# Ehlers transformations as a tool for constructing accelerating NUT black
holes

José Barrientos<EMAIL_ADDRESS>Institute of Mathematics of the Czech
Academy of Sciences, Žitná 25, 11567 Praha 1, Czech Republic  Sede Esmeralda,
Universidad de Tarapacá, Avenida Luis Emilio Recabarren 2477, Iquique, Chile
Adolfo Cisterna<EMAIL_ADDRESS>Sede Esmeralda, Universidad de
Tarapacá, Avenida Luis Emilio Recabarren 2477, Iquique, Chile

###### Abstract

This paper investigates the integrability properties of Einstein’s theory of
gravity in the context of accelerating Newman-Unti-Tamburino (NUT) spacetimes
by utilizing Ernst’s description of stationary and axially symmetric
electrovacuum solutions. We employ Ehlers transformations, Lie point
symmetries of the Einstein field equations, to efficiently endorse
accelerating metrics with a nontrivial NUT charge. Under this context, we
begin by rederiving the known C-metric NUT spacetime described by Chng, Mann,
and Stelea in a straightforward manner, and in the new form of the solution
introduced by Podolský and Vrátný. Next, we construct for the first time an
accelerating NUT black hole dressed with a conformally coupled scalar field.
These solutions belong to the general class of type I spacetimes and,
therefore, cannot be obtained from any limit of the Plebanśki-Demiański family
whatsoever and their integration needs to be carried out independently.
Including Maxwell fields is certainly permitted, however, the use of Ehlers
transformations is subtle and requires further modifications. Ehlers
transformations not only partially rotate the mass parameter such that its
magnetic component appears, but also rotate the corresponding gauge fields.
Notwithstanding, the alignment of the electromagnetic potentials can be
successfully performed via a duality transformation, hence providing a novel
Reissner-Nordström-C-metric NUT black hole that correctly reproduces the
Reissner-Nordström-C-metric and Reissner-Nordström-NUT configurations in the
corresponding limiting cases. We describe the main geometric features of these
solutions and discuss possible embeddings of our geometries in external
electromagnetic and rotating backgrounds.

##  I Introduction

In the understanding of the gravitational interaction exact solutions of
Einstein field equations play a fundamental role. In many cases these
solutions allow one to study, by means of analytic computations, several
phenomena of classical and semiclassical gravity with significant
astrophysical implications. As a matter of fact, the most iconic exact
spacetimes are given by black holes, today widely accepted to be primordial
for the formation of galaxies Cattaneo:2009ub ; Fabian:2012xr  and the
production of gravitational waves as a result of binary black hole mergers
LIGOScientific:2016aoc . In addition, their mechanic and thermodynamic
features are essential in the semiclassical description of gravitational
systems, providing a first glance on the interplay of gravity and quantum
field theory. It is fair to state that the Schwarzschild Schwarzschild:1916uq
and Kerr spacetimes Kerr:1963ud , describing the gravitational field of a
static and a rotating source respectively, are the most famous black hole
spacetimes in general relativity (GR). Along with their charged extensions,
the Reissner-Nordström Reissner ; Nordstrom  and Kerr-Newman Newman:1965my
spacetimes, and the so-called C-metric Kinnersley:1969zz  and Taub-NUT
geometries Taub:1950ez ; Newman:1963yy , known to represent accelerating black
holes and twisted spacetimes with a gravitomagnetic interpretation
respectively, they make part of the general class of spacetimes known as the
Plebanśki-Demiański family Plebanski:1976gy . The Plebanśki-Demiański
spacetime is the most general type D solution of Einstein-Maxwell equations
and it represents a pair of charged rotating black holes, endowed with Newman-
Unti-Tamburino (NUT) charge, that accelerate from each other in opposite
directions. Including the presence of a cosmological constant, this family
depends on seven continuous parameters, which can be (although not
straightforwardly) identified with the mass $m$, angular momentum $a$, NUT
charge $l$, acceleration parameter $\alpha$, and electric $e$ and magnetic $g$
charges. In practice, subfamilies of black holes and their combinations need
to be obtained from the general Plebanśki-Demiański line element
Plebanski:1976gy  by performing special limiting procedures, a process that
differs from being direct and that, on occasions, clouds the interpretation of
the metric parameters Stephani:2003tm . The Plebanśki-Demiański geometry has
been initially studied by several authors, especially the case which
represents a spinning C-metric, a pair of rotating black holes accelerating
from each other in opposite directions Griffiths:2005se ; Griffiths:2005mi ;
Griffiths:2005qp ; Pravda:2002kj ; Bicak:1999sa . Along these lines, Hong and
Teo Hong:2003gx  found a convenient set of coordinates in which the metric
polynomials of the C-metric can be easily factorized. This allowed one not
only to rapidly identify the corresponding Killing horizons but also to
perform a deeper analysis of the causal structure of this spacetime. These
findings motivated Griffiths and Podolský Griffiths:2005se  to perform a
similar construction for the Plebanśki-Demiański metric, which on this new
form111In this paper we focus on the case in which the cosmological constant
vanishes. This is strictly related with the fact that the generating solution
technique we shall use to construct our new geometries does not apply for
$\Lambda\neq 0$. depends on the six independent parameters $m$, $a$, $\alpha$,
$l$, $e$, and $g$ and a seventh parameter, the so-called twisted parameter
$\omega$ that is ultimately related to both, angular momentum and NUT charge.
In this form almost all limits on the Plebanśki-Demiański metric are direct to
take, from which highlight the spinning C-metric and the Kerr-NUT spacetimes
and their charged generalizations. As the Plebanśki-Demiański line element
represents at the end of the day a sort of spinning C-metric with NUT charge,
it seems natural to wonder what the nonrotational limit of this spacetime is.
In this form of the line element, there is no direct obstruction for this
limit to be performed, and actually a simple stationary metric is obtained.
However, this metric is found to not possess any conical singularities, as it
is known, the sources of the acceleration, and, even more, the acceleration
parameter $\alpha$ becomes a redundant parameter that can be removed by a
suitable coordinate transformation. It was then conjectured that an
accelerating NUT line element does not exist, at least in the wide class of
the Plebanśki-Demiański family. Notwithstanding, an accelerating NUT family of
black holes was introduced by Chng, Mann, and Stelea Chng:2006gh  by using a
precise generating solution technique based on the symmetries of the
dimensionally reduced Lagrangian of GR obtained by a timelike compactification
from dimension four to three. This solution, which in fact does represent a
NUT black hole with a nontrivial acceleration, was proven to not belong to the
special type D family given by Plebanśki-Demiański and to have general
algebraic properties and, in fact, is of the general type I. As a consequence,
this metric could have not been obtained by any limiting process acting on the
Plebanśki-Demiański solution. A more convenient form of the metric and a full
geometric analysis of this novel solution was given by Podolský and Vrátný
Podolsky:2020xkf , where Killing horizons, algebraic classification, conformal
completion and causal structures were studied in detail. Recently, the same
authors have found a refined set of coordinates to represent the Plebanśki-
Demiański metric, with and without a cosmological constant Podolsky:2021zwr ;
Podolsky:2022xxd ,222An extension of the Plebanśki-Demiański solutions with a
cosmological constant has been constructed in the framework of metric-affine
gravity Bahamonde:2022meb . and have finally cleaned out the presence of the
nonindependent twist parameter $\omega$. This represents the final form of the
Plebanśki-Demiański spacetime and is constructed in such a manner that all
limiting cases are clearly obtained with no need for further changes of
coordinates; it reads333We follow the same notation given by Podolský and
Vrátný Podolsky:2021zwr ; Podolsky:2022xxd  and constrain ourselves to the
case $\Lambda=0$.

