# AdS-Taub-NUT spacetimes and exact black bounces with scalar hair

José Barrientos<EMAIL_ADDRESS>Institute of Mathematics of the Czech
Academy of Sciences, Žitná 25, 11567 Praha 1, Czech Republic  Departamento de
Enseñanza de las Ciencias Básicas, Universidad Católica del Norte, Larrondo
1281, Coquimbo, Chile  Adolfo Cisterna<EMAIL_ADDRESS>Institute of Mathematics of the Czech Academy of Sciences, Žitná 25, 11567
Praha 1, Czech Republic  Dipartimento di Fisica, Università di Trento, Via
Sommarive 14, 38123 Povo, Trentino, Italy.  TIFPA - INFN, Via Sommarive 14,
38123 Povo, Trentino, Italy.  Nicolás Mora<EMAIL_ADDRESS>Departamento
de Física, Universidad de Concepción, Casilla 160-C, Concepción, Chile
Adriano Viganò<EMAIL_ADDRESS>Istituto Nazionale di Fisica Nucleare
(INFN), Sezione di Milano Via Celoria 16, I-20133 Milano, Italy  Università
degli Studi di Milano Via Celoria 16, I-20133 Milano, Italy

###### Abstract

We present a new family of exact four-dimensional Taub-NUT spacetimes in
Einstein-$\Lambda$ theory supplemented with a conformally coupled scalar field
exhibiting a power-counting superrenormalizable potential. Our configurations
are constructed in the following manner: A solution of a conformally coupled
theory with a conformal potential, henceforth the seed $(g_{\mu\nu},\phi)$, is
transformed by the action of a specific change of frame in addition with a
simultaneous shift of the seed scalar field. The conformal factor of the
transformation and the shift are both affine functions of the original scalar
$\phi$. The new configuration, $(\bar{g}_{\mu\nu},\bar{\phi})$, solves the
field equations of a conformally coupled theory with the extended
aforementioned superrenormalizable potential, this under the presence of an
effective cosmological constant. The new spectrum of solutions is notoriously
enhanced with respect to the original seed containing regular black holes,
wormholes, and bouncing cosmologies. We highlight the existence of two types
of exact black bounces given by de Sitter and anti–de Sitter geometries that
transit across three different configurations each. The de Sitter geometries
transit from a regular black hole with event and cosmological horizons to a
bouncing cosmology that connects two de Sitter Universes with different values
of the asymptotic cosmological constant. An intermediate phase, which might be
represented by two different configurations, takes place. These configurations
are given by a de Sitter wormhole or by a bouncing cosmology that connects two
de Sitter Universes, both under the presence of a cosmological horizon. On the
other hand, the anti–de Sitter geometries transit from a regular black hole
with inner and event horizons to a wormhole that connects two asymptotic
boundaries with different constant curvatures. The intermediate phase is given
in this case by an anti–de Sitter regular black hole with a single event
horizon. This regular black hole might appear in two different configurations.
As a regular anti–de Sitter black hole inside of an anti–de Sitter wormhole or
as an anti–de Sitter regular black hole with a cosmological bounce in its
interior. All these geometries are shown to be smoothly connected by the mass
parameter only. Other standard stationary black holes, bouncing cosmologies
and wormholes are also identified.

##  I Introduction

The Taub-NUT solution of Einstein equations Taub:1950ez ; Newman:1963yy
represents the simplest stationary generalization of the Schwarzschild black
hole and it is well known for having intriguing topological features. Along
with the standard mass parameter it also includes the so-called NUT charge, a
continuous parameter usually regarded as a magnetic mass, basically the dual
of the standard mass, that offers a natural realization of the gravitational
counterpart of the electric-magnetic duality Dowker:1967zz . The spacetime is
thus interpreted, in direct analogy with the Dirac magnetic monopole, as the
gravitational field of a dyon. The causal structure of the solution is wide,
offering regions in which the spacetime is interpreted as a stationary black
hole and regions in which is recognized as a nonsingular anisotropic
cosmological model. These regions are respectively known as NUT and Taub. Due
to the form of the nondiagonal components of the metric the spacetime is a
priori pathological, specially in the NUT regions. It possesses a semi-
infinite line singularity, usually called Misner string (the gravitational
analogue of the Dirac string for the magnetic monopole) Misner:1963fr ;
Astefanesei:2004kn and it contains regions plagued by closed timelike curves
Griffiths:2009dfa ; Hawking:1973uf , both rendering the black hole
interpretation misleading. Despite these undesirable features, this simple
nonradiating spacetime deserves attention, a reason why different
interpretations have emerged in order to study it.

One interpretation, due to Misner Misner:1963fr , regularizes the metric
everywhere by introducing two coordinate patches that cover independently the
north and south hemispheres of the symmetry axis and that smoothly connect at
the equator. These patches are based on a change of the time coordinate that
involves the periodic angular coordinate $\varphi$, i.e $t=t^{(\pm)}\mp
C\varphi$, where $C$ is a constant proportional to the NUT charge that defines
which semiaxis is being covered. Once an observer has access to the
coordinates that cover the north hemisphere the Misner string takes place in
the southern semiaxis and vice versa when the observer belongs to the patch
covering the south hemisphere. The Misner string is then unseen. In the
overlap region both coordinates $t^{(\pm)}$ are related, and due to the
periodicity of the azimuthal angle this ultimately implies that time is
identified with a period proportional to the NUT charge111This approach is
mathematically identical to the one that eliminates the Dirac string in the
vector potential of the Dirac monopole. In there, the periodicity condition
translates into the quantization of the electric charge.. This results into
the existence of closed timelike curves in the NUT regions. Moreover, under
this periodicity condition the analytically extended Taub-NUT spacetime is
either geodesically incomplete Misner:1963fr ; Hawking:1973uf or maximally
extendable to a spacetime with no Haussdorff topology RGeroch ; PHajicek .
Motivated by these considerations the Taub-NUT spacetime is usually studied in
its Euclidean version where it represents a gravitational instanton, a
gravitational cousin of Euclidean solutions of Yang-Mills theory Rajaraman ;
Shifman . It is expected that these gravitational instantons play a similar
role in theories of quantum gravity Esposito:1997wt as the one played by Yang-
Mills instantons in quantum field theory. Several models has been investigated
in this direction Page:1978hdy ; Iwai:1992zu ; Witten:2009xu ; Corral:2019leh
; Arratia:2020hoy ; Corral:2021xsu ; Bueno:2018uoy .

On the other hand, there is Bonnor’s interpretation, in which the Misner
string is not avoided by any identification of the time coordinate and it is
treated as a sort of topological defect usually regarded as a semi-infinite
rod with spin BOnnor ; SAck . In fact, rewriting the Taub-NUT spacetime in
Lewis-Weyl-Papapetrou coordinates it was pointed out in Stephani:2003tm that
NUT regions might represent the exterior field of a rotating source of which
its angular momentum is measured by the NUT charge. The Misner string now
represents an unremovable quasiregular singularity222Quasiregular in the sense
that all curvature polynomials are still bounded. and it naturally sows the
questioning of the geodesic completeness of the spacetime. Moreover, regions
containing closed timelike curves are still present. Nevertheless these
disadvantages, it has been recently shown that the presence of the Misner
string is not that catastrophic. In fact, in Clement:2015cxa ; Clement:2015aka
the authors have shown that the topological defect of the Misner string is
transparent to geodesic observers, and then there is no need to invoke any
compactification of the time coordinate. In consequence, there are no
obstructions for the analytic continuation of the spacetime through the
horizons and then, due to the lack of curvature singularities, the whole Taub-
NUT spacetime is geodesically complete. Moreover, it is also demonstrated that
for certain values of $C$, basically the constant that locates the Misner
string, geodesic observers do not encounter any causality violations, so the
absence of closed timelike geodesics is ensured333For a detailed discussion
about modern interpretations of the Taub-NUT spacetime we refer the readers to
Tesis ..

Among all gravitational solutions black holes represents the most iconic ones.
Formed by gravitational collapse, one of the most astonishing features of
black holes asserts that regardless the type of matter involved in the
collapse they are finally characterized by a restricted set of quantities:
mass, electromagnetic charges and angular momentum, namely, physical
quantities satisfying a Gauss type law and in consequence measurable from
infinity. Any other information about the original matter that formed the
black hole is lost, either eaten up or expelled out from the black hole once
equilibrium is reached. This statement, well-known as the no-hair conjecture
Ruffini:1971bza , is strongly supported in Einstein theory by the uniqueness
of the Kerr-Newman family Carter:1971zc ; Israel:1967wq ; Israel:1967za ;
Wald:1971iw . Hair is used to englobe all those properties of the collapsing
matter that might depart from the mass, electromagnetic charges or angular
momentum of the final configuration and that might represent a nontrivial
trace of the type of matter involved in the gravitational collapse. The no-
hair conjecture is a theory dependent statement and therefore it must be
independently enunciated on all theories in which black holes are worth study.
The simplest type of hair to be explored is the scalar one, basically
complement Einstein theory with a minimally coupled scalar field that inherits
the symmetries of the spacetime under scrutiny. An exact solution for this
system was early known Janis:1969ivo , although unphysical due to the presence
of a naked singularity. Bekenstein no-hair theorem Bekenstein:1995un ;
Bekenstein:1998aw is the first mathematical realization of the no-hair
conjecture and rules out the existence of black hole solutions with nontrivial
scalar hair in Einstein theory with minimally coupled self-interacting scalar
fields. Nonetheless, it also tangentially established the foundations to elude
the absence of scalar hair by reaccommodating the hypothesis associated with
the theorem.

Along these lines the first black hole solution presented as a hairy scalar
black hole, independently found by Bekenstein Bekenstein:1974sf and by
Bronnikov, Melnikov, and Bocharova BBM (BBMB black hole), was proposed for
Einstein theory supplemented with a conformally coupled scalar field. The
solution possesses the causal structure of an extremal Reissner-Nordström
black hole with a scalar field that diverges at the horizon, apparently with
no dramatic consequences Bekenstein:1975ts . It is usually introduced as a
black hole in dimension four only Xanthopoulos:1992fm and it relates to the
one of Janis:1969ivo by a conformal transformation, providing a precise
example of how a change of frame substantially changes the geometric structure
of a given solution444For a recent and detailed review of no-hair theorems and
counterexamples to this conjecture we refer to Herdeiro:2015waa and references
therein.. However, a deeper analysis Sudarsky:1997te reveals that the
divergence of the scalar field is actually harmful and that it is connected
with an ill-defined stress tensor at the horizon. As a result the BBMB
solution does not solves Einstein equations in an extended chart containing
the event horizon and its interpretation as a genuine black hole is obscure.
The situation, however, is relieved by the mere inclusion of a cosmological
constant, which comes into play along with a conformal self-interacting
potential for the scalar. A wider family of solutions is found for this case,
the so-called Martínez-Troncoso-Zanelli (MTZ) spacetimes Martinez:2002ru ;
Martinez:2005di . Here, the scalar field pole is hidden behind the event
horizon and the solutions represent stealths and black holes with different
topologies. Einstein equations are then everywhere solved in the domain of
outer domains of communications, including the horizon and solving the
undesired pathologies of the BBMB solution. Additionally, contrary to the BBMB
black hole these solutions are not pathological in the minimally coupled
frame. The integrability properties of these models have been well-studied
since then, providing solutions for spacetimes with acceleration
Charmousis:2009cm ; Astorino:2013xxa , NUT charge Bardoux:2013swa , rotation
Astorino:2014mda and a full analysis of conformally coupled black holes in the
entire Plebański-Demiański family Anabalon:2009qt ; Cisterna:2021xxq .
Moreover, several solutions have been found also in the presence of external
magnetic fields Astorino:2013xc ; Astorino:2013sfa , in theories with higher
order corrections Caceres:2020myr and in metric-affine geometries
Boudet:2020eyr .

In this context a particularly interesting solution is the one constructed in
Anabalon:2012tu . In there the usual four dimensional conformally coupled
scalar theory is augmented with a self-interaction that deviates from the
conformal potential by allowing the presence of a linear and cubic terms. The
causal structure of the solution is notoriously enhanced with respect to the
MTZ family Martinez:2002ru ; Martinez:2005di including regular black holes,
wormholes and cosmological spacetimes with bounces555The vanishing
cosmological constant case was early discussed in Barcelo:2000zf and it
represents a traversable wormhole.. The inclusion of a Maxwell field was
studied in Ayon-Beato:2015ada where the authors have shown that the MTZ family
and the solutions presented in Anabalon:2012tu are related by a particular set
of transformations; the simultaneous action of a conformal change of frame
plus a specific shift of the initial scalar field configuration. Both, the
conformal factor and the shifted scalar field are affine functions of the
original scalar configuration. In consequence, the solutions contained in
Anabalon:2012tu are the transformed version of the MTZ solutions, this by
means of the transformations provided in Ayon-Beato:2015ada . The scheme,
although different, works in analogy to the one that allows the construction
of the BBMB black hole Bekenstein:1974sf by applying a conformal
transformation onto the originally pathological solution Janis:1969ivo . In
there, a solution of a minimally coupled scalar theory is translated into a
solution of a conformally coupled theory by means of a conformal
transformation. The set of transformations defined in Ayon-Beato:2015ada
transforms a solution of a conformally coupled theory into a solution of a new
conformally coupled theory in which the scalar potential now deviates from the
conformal potential by including all power-counting super-renormalizable
contributions and where a new effective cosmological constant takes place.

In the present work we face the construction of a new family of Taub-NUT
spacetimes in the context of conformally coupled scalar-tensor theories. In a
nutshell we construct the Taub-NUT version of the solutions described in
Anabalon:2012tu ; Ayon-Beato:2015ada . Our starting point is the seed
configuration studied in Bardoux:2013swa , which by means of the
transformations provided in Ayon-Beato:2015ada is upgraded to represents a new
set of geometries given by regular black holes, wormholes and bouncing
cosmologies. Unnoticed in the previous works Anabalon:2012tu ; Ayon-
Beato:2015ada , here we explicitly exploit the relevance of the new self-
interaction: Under certain conditions it provides the existence of exact black
bounce geometries that smoothly transit between regular black holes and
wormholes and bouncing cosmologies. The transitions, as we shall observe, are
controlled by the mass parameter only and do not require the presence of any
new artificial scale into the theory. These type of solutions have recently
attracted considerable attention, see Simpson:2018tsi ; Simpson:2019oft and
references therein, most of them representing candidate spacetimes with
interesting geometric properties but lacking from a well-defined action
principle from which the equations of motion provide the desirable
configurations. Here, we analyze from a heuristic point of view the geometric
structure of these kind of exact spacetimes and we give a detailed analysis of
every new backreaction, this accordingly with the sign of the cosmological
constant and the different topologies of the base manifold.

