drawings in the form
of Carter-Penrose diagrams, given in Fig. 6 and in Fig. 8, respectively. From
a stability point of view we have seen that one universe is stable and the
other unstable, showing thus some form of complementarity.

So, that the two spacetimes, Einstein and Friedmann, demonstrate that they can
be seen as complementary to each other, i.e., they are two sides of the same
concept, is clear. It can be raised that the bubble universe has matter that
obeys the energy conditions, whereas the failed wormhole has matter that does
not obey. This is true, but again there is no real problem. In an early era of
the Universe, when quantum gravity dominates, there is no necessity of
obeyance to the energy conditions and the closed and open universes, created
as bubble universes and failed wormholes. out of the spacetime foam they must
coexist together. Some kind of inflation would grow these objects to
macroscopic dimensions, making bubble universes and traversable wormholes
distinct, but connected, objects, some obeying the energy conditions and being
unstable, like the Eisntein universe, others not obeying the energy conditions
but being stable, like the Friedmann static universe.

##  IV Conclusions: Bubble universes and traversable wormholes, two sides of
one concept

We have analyzed the possible universes that can be built from a junction of
two Minkowski spacetimes through a static, timelike thin matter shell. Taking
aside the trivial Minkowski flat universe with no shell, there are two such
universes. One is a static closed universe with a spherical thin shell with
positive energy density and negative pressure that joins two Minkowski balls,
i.e., it is the Minkowski-Minkowski closed universe, a bubble universe. The
other universe is a static open universe with a spherical thin shell with
negative energy density and positive pressure that joins two Minkowski
asymptotic sheets, it is the Minkowski-Minkowski open universe, or traversable
wormhole. We have seen that they can be seen as complementary to each other.
More specifically, they are two sides of one concept, the concept being the
collection of nontrivial Minkowski-Minkovski spacetimes, with one side given
by $\frac{1}{2}(\xi_{\rm i}-\xi_{\rm e})=1$, the other side given by
$\frac{1}{2}(\xi_{\rm i}-\xi_{\rm e})=-1$, where $\xi_{\rm i}$ and $\xi_{\rm
e}$ are the interior and exterior normals to the shell, respectively.

We have analyzed the possible universes that can be built from static
homegeneous pressurless matter with a cosmological constant. There are two
such universes. One is the static closed Einstein spherical universe. The
other is the static open Friedmann hyperbolic universe. We have seen that they
can be seen as complementary to each other, and, indeed, the idea of the
construction of the static open universe by Friedmann was to find the
complement to the Einstein universe. More specifically, they are two sides of
one concept, the concept being the collection of constant curvature
pressureless universes, with one side given by positive curvature, $k=1$, the
other side given by negative curvature, $k=-1$.

We have also seen that the Minkowski-Minkowski closed universe, a bubble
universe, has resemblances with the static closed Einstein universe with
positive energy density, zero pressure, and positive cosmological constant,
and that the static open Minkowski-Minkowski universe, a traversable wormhole,
has resemblances with the static open Friedmann universe with negative energy
density, zero pressure, and negative cosmological constant, which in turn is a
failed wormhole. The Minkowski-Minkowski universes are both linearly stable,
marginally, and the Einstein and Friedmann static universes are linearly
unstable and stable, respectively. One could think of the Minkowski-Minkowski
universes as being a limit of the homogeneous universes when all the matter of
the thin shell is spread evenly throughout the universes, or vice versa, in
which case the homogeneous universes being a limit of the Minkowski-Minkowski
universes when all the matter of the homogeneous universes is put somehow into
thin shells. For the Einstein universe this would be possible classically,
within general relativity, as the two universes have the same topology, and
so, for constant universe radius, the Minkowski-Minkowski closed universe
could be the end point of the Einstein universe. For the Friedmann static
universe this could not be realized within general relativity, as the two
universes have different topologies and so there is no way of changing
classically, and so continuously, from one into the other, although quantum
jumps of one to the other geometry might be conceivably possible.

The existence of universes and wormholes within the Universe is a tantalizing
possibility allowed by the laws of physics. Indeed, in a early cosmic era when
primordial scalar and gauge fields are dominant and symmetry breaking phase
transitions naturally arise, universes may occur as bubbles within the
Universe, and likewise, wormholes can exist in the form of traversable
shortcuts for distant parts of the Universe or can even connect what would be
distinct universes. Bubble universes and traversable wormholes are distinct
objects. Normally, bubble universes are found as dynamic solutions, whereas,
typically, traversable wormholes are studied as static structures, but of
course they can both be static or dynamic. We have analyzed two static cases,
the two Minkowski-Minkowski spacetimes and the two static homogeneous
universes, and found that these spacetimes demonstrate, in the way of example,
indeed two coupled examples that reinforce each other, that bubble universes
and traversable wormholes can be seen as complementary to each other, i.e.,
they are two sides of some same concept. Dynamical cases where bubble
universes and traversable wormhole are complementary to each other can also be
found and studied. It is plausible that in a quantum gravity scenario or in a
scenario in which quantum gravity is weak but nonnegligible, both bubble
universes and traversable wormholes are dynamically created alike, being as
well two sides of the same concept. In addition, using this duality, one can
infer that, arbitrarily advanced civilizations, with arbitrarily advanced
technology to deal in pratical terms with spacetime features, if they can
build bubble universes, they can also build traversable wormholes, and
conversely, if they are apt to build traversable wormholes, as has been often
suggested, they should be apt to build bubble universes.