$\displaystyle
ds^{2}=\frac{1}{\Omega^{2}}\left(-\frac{Q}{\rho^{2}}\left[dt-\left(a\sin^{2}\theta+4l\sin^{2}\frac{1}{2}\theta\right)d\varphi\right]^{2}+\frac{\rho^{2}}{Q}dr^{2}+\frac{\rho^{2}}{P}d\theta^{2}+\frac{P}{\rho^{2}}\sin^{2}\theta\left[adt-\left(r^{2}+(a+l)^{2}\right)d\varphi\right]^{2}\right),$
(1)

where

$\displaystyle\Omega$ $\displaystyle=1-\frac{\alpha
a}{a^{2}+l^{2}}r(l+a\cos\theta),$ (2) $\displaystyle\rho^{2}$
$\displaystyle=r^{2}+(l+a\cos\theta)^{2},$ (3) $\displaystyle P(\theta)$
$\displaystyle=\left(1-\frac{\alpha
a}{a^{2}+l^{2}}r_{+}(l+a\cos\theta)\right)\left(1-\frac{\alpha
a}{a^{2}+l^{2}}r_{-}(l+a\cos\theta)\right),$ (4) $\displaystyle Q(r)$
$\displaystyle=\left(r-r_{+}\right)\left(r-r_{-}\right)\left(1+\alpha
a\frac{a-l}{a^{2}+l^{2}}r\right)\left(1-\alpha
a\frac{a+l}{a^{2}+l^{2}}r\right).$ (5)

The roots of the metric polynomial $Q(r)$ define the two black hole Killing
horizons

$\displaystyle r_{+}:=m+\sqrt{m^{2}+l^{2}-a^{2}-e^{2}-g^{2}},$ (6)
$\displaystyle r_{-}:=m-\sqrt{m^{2}+l^{2}-a^{2}-e^{2}-g^{2}},$ (7)

that are accompanied by two accelerating horizons located at

$r_{\alpha}:=\pm\frac{1}{\alpha}\frac{a^{2}+l^{2}}{a(a\pm l)}.$ (8)

The $a\rightarrow 0$ limit is, therefore, completely transparent, showing the
outsider behavior of the NUT C-metric black hole with respect to the
Plebanśki-Demiański family. Indeed, $\Omega$ and $P$ tend to one; no conical
singularities emerge and, therefore, no acceleration whatsoever. The metric
simply represents a charged NUT black hole.

In this work, we aim to rederive the Chng, Mann, and Stelea solution
Chng:2006gh  by making use of a different, but theoretically close, solution
generating technique. The Ernst scheme Ernst:1967wx ; Ernst:1967by  describes
stationary and axially symmetric spacetimes in Einstein-Maxwell theory. It is
known that Einstein-Maxwell theory written in terms of the Ernst potentials
enjoys a set of Lie point symmetries known as Ehlers symmetries
Stephani:2003tm , from which the nontrivial, Harrison HARRISON  and Ehlers
Ehlers:1957zz  transformations can be used to obtain novel solutions in
electrovacuum. In their electric and magnetic versions,444Electric and
magnetic versions differ by a double Wick rotation on the corresponding Levis-
Weyl-Papapetrou (LWP) line element, and their names are given by the effect
that, in particular, Harrison transformations exert on a given seed spacetime.
Harrison transformations electrify a given seed spacetime or embed the seed on
an electromagnetic background, respectively Ernst:1976mzr ; Vigano:2022hrg ;
Siahaan:2021ags ; Ghezelbash:2021lcf . On the other hand, Ehlers
transformations are known to add NUT charge or to embed the seed on a rotating
(swirling) background Astorino:2022aam . We shall apply an Ehlers
transformation on the standard C-metric, written in intuitive spherical
coordinates, to find in a few steps the convenient form of the Chng, Mann, and
Stelea solution Chng:2006gh  provided by Podolský and Vrátný Podolsky:2020xkf
. Then we present, for the first time, an accelerating NUT black hole dressed
by a conformally coupled scalar field. This requires a specific treatment in
order for the Ernst scheme to be applicable Astorino:2013xc . Several
solutions of this kind have been previously reported Astorino:2012zm ;
Astorino:2013sfa ; Astorino:2013xxa ; Astorino:2014mda ; Astorino:2022prj ;
however, none of them are in the context of accelerating NUT spacetimes. In
technical terms, this is due to the algebraically general nature of
accelerating NUT metrics. A natural step forward is to endorse these
spacetimes with electromagnetic charges. At first glance, this could be
achieved by simply adding NUT charge via an Ehlers transformation acting onto
a Reissner-Nordström C-metric. Nonetheless, it has been shown that in order to
add NUT parameter on a given charged spacetime an enhanced Ehlers
transformation it is requiered that, along with the usual Ehlers map, includes
an extra duality transformation that restores the alignment of the
corresponding gauge fields Astorino:2019ljy . As a matter of fact, Ehlers
transformations rotate the mass parameter in such a manner that its magnetic
component becomes nontrivial but rotating at the same time the electromagnetic
potentials. Here, we construct the NUT extension of the Reissner-Nordström
C-metric spacetime, showing how the corresponding line element correctly
reproduces the NUT Reissner-Nordström limit with an aligned gauge potential if
the naive Ehlers map used in the uncharged case is improved with a simple
duality transformation.