Our results organize as follows: Sec. II is devoted to provide preliminary
results concerning with the construction of the solutions. First, we summarize
the transformations presented in Ayon-Beato:2015ada showing how a seed
solution of a conformally coupled scalar theory is transformed into a solution
of a conformally coupled scalar theory with an extended self-interacting
potential. Then, we continue by giving a robust analysis of the seed solution
Bardoux:2013swa , this in order to explicitly determine how its original
causal structures are enhanced by the action of the transformations. In Sec.
III we present our novel solutions, discuss their main general properties and
give a detailed analysis of every of these new geometries. We segment our
analysis according to the sign of the cosmological constant and we describe
every possible horizon geometry, namely, spherical, hyperbolic or planar.
Several families of regular black holes, wormholes and bouncing cosmologies
are described paying particular attention to those cases representing black
bounces that transit across different geometries. We finally conclude in Sec.
IV where we summarize our principal findings and where concrete extensions of
the ideas proposed here are debated. The vanishing cosmological constant case
is briefly described in the appendix.

##  II Preliminaries

Before presenting our new solutions we start by providing a concise review of
the technique developed in Ayon-Beato:2015ada and the main features of the
seed configuration Bardoux:2013swa . As stated before, the authors have
developed a technique by which it is possible to obtain new exact solutions in
conformally coupled theories where the scalar self-interaction is no longer
conformal invariant, this by simply applying a specific set of transformations
onto a known solution, dubbed the seed, of a fully conformal invariant scalar
theory. The transformations are nothing else than a change of frame, for which
the conformal factor is an affine function of the scalar field seed
configuration. A specific shift of the seed scalar field is simultaneously
performed. The combined effect of both transformations results into an upgrade
of the original action, the cosmological constant acquires an effective value
while the scalar field self-interaction is enhanced (deviating from conformal
invariance), containing for some specific spacetime dimensions all power-
counting super-renormalizable contributions. The new solutions, which solve
the field equations coming from the transformed action, contain a much richer
causal structure as it has been already implicitly observed in Anabalon:2012tu
. The technique works in direct analogy with the usual conformal
transformations that permit the integration of solutions in a standard
conformally coupled scalar action starting from a minimally coupled scalar
field theory Bekenstein:1974sf . In what is next the transformations are
specifically given and the transformed action is individualized. After the
scheme behind the transformations is clear we proceed with the analysis of the
Taub-NUT seed solution Bardoux:2013swa , thus paving the road to understand
how our novel solutions upgrade the geometric structure of the initial seed.

###  II.1 The transformation

The starting point is the action of Einstein-$\Lambda$ theory supplemented by
a self-interacting conformally coupled scalar field,

$S[g_{\mu\nu},\phi,A_{\mu}]=\int
d^{D}x\sqrt{-g}\left[\frac{R-2\Lambda}{2\kappa}-\frac{1}{2}\left(\partial\phi\right)^{2}-\frac{\xi_{D}}{2}\phi^{2}R-V(\phi)-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\right].$
(1)

A Maxwell field is also considered. We define $\kappa=8\pi G$ and the
nonminimal coupling with the curvature, as usual, is identified in arbitrary
dimensions to the factor $\xi_{D}:=\frac{1}{4}\left(\frac{D-2}{D-1}\right)$.
This value, up to a boundary term, dresses the scalar field action with
invariance under the conformal transformations

$g_{\mu\nu}\rightarrow\Omega^{2}g_{\mu\nu},\hskip
19.91684pt\phi\rightarrow\frac{\phi}{\Omega^{(D-2)/2}},$ (2)

with $\Omega$ a local function of the spacetime coordinates. The self-
interaction is given by the conformal potential, which in arbitrary dimensions
reads

$V(\phi)=\alpha\phi^{\frac{2D}{D-2}}.$ (3)

Here $\alpha$ stands for a dimensionless coupling constant. The respective
field equations,

$\displaystyle G_{\mu\nu}+\Lambda g_{\mu\nu}$
$\displaystyle=\kappa\left[\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{2}g_{\mu\nu}(\partial\phi)^{2}-\alpha
g_{\mu\nu}\phi^{\frac{2D}{D-2}}+\xi_{D}(g_{\mu\nu}\Box-\nabla_{\mu}\nabla_{\nu}+G_{\mu\nu})\phi^{2}\right]+\kappa\left(F_{\mu\lambda}F_{\nu}^{\lambda}-\frac{1}{4}g_{\mu\nu}F_{\lambda\rho}F^{\lambda\rho}\right)$
$\displaystyle\Box\phi-\xi_{D}R\phi$
$\displaystyle=\alpha\frac{2D}{D-2}\phi^{\frac{D+2}{D-2}}$
$\displaystyle\nabla_{\mu}F^{\mu\nu}$ $\displaystyle=0,$ (4)

once solved, provide the corresponding seed solution
$(g_{\mu\nu},\phi,A_{\mu})$, which by means of the set of transformations
defined in Ayon-Beato:2015ada

$\displaystyle\bar{g}_{\mu\nu}$
$\displaystyle=\left(a\sqrt{\kappa\xi_{D}}\phi+1\right)^{\frac{4}{D-2}}g_{\mu\nu},$
(5a) $\displaystyle\bar{\phi}$
$\displaystyle=\frac{1}{\sqrt{\kappa\xi_{D}}}\frac{\sqrt{\kappa\xi_{D}}\phi+a}{a\sqrt{\kappa\xi_{D}}\phi+1},$
(5b) $\displaystyle\bar{A}_{\mu}$ $\displaystyle=\sqrt{(1-a^{2})}A_{\mu},$
(5c)

delivers a new conformally related solution
$(\bar{g}_{\mu\nu},\bar{\phi},\bar{A_{\mu}})$ that solves the transformed
field equations

$\displaystyle\bar{G}_{\mu\nu}+\lambda\bar{g}_{\mu\nu}$
$\displaystyle=\kappa\left[\partial_{\mu}\bar{\phi}\partial_{\nu}\bar{\phi}-\frac{1}{2}\bar{g}_{\mu\nu}(\partial\bar{\phi})^{2}-\bar{g}_{\mu\nu}\bar{V}(\bar{\phi})+\xi_{D}(\bar{g}_{\mu\nu}\bar{\Box}-\bar{\nabla}_{\mu}\bar{\nabla}_{\nu}+\bar{G}_{\mu\nu})\bar{\phi}^{2}\right]+\kappa\left(\bar{F}_{\mu\lambda}\bar{F}_{\nu}^{\lambda}-\frac{1}{4}g_{\mu\nu}\bar{F}_{\lambda\rho}\bar{F}^{\lambda\rho}\right)$
$\displaystyle\bar{\Box}\bar{\phi}-\xi_{D}\bar{R}\bar{\phi}$
$\displaystyle=\frac{\partial\bar{V}(\bar{\phi})}{\partial\bar{\phi}}$
$\displaystyle\bar{\nabla}_{\mu}\bar{F}^{\mu\nu}$ $\displaystyle=0.$ (6)

These equations of motion arise from the redefined action principle

$\bar{S}[\bar{g}_{\mu\nu},\bar{\phi},\bar{A}_{\mu}]=\int
d^{D}x\sqrt{-\bar{g}}\left[\frac{\bar{R}-2\lambda}{2\kappa}-\frac{1}{2}\left(\partial\bar{\phi}\right)^{2}-\frac{\xi_{D}}{2}\bar{\phi}^{2}\bar{R}-\bar{V}(\bar{\phi})-\frac{1}{4}\bar{F}_{\mu\nu}\bar{F}^{\mu\nu}\right],$
(7)

which is a consequence of transformations (5) acting on (1). The new action
relates to the seed action by means of
$\bar{S}[\bar{g},\bar{\phi},\bar{A}_{\mu}]=(1-a^{2})S[g_{\mu\nu},\phi,A_{\mu}]$,
maintaining unitarity as long as the shift parameter is constrained to
$|a|<1$. It must be noticed that the neat effect of the transformations on the
seed action translates into the presence of a redefined cosmological constant

$\displaystyle\lambda$
$\displaystyle=\frac{\kappa}{\left(1-a^{2}\right)^{\frac{D+2}{D-2}}}\left[\frac{\Lambda}{\kappa}+\alpha\left(-\frac{a}{\sqrt{\kappa\xi_{D}}}\right)^{\frac{2D}{D-2}}\right],$
(8)

along with an enhanced scalar potential defined by

$\displaystyle\bar{V}(\bar{\phi})$
$\displaystyle={}\frac{1}{\left(1-a^{2}\right)^{\frac{D+2}{D-2}}}\Bigg{\\{}\frac{\Lambda}{\kappa}\left[\left(1-a\sqrt{\kappa\xi_{D}}\bar{\phi}\right)^{\frac{2D}{D-2}}-1\right]+\alpha\left[\left(\bar{\phi}-\frac{a}{\sqrt{\kappa\xi_{D}}}\right)^{\frac{2D}{D-2}}-\left(-\frac{a}{\sqrt{\kappa\xi_{D}}}\right)^{\frac{2D}{D-2}}\right]\Bigg{\\}}.$
(9)

This new potential, which ceases to be conformally invariant, is
superrenormalizable for integer values of the conformal power $2D/(D-2)$. This
only occurs in three, four, and six dimensions. In addition, by construction
this potential is defined such that no zeroth-order term appears, and then the
new cosmological constant is precisely the one of (8). As we will shortly
observe, every new coupling constant $\alpha_{2D/(D-2)}$ will be related to
both, the original conformal coupling $\alpha$ and the corresponding seed
cosmological constant $\Lambda$.

###  II.2 The seed configuration

We are interested on the study of new Taub-NUT solutions, in consequence we
fix ourselves to dimension four where the seed solution for the conformally
coupled theory (1) has been previously reported in Bardoux:2013swa . In order
to properly study the new solutions we shall present in the next section, we
first provide a summary of the principal features of the seed metric, and in
this way we pave the road to understand the causal structure of our novel
solutions and their main differences with respect to solutions previously
reported in the literature. The field equations (4) are solved by the line
element

$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=-F(r)(dt+\mathcal{B})^{2}+\frac{dr^{2}}{F(r)}+(r^{2}+n^{2})d\Sigma_{K}^{2},$
(10)

with

$F(r)=-\frac{\Lambda}{3}(r^{2}+n^{2})+\frac{(K-\frac{4}{3}n^{2}\Lambda)(r-M)^{2}}{(r^{2}+n^{2})},$
(11)

and where each Kähler potential and the corresponding Kähler manifolds are
given by

$\mathcal{B}=\left\\{\begin{aligned}
4n\sin^{2}\frac{\theta}{2}d\varphi\qquad&\text{when}\qquad
d\Sigma^{2}_{(K=1)}=d\theta^{2}+\sin^{2}\theta d\varphi^{2}\\\
n\theta^{2}d\varphi\qquad&\text{when}\qquad
d\Sigma^{2}_{(K=0)}=d\theta^{2}+\theta^{2}d\varphi^{2}\\\
4n\sinh^{2}\frac{\theta}{2}d\varphi\qquad&\text{when}\qquad
d\Sigma^{2}_{(K=-1)}=d\theta^{2}+\sinh^{2}\theta d\varphi^{2}.\\\
\end{aligned}\right.$ (12)

The configuration is completed by the following scalar and electromagnetic
fields

$\phi(r)=\sqrt{-\frac{\Lambda}{6\alpha}}\frac{\sqrt{n^{2}+M^{2}}}{r-M},\hskip
14.22636pt\mathcal{A}=\frac{Qr}{r^{2}+n^{2}}(dt+\mathcal{B}).$ (13)

$Q$666The uncharged case will imply the tuning of the cosmological constant in
terms of the coupling $\alpha$. In addition, the $\Lambda=\alpha=0$ case will
represent the Taub-NUT extension of the BBMB black hole Bekenstein:1974sf ;
BBM . stands for the electric charge and two conditions must be met,
$\Lambda\alpha<0$ and

$Q^{2}=\frac{1}{18}\frac{(n^{2}+M^{2})(K-\frac{4}{3}n^{2}\Lambda)(\kappa\Lambda+36\alpha)}{\kappa\alpha}.$
(14)

Reality of the electric charge finally requires

$\displaystyle(K-\frac{4}{3}n^{2}\Lambda)>0\hskip 5.69046pt\text{and}\hskip
5.69046pt\frac{(\kappa\Lambda+36\alpha)}{\kappa\alpha}>0,\hskip 2.84544ptor$
(15) $\displaystyle(K-\frac{4}{3}n^{2}\Lambda)<0\hskip
5.69046pt\text{and}\hskip
5.69046pt\frac{(\kappa\Lambda+36\alpha)}{\kappa\alpha}<0.$ (16)

Notice that we have applied the coordinate transformation $t\rightarrow
t+2n\varphi$ onto the original solution presented in Bardoux:2013swa . This
allows to remove one of the singular semiaxis and localize the Misner string
at the southern semiaxis $(\theta=\pi)$ only. Moreover, these coordinates
provides a simpler setup in which to study the appearance of generic closed
timelike curves associated with the change of sign of the metric component
$g_{\varphi\varphi}$. From hereon we assume a combination of $\alpha$ and
$\Lambda$ such that the gauge and scalar fields remain real.