## Acknowledgments

JPSL acknowledges Fundação para a Ciência e Tecnologia - FCT, Portugal, for
financial support through Project No. UIDB/00099/2020. PL thanks Center for
Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior
Técnico and Centro de Matemática, Universidade do Minho, where part of this
work has been performed, for the hospitality.

## References

  * (1) W. Israel, “Singular hypersurfaces and thin shells in general relativity”, Il Nuovo Cimento 44, 10 (1966).
  * (2) J. Katz and D. Lynden-Bell, “Tension shells and tension stars”, Classical Quantum Gravity 8, 2231 (1991).
  * (3) J. P. S. Lemos and P. Luz, “All fundamental electrically charged thin shells in general relativity: From star shells to tension shell black holes, regular black holes, and beyond”, Phys. Rev. D 103, 104046 (2021); arXiv:2103.15832 [gr-qc].
  * (4) A. Einstein, “Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie”, Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften (Berlin) 1917, 142 (1917); English translation see, e.g., A. Einstein, “Cosmological considerations on the general theory of relativity”, in H. A. Lorentz, A. Einstein, H. Minkowski, and H. Weyl, The Principle of Relativity (Methuen and Company, London, 1923), p. 175.
  * (5) W. de Sitter, “On Einstein’s theory of gravitation and its astronomical consequences”, Monthly Notices Royal Astronomical Society 78, 3 (1917).
  * (6) A. Friedmann, “Über die Krümmung des Raumes”, Zeitschrift für Physik 10 377 (1922); English translation see, e.g., A. Friedmann, “On the curvature of space”, General Relativity and Gravitation 31, 1991 (1999).
  * (7) G. Lemaître, “Un univers homogène de masse constant et de rayon croissant, rendant compte de la vitesse radiale des nébeleuses extra-galactiques”, Annales Societé Scientifique Bruxelles A 47, 49 (1927); English translation see, e.g., G. Lemaître, “A homogeneous universe of constant mass and increasing radius accounting for the radial velocity of extra-galactic nebulae”, General Relativity and Gravitation 45, 1635 (2013).
  * (8) A. S. Eddington, “On the stability of Einstein’s spherical world”, Monthly Notices Royal Astronomical Society 90, 668 (1930).
  * (9) W. B. Bonnor, “The instability of the Einstein universe”, Monthly Notices Royal Astronomical Society 115, 310 (1955).
  * (10) E. R. Harrison, “Normal modes of vibrations of the Universe”, Review Modern Physics 39, 862 (1967).
  * (11) G. W. Gibbons, “The entropy and stability of the Universe”, Nuclear Physics B 292, 784 (1987).
  * (12) J. D. Barrow, G. F. R. Ellis, R. Maartens, and C. G. Tsagas, “On the stability of the Einstein static universe”, Classical Quantum Gravity 20, L155 (2003).
  * (13) C. Barceló and G. E. Volovik, “A stable static Universe?”, JETP Letters 88, 209 (2004).
  * (14) A. Friedmann, “Über die Möglichkeit einer Welt mit konstanter negativer Krümmung des Raumes”, Zeitschrift für Physik 21, 326 (1924); English translation see, e.g., A. Friedmann, “On the possibility of a world with constant negative curvature of space”, General Relativity and Gravitation 31, 2001 (1999).
  * (15) E. R. Harrison, “Classification of uniform cosmological models”, Monthly Notices Royal Astronomical Societry 137, 69 (1967).
  * (16) S. Coleman and F. de Luccia, “Gravitational effects on and of vacuum decay”, Phys. Rev. D 21, 3305 (1980).
  * (17) A. H. Guth, “Inflationary universe: A possible solution to the horizon and flatness problems”, Phys. Rev. D 23, 348 (1981).
  * (18) S. K. Blau, E. I. Guendelman, and A. H. Guth, “Dynamics of false-vacuum bubbles”, Phys. Rev. D 35, 1747 (1987).
  * (19) J. R. Ipser, “Double-bubble spacetimes”, Phys. Rev. D 36, 1933 (1987).
  * (20) V. A. Berezin, V. A. Kuzmin, and I. I. Tkachev, “Dynamics of bubbles in general relativity”, Phys. Rev. D 36, 2919 (1987).