This paper is structured as follows: In Sec. II, we shortly review the Ernst
scheme for stationary and axially symmetric solutions in Einstein-Maxwell
theory. We put particular emphasis in explaining how to produce new solutions
by means of Harrison and Ehlers symmetry transformations. Section III is
devoted to rederive the Chng, Mann, and Stelea solution à la Podolský and
Vrátný by making use of an electric Ehlers transformation. We continue with
Sec. IV extending our findings to the case in which a conformally coupled
scalar field is introduced in the matter sector. We explain how the Ernst
scheme is extended to this theory and how to integrate such a solution.
Section V is destined to the construction of NUT Reissner-Nordström C-metric
configurations, where we explicitly show the line element, its corresponding
nonaccelerating limit, and how the alignment of the gauge field can be
retrieved by a duality transformation. Finally, we conclude in Sec. VI with a
discussion of our results and with a list of possible avenues that can be
explored to continue studying these geometries.

##  II Ernst scheme

The Ernst scheme Ernst:1967wx ; Ernst:1967by  provides an elegant framework in
which to study stationary and axially symmetric spacetimes within Einstein-
Maxwell theory. It usefulness relies in the ability it has to disclose certain
symmetries of the electrovacuum theory that allow the generation of nontrivial
solutions starting from a known seed. In concrete terms, Einstein-Maxwell
field equations for a general stationary and axisymmetric spacetime given by
the (LWP) metric and a stationary and axially symmetric Maxwell field

$\displaystyle ds^{2}$ $\displaystyle=-f(dt-\omega
d\varphi)^{2}+f^{-1}\left[\rho^{2}d\varphi^{2}+e^{2\gamma}\left(d\rho^{2}+dz^{2}\right)\right],$
(9) $\displaystyle A$ $\displaystyle=A_{t}dt+A_{\varphi}d\varphi,$ (10)

can be cast in a completely equivalent form, the Ernst equations

$\displaystyle\left(\operatorname{Re}\mathcal{E}+|\Phi|^{2}\right)\nabla^{2}\mathcal{E}=\nabla\mathcal{E}\cdot\left(\nabla\mathcal{E}+2\Phi^{*}\nabla\Phi\right),$
(11a)
$\displaystyle\left(\operatorname{Re}\mathcal{E}+|\Phi|^{2}\right)\nabla^{2}\Phi=\nabla\Phi\cdot\left(\nabla\mathcal{E}+2\Phi^{*}\nabla\Phi\right),$
(11b)

where $\vec{\nabla}$ and the various vectorial quantities are understood in
Euclidean space with cylindrical coordinates $(\rho,\varphi,z)$. The so-called
Ernst potentials are defined as

$\mathcal{E}=f-|\Phi|^{2}+i\chi,\quad\Phi=A_{t}+i\tilde{A}_{\varphi},$ (12)

where $\tilde{A}_{\varphi}$ and $\chi$, known as twisted potentials, are
characterized via the differential equations

$\hat{\varphi}\times\nabla\tilde{A}_{\varphi}=\rho^{-1}f\left(\nabla
A_{\varphi}+\omega\nabla A_{t}\right)$ (13)

and

$\hat{\varphi}\times\nabla\chi=-\rho^{-1}f^{2}\nabla\omega-2\hat{\varphi}\times\operatorname{Im}\left(\Phi^{*}\nabla\Phi\right).$
(14)

It is important to remark that, due to the integrability features of the
Einstein-Maxwell system, the function $\gamma(\rho,z)$ decouples, and it is
uniquely determined by the functions $f(\rho,z)$ and $\omega(\rho,z)$. This
form of the electrovacuum field equations can be proven to enjoy a set of Lie
point symmetries, known as Ehlers symmetries Stephani:2003tm , given by

$\displaystyle\mathcal{E}$
$\displaystyle=|\lambda|^{2}\mathcal{E}_{0}\,,\qquad\qquad\qquad\quad\;\;\Phi=\lambda\Phi_{0}\,,$
(15a) $\displaystyle\mathcal{E}$
$\displaystyle=\mathcal{E}_{0}+ib\,,\qquad\qquad\qquad\quad\,\Phi=\Phi_{0}\,,$
(15b) $\displaystyle\mathcal{E}$
$\displaystyle=\frac{\mathcal{E}_{0}}{1+i\jmath\mathcal{E}_{0}}\,,\qquad\qquad\qquad\;\,\Phi=\frac{\Phi_{0}}{1+i\jmath\mathcal{E}_{0}}\,,$
(15c) $\displaystyle\mathcal{E}$
$\displaystyle=\mathcal{E}_{0}-2\beta^{*}\Phi_{0}-|\beta|^{2}\,,\quad\quad\hskip
2.84544pt\Phi=\Phi_{0}+\beta\,,$ (15d) $\displaystyle\mathcal{E}$
$\displaystyle=\frac{\mathcal{E}_{0}}{1-2\alpha^{*}\Phi_{0}-|\alpha|^{2}\mathcal{E}_{0}}\,,\quad\hskip
5.69046pt\Phi=\frac{\alpha\mathcal{E}_{0}+\Phi_{0}}{1-2\alpha^{*}\Phi_{0}-|\alpha|^{2}\mathcal{E}_{0}}\,,$
(15e)

where $\alpha,\beta$, and $\lambda$ are complex parameters, while $b$ and
$\jmath$ are real. After reducing the pure gauge transformations we are left
with (15c) and (15e), the so-called Ehlers Ehlers:1957zz  and Harrison
HARRISON  transformations. Because of their Lie point symmetry nature, the
action of the transformations leaves the Ernst equations unchanged while at
the same time producing new nonequivalent geometries. Notice that LWP metric
(9) is not the unique choice for a general stationary and axisymmetric line
element. As a matter of fact, we can act on it with a discrete double-Wick
rotation producing the nonequivalent configuration

$\displaystyle ds^{2}$ $\displaystyle=f(d\varphi-\omega
dt)^{2}+f^{-1}\left[e^{2\gamma}\left(d\rho^{2}+dz^{2}\right)-\rho^{2}dt^{2}\right],$
(16) $\displaystyle A$ $\displaystyle=A_{t}dt+A_{\varphi}d\varphi,$ (17)

that gives rise to the very same Ernst equations (11). Here,
$\Phi=A_{\varphi}+i\tilde{A}_{t}$ and

$\hat{\varphi}\times\nabla\tilde{A}_{t}=\rho^{-1}f\left(\nabla
A_{t}+\omega\nabla A_{\varphi}\right).$ (18)

This ansatz is usually referred to as the magnetic LWP ansatz. Then, the
effect of Harrison and Ehlers transformations on a given seed spacetime
crucially depends on which ansatz they act Vigano:2022hrg , i.e., (9) or (16).
The Harrison map is known to electrify a given seed when acting on (9), while
it embeds a seed spacetime of the form (16) in an electromagnetic universe
Ernst:1976mzr . On the other hand, Ehlers transformations add a NUT parameter
to a given seed when acting on (9) while embedding the initial spacetime in a
rotating (swirling) background when acting on (16) Astorino:2022aam .