The spacetime inherits the asymptotic behavior of Taub-NUT spaces, namely,
with or without a cosmological constant the spacetime is not globally
asymptotically (A)dS neither globally asymptotically flat. This, as it is
known, is due to the presence of the nondiagonal components of the metric,
basically the presence of the NUT parameter that turns on the Riemann tensor
components associated with the magnetic part of the Weyl tensor. Nevertheless,
the electric part of the Weyl tensor provides the associated Riemann tensor
components, $\bar{R}^{\mu\nu}_{\,\,\sigma\rho}$, with the usual asymptotic
$\lim_{r\rightarrow\infty}\bar{R}^{\mu\nu}_{\,\,\sigma\rho}=\frac{\Lambda}{3}\delta^{\mu\nu}_{\,\,\sigma\rho}$.
The spacetime is free of curvature singularities, however the Misner string
defect persists for a spherical foliation. Notwithstanding this, leaning on
Bonnor’s interpretation BOnnor ; SAck and the recent works Clement:2015cxa ;
Clement:2015aka the spacetime does not exhibits any pathologies for free
falling observers whose backreaction is negligible. The spacetime is taken as
geodesically complete, and in consequence the radial coordinate takes values
on the full range from $-\infty$ to $+\infty$. The mass parameter is then
positive defined, negative values are englobed in the change of coordinate
$r\rightarrow-r$. Despite the presence of the Misner string we analyze the
causal structure of the seed metric for each of the horizon geometries. The
metric function (11) possesses four roots

$\displaystyle r_{--}$
$\displaystyle=\frac{\bar{l}}{2}\left(-1-\sqrt{1-4\frac{n^{2}}{\bar{l}^{2}}+4\frac{M}{\bar{l}}}\right)$
(17) $\displaystyle r_{-}$
$\displaystyle=\frac{\bar{l}}{2}\left(-1+\sqrt{1-4\frac{n^{2}}{\bar{l}^{2}}+4\frac{M}{\bar{l}}}\right)$
(18) $\displaystyle r_{+}$
$\displaystyle=\frac{\bar{l}}{2}\left(1-\sqrt{1-4\frac{n^{2}}{\bar{l}^{2}}-4\frac{M}{\bar{l}}}\right)$
(19) $\displaystyle r_{++}$
$\displaystyle=\frac{\bar{l}}{2}\left(1+\sqrt{1-4\frac{n^{2}}{\bar{l}^{2}}-4\frac{M}{\bar{l}}}\right),$
(20)

which might define up to four Killing horizons. Here, accordingly with the
sign of the cosmological constant we have made use of the following
conventions

$\displaystyle\Lambda>0$
$\displaystyle\rightarrow\bar{l}=\sqrt{3K/\Lambda-4n^{2}}$ (21)
$\displaystyle\Lambda<0$
$\displaystyle\rightarrow\bar{l}=\sqrt{-(-3K/\Lambda+4n^{2})}.$ (22)

The location of these horizons satisfies $r_{--}<0<r_{-}<M<r_{+}<r_{++}$,
where $r=M$ defines the point in which the scalar field diverges. The presence
of the scalar field pole is a common feature in solutions of conformally
coupled theories of the kind considered here for scalar fields that are
radially dependent only. All roots of the lapse function $F(r)$ remain real as
long as the mass parameter obeys
$0<M<\frac{\bar{l}}{4}-\frac{n^{2}}{\bar{l}}$, reality condition that is
subjected to the reality of $\bar{l}$. In addition, the mass parameter is
taken to be always positive and then
$\frac{\bar{l}}{4}-\frac{n^{2}}{\bar{l}}>0$ must be met. This case is the most
generic one and implies the existence of four Killing horizons. One direct
difference with respect to the static counterpart is the presence of a
solution with a planar horizon, which in the absence of the NUT parameter
requires the presence of extra matter fields Bardoux:2012aw . For each of the
possible horizon geometries the following features are sketched out:

  1. i)

$K=1$, spherical foliation:

$\Lambda>0$: The reality of $\bar{l}$ and the condition
$\frac{\bar{l}}{4}-\frac{n^{2}}{\bar{l}}>0$ are simultaneously achieved by
considering $\Lambda<\frac{3}{8n^{2}}$. This allows the generic existence of
Killing horizons. In addition, $\alpha<0$ provides a real scalar field
configuration. Thus, for $0<M<\frac{\bar{l}}{4}-\frac{n^{2}}{\bar{l}}$ there
are four Killing horizons. Both, $r_{--}$ and $r_{++}$, correspond to
cosmological horizons beyond which the metric becomes time dependent. The
scalar field pole is hidden behind the event horizon $r_{+}$. The region
between $r_{+}$ and $r_{++}$ is recognized as the exterior of a regular
stationary black hole, however under the presence of a pathological semiaxis
of rotation at $\theta=\pi$ and due to the generic existence of closed
timelike curves its interpretation is questionable. Notwithstanding this, the
existence of closed timelike curves is more subtle under the presence of the
conformally coupled scalar field. These might be avoided by restricting the
parameter space determined by the cosmological constant and the NUT charge,
this at the price of changing the interpretation of the spacetime metric. It
can be seen as follows: the $g_{\varphi\varphi}$ metric component is given by

$g_{\varphi\varphi}=\left[(r^{2}+n^{2})\sin^{2}\theta-16n^{2}F(r)\sin^{4}\frac{\theta}{2}\right].$
(23)

Then, in order to maintain the Riemannian nature of the two dimensional metric
given by $t=r=constant$ it is required to hold

$\frac{4n^{2}F(r)}{r^{2}+n^{2}}<\ -1.$ (24)

After some algebraic manipulations it is possible to show that777This is an
exclusive property of the conformally coupled scalar case we consider here. It
is mostly due to the quasiextremal form that the scalar field provides to the
metric function.

$\left(1-\frac{4}{3}n^{2}\Lambda\right)\left[1+4n^{2}\left(\frac{r-M}{r^{2}+n^{2}}\right)^{2}\right]<0,$
(25)

providing the complete absence of closed timelike curves as soon as
$\Lambda>3/4n^{2}$. This condition does not overlap with the requirement of a
real $\bar{l}$ and in consequence under this condition the metric function is
devoid of Killing horizons, all roots of $F(r)$ become complex. Therefore, the
metric function $F(r)$ is everywhere negative, reason why the metric component
$g_{\varphi\varphi}$ remains always positive. Then, the spacetime represents a
cosmological model in which the scalar field pole is uncovered.
It is instructive to analyze the case in which the mass parameter satisfies
$M>\frac{\bar{l}}{4}-\frac{n^{2}}{\bar{l}}$. In this case the roots $r_{+}$
and $r_{++}$ become complex while $r_{-}$ and $r_{--}$ remain real. It is not
possible to make disappear all Killing horizons at the same time. Moreover,
the scalar field pole stays uncovered. The metric function $F(r)$ is
everywhere negative outside $r_{-}$ and the spacetime there, although
pathological, possesses a cosmological behavior. This will be particularly
interesting when presenting our novel solutions. We will see that after
applying the transformations previously explained this spacetime will become a
bouncing cosmology with no pole for the scalar field and with no pathologies
associated to the presence of closed timelike curves. In addition, under
specific circumstances a black bounce transition in between a regular black
hole and a bouncing cosmology will be observable.
It is important to stress that in spite of this the Misner string is still
present and that ultimately this obstruction, as shown in Clement:2015cxa ;
Clement:2015aka , can be circumvented by considering geodesic observers only.
More appealing for spherical foliation is the Euclidean case, for which the
Misner string can be completely removed by the Misner approach and for which
the presence of closed timelike curves is not a pathology. This case will be
further analyzed in eu-case .
$\Lambda<0$: For a negative cosmological constant relation (22) implies always
a complex $\bar{l}$, and in consequence the complete absence of Killing
horizons no matter the values the mass parameter can take. The scalar field is
real for $\alpha>0$ and its pole remains uncovered. The metric function $F(r)$
is always positive and the radial coordinate remains spacelike. Both, a
singular semiaxis and closed timelike curves are present.

  2. ii)

$K=-1$, hyperbolic foliation:

$\Lambda>0$: In this case a positive cosmological constant implies the absence
of Killing horizons, no real roots for $F(r)$. The metric function is
everywhere negative, and in consequence the spacetime inherits a cosmological
nature. The scalar field is real for $\alpha<0$ and its pole is uncovered. The
hyperbolic geometry, in contrast to the previous spherical case, does not deal
with the presence of the singular semiaxes, namely, Misner strings are not
present Astefanesei:2004kn . Then, only the presence of closed timelike curves
must be analyzed. The azimuthal angle component of the metric now reads

$g_{\varphi\varphi}=\left[(r^{2}+n^{2})\sinh^{2}\theta-16n^{2}F(r)\sinh^{4}\frac{\theta}{2}\right].$
(26)

Following a similar computation to the one explained for the spherical case it
is shown that the absence of closed timelike curves is ensured by satisfying

$-\left(1+\frac{4}{3}n^{2}\Lambda\right)\left[1+4n^{2}\left(\frac{r-M}{r^{2}+n^{2}}\right)^{2}\right]<0,$
(27)

condition always achieved by positive cosmological constant.

$\Lambda<0$: For a negative cosmological constant the reality of $\bar{l}$ and
the positivity of the combination $\frac{\bar{l}}{4}-\frac{n^{2}}{\bar{l}}$
are ensured by $|\Lambda|<\frac{3}{8n^{2}}$, where we have considered
$\Lambda=-|\Lambda|$. $\alpha$ is taken to be positive. Hence, for
$0<M<\frac{\bar{l}}{4}-\frac{n^{2}}{\bar{l}}$ four Killing horizons take
place, none of them of cosmological nature. The spacetime between the event
horizon $r_{++}$ and asymptotic infinity is identified as the exterior of a
regular stationary topological black hole where no Misner string takes place.
Moreover, this case offers the complete absence of closed timelike curves as
soon as $|\Lambda|<3/4n^{2}$, condition that overlaps with the generic
existence of Killing horizons. This spacetime represents a well-behaved
stationary black hole, free of Misner strings as well as free of causality
violations for any observer. On the other hand, for
$M>\frac{\bar{l}}{4}-\frac{n^{2}}{\bar{l}}$ both roots $r_{+}$ and $r_{++}$
become complex and the metric function $F(r)$ is everywhere positive outside
$r_{-}$. The scalar field pole is uncovered. In the next section we will
observe how this geometry transforms into a wormhole spacetime with no
associated pathologies, namely, Misner strings nor closed timelike curves, and
how under specific conditions provides a black bounce that smoothly connects a
regular black hole with an anti–de Sitter wormhole.

  3. iii)

$K=0$, planar foliation:

This case is particular, despite the value of the cosmological constant $F(r)$
is always devoid of real roots. Then, the spacetime is either radially
dependent or time dependent according to $\Lambda<0$ or $\Lambda>0$,
respectively. Accordingly, $\alpha>0$ or $\alpha<0$ and in both cases the
scalar field pole is uncovered. Misner strings are pulled out to infinity,
however closed timelike curves generically occur.

The case of vanishing cosmological constant delivers, for $K=1$, the well-
known Taub-NUT extension of the BBMB black hole. Its structure and further
analysis can be extracted from Bardoux:2013swa . We will briefly comment on
this case in the appendix. In there, we will explicitly show how it will
represent the charged Taub-NUT extension of Barcelo’s wormhole Barcelo:2000zf
.

It must be noticed that the presence of Misner strings and of the causal
pathologies associated with closed timelike curves can be generically
circumvented only when the horizon is locally isomorphic to $\mathcal{H}^{2}$,
as long as condition (27) is met. This behavior is due to the presence of the
conformally coupled scalar field and the extremal form it provides to the
metric. However, notice that according to Bonnor’s interpretation of the
Misner string the spherical and planar solutions are not _a priori_
pathological for geodesic observers. As it was previously explained, by a
simple change of coordinates and a proper constraint on the constant $C$
Clement:2015cxa ; Clement:2015aka , the Misner string becomes transparent for
geodesic observers and even more no closed timelike geodesics take place. This
open a window in which all cases, no matter the presence of these pathologies,
might be interpreted as black holes when it corresponds. Accordingly, the
spherical and planar cases are worth study. We follow this interpretation and
in consequence we will provide a detailed description of all cases, $K=\pm
1,0$.

##  III solutions

Once the main properties of the seed solution (11) are known it becomes
straightforward to understand how the new solutions, the ones obtained by
applying the set of transformations (5), upgrade the seed spacetime to a wider
family of geometries. Obtained by applying the map (5) into the seed
configuration (11) and (13) and solving in consequence the field equations
coming from (7), our new spacetime takes the form

$\displaystyle d\bar{s}^{2}$
$\displaystyle=\frac{(ab\sqrt{n^{2}+M^{2}}+r-M)^{2}}{(r-M)^{2}}\left[-\left(-\frac{\Lambda}{3}(r^{2}+n^{2})+\frac{(K-\frac{4}{3}n^{2}\Lambda)(r-M)^{2}}{(r^{2}+n^{2})}\right)(dt+\mathcal{B})^{2}\right.$
(28)
$\displaystyle\qquad\left.+\frac{dr^{2}}{\left(-\frac{\Lambda}{3}(r^{2}+n^{2})+\frac{(K-\frac{4}{3}n^{2}\Lambda)(r-M)^{2}}{(r^{2}+n^{2})}\right)}+(r^{2}+n^{2})d\Sigma_{K}^{2}\right],\hskip
11.38092pt\bar{A}=\frac{\bar{Q}r}{r^{2}+n^{2}}(dt+\mathcal{B}),$
$\displaystyle\bar{\phi}(r)$
$\displaystyle=\left(\frac{6}{\kappa}\right)^{1/2}\frac{b\sqrt{n^{2}+M^{2}}+a(r-M)}{ab\sqrt{n^{2}+M^{2}}+r-M},\hskip
22.76228ptb=\sqrt{\frac{-\kappa\Lambda}{36\alpha}},\hskip
22.76228pt\lambda=\frac{\kappa\Lambda+36\alpha a^{4}}{\kappa(1-a^{2})^{3}}.$

Notice that the condition $\alpha\Lambda<0$ ensures the reality of the
parameter $b$. In four dimensions the conformal power is an integer and in
consequence the scalar potential (9) contains all power-counting
superrenormalizable contributions

$\bar{V}(\bar{\phi})=\alpha_{1}\bar{\phi}+\alpha_{2}\bar{\phi}^{2}+\alpha_{3}\bar{\phi}^{3}+\alpha_{4}\bar{\phi}^{4},$
(29)

where consistence with all field equations requires to fix the coupling
constants as

$\displaystyle\alpha_{1}=-\frac{2\sqrt{6}}{3}\frac{a(\Lambda\kappa+36a^{2}\alpha)}{\kappa^{\frac{3}{2}}(1-a^{2})^{3}},\hskip
2.84544pt\alpha_{2}=\frac{a^{2}(\Lambda\kappa+36\alpha)}{\kappa(1-a^{2})^{3}},\hskip
2.84544pt\alpha_{3}=-\frac{\sqrt{6}}{9}\frac{a(\Lambda
a^{2}\kappa+36\alpha)}{\sqrt{\kappa}(1-a^{2})^{3}},\hskip
2.84544pt\alpha_{4}=\frac{1}{36}\frac{(a^{4}\kappa\Lambda+36\alpha)}{(1-a^{2})^{3}}.$
(30)

Notice that, beyond the superrenormalizable nature of the potential, each of
the contributions make it worth studying. The conformal potential is itself
interesting due to its Weyl invariance while on the other hand the quadratic
contribution represents a mass term. More unconventional are the presence of
the linear and cubic terms. Nonetheless, realistic models of a scalar field
theory, as it is the case of the neutral pion $\pi_{0}$ and its sigma model
description, do include these type of potentials, see for instance
Donoghue:1992dd . Finally and in analogy with the seed configuration, the
solution is closed by imposing the relation

$\bar{Q}^{2}=\frac{1}{18}\frac{(n^{2}+M^{2})(1-a^{2})(K-\frac{4n^{2}\Lambda}{3})(\Lambda\kappa+36\alpha)}{\alpha\kappa},$
(31)

which in agreement with transformation (5c) is nothing else than consider
$\bar{Q}^{2}=(1-a^{2})Q^{2}$. In consequence $\bar{Q}$ remains real as long as
any of the constraints (15) and (16) is respected.