  * (21) Y. Takamizu and K. Maeda, “Bubble universes with different gravitational constants”, Phys. Rev. D 92, 023514 (2015); arXiv:1504.05788 [gr-qc].
  * (22) J. A. González, A. Bellorín, M. A. García-Ñustes, L. E. Guerrero, S. Jiménez, J. F. Marín, and L. Vázquez, “Fate of the true-vacuum bubbles”, Journal of Cosmology and Astroparticle Physics JCAP 06 (2018) 033; arXiv:1710.05334 [hep-th].
  * (23) H. Deng and A. Vilenkin, “Primordial black hole formation by vacuum bubbles”, Journal of Cosmology and Astroparticle Physics JCAP 12 (2017) 044; arXiv:1710.02865 [gr-qc].
  * (24) H. Deng, “Primordial black hole formation by vacuum bubbles. Part II”, Journal of Cosmology and Astroparticle Physics JCAP 09 (2020) 023; arXiv: 2006.11907 [astro-ph.CO].
  * (25) A. Einstein and N. Rosen, “The particle problem in the general theory of relativity”, Phys. Rev. 48, 73 (1935).
  * (26) C. Misner and J. A. Wheeler, “Classical physics as geometry”, Annals of Physics 2, 525 (1957).
  * (27) M. Kruskal, “Maximal extension of Schwarzschild metric”, Phys. Rev. 119, 1743 (1959).
  * (28) R. W. Fuller and J. A. Wheeler, “Causality and multiply connected space-time”, Phys. Rev. 128, 919 (1962).
  * (29) H. G. Ellis, “Ether flow through a drainhole: A particle model in general relativity”, Journal of Mathematical Physics 14, 104 (1973).
  * (30) K. A. Bronnikov, “Scalar-tensor theory and scalar charge”, Acta Physica Polonica B 4, 251 (1973).
  * (31) M. Morris and K. S. Thorne, “Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativity”, American Journal of Physics 56, 395 (1988).
  * (32) T. A. Roman, “Inflating Lorentzian wormholes”, Phys. Rev. D 47, 1370 (1993); arXiv:gr-qc/9211012.
  * (33) M. Visser, Lorentzian Wormholes (Springer-Verlag, New York 1996).
  * (34) D. Hochberg and M. Visser, “Dynamic wormholes, anti-trapped surfaces, and energy conditions”, Phys. Rev. D 58, 044021 (1998); arXiv:gr-qc/9802046.
  * (35) S. V. Krasnikov, “A traversable wormhole”, Phys. Rev. D 62, 084028 (2000); arXiv:gr-qc/9909016.
  * (36) J. P. S. Lemos, F. S. N. Lobo, and S. Quinet Oliveira, “Morris -Thorne wormholes with a cosmological constant”, Phys. Rev. D 68, 064004 (2003); arXiv:gr-qc/0302049.
  * (37) O. B. Zaslavskii, “Traversable wormholes: minimum violation of null energy condition revisited”, Phys. Rev. D 76, 044017 (2007); arXiv:0707.1487 [gr-qc].
  * (38) K. A. Bronnikov, R. A. Konoplya, and T. D. Pappas, “General parametrization of wormhole spacetimes and its application to shadows and quasinormal modes”, Phys. Rev. D 103, 124062 (2021); arXiv:2102.10679 [gr-qc].
  * (39) J. C. Feng, J. P. S. Lemos, and R. A. Matzner, “Self collision of a portal wormhole”, Phys. Rev. D 104, 121212 (2021); arXiv:2104.12121 [gr-qc].
  * (40) H.-A Shinkai and S. A. Hayward, “Fate of the first traversible wormhole: Black hole collapse or inflationary expansion”, Phys. Rev. D 66, 044005 (2002); arXiv:gr-qc/0205041.
  * (41) S. V. Sushkov and Y.-Z. Zhang, “Scalar wormholes in cosmological setting and their instability”, Phys. Rev. D 77, 024042 (2008); arXiv:0712.1727 [gr-qc].
  * (42) J. P. S. Lemos and F. S. N. Lobo, “Plane symmetric thin-shell wormholes: Solutions and stability”, Phys. Rev. D 78, 044030 (2008); arXiv:0806.4459 [gr-qc].
  * (43) A. A. Kirillov and E. P. Savelova, “Density perturbations in a gas of wormholes”, Monthly Notices of the Royal Astronomical Society 412, 1710 (2011); arXiv:1006.3230 [astro-ph.CO].
  * (44) A. 0̈vgün, G. Gyulchev, and K. Jusufi, “Weak gravitational lensing by phantom black holes and phantom wormholes using the Gauss-Bonnet theorem”, Annals Physics 406, 152 (2019); arXiv:1806.03719 [gr-qc].
  * (45) M. Y. Piotrovich, S. V. Krasnikov, S. D. Buliga, and T. M. Natsvlishvili, “Search for wormhole candidates in active galactic nuclei: radiation from colliding accreting flows”, Monthly Notices of the Royal Astronomical Society 498, 3684 (2020); arXiv:2008.09411 [astro-ph.HE].