For our purposes, it is useful to consider the enhancement in which a
minimally coupled scalar field is introduced in the matter sector. It was
proven in Ref. Astorino:2013xc  that the Ernst scheme survives the addition of
such a field; in fact, the reduced action principle in terms of the Ernst
potentials reveals that in the Einstein-Maxwell-scalar case the set of Ehlers
symmetries (15) needs to be complemented by the trivial transformation
$\Psi\rightarrow\Psi^{\prime}=\Psi$, while Ernst equations are complemented
with the corresponding Klein-Gordon equation for $\Psi$. This ensures the
feasibility of Ehlers transformations as an effective solution generating
technique when dealing with black holes with scalar hair. Most of spacetimes
with scalar hair are easily found in the so-called Einstein-Maxwell-conformal-
scalar theory; this means in the Jordan frame in which the scalar field is
conformally coupled to the curvature scalar Bocharova:1970skc ;
Bekenstein:1975ts ; Charmousis:2009cm ; Anabalon:2009qt ; Bhattacharya:2013hvm
; Bardoux:2013swa ; Cisterna:2021xxq . Both theories, with minimally and
conformally coupled scalar fields, are related by means of the conformal map
known as Bekenstein transformations Bekenstein:1974sf . These transformations
provide a vehicle to transport all the machinery of the Ernst scheme to
theories with conformally coupled scalars. It is then enough to consider a
seed spacetime in the Jordan frame, transform it to the minimally coupled
frame (in which Ehlers transformations can be applied) and move back to the
conformally couple theory to obtain new solutions in the Einstein-Maxwell-
conformal-scalar sector.

With these tools at hand, in the next sections we will rederive the Chng,
Mann, and Stelea solution Chng:2006gh , apply these results to the case in
which a conformally coupled scalar field is considered in the matter sector,
and explore the construction of charged accelerating NUT geometries.

##  III Dressing the C-metric with NUT

The aim of this section is to construct the accelerating NUT black hole of
Chng, Mann, and Stelea Chng:2006gh , written in the coordinates provided by
Podolský and Vrátný Podolsky:2020xkf . The steps are very clear. We take the
known C-metric in spherical-like coordinates, and we dress it with a NUT
charge via an electric Ehlers transformation Ehlers:1957zz . Thus, we start
from

$ds^{2}=\frac{1}{\Omega(R,\theta)^{2}}\left[-Q(R)dt^{2}+\frac{dR^{2}}{Q(R)}+\frac{R^{2}d\theta^{2}}{P(\theta)}+R^{2}P(\theta)\sin^{2}\theta
d\varphi^{2}\right],$ (19)

where

$\displaystyle\Omega(R,\theta)$ $\displaystyle=1+AR\cos\theta,$ (20)
$\displaystyle Q(R)$
$\displaystyle=\left(1-A^{2}R^{2}\right)\left(1-\frac{2M}{R}\right),$ (21)
$\displaystyle P(\theta)$ $\displaystyle=1+2AM\cos\theta.$ (22)

Here, $A$ and $M$ stand for the acceleration parameter and the mass of the
black hole, respectively Griffiths:2009dfa . The standard use of the Ernst
scheme requires one to compare metric (19) with the electric LWP ansatz (9),
from which it is direct to recognize the seed functions

$\displaystyle f_{0}$ $\displaystyle=\frac{Q}{\Omega^{2}},$ (23a)
$\displaystyle\rho$ $\displaystyle=\frac{\sqrt{PQ}R\sin\theta}{\Omega^{2}},$
(23b) $\displaystyle\omega_{0}$ $\displaystyle=0.$ (23c)

This is done in order to write the seed Ernst potentials. Note that seed
quantities are denoted by the subindex 0. Therefore, the seed potentials are
given by

$\mathcal{E}_{0}=\frac{Q}{\Omega^{2}},\quad\Phi_{0}=0,$ (24)

where indeed the potential $\Phi_{0}$ vanishes and the seed is uncharged. In
addition, $\mathcal{E}_{0}$ is real, corresponding with the static character
of the seed metric. With these at hand, we apply the Ehlers transformation
555Aiming to maintain the notation given by Podolský and Vrátný, we change the
real parameter $\jmath$ to $c$. (15c) and obtain the transformed Ernst
potential (free of any subindex)

$\mathcal{E}=\frac{\mathcal{E}_{0}}{1+ic\mathcal{E}_{0}}=\frac{Q/\Omega^{2}}{\Lambda},$
(25)

where we have defined the quantity

$\Lambda(R,\theta)=1+ic\frac{Q}{\Omega^{2}}.$ (26)

The electromagnetic potential remains null: $\Phi=0$. To read the transformed
metric components, we make use of the definition of the Ernst potential, which
now is composed by real and imaginary parts. From the real part, we identify

$f=\operatorname{Re}(\mathcal{E})=\frac{Q/\Omega^{2}}{|\Lambda|^{2}}=\frac{f_{0}}{|\Lambda|^{2}},$
(27)

where $|\Lambda|^{2}=\Lambda\bar{\Lambda}$. The imaginary part of
$\mathcal{E}$,

$\chi=\operatorname{Im}(\mathcal{E})=-c\frac{Q^{2}/\Omega^{4}}{|\Lambda|^{2}},$
(28)

provides us, via Eq. (14),666We use the orthonormal frame defined by the
ordered triad $\left(\vec{e}_{R},\vec{e}_{\varphi},\vec{e}_{\theta}\right)$,
so then $\vec{e}_{\varphi}\times\vec{e}_{R}=-\vec{e}_{\theta}$ and
$\vec{e}_{\varphi}\times\vec{e}_{\theta}=\vec{e}_{R}$. The gradient operator
is then defined as $\vec{\nabla}\propto\sqrt{Q}R\frac{\partial}{\partial
R}\vec{e}_{R}+\sqrt{P}\frac{\partial}{\partial\theta}\vec{e}_{\theta}+\frac{1}{\sqrt{P}\sin\theta}\frac{\partial}{\partial\varphi}\vec{e}_{\varphi}$.
with the form of the rotating function $\omega$:

$\omega(R,\theta)=2c\left(2M\cos\theta+\frac{AP(\theta)R^{2}\sin^{2}\theta}{\Omega(R,\theta)^{2}}\right).$
(29)

Ehlers transformations always endorse an imaginary part on the transformed
Ernst potential. Therefore, it always implies the appearance of some sort of
rotation [via (14)], NUT charge in the electric case, or swirling rotation in
the magnetic one Vigano:2022hrg . Recalling that Ehlers transformations do not
affect the metric function $\gamma,\gamma=\gamma_{0}$, it is possible to
verify that

$e^{2\gamma_{0}}\left(d\rho^{2}+dz^{2}\right)=e^{2\gamma}\left(d\rho^{2}+dz^{2}\right)=\left(\frac{dR^{2}}{Q}+\frac{R^{2}}{P}d\theta^{2}\right)f_{0},$
(30)

a result that, along with the definitions for $f$ and $\omega$, can be plugged
into the LWP ansatz (9) to deliver the transformed metric

$\displaystyle ds^{2}=$
$\displaystyle\frac{1}{\Omega(R,\theta)^{2}}\left[-\frac{Q(R)}{|\Lambda(R,\theta)|^{2}}\left[dt-2c\left(2M\cos\theta+\frac{AP(\theta)R^{2}}{\Omega(R,\theta)^{2}}\sin^{2}\theta\right)d\varphi\right]^{2}\right.$
$\displaystyle\left.+|\Lambda(R,\theta)|^{2}\left(R^{2}P(\theta)\sin^{2}\theta
d\varphi^{2}+\frac{dR^{2}}{Q(R)}+R^{2}\frac{d\theta^{2}}{P(\theta)}\right)\right].$
(31)

At this stage, the construction is completed, and it remains only to establish
the connection between the real parameter $c$ and the NUT charge $l$. For this
to be done, we define a new mass parameter via the relation

$m=\sqrt{M^{2}-l^{2}},$ (32)

which together with the definitions

$\displaystyle r_{+}\equiv m+\sqrt{m^{2}+l^{2}},$ (33) $\displaystyle
r_{-}\equiv m-\sqrt{m^{2}+l^{2}}$ (34)

allows one to find a proper change of coordinates to bring down the metric in
an explicit NUT form. Keeping in mind that

$\displaystyle r_{+}+r_{-}$ $\displaystyle=2m,$ (35) $\displaystyle
r_{+}-r_{-}$ $\displaystyle=2\sqrt{m^{2}+l^{2}},$ (36) $\displaystyle
r_{+}\left(r_{+}-r_{-}\right)$ $\displaystyle=r_{+}^{2}+l^{2},$ (37)

we perform the change of coordinates $R\rightarrow\left(r-r_{-}\right)$ and
$t\rightarrow\frac{\left(r_{+}-r_{-}\right)}{r_{+}}\tau$ together with the
reparametrization $c\rightarrow l/r_{+}$. The metric terms change as

$\displaystyle-\frac{Q(R)}{|\Lambda(R,\theta)|^{2}}{\left[dt-2c\left(2M\cos\theta+\frac{AP(\theta)R^{2}}{\Omega(R,\theta)^{2}}\sin^{2}\theta\right)d\varphi\right]^{2}}\rightarrow-\frac{r_{+}-r_{-}}{r_{+}}\frac{\mathcal{Q}(r)}{\mathcal{R}^{2}(r,\theta)}\left[d\tau-2l\left(\cos\theta+A\mathcal{T}(r,\theta)\sin^{2}\theta\right)d\varphi\right]^{2},$
(38)
$\displaystyle|\Lambda(R,\theta)|^{2}R^{2}\rightarrow\frac{r_{+}-r_{-}}{r_{+}}\mathcal{R}^{2}(r,\theta),\quad\frac{|\Lambda(R,\theta)|^{2}}{Q(R)}\rightarrow\frac{r_{+}-r_{-}}{r_{+}}\frac{\mathcal{R}^{2}(r,\theta)}{\mathcal{Q}(r)},$
(39)

where we have defined the new metric functions

$\displaystyle\Omega(r,\theta)$
$\displaystyle=1+A\left(r-r_{-}\right)\cos\theta,$ (40)
$\displaystyle\mathcal{P}(\theta)$
$\displaystyle=1+A\left(r_{+}-r_{-}\right)\cos\theta,$ (41)
$\displaystyle\mathcal{Q}(r)$
$\displaystyle=\left(1-A^{2}\left(r-r_{-}\right)^{2}\right)\left(r-r_{-}\right)\left(r-r_{+}\right),$
(42) $\displaystyle\mathcal{T}(r,\theta)$
$\displaystyle=\frac{P(\theta)\left(r-r_{-}\right)^{2}}{\left(r_{+}-r_{-}\right)\Omega(r,\theta)^{2}},$
(43) $\displaystyle\mathcal{R}^{2}(r,\theta)$
$\displaystyle=\frac{1}{r_{+}^{2}+l^{2}}\left[r_{+}^{2}\left(r-r_{-}\right)^{2}+l^{2}\frac{\left[1-A^{2}\left(r-r_{-}\right)^{2}\right]^{2}}{\Omega(r,\theta)^{4}}\left(r-r_{+}\right)^{2}\right].$
(44)

Up to a conformal factor that can be disregarded by a conformal rescaling of
the line element, we find the final form of the spacetime to be

$\displaystyle ds^{2}=$
$\displaystyle\frac{1}{\Omega(r,\theta)^{2}}\left[-\frac{\mathcal{Q}(r)}{\mathcal{R}^{2}(r,\theta)}\left[d\tau-2l\left(\cos\theta+A\mathcal{T}(r,\theta)\sin^{2}\theta\right)d\varphi\right]^{2}+\mathcal{R}^{2}(r,\theta)\left(\mathcal{P}(\theta)\sin^{2}\theta
d\varphi^{2}+\frac{dr^{2}}{\mathcal{Q}(r)}+\frac{d\theta^{2}}{\mathcal{P}(\theta)}\right)\right].$
(45)

This precisely coincides with the accelerating NUT spacetime presented by
Podolský and Vrátný Podolsky:2020xkf . The corresponding limits are clean,
either $A\rightarrow 0$ or $l\rightarrow 0$; thus, we recover the NUT or
C-metric black holes with no further changes of coordinates. We do not intend
to provide a detailed description of the geometric features of this solution,
as it has been already done in Ref. Podolsky:2020xkf .