The analysis of these new geometries starts by noticing two important
differences with respect to the seed solution. These are the appearance of a
curvature singularity and the emergence of new asymptotic regions, due to the
pole of the conformal factor, that might split the spacetime into two causally
disconnected different regions. In contrast with the seed metric and despite
the presence of the NUT charge, which usually softens the geometry of Taub-NUT
spacetimes, a curvature singularity located at

$r_{0}=M-ab\sqrt{n^{2}+M^{2}},$ (32)

emerges. This singularity arises due to the presence of the conformal factor
and it precisely corresponds to the point in which the latter vanishes. Thus,
beyond the interpretation of the Misner string the spacetime ends at the
singularity $r_{0}$ which might be located either in the positive or negative
range of the radial coordinate, this depending on the signs of $M$, $a$ and
$b$. For the curvature singularity to be located at positives values of $r$ we
observe,

$r_{0}^{+}>0\rightarrow\left\\{\begin{aligned}
M>\frac{abn}{\sqrt{1-a^{2}b^{2}}}\qquad&\text{and}\qquad 0<ab<1\\\
M>0\qquad&\text{and}\qquad ab<0\\\
0<|M|<\frac{|ab|n}{\sqrt{1-|ab|^{2}}}\qquad&\text{and}\qquad
0<|ab|<1\qquad\text{where}\qquad M=-|M|,ab=-|ab|.\\\ \end{aligned}\right.$
(33)

On the other hand, the singularity takes place at negative values of the
radial coordinate if,

$r_{0}^{-}<0\rightarrow\left\\{\begin{aligned}
0<M<\frac{abn}{\sqrt{1-a^{2}b^{2}}}\qquad&\text{and}\qquad 0<ab<1\\\
|M|>\frac{|ab|n}{\sqrt{1-|ab|^{2}}}\qquad&\text{and}\qquad
0<|ab|<1\qquad\text{where}\qquad M=-|M|,ab=-|ab|.\\\
|M|>0\qquad&\text{and}\qquad ab>0.\\\ \end{aligned}\right.$ (34)

Thus, the range of the radial coordinate is given by $(r_{0}^{\pm},\infty)$,
where we have dubbed $r_{0}^{\pm}$ the positive and negative possible
locations of the singularity.

While the curvature singularity defines the domain of the radial coordinate,
and basically the extension of the spacetime, the conformal factor defines the
structure of the asymptotic regions. The presence of the asymptotic region at
plus infinity is clear, nevertheless as already mentioned the conformal factor
may exhibits a pole which eventually would include a new asymptotic region
that drastically changes the causal structure of the solutions. It is then
necessary to analyze the position of the conformal factor pole for both cases,
whenever $r_{0}^{+}$ or $r_{0}^{-}$ takes place. Then, it is mandatory to
understand how the singularity, the eventual new asymptotic region, and the
Killing horizons are located with respect to each other.

According to these observations, and before giving a detailed analysis of
every of the geometries contained in (28), a few comments are in order:

  1. i)

Although the new metric $\bar{g}_{\mu\nu}$ is defined through a change of
frame of the seed metric $g_{\mu\nu}$ and in consequence inherits most of its
features, the conformal factor dramatically changes the causal structure of
the spacetime. As already mentioned its root introduces a curvature
singularity (32), while on the other hand may drastically change the causal
structure of the spacetime including a new asymptotic region at the point in
which it diverges. The position of this pole depends on the sign of the mass
parameter and accordingly to the location of the curvature singularity
provides the following spacetimes:

     * –

$r_{0}^{+}$: The domain of the radial coordinate is given by
$(r_{0}^{+},\infty)$. On the other hand, the conformal factor exhibits a pole
either at $r=M$ or $r=-|M|$ depending if the mass parameter is positive or
negative. For $M>0$ and, as long as $r_{0}^{+}<r=M$, a new asymptotic region
emerges dividing the spacetime into two regions we have dubbed
$\mathcal{R}^{-}\in(r^{+}_{0},M)$ and $\mathcal{R}^{+}\in(M,\infty)$. For
observers located in $\mathcal{R}^{+}$ the access to region $\mathcal{R}^{-}$
is not allowed, being this portion of the spacetime replaced by a new
asymptotic region with its own constant curvature. Similarly occurs for
observers located in $\mathcal{R}^{-}$, their access to $\mathcal{R}^{+}$ is
not permited. This happens for the first of the conditions in (33). For the
second case contained in (33) we observe that $r_{0}^{+}>r=M$, and in
consequence the pole of the conformal factor does not belong to the domain of
the radial coordinate. Similarly occurs for $M<0$, then the pole is located at
$r=-|M|$ and again does not belong to the possible values the radial
coordinate can take. Thus, for the last two cases exposed in (33) no extra
asymptotic region takes place and the region $(r_{0}^{+},\infty)$ is not
divided.

     * –

$r_{0}^{-}$: The domain of the radial coordinate is given by
$(r_{0}^{-},\infty)$. Again, the pole of the conformal factor depends on the
sign of the mass parameter, given either by $r=M$ or $r=-|M|$. For the first
of the conditions contained in (34) the mass parameter is positive and in
consequence the pole located at $r=M$ satisfies $r=M>r_{0}^{-}$. This pole
induces the presence of a new asymptotic region which divides the spacetime
into the regions $\mathcal{R}^{-}\in(r^{-}_{0},M)$ and
$\mathcal{R}^{+}\in(M,\infty)$, which as in the previous case are causally
disconnected. For the second case of (34) we observe that $r_{0}^{-}>r=-|M|$
and then the pole of the conformal factor does not belong to the domain
$(r_{0}^{-},\infty)$. The spacetime is not divided in this case. Finally, for
the third condition we observe that the singularity satisfies
$r_{0}^{-}<r=-|M|$, and then the pole of the conformal factor induces a new
asymptotic region, this time located at $r=-|M|$, that splits the spacetime
into the regions $\mathcal{R}^{-}\in(r^{-}_{0},-|M|)$ and
$\mathcal{R}^{+}\in(-|M|,\infty)$.

     * –

These spacetimes might have different geometric interpretations depending on
the space of parameters under consideration and on the relative position of
the Killing horizons that this set of parameters defines. As matter of fact,
the position of the Killing horizons is affected by the sign of the mass
parameter, and in consequence their distribution varies from case to case
accordingly to the position of the singularity and the position of an eventual
new asymptotic region.

  2. ii)

For the cases in which the conformal factor induces a splitting of the
spacetime we observe that the new asymptotic regions possess their own
constant curvature with respect to asymptotic infinity. As inherited from the
seed solution, the asymptotic behavior of the metric is of the Taub-NUT type
and then it is only locally asymptotically equivalent to (anti)-de Sitter
spacetime. The Riemann tensor components associated to the electric part of
the Weyl tensor behave as

$\displaystyle\lim_{\stackrel{{\scriptstyle
r\rightarrow\infty}}{{M>0}}}\bar{R}^{\mu\nu}_{\,\,\sigma\rho}$
$\displaystyle=\lim_{\stackrel{{\scriptstyle
r\rightarrow\infty}}{{M<0}}}\bar{R}^{\mu\nu}_{\,\,\sigma\rho}=\frac{\Lambda}{3}\delta^{\mu\nu}_{\,\,\sigma\rho},$
$\displaystyle\lim_{r\rightarrow M>0}\bar{R}^{\mu\nu}_{\,\,\sigma\rho}$
$\displaystyle=\lim_{r\rightarrow
M<0}\bar{R}^{\mu\nu}_{\,\,\sigma\rho}=\frac{\Lambda}{3a^{2}b^{2}}\delta^{\mu\nu}_{\,\,\sigma\rho}.$
(35)

  3. iii)

The scalar field configuration diverges precisely at the curvature
singularity, namely, at the zero of the conformal factor. Then, the scalar
field is unbounded either at $r_{0}^{-}$ or $r_{0}^{+}$, depending on the case
under consideration. The expectation value of the scalar on each of the
possible boundaries, for positive and negative mass parameter, is given by
$\bar{\phi}(M)\sim\sqrt{\frac{6}{\kappa}}\frac{1}{a}$ and
$\bar{\phi}(\infty)\sim\sqrt{\frac{6}{\kappa}}a$.

  4. iv)

The Killing horizons of these geometries are still given by (20), namely, the
Killing horizons of the seed metric. However, their relative locations as well
as their location with respect to the new asymptotic region and the curvature
singularities depend now on the sign of the mass parameter.

  5. v)

The $a\rightarrow 0$ limit, of course, brings the solutions into the seed
Bardoux:2013swa . On the other hand, keeping $a$ but going to the vanishing
NUT charge case, the families of solutions described in Anabalon:2012tu and
Ayon-Beato:2015ada are recovered depending on whether or not the electric
charge is present. In the vanishing $\Lambda$ case the latter two solutions
corresponds to Barcelo’s solution Barcelo:2000zf with or without charge.
Finally, for $n=a=0$, the BBMB Bekenstein:1974sf ; BBM and MTZ Martinez:2002ru
; Martinez:2005di solutions are recovered depending on the absence or the
presence of a cosmological constant and the corresponding self-interaction.

Some of these general considerations are holistically shared by the
$n\rightarrow 0$ solutions Anabalon:2012tu and Ayon-Beato:2015ada . We refer
to Anabalon:2012tu for more details regarding generic features of these kind
of geometries in a more simple setup. In the next subsections we proceed with
the analysis of the causal structures contained in our new solution (28), for
positive and negative cosmological constant and for spherical, hyperbolic and
planar base manifolds.

###  III.1 $\Lambda\neq 0$ solutions

In agreement with the analysis performed on the seed configuration, here we
connect with a specific analysis of each of the new geometries pertaining to
(28). The analysis is carried out in the following fashion: First it is
defined the sign of the seed cosmological constant $\Lambda$ and the topology
of the black hole horizon $K$. Then, for every $(\Lambda,K)$ combination we
separately describe the cases with curvature singularities at $r_{0}^{+}$ and
$r_{0}^{-}$. We explore all the possibilities contained in (33) and (34). Due
to the presence of these curvature singularities the mass parameter takes
positive and negatives values. This implies that the appearance of all Killing
horizons is more involved. As stated for the seed solution it is needed to
ensure the reality of the quantity $\bar{l}$, then the generic existence of
these horizons will depend on the values the mass parameter can take. For
positive and negative values of $M$ we have:

$M>0$: In this case it is observed that all four Killing horizons appear as
long as the mass parameter satisfies
$0<M<\frac{\bar{l}}{4}-\frac{n^{2}}{\bar{l}}$. This condition must be
complemented with two requirements: $\bar{l}\in\mathbb{R}$ and
$\frac{\bar{l}}{4}-\frac{n^{2}}{\bar{l}}>0$. Both requirements are fulfilled
by $0<|\Lambda|<3/8n^{2}$ for positive and negative cosmological constant.

$M<0$: In this case the situation is more subtle. Not all the roots for $F(r)$
are real simultaneously, thus not all Killing horizons appear at the same
time. It is observed that both horizons $r_{+}$ and $r_{++}$ exist for
$|M|>\frac{n^{2}}{\bar{l}}-\frac{\bar{l}}{4}$. Then conditions
$\bar{l}\in\mathbb{R}$ and $\frac{n^{2}}{\bar{l}}-\frac{\bar{l}}{4}>0$ are
simultaneously accomplished by requiring $3/8n^{2}<|\Lambda|<3/4n^{2}$ for
positive and negative cosmological. In this case the horizons $r_{-}$ and
$r_{--}$ do not emerge. Now, for the horizons $r_{-}$ and $r_{--}$ to exist it
is observed that the mass parameter needs to satisfy
$0<|M|<\frac{\bar{l}}{4}-\frac{n^{2}}{\bar{l}}$. The conditions
$\bar{l}\in\mathbb{R}$ and $\frac{\bar{l}}{4}-\frac{n^{2}}{\bar{l}}>0$ are
then satisfied by $0<|\Lambda|<3/8n^{2}$ for positive and negative
cosmological constant. Here, both horizons $r_{+}$ and $r_{++}$ do not emerge.

In the next subsections the cosmological constant is accordingly subjected to
these conditions. We explicitly determine the existence or absence of Killing
horizons controlling the mass parameter only.

####  III.1.1 $\Lambda>0$ and $K=1$

The curvature singularities depend on the mass parameter and on the values the
combination $ab$ can take. We tackle first the $r_{0}^{+}$ cases.

$r_{0}^{+}$: When the curvature singularity $r_{0}^{+}$ takes place the radial
coordinate assumes positive values in the range $(r_{0}^{+},\infty)$. Hence,
according to (33) three cases arise.

Case I: Here, the mass parameter is positive and it is bounded from below
$M>\frac{abn}{\sqrt{1-a^{2}b^{2}}}$, with $0<ab<1$. This implies that the pole
of the conformal factor located at $r=M$ satisfies
$r=M>r_{0}^{+}=M-ab\sqrt{n^{2}+M^{2}}$. Hence, the two regions
$\mathcal{R}^{-}\in(r^{+}_{0},M)$ and $\mathcal{R}^{+}\in(M,\infty)$ emerge,
dividing the spacetime into two causally disconnected regions with different
geometric properties according with the relative position of the Killing
horizons. The number of possible Killing horizons depends on the mass
parameter. The most generic case corresponds to
$\frac{abn}{\sqrt{1-a^{2}b^{2}}}<M<\frac{\bar{l}}{4}-\frac{n^{2}}{\bar{l}}$
which ensures the existence of the four Killing horizons (20), three of them
in the positive range of the radial coordinate. These obey
$r_{0}^{+}<r_{-}<M<r_{+}<r_{++}$. The interpretation of each of the regions
proceeds as follows:
$\mathcal{R}^{+}$: This region is limited by the boundaries $r=M$ and
$r=\infty$, whose constant curvatures are given by (35). Two Killing horizons
are present, the event horizon $r_{+}$ and the cosmological horizon $r_{++}$.
The spacetime region lying in between is recognized as the exterior of a
regular stationary black hole, with no pole for the scalar field neither a
curvature singularity. This is a consequence of the presence of the new
boundary at $r=M$ that forbids the access to $\mathcal{R}^{-}$ where
$r_{0}^{+}$ lies. The causal structure of this black hole resembles the one
encountered in the Reissner-Nordström de Sitter black hole but with the
timelike singularity replaced by the new asymptotic region at $r=M$ and with a
different distribution of the Killing horizons, see Fig. 1. Instead of the
appearance of a cosmological, event and Cauchy horizons, only the first two
emerge. For instance, the outer region VI and the inner region I that is found
when the cosmological horizon is crossed are identical to the ones found for
the Reissner-Nordström de Sitter black hole. Nevertheless, once the event
horizon of solution (37) is crossed an observer will encounter the new
asymptotic region at $r=M$ instead of the region lying between the event and
Cauchy horizons that would face in the Reissner-Nordström de Sitter geometry.
In the context of regular black holes this is an interesting feature, Cauchy
horizons are known to be unstable Maeda:2005yd or for suffering from mass
inflation Dokuchaev:2013uda . It is then appealing to investigate how the
event horizon behaves in this solution and to test how robust it is under
perturbations.

Figure 1: Penrose diagram of the de Sitter regular black hole with event and
cosmological horizons contained in solution (37). The new asymptotic region
$r=M$ is specially highlighted.