##  IV Accelerating NUT black holes with a conformally coupled scalar dress

The Ernst solution generating technique is, in principle, intrinsic to four-
dimensional Einstein-Maxwell theory; then it is expected that Ehlers
transformations apply only in the context of black holes with scalar hair.
However, a few counterexamples have been found. A particular symmetry of the
Einstein-dilaton-Maxwell theory has been identified, of which the effect is to
embed a given seed on an external magnetic field Dowker:1993bt . Moreover,
this system has been shown to exhibit internal symmetries allowing for a
deeper exploration of its black hole spectrum Galtsov:1994pd ; Galtsov:1994sjr
. On the other hand, in higher dimensions, a magnetizing transformation has
been also identified, providing Melvin solutions in arbitrary dimensions
Ortaggio:2004kr . A particularly appealing extension of the Ernst scheme is
the one in which a minimally coupled scalar field is included in the matter
sector Astorino:2013xc . As explained before, Harrison and Ehlers symmetries
remain as valid Lie point transformations of this theory and, therefore, open
the road for the exploration of the integrability properties of Einstein-
Maxwell-scalar theory in a wide variety of cases. Making use of Bekenstein
transformations Bekenstein:1974sf , all these new solutions can be mapped to
the Einstein-Maxwell-conformal-scalar scenario, in which they often represent
black hole spacetimes. In this section, we apply all this machinery to
construct an accelerating NUT black hole with a conformally coupled scalar
dress. Black holes with a conformally coupled scalar field have been deeply
investigated in many contexts, but in the case where acceleration and NUT
charge are combined in the absence of rotation. This is, again, due to the
fact that when doing so we depart from the Plebanśki-Demiański family. The
strategy follows from the previous section, now augmented with the use of the
Bekenstein map.

Thus, we start by considering a C-metric spacetime in the context of a
conformally coupled scalar theory. We proceed by translating the solution to
the Einstein frame by means of an inverse Bekenstein map and by applying an
electric Ehlers transformation to add the corresponding NUT charge. The final
form of the solution is obtained by moving back to the Jordan frame. Our guide
action is

$I=\frac{1}{2\kappa}\int
d^{4}x\sqrt{-g}\left[R-\kappa\left(g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi+\frac{1}{6}R\phi^{2}\right)\right],$
(46)

for which C-metric black holes with conformal scalar hair have been identified
in Refs. Charmousis:2009cm ; Anabalon:2009qt . In terms of the acceleration
conformal factor, the metric and scalar are written, respectively, as

$\displaystyle ds^{2}$
$\displaystyle=\frac{1}{\Omega(R,\theta)^{2}}\left[-\frac{Q(R)}{R^{2}}dt^{2}+\frac{R^{2}}{Q(R)}dR^{2}+\frac{R^{2}}{P(\theta)}d\theta^{2}+P(\theta)R^{2}\sin^{2}\theta
d\varphi^{2}\right],$ (47) $\displaystyle\phi(R,\theta)$
$\displaystyle=\sqrt{\frac{6}{\kappa}}\frac{k\Omega(R,\theta)}{R+k(\Omega(R,\theta)-2)},$
(48)

where

$\displaystyle\Omega(R,\theta)$ $\displaystyle=1+AR\cos\theta,$ (49a)
$\displaystyle Q(R)$
$\displaystyle=\left(1-A^{2}R^{2}\right)(R-M)\left(R-\frac{M}{1+2AM}\right),$
(49b) $\displaystyle P(\theta)$
$\displaystyle=(1+AM\cos\theta)\left(1+\frac{AM}{1+2AM}\cos\theta\right),$
(49c) $\displaystyle k$ $\displaystyle=\frac{M}{1+AM}.$ (49d)

In order to proceed, we move to the Einstein frame

$I=\frac{1}{2\kappa}\int
d^{4}x\sqrt{-\tilde{g}}\left[\tilde{R}-\kappa\tilde{g}^{\mu\nu}\partial_{\mu}\Psi\partial_{\nu}\Psi\right],$
(50)

where the metric and scalar read, respectively,

$\displaystyle ds^{2}$
$\displaystyle=\frac{\Omega_{B}(R,\theta)}{\Omega(R,\theta)^{2}}\left[-\frac{Q(R)}{R^{2}}dt^{2}+\frac{R^{2}}{Q(R)}dR^{2}+\frac{R^{2}}{P(\theta)}d\theta^{2}+P(\theta)R^{2}\sin^{2}\theta
d\varphi^{2}\right],$ (51) $\displaystyle\Psi(R,\theta)$
$\displaystyle=\sqrt{\frac{6}{\kappa}}\tanh^{-1}\left(\sqrt{\frac{\kappa}{6}}\phi(R,\theta)\right).$
(52)

Here, the Bekenstein factor is given by

$\Omega_{B}(R,\theta)=1-\frac{\kappa}{6}\phi(R,\theta)^{2}.$ (53)

Now, we apply an electric Ehlers transformation in order to add the NUT
charge. We do not repeat every step, as they are identical to the previous
case, no electromagnetic charges are here included, and the identification of
the seed and transformed Ernst potentials follows in full analogy with those
of the previous section. Thus, we get

$ds^{2}=\frac{\Omega_{B}(R,\theta)}{\Omega(R,\theta)^{2}}\left[-\frac{Q(R)}{|\Lambda(R,\theta)|^{2}R^{2}}(dt-\omega(R,\theta)d\varphi)^{2}+|\Lambda(R,\theta)|^{2}R^{2}\left(P(\theta)\sin^{2}\theta
d\varphi^{2}+\frac{d\theta^{2}}{P(\theta)}+\frac{dR^{2}}{Q(R)}\right)\right],$
(54)

with

$|\Lambda(R,\theta)|^{2}=1+c^{2}\frac{\Omega_{B}(R,\theta)^{2}Q(R)^{2}}{\Omega(R,\theta)^{4}R^{4}},\quad\omega(R,\theta)=2c(2M\cos\theta-A\mathcal{T}(R,\theta)),$
(55)