Now, for $M>\frac{\bar{l}}{4}-\frac{n^{2}}{\bar{l}}$, $r_{+}$ and $r_{++}$ do
not emerge and $\mathcal{R}^{+}$ is devoid of Killing horizons, indeed $r_{-}$
belongs to $\mathcal{R}^{-}$. The metric is everywhere regular and the scalar
field has no pole in contrast with the seed counterpart. Due to the absence of
horizons the metric function $F(r)$ is always negative, the $r$ coordinate is
timelike and so the spacetime is cosmological. This case is particularly
interesting and it offers an explicit example in which the effect of the
transformation is notorious. It transforms the seed solution, which behaves as
a cosmological model outside of $r_{-}$ only, into a fully regular bouncing
cosmology with no horizons whatsoever and with no divergences for the scalar
field. This can be seen by investigating the two dimensional determinant for
$t=r=constant$ surfaces

$g_{2}=\Omega(r)^{2}(r^{2}+n^{2})\left[(r^{2}+n^{2})\sin^{2}\theta-16n^{2}F(r)\sin^{4}\frac{\theta}{2}\right],$
(36)

which provides a measurement of the 2-dimensional volume of the transverse
manifold. Due to the fact that $F(r)$ is everywhere negative this volume never
vanishes, no matter the values of the radial and polar coordinates. This
provides a bouncing cosmology that interpolates between two de Sitter
Universes with cosmological constants given by $\frac{\Lambda}{3a^{2}b^{2}}$
and $\frac{\Lambda}{3}$, respectively (35). The location of the bounce
corresponds to a minimum of $g_{2}$ which can be graphically observed to occur
for an open set of parameter $\Lambda$, $M$ and $n$, such that all the
conditions that defines this spacetime are satisfied, see Fig. 2.

Figure 2: Position of the bounce for different set of parameters $\Lambda$,
$M$ and $n$ and for several values of the polar coordinate $\theta$.
$r_{0}^{+}$ Case I.

Let us now notice that the mass parameter might saturate the previous
inequalities, namely, $M=\frac{\bar{l}}{4}-\frac{n^{2}}{\bar{l}}$. For such
scenario the cosmological and event horizons merge and only one cosmological
horizon $r_{+}=r_{++}:=r_{c}$ takes place. This case is particularly
interesting, it might define either a de Sitter wormhole that connects the two
asymptotic regions $r=M$ and $r=\infty$ when the minimum of $g_{2}$, the
throat, is located behind the cosmological horizon $r_{c}$ or a bouncing
cosmology connecting two de Sitter Universes when the minimum of $g_{2}$, in
this case the bounce, is located outside the cosmological horizon. The
bouncing cosmology is clearer, $F(r)$ is always negative outside the
cosmological horizon an so the bounce is guaranteed. However, for the wormhole
case the situation is less trivial due to the fact that inside the
cosmological horizon $F(r)$ is actually always positive. Nonetheless, in the
space parameter that defines this solution it is possible to find a set in
which the nontrivial minimum volume is preserved in the wormhole case as well.
Both cases are graphically shown to occur, see Fig. 3.

Figure 3: The first two diagrams on the left hand side represent locations of
the bounces for the bouncing cosmology. On the other hand, the next two
diagrams on the right hand side represent positions for the wormhole throat.
In both cases different values of the parameters $\Lambda$, $M$ and $n$ and
several values of the polar coordinate $\theta$ are shown. The location of the
cosmological horizon is explicitly indicated.

The conformal diagram of these solutions is given in Fig. 4.

Figure 4: Conformal diagrams for the bouncing cosmology and the de Sitter
wormhole configurations. The asymptotic region $r=M$, and the bounce and
throat locations are specially highlighted.

The diagram on the left side shows the conformal structure of the bouncing
cosmology. The bounce, represented by the dashed line, connects the time
dependent region I with the stationary region II. Once the bounce is surpassed
an observer will encounter the cosmological horizon $r_{c}$, which after being
crossed gives access to the stationary region II, latter bounded by the
asymptotic region $r=M$. On the other hand, the diagram on the right side
shows the de Sitter wormhole. The throat is located in region II and connects
the two asymptotic boundaries at $r=\infty$ and $r=M$.

Finally, region $\mathcal{R}^{+}$ represents an exact de Sitter black bounce
which smoothly transits from a regular stationary black hole to a bouncing
cosmology that interpolates between two de Sitter Universes. An intermediate
phase represented by two different configurations is observed. A bouncing
cosmology or a de Sitter wormhole, both connecting the asymptotic regions
$r=M$ and $r=\infty$.

$dS\hskip 2.84544ptblack\hskip
2.84544ptbounce\rightarrow\left\\{\begin{aligned}
\frac{abn}{\sqrt{1-a^{2}b^{2}}}<&M<\frac{\bar{l}}{4}-\frac{n^{2}}{\bar{l}}\hskip
5.69046pt\rightarrow\hskip 8.5359ptregular\hskip 2.84544ptstationary\hskip
2.84544ptblack\hskip 2.84544pthole\\\
&M=\frac{\bar{l}}{4}-\frac{n^{2}}{\bar{l}}\hskip 5.69046pt\rightarrow\hskip
8.5359ptbouncing\hskip 2.84544ptcosmology\hskip 2.84544ptor\hskip
2.84544ptwormhole\\\ \frac{\bar{l}}{4}-\frac{n^{2}}{\bar{l}}<&M<\infty\hskip
28.45274pt\rightarrow\hskip 8.5359ptbouncing\hskip
2.84544ptcosmology.\end{aligned}\right.$ (37)

$\mathcal{R}^{-}$: This region lies between $r_{0}^{+}$ and the asymptotic
region $r=M$. Only the cosmological horizon $r_{-}$ might be present when
$M<\frac{\bar{l}}{4}-\frac{n^{2}}{\bar{l}}$. Two cases arise,
$r_{0}^{+}<r_{-}$ and $r_{0}^{+}>r_{-}$, both representing naked
singularities.

Case II: This case is defined by $M>0$ and $ab<0$. The curvature singularity
satisfies $r_{0}^{+}=M+|ab|\sqrt{n^{2}+M^{2}}>r=M$ and in consequence the
conformal factor pole does not belong to the domain of the radial coordinate
$(r_{0}^{+},\infty)$. Hence, only one region takes place which might be
dressed by up to three Killing horizons depending on whether or not the mass
parameter satisfies $0<M<\frac{\bar{l}}{4}-\frac{n^{2}}{\bar{l}}$. The
location of the curvature singularity with respect to the Killing horizons
opens three possible cases: (i) $r_{0}^{+}<r_{+}<r_{++}$, (ii)
$r_{0}^{+}<r_{++}$, and (iii) $r_{++}<r_{0}^{+}$. It is always observed that
$r_{-}<r_{0}^{+}$. (i) represents a black hole with two horizons, the event
horizon $r_{+}$ and the cosmological horizon $r_{++}$. (ii) exhibits only one
horizon, the cosmological horizon $r_{++}$, while (iii) is free of horizons.
The last two cases correspond to spacetimes with naked singularities. Finally,
for $M>\frac{\bar{l}}{4}-\frac{n^{2}}{\bar{l}}$ no Killing horizons appear and
all cases represent naked singularities.

Case III: This case possesses a negative mass satisfying
$|M|<\frac{|ab|n}{\sqrt{1-|ab|^{2}}}$, provided that $0<|ab|<1$ holds. Once
again, the conformal factor diverges outside of the domain of the radial
coordinate $r=-|M|<r_{0}^{+}=-|M|+|ab|\sqrt{n^{2}+M^{2}}$. We rewrite the
Killing horizons by performing the change $M\rightarrow-|M|$ on (20)

$\displaystyle r_{--}$
$\displaystyle=\frac{\bar{l}}{2}\left(-1-\sqrt{1-4\frac{n^{2}}{\bar{l}^{2}}-4\frac{|M|}{\bar{l}}}\right)$
(38) $\displaystyle r_{-}$
$\displaystyle=\frac{\bar{l}}{2}\left(-1+\sqrt{1-4\frac{n^{2}}{\bar{l}^{2}}-4\frac{|M|}{\bar{l}}}\right)$
(39) $\displaystyle r_{+}$
$\displaystyle=\frac{\bar{l}}{2}\left(1-\sqrt{1-4\frac{n^{2}}{\bar{l}^{2}}+4\frac{|M|}{\bar{l}}}\right)$
(40) $\displaystyle r_{++}$
$\displaystyle=\frac{\bar{l}}{2}\left(1+\sqrt{1-4\frac{n^{2}}{\bar{l}^{2}}+4\frac{|M|}{\bar{l}}}\right).$
(41)

It is observed that the Killing horizons $r_{--}$ and $r_{-}$ always lie
outside of the domain $(r_{0}^{+},\infty)$. Thus, the spacetime contains a
curvature singularity at $r_{0}^{+}$ which might be covered by up to two
horizons $r_{+}$ and $r_{++}$ as long as
$|M|>\frac{n^{2}}{\bar{l}}-\frac{\bar{l}}{4}$. Two cases arise: (i)
$r_{0}^{+}<r_{+}<r_{++}$ and (ii) $r_{0}^{+}<r_{++}$. Again, the first case
represents a stationary black hole with two horizons, the event horizon
$r_{+}$ and the cosmological horizon $r_{++}$. On the other hand, the second
case exhibits only the cosmological horizon and the singularity remains naked.
Naked singularities are obtained for
$0<|M|<\frac{n^{2}}{\bar{l}}-\frac{\bar{l}}{4}$.

$r_{0}^{-}$: When the curvature singularity $r_{0}^{-}$ takes place the radial
coordinate assumes negative and positive values in the range
$(r_{0}^{-},\infty)$. Hence, according to (34) three cases arises.

Case I: This case is defined by $0<M<\frac{abn}{\sqrt{1-a^{2}b^{2}}}$ with
$0<ab<1$. The pole of the conformal factor satisfies
$r=M>r_{0}^{-}=M-ab\sqrt{n^{2}+M^{2}}$, and in consequence belongs to the
allowed domain of the radial coordinate. The spacetime is thus divided into
the two regions $\mathcal{R}^{-}\in(r^{-}_{0},M)$ and
$\mathcal{R}^{+}\in(M,\infty)$, which are again causally disconnected. All
Killing horizons exist for $M<\frac{\bar{l}}{4}-\frac{n^{2}}{\bar{l}}$. In
contrast to the first case in (33), here the radial coordinate also takes
negative values and in principle up to four Killing horizons might take part
in the causal structure of this solution. However, it is observed that only
$r_{-}$, $r_{+}$, and $r_{++}$ belong to the domain of the radial coordinate.
For each of the regions we have:
$\mathcal{R}^{+}$: This region is composed by the boundaries $r=M$ and
$r=\infty$. The event $r_{+}$ and the cosmological $r_{++}$ horizons are
englobed, and a regular stationary black hole with a similar horizon structure
to the one of the Reissner-Nordström de Sitter black hole is again obtained.
The causal structure is fairly represented by the conformal diagram of Fig. 1.
Notice that in this case to make $r_{+}$ and $r_{++}$ disappear the mass
parameter must satisfies
$\frac{\bar{l}}{4}-\frac{n^{2}}{\bar{l}}<M<\frac{abn}{\sqrt{1-a^{2}b^{2}}}$. A
new bouncing cosmology similar to the one previously reported is obtained in
this case as well, now in a different subset of the parameter space. See Fig.
5. Again, the two configurations explained in the previous black bounce are
obtained when saturating the value of the mass parameter. We do not provide a
new plot for this case. The following exact de Sitter black bounce is then
identified

$dS\hskip 2.84544ptblack\hskip
2.84544ptbounce\rightarrow\left\\{\begin{aligned}
0<&M<\frac{\bar{l}}{4}-\frac{n^{2}}{\bar{l}}\leq\frac{abn}{\sqrt{1-a^{2}b^{2}}}\hskip
9.38945pt\rightarrow\hskip 8.5359ptregular\hskip 2.84544ptstationary\hskip
2.84544ptblack\hskip 2.84544pthole\\\
&M=\frac{\bar{l}}{4}-\frac{n^{2}}{\bar{l}}\hskip 68.28644pt\rightarrow\hskip
8.5359ptbouncing\hskip 2.84544ptcosmology\hskip 2.84544ptor\hskip
2.84544ptwormhole\\\
\frac{\bar{l}}{4}-\frac{n^{2}}{\bar{l}}<&M<\frac{abn}{\sqrt{1-a^{2}b^{2}}}\hskip
54.06006pt\rightarrow\hskip 8.5359ptbouncing\hskip
2.84544ptcosmology.\end{aligned}\right.$ (42)

Figure 5: Position of the bounce for different sets of parameters $\Lambda$,
$M$ and $n$ and for several values of the polar coordinate $\theta$, now for
Case I contained in $r_{0}^{-}$.

$\mathcal{R}^{-}$: This region lies between $r_{0}^{-}$ and the asymptotic
region $r=M$. Two cases arise: $r_{0}^{-}<r_{-}<0<M$ and
$r_{0}^{-}<0<r_{-}<M$. In both scenarios the curvature singularity is covered
by the cosmological horizon $r_{-}$, then observers outside $r_{-}$ are not
protected from the singularity.

Case II: The mass parameter satisfies $|M|>\frac{|ab|n}{\sqrt{1-|ab|^{2}}}$,
where $0<|ab|<1$. The pole of the conformal factor is situated at $r=-|M|$,
and it satisfies $r=-|M|<r_{0}^{-}=-|M|+|ab|\sqrt{n^{2}+|M|^{2}}$. It does not
belong to the domain $(r_{0}^{-},\infty)$ and no split of the spacetime
occurs. Only the horizons $r_{+}$ and $r_{++}$ take place as soon as the mass
parameter obeys
$\frac{|ab|n}{\sqrt{1-|ab|^{2}}}\leq\frac{n^{2}}{\bar{l}}-\frac{\bar{l}}{4}<|M|$.
Hence, the following cases arise: A stationary black hole with a curvature
singularity covered by the event $r_{+}$ and cosmological $r_{++}$ horizons,
where $r_{0}^{-}<r_{+}<0<r_{++}$ or $r_{0}^{-}<0<r_{+}<r_{++}$ and a curvature
singularity covered by the cosmological horizon $r_{++}$ only, the latter
representing a naked singularity. For
$\frac{|ab|n}{\sqrt{1-|ab|^{2}}}<|M|<\frac{n^{2}}{\bar{l}}-\frac{\bar{l}}{4}$
both horizons disappear and only naked singularities take place.

Case III: The mass parameter is negative $M<0$ and $ab>0$. In this case the
pole of the conformal factor satisfies
$r=-|M|>r_{0}^{-}=-|M|-ab\sqrt{n^{2}+M^{2}}$, belonging to the domain
$(r_{0}^{-},\infty)$ and thus dividing the spacetime into the regions
$\mathcal{R}^{-}\in(r^{-}_{0},-|M|)$ and $\mathcal{R}^{+}\in(-|M|,\infty)$.
Each region proceeds as follows.
$\mathcal{R}^{+}$: This region is limited by $r=-|M|$ and $r=\infty$. For
values of the mass such that $|M|>\frac{n^{2}}{\bar{l}}-\frac{\bar{l}}{4}$ the
two Killing horizons $r_{+}$ and $r_{++}$ emerge, which satisfy either
$-|M|<0<r_{+}<r_{++}$ or $-|M|<r_{+}<0<r_{++}$. The spacetime contained in
between the event and cosmological horizons is recognized once again as the
exterior of a stationary regular black hole with a similar structure to the
one of the Reissner-Nordström de Sitter geometry, see Fig. 2. Moving the mass
parameter such that $0<|M|<\frac{n^{2}}{\bar{l}}-\frac{\bar{l}}{4}$ is
achieved, then both horizons disappear and once again a bouncing cosmology
interpolating between two de Sitter Universes is obtained. See Fig. 6.
Saturating the mass inequality the bouncing cosmology or the de Sitter
wormhole previously explained emerge. Ultimately, the spacetime describes the
de Sitter black bounce

$dS\hskip 2.84544ptblack\hskip
2.84544ptbounce\rightarrow\left\\{\begin{aligned}
\frac{n^{2}}{\bar{l}}-\frac{\bar{l}}{4}<&|M|<\infty\hskip
28.45274pt\rightarrow\hskip 8.5359ptregular\hskip 2.84544ptstationary\hskip
2.84544ptblack\hskip 2.84544pthole\\\
&|M|=\frac{n^{2}}{\bar{l}}-\frac{\bar{l}}{4}\hskip 5.69046pt\rightarrow\hskip
8.5359ptbouncing\hskip 2.84544ptcosmology\hskip 2.84544ptor\hskip
2.84544ptwormhole\\\ 0<&|M|<\frac{n^{2}}{\bar{l}}-\frac{\bar{l}}{4}\hskip
7.11317pt\rightarrow\hskip 8.5359ptbouncing\hskip
2.84544ptcosmology.\end{aligned}\right.$ (43)

Figure 6: Position of the bounce for different set of parameters $\Lambda$,
$M$ and $n$ and for several values of the polar coordinate $\theta$, now for
Case III contained in $r_{0}^{-}$.

$\mathcal{R}^{-}$: This region is limited by $r=r_{0}^{-}$ and $r=-|M|$. For a
mass parameter $|M|<\frac{\bar{l}}{4}-\frac{n^{2}}{\bar{l}}$ the two Killing
horizons $r_{--}$ and $r_{-}$ take place defining two possible cases: (i)
$r_{0}^{-}<r_{--}<r_{-}<-|M|$ and (ii) $r_{0}^{-}<r_{-}<-|M|$. In the first
case the region between the event horizon $r_{--}$ and the cosmological
horizon $r_{-}$ defines the exterior of a stationary black hole, while on the
other hand the second case in which only $r_{-}$ is present defines a naked
singularity.

####  III.1.2 $\Lambda<0$ and $K=-1$

$r_{0}^{+}$: The radial coordinate assumes positive values in the range
$(r_{0}^{+},\infty)$. We consider each of the cases contained in (33). We do
not repeat here all constraints on $M$ and $ab$ regarding the position of this
singularity.

Case I: The pole of the conformal factor satisfies
$r=M>r_{0}^{+}=M-ab\sqrt{n^{2}+M^{2}}$. The spacetime splits into the regions
$\mathcal{R}^{-}\in(r^{+}_{0},M)$ and $\mathcal{R}^{+}\in(M,\infty)$. For mass
parameter values such that $0<M<\frac{\bar{l}}{4}-\frac{n^{2}}{\bar{l}}$ three
horizons in the domain of the radial coordinate take place,
$r_{0}^{+}<r_{-}<M<r_{+}<r_{++}$. These horizons are now free of cosmological
behavior, the cosmological constant is negative.
$\mathcal{R}^{+}$: This region is limited by $r=M$ and $r=\infty$ and it
encloses the inner and event horizons $r_{+}$ and $r_{++}$. Again, the
asymptotic region $r=M$ replaces the singularity and provides a regular metric
with no poles for the scalar field. The spacetime is interpreted as a regular
stationary black hole with a similar causal structure to the one of the
Reissner-Nordström anti–de Sitter black hole. Its conformal diagram is
depicted in Fig. 7.

Figure 7: Penrose diagram of the anti–de Sitter regular black hole with inner
and event horizons contained in solution (46). The new asymptotic region $r=M$
is specially highlighted.

From Fig. 7 it is again observed how the asymptotic region at $r=M$ replaces
the singularity. This diagram does not radically differ from the one
describing the de Sitter case. The main differences are two: First, the outer
region is now stationary, $r_{++}$ plays the role of an event horizon. Second,
the inner horizon $r_{+}$ appears beyond which the asymptotic region is
encountered.

Particularly appealing is the case in which the mass obeys the bound
$M=\frac{\bar{l}}{4}-\frac{n^{2}}{\bar{l}}$, for which both horizons merge in
a single event horizon $r_{+}=r_{++}:=r_{H}$. It is observed that $r_{H}$
takes place at the right-hand side of the asymptotic region $r=M$ and so it
belongs to $\mathcal{R}^{+}$. This occurs for several open sets of the
corresponding parameters $\Lambda$, $M$, and $n$. Therefore, two possible
configurations might emerge. The first possible configuration corresponds to a
regular anti–de Sitter black hole with a single event horizon which lies
inside of an anti–de Sitter wormhole. On the other hand, the second
configuration represents a bouncing cosmology lying at the interior of a
regular anti–de Sitter black hole with a single event horizon. The location of
the bounce for the latter configuration is trivial to obtain, namely, $F(r)$
is everywhere negative inside the event horizon $r_{H}$. Conversely, due to
the positivity of $F(r)$ outside $r_{H}$ the appearance of the wormhole throat
is less trivial, see the discussion below Eq. (44). Nonetheless, in Fig. 8 we
show the existence of both, the bounce and the throat, for different sets of
the parameters $\Lambda$, $M$, and $n$.

Figure 8: The first two diagrams in the left hand side represent locations for
the bounce of the bouncing cosmology inside the anti–de Sitter black hole. On
the other hand, the next two diagrams in the right hand side represent the
position of the wormhole throat inside of which lies the anti–de Sitter black
hole. In both cases different values of the parameters $\Lambda$, $M$, and $n$
and several values of the $\theta$ coordinate are shown. The location of the
event horizon is explicitly indicated.

The conformal diagram of these solutions is depicted in Fig. 9. For both cases
the general picture is very similar and it resembles the causal structure
encountered in the Schwarzschild anti–de Sitter black hole however with the
central singularity replaced by the asymptotic region $r=M$. This converts the
solution into a fully regular black hole with one single event horizon. The
latter is a very exciting feature for a regular black hole, indeed most of
regular black hole solutions include Cauchy horizons as well which are known
for being subjected to different types of instabilities. This solution
represents an excellent candidate for a fully regular black hole with no
perturbative instabilities. The diagram on the left-hand side of Fig. 9 shows
the case in which the regular black hole is immersed in an anti–de Sitter
wormhole. Thus, an observer coming from asymptotic infinity in region I first
observe a wormhole which once is crossed leads to an anti–de Sitter regular
black hole. Region II represents the interior of this black hole where no
singularity takes place. Such an observer stays trapped in region I. The
diagram on the right-hand side of Fig. 9 represents the same regular black
hole but with a cosmological bounce in its interior. Thus, after the observer
has crossed the event horizon the interior of the black hole is given by a
bouncing cosmology that leads to the asymptotic region $r=M$, a region from
which the observer cannot escape.

Figure 9: Conformal diagrams for the bouncing cosmology inside of the anti–de
Sitter regular black hole and for the anti–de Sitter regular black hole inside
of the anti–de Sitter wormhole. The asymptotic region $r=M$, and the bounce
and throat locations are especially highlighted.

Now, as for the de Sitter solutions previously analyzed it is desirable to
understand what occurs for $M>\frac{\bar{l}}{4}-\frac{n^{2}}{\bar{l}}$. In
such a case both horizons $r_{+}$ and $r_{++}$ disappear and the spacetime
represents a regular gravitational soliton. The metric function is everywhere
positive and the spacetime remains stationary. In this case the effect of the
transformations is also decisive. The two dimensional determinant $g_{2}$ is
now given by

$g_{2}=\Omega(r)^{2}(r^{2}+n^{2})\left[(r^{2}+n^{2})\sinh^{2}\theta-16n^{2}F(r)\sinh^{4}\frac{\theta}{2}\right].$
(44)

Contrary to the cases with spherical foliation, the metric function $F(r)$ is
everywhere positive and in principle $g_{2}$ may vanish and even more take
negative values. A negative $g_{2}$ is a consequence of the appearance of
closed timelike curves as can be seen from

$\bar{g}_{\varphi\varphi}=\Omega(r)\left[(r^{2}+n^{2})\sinh^{2}\theta-16n^{2}F(r)\sinh^{4}\frac{\theta}{2}\right].$
(45)

In spite of this, $\bar{g}_{\varphi\varphi}$ will always remain positive if
the cosmological constant is restricted to $|\Lambda|<3/4n^{2}$ (see the seed
case), implying at the same time a nontrivial minimum for the 2-dimensional
volume of the transverse manifold for all values of the coordinates $r$ and
$\theta$. Our spacetime then represents a hyperbolic wormhole connecting two
asymptotically locally anti–de Sitter boundaries at $r=M$ and $r=\infty$ with
asymptotic constant curvatures given by (35). The throat of the wormhole can
be graphically observed for a wide set of parameters, see Fig. 10.

Figure 10: Position of the wormhole throat for different set of parameters
$\Lambda$, $M$ and $n$ and for several values of the coordinate $\theta$.

Finally, region $\mathcal{R}^{+}$ provides an exact anti–de Sitter black
bounce that interpolates between a regular black hole with inner $r_{+}$ and
event $r_{++}$ horizons and a wormhole. The intermediate phase is given by a
regular black hole with a single event horizon $r_{H}$ which might appear in
two different configurations, inside an anti–de Sitter wormhole or with an
interior given by a cosmological bounce. Again, all these transitions are
smoothly connected by the mass parameter only.

$AdS\hskip 2.84544ptblack\hskip
2.84544ptbounce\rightarrow\left\\{\begin{aligned}
0<&M<\frac{\bar{l}}{4}-\frac{n^{2}}{\bar{l}}\hskip 19.91684pt\rightarrow\hskip
8.5359ptregular\hskip 2.84544ptblack\hskip 2.84544pthole\hskip
2.84544ptwith\hskip 2.84544ptinner\hskip 2.84544ptand\hskip
2.84544ptevent\hskip 2.84544pthorizons\\\
&M=\frac{\bar{l}}{4}-\frac{n^{2}}{\bar{l}}\hskip 18.77892pt\rightarrow\hskip
8.5359ptregular\hskip 2.84544ptblack\hskip 2.84544pthole\hskip
2.84544ptwith\hskip 2.84544ptone\hskip 2.84544ptevent\hskip
2.84544pthorizon\\\ &M>\frac{\bar{l}}{4}-\frac{n^{2}}{\bar{l}}\hskip
18.77892pt\rightarrow\hskip 8.5359ptwormhole.\end{aligned}\right.$ (46)

$\mathcal{R}^{-}$: This region is bounded by $r_{0}^{+}$ and the asymptotic
region $r=M$. For mass parameter satisfying
$0<M<\frac{\bar{l}}{4}-\frac{n^{2}}{\bar{l}}$ the singularity is covered by
the event horizon $r_{-}$. The spacetime represents an anti–de Sitter black
hole with a single event horizon which is locally isomorphic to
$\mathcal{H}^{2}$.

Case II: Here, the curvature singularity satisfies
$r_{0}^{+}=M+|ab|\sqrt{n^{2}+M^{2}}>r=M$ and then no division of the spacetime
is performed. The region $(r_{0}^{+},\infty)$ might be then dressed by up to
three horizons as soon as the mass parameter follows
$0<M<\frac{\bar{l}}{4}-\frac{n^{2}}{\bar{l}}$. The distribution of the inner
and event horizons with respect to the singularity opens three cases: (i)
$r_{0}^{+}<r_{+}<r_{++}$, (ii) $r_{0}^{+}<r_{++}$, and (iii)
$r_{++}<r_{0}^{+}$. $r_{-}<r_{0}^{+}$ always holds. (i) represents a black
hole with two horizons, the inner horizon $r_{+}$ and the event horizon
$r_{++}$. (ii) provides a black hole with a single horizon, the event horizon
$r_{++}$, while (iii) is free of horizons. The last case corresponds to a
naked singularity. Finally, for $M>\frac{\bar{l}}{4}-\frac{n^{2}}{\bar{l}}$
the horizons $r_{+}$ and $r_{++}$ are not present and $r_{0}^{+}$ remains
naked.

Case III: In this case we have $r=-|M|<r_{0}^{+}=-|M|+|ab|\sqrt{n^{2}+M^{2}}$,
thus the conformal factor diverges outside of the domain of the radial
coordinate. No new asymptotic region takes place. All Killing horizons are
given by (41). The Killing horizons $r_{-}$ and $r_{--}$ always lie outside
the domain $(r_{0}^{+},\infty)$. For
$|M|>\frac{n^{2}}{\bar{l}}-\frac{\bar{l}}{4}$ two cases arise: (i)
$r_{0}^{+}<r_{+}<r_{++}$ and (ii) $r_{0}^{+}<r_{++}$. The first case
corresponds to a black hole with two horizons, the internal and event horizons
$r_{+}$ and $r_{++}$ while the second case represents a black hole with one
single event horizon $r_{++}$. As soon as
$0<|M|<\frac{n^{2}}{\bar{l}}-\frac{\bar{l}}{4}$ both horizons disappear and
the singularity remains naked.

$r_{0}^{-}$: The radial coordinate assumes negative values in the range
$(r_{0}^{-},\infty)$. We consider each of the cases contained in (34). Again,
we do not repeat all constraints on $M$ and $ab$.

Case I: Here we have $r=M>r_{0}^{-}=M-ab\sqrt{n^{2}+M^{2}}$, as a consequence
the pole of the conformal factor belongs to the domain of the radial
coordinate. Once again our spacetime splits into the causally disconnected
regions $\mathcal{R}^{-}\in(r^{-}_{0},M)$ and $\mathcal{R}^{+}\in(M,\infty)$.
Three Killing horizons might take part of the causal structure, as long as
$M<\frac{\bar{l}}{4}-\frac{n^{2}}{\bar{l}}$. It is always observed that
$r_{--}<r_{0}^{-}$. Thus:
$\mathcal{R}^{+}$: The boundaries $r=M$ and $r=\infty$ encompass two Killing
horizons, the inner and event horizons $r_{+}$ and $r_{++}$, giving the
spacetime the interpretation of a stationary regular black hole with a similar
causal structure to the Reissner-Nordström anti–de Sitter black hole. Its
causal structure can be seen from Fig. 7. For
$M=\frac{\bar{l}}{4}-\frac{n^{2}}{\bar{l}}$ the aforementioned horizons merge
and a black hole with one single horizon $r_{H}$ appears. Again, this horizon
satisfies $r_{H}>r=M$. The same configurations previously discussed in this
context are again present, see Fig. 9. Finally, for
$\frac{\bar{l}}{4}-\frac{n^{2}}{\bar{l}}<M<\frac{abn}{\sqrt{1-a^{2}b^{2}}}$
both horizons disappear and the spacetime represents a wormhole connecting the
two locally asymptotically anti–de Sitter boundaries with constant curvatures
$\frac{\Lambda}{3a^{2}b^{2}}$ and $\frac{\Lambda}{3}$, respectively. The
wormhole throat is graphically observable in Fig. 11. This region represents
an anti–de Sitter black bounce smoothly connecting, through the mass
parameter, a regular black hole with two horizons, a single event horizon
black hole and a hyperbolic wormhole.