and where the function $\mathcal{T}(R,\theta)$ is given by

$\displaystyle\mathcal{T}(R,\theta)$
$\displaystyle=\frac{1}{(1+AM)^{2}(1+2AM)\Omega(R,\theta)^{2}(R+k(\Omega(R,\theta)-2))^{2}}\left[2A^{3}M^{3}R^{4}(1+2AM)\cos^{5}\theta\right.$
(56)
$\displaystyle\quad\left.+A^{2}M^{2}R^{2}(8A^{2}M^{2}R^{2}-6AM^{2}R+14AMR^{2}+6M^{2}-6MR+5R^{2})\cos^{4}\theta\right.$
$\displaystyle\quad\left.+2AMR^{2}(2+AM)(2A^{2}M^{2}R^{2}-2AM^{2}R+3AMR^{2}+2M^{2}-2MR+R^{2})\cos^{3}\theta\right.$
$\displaystyle\quad\left.+\left(12A^{3}M^{4}R^{3}-2A^{4}M^{4}R^{4}-6A^{3}M^{3}R^{4}-14A^{2}M^{4}R^{2}+12A^{2}M^{3}R^{3}+A^{2}M^{2}R^{4}\right.\right.$
$\displaystyle\quad\left.\left.+4AM^{4}R-4AM^{3}R^{2}-2AM^{2}R^{3}+4AMR^{4}-2M^{4}+4M^{3}R-2MR^{3}+R^{4}\right)\right.\cos^{2}\theta$
$\displaystyle\quad\left.-2M\left(A^{4}M^{3}R^{4}-5A^{3}M^{3}R^{3}+5A^{3}M^{2}R^{4}+4A^{2}M^{3}R^{2}-10A^{2}M^{2}R^{3}+6A^{2}MR^{4}\right.\right.$
$\displaystyle\quad\left.\left.+2AM^{3}R+3AM^{2}R^{2}-5AMR^{3}+2AR^{4}-M^{3}+2M^{2}R-MR^{2}\right)\cos\theta\right.$
$\displaystyle\quad\left.-2A^{3}M^{3}R^{4}+3A^{2}M^{4}R^{2}+4A^{2}M^{3}R^{3}-5A^{2}M^{2}R^{4}-6AM^{4}R+2AM^{3}R^{2}+6AM^{2}R^{3}\right.$
$\displaystyle\quad\left.-4AMR^{4}+3M^{4}-6M^{3}R+M^{2}R^{2}+2MR^{3}-R^{4}\right].$

Contrary to the previous construction, the function $\chi$ is more involved,
and usually solving (14) becomes cumbersome. To circumvent these
technicalities, we have solved $\omega$ from the different, but equivalent
expression

$\vec{\nabla}\omega=|\Lambda|^{2}\vec{\nabla}\omega_{0}-i\hat{\varphi}\times\frac{\rho}{f_{0}}\left(\bar{\Lambda}\vec{\nabla}\Lambda-\Lambda\vec{\nabla}\bar{\Lambda}\right).$
(57)

Hence, back to the conformal frame, the solution takes the form

$ds^{2}=\frac{1}{\Omega(R,\theta)^{2}}\left[-\frac{Q(R)}{|\Lambda(R,\theta)|^{2}R^{2}}(dt-\omega(R,\theta)d\varphi)^{2}+|\Lambda(R,\theta)|^{2}R^{2}\left(P(\theta)\sin^{2}\theta
d\varphi^{2}+\frac{d\theta^{2}}{P(\theta)}+\frac{dR^{2}}{Q(R)}\right)\right],$
(58)

where the scalar field and metric functions are given by (48) and (49). As a
final step, it remains only to make up the metric in such a way that the NUT
parameter naturally emerges. This proceeds in analogy with the case of
accelerating NUT in GR, yielding

$\displaystyle
ds^{2}=\frac{1}{\bar{\Omega}(r,\theta)^{2}}\left[-\frac{\bar{\mathcal{Q}}(r)}{\overline{\mathcal{R}}^{2}(r,\theta)}[d\tau-2l(\cos\theta-A\overline{\mathcal{T}}(r,\theta))d\varphi]^{2}+\overline{\mathcal{R}}^{2}(r,\theta)\left(\bar{\mathcal{P}}(\theta)\sin^{2}\theta
d\varphi^{2}+\frac{d\theta^{2}}{\bar{\mathcal{P}}(\theta)}+\frac{dr^{2}}{\bar{\mathcal{Q}}(r)}\right)\right],$
(59a)
$\displaystyle\bar{\phi}(r,\theta)=\sqrt{\frac{6}{\kappa}}\frac{k\bar{\Omega}(r,\theta)}{r-r_{-}+k(\bar{\Omega}(r,\theta)-2)},$
(59b)

where

$\displaystyle\bar{\Omega}(r,\theta)=1+A\left(r-r_{-}\right)\cos\theta,$ (60a)
$\displaystyle\bar{\mathcal{Q}}(r)=\left(1-A^{2}\left(r-r_{-}\right)^{2}\right)\left(r-r_{-}-\frac{r_{+}-r_{-}}{2}\right)\left(r-r_{-}-\frac{r_{+}-r_{-}}{2\left(1+A\left(r_{+}-r_{-}\right)\right)}\right),$
(60b)
$\displaystyle\bar{\mathcal{P}}(\theta)=\left(1+\frac{A\left(r_{+}-r_{-}\right)}{2}\cos\theta\right)\left(1+\frac{A\left(r_{+}-r_{-}\right)}{2\left(1+A\left(r_{+}-r_{-}\right)\right)}\cos\theta\right),$
(60c)
$\displaystyle\overline{\mathcal{T}}(r,\theta)=\frac{\mathcal{T}(r,\theta)}{r_{+}-r_{-}},$
(60d)
$\displaystyle\overline{\mathcal{R}}^{2}(r,\theta)=\frac{1}{r_{+}^{2}+l^{2}}\left[r_{+}^{2}\left(r-r_{-}\right)^{2}+l^{2}\frac{\bar{\Omega}_{B}(r,\theta)^{2}}{\bar{\Omega}(r,\theta)^{4}\left(r-r_{-}\right)^{2}}\bar{\mathcal{Q}}(r)^{2}\right]\text{.
}$ (60e)

Let us now briefly analyze the main features of this solution.777A detailed
study of this geometry will be given in Ref.barrientosetall . It is
instructive to start by taking the $l=0$ and $A=0$ limits. The former case,
indeed, allows us to recover the original seed, namely, the accelerating
solution with a conformally coupled scalar dress Charmousis:2009cm ;
Anabalon:2009qt . Recalling that in the limit $l=0$ we have $r_{+}=2m$ and
$r_{-}=0$, we can obtain

$\displaystyle\bar{\Omega}(r,\theta)=1+Ar\cos\theta,$ (61a)
$\displaystyle\bar{P}(\theta)=(1+2Am\cos\theta)\left(1+\frac{Am\cos\theta}{1+2Am}\right),$
(61b)
$\displaystyle\bar{Q}(r)=(1-A^{2}r^{2})(r-m)\left(r-\frac{m}{1+2Am}\right),$
(61c) $\displaystyle\bar{\mathcal{R}}(r,\theta)=r^{2},$ (61d)

on which we can replace $m$ by $M$ and $r$ by $R$, providing the seed metric
configuration