$AdS\hskip 2.84544ptblack\hskip
2.84544ptbounce\rightarrow\left\\{\begin{aligned}
0<&M<\frac{\bar{l}}{4}-\frac{n^{2}}{\bar{l}}\leq\frac{abn}{\sqrt{1-a^{2}b^{2}}}\hskip
9.38945pt\rightarrow\hskip 8.5359ptregular\hskip 2.84544ptblack\hskip
2.84544pthole\hskip 2.84544ptwith\hskip 2.84544ptinner\hskip
2.84544ptand\hskip 2.84544ptevent\hskip 2.84544pthorizons\\\
&M=\frac{\bar{l}}{4}-\frac{n^{2}}{\bar{l}}\hskip 68.28644pt\rightarrow\hskip
8.5359ptregular\hskip 2.84544ptblack\hskip 2.84544pthole\hskip
2.84544ptwith\hskip 2.84544ptone\hskip 2.84544ptevent\hskip
2.84544pthorizon\\\
\frac{\bar{l}}{4}-\frac{n^{2}}{\bar{l}}<&M<\frac{abn}{\sqrt{1-a^{2}b^{2}}}\hskip
54.06006pt\rightarrow\hskip 8.5359ptwormhole.\end{aligned}\right.$ (47)

Figure 11: Position of the wormhole throat for different set of parameters
$\Lambda$, $M$ and $n$ and for several values of the coordinate $\theta$.

$\mathcal{R}^{-}$: For
$0<M<\frac{\bar{l}}{4}-\frac{n^{2}}{\bar{l}}\leq\frac{abn}{\sqrt{1-a^{2}b^{2}}}$
two cases may arise, $r_{0}^{-}<r_{-}<0<M$ and $r_{0}^{-}<0<r_{-}<M$. In both
scenarios the curvature singularity is covered by the event horizon $r_{-}$,
then a black hole with one event horizon locally isomorphic to
$\mathcal{H}^{2}$ is identified.

Case II: The pole of the conformal factor does not belong to the radial
coordinate domain $(r_{0}^{-},\infty)$,
$r=-|M|<r_{0}^{-}=-|M|+|ab|\sqrt{n^{2}+|M|^{2}}$. $r_{0}^{-}$ is always
greater than $r_{-}$ and $r_{--}$. Then, for values of the mass parameter
$\frac{|ab|n}{\sqrt{1-|ab|^{2}}}\leq\frac{n^{2}}{\bar{l}}-\frac{\bar{l}}{4}<|M|$
the two Killing horizons $r_{+}$ and $r_{++}$ emerge defining three cases: (i)
$r_{0}^{-}<r_{+}<0<r_{++}$, (ii) $r_{0}^{-}<0<r_{+}<r_{++}$, and (iii)
$r_{0}^{-}<r_{++}$. The first two cases define a stationary black hole with
inner and event horizons while the third case represents a single event
horizon black hole.

Case III: Here we have $r=-|M|>r_{0}^{-}=-|M|-ab\sqrt{n^{2}+M^{2}}$ and thus
the spacetime divides into the regions $\mathcal{R}^{-}\in(r^{-}_{0},-|M|)$
and $\mathcal{R}^{+}\in(-|M|,\infty)$. Therefore:
$\mathcal{R}^{+}$: This region is bounded by $r=-|M|$ and $r=\infty$. For
$|M|>\frac{n^{2}}{\bar{l}}-\frac{\bar{l}}{4}$ the following two cases display:
(i) $-|M|<0<r_{+}<r_{++}$ and (ii) $-|M|<r_{+}<0<r_{++}$, both representing
stationary regular black holes with a similar causal structure to the one of
the Reissner-Nordström anti–de Sitter black hole, see Fig. 7. By saturating
the mass bound both horizons merge and the solution represents a regular black
hole with one event horizon, Fig. 9. Finally, if
$0<|M|<\frac{n^{2}}{\bar{l}}-\frac{\bar{l}}{4}$ the spacetime is devoid of
Killing horizons and it transforms into a wormhole connecting two anti–de
Sitter boundaries. See Fig. 12. The following anti–de Sitter black bounce is
thus observed

$AdS\hskip 2.84544ptblack\hskip
2.84544ptbounce\rightarrow\left\\{\begin{aligned}
\frac{n^{2}}{\bar{l}}-\frac{\bar{l}}{4}<&|M|<\infty\hskip
41.25641pt\rightarrow\hskip 8.5359ptregular\hskip 2.84544ptblack\hskip
2.84544pthole\hskip 2.84544ptwith\hskip 2.84544ptinner\hskip
2.84544ptand\hskip 2.84544ptevent\hskip 2.84544pthorizons\\\
&|M|=\frac{n^{2}}{\bar{l}}-\frac{\bar{l}}{4}\hskip 18.77892pt\rightarrow\hskip
8.5359ptregular\hskip 2.84544ptblack\hskip 2.84544pthole\hskip
2.84544ptwith\hskip 2.84544ptone\hskip 2.84544ptevent\hskip
2.84544pthorizon\\\ 0<&|M|<\frac{n^{2}}{\bar{l}}-\frac{\bar{l}}{4}\hskip
18.77892pt\rightarrow\hskip 8.5359ptwormhole.\end{aligned}\right.$ (48)

Figure 12: Position of the wormhole throats for different set of parameters
$\Lambda$, $M$, and $n$ and for several values of the coordinate $\theta$.

$\mathcal{R}^{-}$: If the mass parameter satisfies
$|M|<\frac{\bar{l}}{4}-\frac{n^{2}}{\bar{l}}$ then both horizons $r_{-}$ and
$r_{--}$ appear. The following two cases are displayed: (i)
$r_{0}^{-}<r_{--}<r_{-}<-|M|$ and (ii) $r_{0}^{-}<r_{-}<-|M|$. Both cases
represent black holes, with inner and event or only event horizons,
respectively.

####  III.1.3 $\Lambda>0$ and $K=-1$

The combination $\Lambda>0$ and $K=-1$ always provides a complex $\bar{l}$.
Therefore, the spacetime is devoid of Killing horizons and the curvature
singularity as well as the pole of the conformal factor are, in principle,
naked. This situation will occur for all the cases contained in (33) and (34)
in which the pole of the conformal factor does not belong to the corresponding
domains of the radial coordinate. On the other hand, for all cases in which
the spacetime is divided into the regions $\mathcal{R}^{-}$ and
$\mathcal{R}^{+}$ the situation differs. As a matter of fact, the curvature
singularity and the pole of the scalar field will always belong to the region
$\mathcal{R}^{-}$ and in consequence region $\mathcal{R}^{+}$ will be free of
pathologies. There is only one possible configuration, a bouncing cosmology
connecting two de Sitter Universes with cosmological constants
$\frac{\Lambda}{3a^{2}b^{2}}$ and $\frac{\Lambda}{3}$, no matter the value of
the mass parameter. This can be directly observed from the determinant of the
two dimensional $t=r=constant$ sections (44). The metric function is
everywhere negative and in consequence this determinant never vanishes. This
will be observed in three of the cases previously studied, Case I when
$r_{0}^{+}$ takes places and Cases I and III when $r_{0}^{-}$ appears. Here we
provide a graphical proof of the bounce for the case in which $r_{0}^{+}$
takes place. See Fig. 13. Due to the hyperbolic nature of the transverse
section this geometry is free of Misner strings, and moreover does not possess
any kind of closed timelike curves as long as $\Lambda>3/4n^{2}$, see the seed
analysis.

Figure 13: Position of the bounce for different set of parameters $\Lambda$,
$M$ and $n$ and for several values of the coordinate $\theta$.

####  III.1.4 $\Lambda<0$ and $K=1$

This combination possesses complex $\bar{l}$ as well as in the previous case.
No Killing horizons take place and the curvature singularity and scalar field
pole will be naked in all those cases in which the conformal factor pole does
not belong to the domain of the radial coordinate. For those cases in which
the spacetime is divided into the regions $\mathcal{R}^{-}$ and
$\mathcal{R}^{+}$ the spacetime $\mathcal{R}^{+}$ will represent an anti–de
Sitter wormhole connecting to asymptotic boundaries with constant curvatures
$\frac{\Lambda}{3a^{2}b^{2}}$ and $\frac{\Lambda}{3}$. For a non geodesic
observer the Misner string is unavoidable as well as the occurrence of closed
timelike curves. As a consequence of the existence of closed timelike curves
the throat of the wormhole can be graphically observed only for a narrow set
of parameters. See Fig. 14 for Case I with $r_{0}^{+}$.

Figure 14: Position of the wormhole throat for different set of parameters and
for different values of the polar coordinate. Notice that for this case it is
not possible to access the wormhole throat for $\theta=\pi$.

####  III.1.5 $K=0$ cases

For the cases in which the base manifold is flat the absence of Killing
horizons always takes place. Then, the metric function is either everywhere
negative or everywhere positive depending if the cosmological constant is
positive or negative. Again, generically the curvature singularity and the
scalar field pole might be naked unless the conformal factor includes a new
asymptotic region that provides a spacetime $\mathcal{R}^{+}$ for which no
singularities appear. In such cases the solution is interpreted as a bouncing
cosmology connecting to de Sitter Universes or as a wormhole connecting two
locally anti–de Sitter boundaries. The Misner string is not present, for
planar base manifold geometries these are pulled out to infinity. The presence
of closed timelike curves is a generic feature for nongeodesic observers.
These are not going to affect the cosmological solution, nevertheless
severally restricts the existence of the throat in the wormhole case,
narrowing the set of parameters in which the latter might take place.

##  IV Conclusions and further developments

In the present paper we have constructed and analyzed a new family of Taub-NUT
spacetimes representing regular black holes, wormholes, and bouncing
cosmologies in Einstein-$\Lambda$ theory in which a self-interacting
conformally coupled scalar field is considered. The self-interaction deviates
from the usual conformal potential by including all power-counting super-
renormalizable contributions. The construction is based on the use of a
specific set of transformations Ayon-Beato:2015ada , that once applied to a
seed solution that solves the field equations of a conformally coupled scalar-
tensor theory with a conformal potential (4), provides a new set of geometries
that drastically enlarge the causal structure of the spacetime. These
solutions solves the new set of field equations (6) where the scalar field
action enjoys conformal invariance up to the presence of a nonconformal
potential that exhibits all power-counting superrenormalizable contributions.
Additionally, the cosmological constant is also modified. Comparison with the
seed spacetime is direct. Table I summarizes our most relevant findings.

The first four solutions contained in Table I possess a spherical foliation
and the seed cosmological constant is positive. In this scenario the seed
solution represents a Taub-NUT black hole in de Sitter space with three
Killing horizons. Its black hole interpretation is subjected to the arguments
of Clement:2015cxa ; Clement:2015aka , this due to the presence of the Misner
string and closed timelike curves which might generically appear. Then, we
adopted this interpretation for geodesic observers only. Our new solutions
drastically differ from this causal structure. The presence of the conformal
factor pole divides the spacetime into two regions from which the most
relevant is $\mathcal{R}^{+}$. This region is interpreted as a regular black
hole with two horizons only, the event horizon $r_{+}$ and the cosmological
horizon $r_{++}$, which by smoothly moving the mass parameter transits to a
bouncing cosmology connecting two de Sitter Universes. It is shown the
existence of an intermediate phase in which a de Sitter wormhole or a de
Sitter bouncing cosmology might appear, this under the presence of a
cosmological horizon. Less attractive is region $\mathcal{R}^{-}$, which
mostly represents naked singularities with the exception of Case III in the
presence of $r_{0}^{-}$. This case represents a de Sitter black hole with
event $r_{+}$ and cosmological $r_{++}$ horizons. It is important to stress
that for all the cosmological configurations the absence of closed timelike
curves is ensured beyond geodesic observers. Nonetheless, the Misner string is
still visible for them. All these solutions are completely regular in terms of
their curvature invariants and the scalar field profile is everywhere
nondivergent.

The next six solutions contained in Table I possess a hyperbolic foliation and
the seed cosmological constant is negative. Among all the solutions contained
in the seed metric this is the most relevant. For hyperbolic foliation this
spacetime does not contain a Misner string, and moreover it is shown that by
properly restricting the cosmological constant in terms of the NUT parameter
it is possible to remove any trace of closed timelike curves. This geometry
represents an anti–de Sitter regular black hole with three horizons. Our
solutions enhance this geometry in the following manner: again, the presence
of a pole in the conformal factor implies the division of the spacetime into
the regions $\mathcal{R}^{-}$ and $\mathcal{R}^{+}$. Region $\mathcal{R}^{+}$
represents a regular black hole with inner and event horizons $r_{+}$ and
$r_{++}$, which by smoothly moving the mass parameter transits to a wormhole
configuration that connects two anti–de Sitter asymptotic boundaries with
different constant curvatures. An intermediate phase given by a regular black
hole with a single event horizon $r_{H}$ is observed. This phase might appear
in two different setups, either as a regular black hole inside an anti–de
Sitter wormhole or as a regular black hole of which its interior is given by a
cosmological bounce. In both scenarios this phase shows an interesting
characteristic: it possesses only one horizon, the event horizon, and in
consequence is free of Cauchy horizons which are known for suffering from
instabilities. On the other hand, region $\mathcal{R}^{-}$ provides anti–de
Sitter black holes with inner and event horizons. It is important to stress
that all these solutions are free of Misner strings and of any closed timelike
curves for any observer, geodesic or nongeodesic.

Other solutions might be highlighted as well, nonetheless none of them
representing black bounce transitions. These are the case with positive
cosmological constant and hyperbolic base manifold that represents a bouncing
cosmology and the case with negative cosmological constant and spherical base
manifold that represents a wormhole. The latter case being valid for a
restricted set of the parameter space. Additionally, several cases represent
black holes with different horizons distributions when no pole of the
conformal factor takes place. These latter are not included in Table I.

An important aspect to be discussed in all these solutions is whether or not
they satisfy reasonable energy conditions. Due to the complexity of our
solutions explicit expressions for the energy density or for the pressures are
rather involved, nevertheless some general conclusions can be drawn by casting
the energy momentum tensor of our matter fields in the form of a perfect
fluid, in this case an anisotropic fluid which in an orthonormal frame is
given by $T^{ab}:=(\rho,p_{1},p_{2},p_{2})$. It is realized to be useful to
consider the case in which the electric charge vanishes. From the point of
view of the solutions this does not produce any substantial change, besides a
fine tune between the cosmological constant $\Lambda$ and the conformal
coupling $\alpha$. It is observed that the null energy condition given by
$\rho+p_{1}\geq 0$ and $\rho+p_{2}\geq 0$ is always satisfy for bouncing
cosmologies. On the other hand, wormhole configurations requires, as usual,
the presence of exotic matter, which in our cases is represented by our scalar
fields not obeying the aforementioned energy condition. For black holes energy
conditions are not satisfied everywhere due to the change of sign of the
metric function $F(r)$. However, several cases respect the null energy
condition in the domain of outer communications.

Many appealing directions might be further explored. Remaining in the context
of Taub-NUT spacetimes it would be interesting to study the Euclidean version
of our solutions. These gravitational instantons will certainly exhibit
interesting new geometric and topological features healing the pathologies
associated with the Misner string and the closed timelike curves eu-case .
Additionally, either in their Lorentzian or Euclidean versions, our solutions
offer an interesting arena to study black hole thermodynamic Mann:1999pc ;
Garfinkle:2000ms ; Ciambelli:2020qny ; Hennigar:2019ive ; Durka:2019ajz ;
Frodden:2021ces . It would be worth studying the existence of phase
transitions in the presence of scalar fields not only in standard black hole
thermodynamic but also in the context of black hole chemistry Simovic:2020dke
; Johnson:2014pwa and to understand how the presence or absence of the Misner
string affects the usual charges. On the other hand it is clear that the set
of transformations defined in Ayon-Beato:2015ada do not depend on the
particular symmetries the metric or the scalar field enjoy. Namely, for any
solution of a conformally coupled theory with a conformal potential a new
conformally related family of the type found here can be constructed. A
concrete example is to take the C-metric solution with a conformally coupled
scalar found in Charmousis:2009cm and construct a new family of accelerating
geometries. The transformations affect in a highly nontrivial manner the
geometric structure of this spacetime providing interesting new phenomena to
be explored dolphiprogress . Even more, a full new Plebański-Demiański family
of solutions can be constructed along the lines of Anabalon:2009qt to explore
how the transformation modifies the corresponding limits that provide specific
spacetimes with a desirable specific symmetry. It is known that the Plebański-
Demiański solutions found in Anabalon:2009qt does not provide a healthy limit
when turning off the acceleration while keeping rotation, this in the presence
of the cosmological constant. An interesting path to follow would be to
investigate whether or not this limit might be cured by the effect of the
transformations. This can be pushed further by including higher order
corrections to the original action that do not spoil the initial conformal
symmetry of the seed theory Cisterna:2021xxq . The same conclusions are valid
for solutions with external magnetic fields Astorino:2013xc ; Astorino:2013sfa
, for pp/AdS-waves metrics Ayon-Beato:2005gdo or conformally coupled solutions
in higher or lower dimensions Oliva:2011np ; Giribet:2014bva ; Galante:2015voa
.

Finally, specifically in the context of the black bounce solutions described
here it would be interesting to explore all kinds of applications. The most
obvious, again, black hole thermodynamic since now they represent solutions
with a sensible action principle with a clear field content888Recently,
following a bottom up approach these type of models have been embedded in
higher order scalar-tensor theories Chatzifotis:2021hpg . In there, the
parameter that controls the bounce emerges in the action principle when
integrating out the theory that supports the desired solution. Similar
strategy has been also applied in the context of nonlinear electrodynamics
Canate:2022gpy .. On the other hand, it is possible to study several
astrophysically relevant aspects of these black holes, as for example black
hole shadows Cunha:2017eoe ; Lima:2021las , gravitational lensing
Virbhadra:1999nm ; Tsukamoto:2020bjm , black hole mimickers Johnson-
Mcdaniel:2018cdu ; Mazza:2021rgq , quasinormal modes and echoes
Churilova:2019cyt ; Yang:2021cvh ; Wu:2022eiv ; Cano:2021qzp , to mention a
few examples. We expect to report along these lines in our upcoming
contributions.

##  V Acknowledgments

We appreciate interesting discussions with Marco Astorino, Cristóbal Corral,
Gaston Giribet, Julio Oliva, Julio Méndez-Zavaleta and Marcello Ortaggio. A.C
would like to acknowledge the Institute of Mathematics of the Czech Academy of
Science for its kind hospitality during the development of this project. A.C
work is funded by FONDECYT Regular grants No. 1210500, No. 1181047 and Beca
Chile de Postdoctorado Grant No. 74200012. J.B is supported by the “Programme
to support prospective human resources – post Ph.D. candidates” of the Czech
Academy of Sciences, project No. L100192101. N.M is funded by Beca ANID de
Doctorado Nacional 2021 Grant No. 21212393. A.V is supported in part by by
MIUR-PRIN contract No. 2017CC72MK003.

$(K,\Lambda)$ | Singularity | Region/Mass |  Interpretation
---|---|---|---
$\begin{matrix}K=1\\\ 0<\Lambda<\dfrac{3}{8n^{2}}\end{matrix}$ |  $r_{0}^{+}$ Case I | $\mathcal{R}^{+}:\left\\{\begin{matrix}\dfrac{abn}{\sqrt{1-a^{2}b^{2}}}<M<\dfrac{\bar{l}}{4}-\dfrac{n^{2}}{\bar{l}}\\\ M=\dfrac{\bar{l}}{4}-\dfrac{n^{2}}{\bar{l}}\\\ \dfrac{\bar{l}}{4}-\dfrac{n^{2}}{\bar{l}}<M<\infty\end{matrix}\right.$ |  de Sitter black bounce: regular black hole with event $r_{+}$ and cosmological $r_{++}$ horizons, de Sitter wormhole or bouncing cosmology with a single cosmological horizon $r_{c}$ and free horizons bouncing cosmology.
$\begin{matrix}K=1\\\ 0<\Lambda<\dfrac{3}{8n^{2}}\end{matrix}$ |  $r_{0}^{-}$ Case I | $\mathcal{R}^{+}:\left\\{\begin{matrix}0<M<\dfrac{\bar{l}}{4}-\dfrac{n^{2}}{\bar{l}}\leq\dfrac{abn}{\sqrt{1-a^{2}b^{2}}}\\\ M=\dfrac{\bar{l}}{4}-\dfrac{n^{2}}{\bar{l}}\\\ \dfrac{\bar{l}}{4}-\dfrac{n^{2}}{\bar{l}}<M<\dfrac{abn}{\sqrt{1-a^{2}b^{2}}}\end{matrix}\right.$ |  de Sitter black bounce: regular black hole with event $r_{+}$ and cosmological $r_{++}$ horizons, de Sitter wormhole or bouncing cosmology with a single cosmological horizon $r_{c}$ and free horizons bouncing cosmology.
$\begin{matrix}K=1\\\ \dfrac{3}{8n^{2}}<\Lambda<\dfrac{3}{4n^{2}}\end{matrix}$ |  $r_{0}^{-}$ Case III | $\mathcal{R}^{+}:\left\\{\begin{matrix}\dfrac{n^{2}}{\bar{l}}-\dfrac{\bar{l}}{4}<\left|M\right|<\infty\\\ \left|M\right|=\dfrac{n^{2}}{\bar{l}}-\dfrac{\bar{l}}{4}\\\ 0<\left|M\right|<\dfrac{n^{2}}{\bar{l}}-\dfrac{\bar{l}}{4}\end{matrix}\right.$ |  de Sitter black bounce : regular black hole with event $r_{+}$ and cosmological $r_{++}$ horizons, de Sitter wormhole or bouncing cosmology with a single cosmological horizon $r_{c}$ and free horizons bouncing cosmology.
$\begin{matrix}K=1\\\ 0<\left|\Lambda\right|<\dfrac{3}{8n^{2}}\end{matrix}$ |  $r_{0}^{-}$ Case III | $\mathcal{R}^{-}:\left\\{\left|M\right|<\dfrac{\bar{l}}{4}-\dfrac{n^{2}}{\bar{l}}\right.$ |  de Sitter black hole with two horizons, event $r_{+}$ and cosmological $r_{++}.$
$\begin{matrix}K=-1\\\ 0<\left|\Lambda\right|<\dfrac{3}{8n^{2}}\end{matrix}$ |  $r_{0}^{+}$ Case I | $\mathcal{R}^{+}:\left\\{\begin{matrix}0<M<\dfrac{\bar{l}}{4}-\dfrac{n^{2}}{\bar{l}}\\\ M=\dfrac{\bar{l}}{4}-\dfrac{n^{2}}{\bar{l}}\\\ M>\dfrac{\bar{l}}{4}-\dfrac{n^{2}}{\bar{l}}\end{matrix}\right.$ |  anti–de Sitter black bounce: regular black hole with inner $r_{+}$ and event $r_{++}$ horizons, regular black hole with one event horizon $r_{H}$ either inside of a wormhole or with an interior cosmological bounce, and anti–de Sitter wormhole.
$\begin{matrix}K=-1\\\ 0<\left|\Lambda\right|<\dfrac{3}{8n^{2}}\end{matrix}$ |  $r_{0}^{-}$ Case I | $\mathcal{R}^{+}:\left\\{\begin{matrix}0<M<\dfrac{\bar{l}}{4}-\dfrac{n^{2}}{\bar{l}}\leq\dfrac{abn}{\sqrt{1-a^{2}b^{2}}}\\\ M=\dfrac{\bar{l}}{4}-\dfrac{n^{2}}{\bar{l}}\\\ \dfrac{\bar{l}}{4}-\dfrac{n^{2}}{\bar{l}}<M<\dfrac{abn}{\sqrt{1-a^{2}b^{2}}}\end{matrix}\right.$ |  anti–de Sitter black bounce: regular black hole with inner $r_{+}$ and event $r_{++}$ horizons, regular black hole with one event horizon $r_{H}$ either inside of a wormhole or with an interior cosmological bounce, and anti–de Sitter wormhole.
$\begin{matrix}K=-1\\\ \dfrac{3}{8n^{2}}<\left|\Lambda\right|<\dfrac{3}{4n^{2}}\end{matrix}$ |  $r_{0}^{-}$ Case III | $\mathcal{R}^{+}:\left\\{\begin{matrix}\dfrac{n^{2}}{\bar{l}}-\dfrac{\bar{l}}{4}<\left|M\right|<\infty\\\ \left|M\right|=\dfrac{n^{2}}{\bar{l}}-\dfrac{\bar{l}}{4}\\\ 0<\left|M\right|<\dfrac{n^{2}}{\bar{l}}-\dfrac{\bar{l}}{4}\end{matrix}\right.$ |  anti–de Sitter black bounce: regular black hole with inner $r_{+}$ and event $r_{++}$ horizons, regular black hole with one event horizon $r_{H}$ either inside of a wormhole or with an interior cosmological bounce, and anti–de Sitter wormhole.
$\begin{matrix}K=-1\\\ 0<\left|\Lambda\right|<\dfrac{3}{8n^{2}}\end{matrix}$ |  $r_{0}^{+}$ Case I | $\mathcal{R}^{-}:\left\\{0<M<\dfrac{\bar{l}}{4}-\dfrac{n^{2}}{\bar{l}}\right.$ |  anti–de Sitter black hole with a single event horizon locally isomorphic to $\mathcal{H}^{2}$: $r^{-}0<r-<0<M$.
$\begin{matrix}K=-1\\\ 0<\left|\Lambda\right|<\dfrac{3}{8n^{2}}\end{matrix}$ |  $r_{0}^{-}$ Case I | $\mathcal{R}^{-}:\left\\{0<M<\dfrac{\bar{l}}{4}-\dfrac{n^{2}}{\bar{l}}\leq\dfrac{abn}{\sqrt{1-a^{2}b^{2}}}\right.$ |  anti–de Sitter black hole with a single event horizon locally isomorphic to $\mathcal{H}^{2}$. Two cases: $r^{-}0<r-<0<M$ and $r^{-}0<0<r-<M$.
$\begin{matrix}K=-1\\\ 0<\left|\Lambda\right|<\dfrac{3}{8n^{2}}\end{matrix}$ |  $r_{0}^{-}$ Case III | $\mathcal{R}^{-}:\left\\{\left|M\right|<\dfrac{\bar{l}}{4}-\dfrac{n^{2}}{\bar{l}}\right.$ |  anti–de Sitter black hole with one or two horizons locally isomorphics to $\mathcal{H}^{2}$. Two cases: $r^{-}0<r{--}<r_{-}<-\left|M\right|$ and $r^{-}0<r-<-\left|M\right|$.
$\begin{matrix}K=-1,\Lambda>\dfrac{3}{4n^{2}}\end{matrix}$ |  $r_{0}^{+}$ Case I | $\mathcal{R}^{+}:\left\\{M>0\right.$ |  de Sitter bouncing cosmology
$\begin{matrix}K=1,\Lambda<0\end{matrix}$ |  $r_{0}^{+}$ Case I | $\mathcal{R}^{-}:\left\\{M>0\right.$ |  anti–de Sitter wormhole.
Table 1: Summary of the most significant results for different values of $K$
and $\Lambda$.

##  VI Appendix: $\Lambda=0$ case

When the seed cosmological constant vanishes consistence of the seed solution
(11) requires the absence of the seed quartic self-interaction, $\alpha=0$.
The seed scalar field integrates in a slightly different manner. In this case
no potential $V(\bar{\phi})$ appears after the transformation neither an
effective cosmological constant $\lambda$. The lapse function takes a very
simple form possessing one single root at $r=M$, same point at which the
conformal factor has a pole. The solution reads

$\displaystyle d\bar{s}^{2}$
$\displaystyle=\frac{(a\sqrt{n^{2}+M^{2}-\frac{\kappa\bar{Q}^{2}}{2(1-a^{2})}}+r-M)^{2}}{(r-M)^{2}}\left[-\frac{K(r-M)^{2}}{r^{2}+n^{2}}(dt+\mathcal{B})^{2}+\frac{dr^{2}}{\frac{K(r-M)^{2}}{r^{2}+n^{2}}}+(r^{2}+n^{2})d\Sigma_{K}^{2}\right],\hskip
2.84544pt\bar{A}=\frac{\bar{Q}r}{r^{2}+n^{2}}(dt+\mathcal{B})$ (49)
$\displaystyle\bar{\phi}(r)$
$\displaystyle=\left(\frac{6}{\kappa}\right)^{1/2}\frac{\sqrt{n^{2}+M^{2}-\frac{\kappa\bar{Q}^{2}}{2(1-a^{2})}}+a(r-M)}{a\sqrt{n^{2}+M^{2}-\frac{\kappa\bar{Q}^{2}}{2(1-a^{2})}}+r-M}.$

As explained before the zero of the conformal factor induces a curvature
singularity, now located at

$r_{0}=M-a\sqrt{n^{2}+M^{2}-\frac{\kappa\bar{Q}^{2}}{2(1-a^{2})}},$ (50)

which, depending on the signs of $M$ and $a$ will be either located in the
positive or negative range of the radial coordinate. The electric charge must
accomplish $\bar{Q}^{2}<\frac{2(1-a^{2})(n^{2}+M^{2})}{\kappa}$ such that the
conformal factor remains real. In comparison with the previous cases in which
the cosmological constant was nonvanishing here the presence of the conformal
factor pole is fundamental, otherwise the spacetime will always exhibit a
naked singularity. The region $\mathcal{R}^{+}$ will represent either a
wormhole or a cosmological bounce depending on the horizon geometry. No black
hole horizons will exists. This solution represents the charged Taub-NUT
extension of Barcelo’s wormhole Barcelo:2000zf for $K=1$. Region
$\mathcal{R}^{-}$ contains a naked singularity.

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