$\displaystyle
ds^{2}=\frac{1}{(1+AR\cos\theta)^{2}}\left[-\frac{\bar{Q}(R)}{R^{2}}dt^{2}+\frac{R^{2}dR^{2}}{\bar{Q}(R)}+\frac{d\theta^{2}}{\bar{P}(\theta)}+\bar{P}(\theta)r^{2}\sin^{2}\theta
d\varphi\right],$ (62)
$\displaystyle\phi(R,\theta)=\sqrt{\frac{6}{\kappa}}\frac{k(1+AR\cos\theta)}{R-k(1-AR\cos\theta)}.$
(63)

More informative in terms of the NUT nature of our solution is the
nonaccelerating limit. Taking $A\rightarrow 0$ and recalling that
$r_{+}+r_{-}=2m$, we get

$\displaystyle\bar{\Omega}(r,\theta)=1,$ (64a)
$\displaystyle\bar{P}(\theta)=1,$ (64b) $\displaystyle\bar{Q}(r)=(r-m)^{2},$
(64c) $\displaystyle\bar{\mathcal{R}}(r,\theta)=r^{2}+l^{2},$ (64d)

that results into the NUT black hole with scalar conformal dress presented in
Refs. Bhattacharya:2013hvm ; Bardoux:2013swa :

$\displaystyle ds^{2}=-\frac{(r-m)^{2}}{(r^{2}+l^{2})}(dt-2l\cos\theta
d\varphi)^{2}+\frac{(r^{2}+l^{2})}{(r-m)^{2}}dr^{2}+(r^{2}+l^{2})(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}),$ (65)
$\displaystyle\phi(r)=\sqrt{\frac{6}{\kappa}}\frac{\sqrt{m^{2}+l^{2}}}{r-m}.$
(66)

This ensures that our NUT charge has been properly added into the original
C-metric seed. As a matter of fact, both solutions, the seed and the NUT
conformally coupled configuration, retrieve the well-known Bocharova-
Bronnikov-Melnikov-Bekenstein (BBMB) black hole Bocharova:1970skc ;
Bekenstein:1975ts  in the corresponding $A=0$ and $l=0$ limits, a solution
that finally lands on the Minkowski spacetime in the vanishing mass case. As a
matter of fact, the BBMB solution is not connected with the Schwarzschild
black hole; there is no continuous limit that renders the spacetime free from
the scalar field profile while keeping the mass. This hierarchy of scalar
dressed solutions is depicted in FIG. 1.

Figure 1: This diagram represents a hierarchy of solutions with a scalar
conformal dress that continuously connect with the solution described in this
section. On top, there is a spacetime that includes rotation, acceleration,
and NUT charge and that might be obtained by adding rotation on our spacetime
(59). As discussed in our conclusions, it is expected to be of type I and,
therefore has not been yet found. The corresponding limits $l=0$ and/or $A=0$
bring us to the well-known type D solutions with scalar conformal dress.

The horizon structure of our accelerating NUT solution (59) is easily
obtainable due to the factorizable form of $\bar{Q}(r)$. Four Killing horizons
are identified:

$\displaystyle r_{a}^{-}=r_{-}-\frac{1}{A}=m-\sqrt{m^{2}+l^{2}}-\frac{1}{A},$
(67a) $\displaystyle
r_{a}^{+}=r_{-}+\frac{1}{A}=m-\sqrt{m^{2}+l^{2}}+\frac{1}{A},$ (67b)
$\displaystyle r_{H}^{-}=m-\frac{2A(m^{2}+l^{2})}{1+2A\sqrt{m^{2}+l^{2}}},$
(67c) $\displaystyle r_{H}^{+}=m,$ (67d)

which are given in two pairs, two acceleration horizons and two black hole
horizons. Their appearance on the causal structure of (59) is directly
dependent on the range of the radial coordinate $r$. A brute-force inspection
of the Kretschmann curvature scalar reveals, through a particularly lengthly
and ugly expression, that the spacetime is free of curvature singularities. As
a matter of fact, the initial singularity at $r=0$ of the seed configuration
is completely softened by the addition of the NUT parameter, rendering our
metric fully regular. In consequence, the four Killing horizons (67) are
present in our geometry. It is direct to observe that $r_{a}^{+}>r_{a}^{-}$
and $r_{H}^{+}>r_{H}^{-}$; however, to completely determine the relative
position of all Killing horizons with respect to themselves, an extra
assumption is needed. A physically plausible condition is to consider the
slowly acceleration regime, which, in turn, restricts the parameters $A$, $m$,
and $l$ according with

$A<\frac{1}{\sqrt{m^{2}+l^{2}}}=\frac{1}{M}.$ (68)

Hence, the horizons satisfy $r_{a}^{-}<r_{H}^{-}<r_{H}^{+}<r_{a}^{+}$. It is
interesting to observe that, contrary to what occurs in the hairless case,
here $r_{H}^{-}$ is not necessarily negative, and a causal structure might
arise with three positive Killing horizons $r_{H}^{-}$, $r_{H}^{+}$, and
$r_{a}^{+}$, namely, one acceleration horizon and a black hole and internal
horizon.

On the other hand, it is important to establish a possible pole of the scalar
field profile ($r_{0}$), a pole that could render the scalar field profile
singular on the outer domain of communications. In the case of (59), the
$\theta$ dependence in the scalar field profile transforms such a locus in a
region. In fact, it lies at

$r_{0}(\theta)=m-\frac{A(m^{2}+l^{2})(\cos\theta+1)}{A\sqrt{m^{2}+l^{2}}(\cos\theta+1)+1}.$
(69)

We observe that for any value $\theta\neq\pi$ such a divergence of the scalar
field profile always lies behind the event horizon $r_{H}^{+}$. However, the
conflictive locus remains located at the south pole. This is an awkwardness
that has been inherited from the seed ancestor and that can be removed by
applying a conformal transformation to a very specific frame barrientosetall .

Accelerating and NUT spacetimes are, from the scratch, geometrically intricate
solutions. It is known that the acceleration in the former case is due to the
presence of conical defects, defects that are better represented by either
cosmic strings or struts that pull or push the black hole to infinity, causing
its acceleration. On the other hand, the NUT spacetime possesses the so-called
Misner string topological defect, a sort of topological defect usually
regarded as a semi-infinite rod with spin Bonnor:1969ala . Solution (59), as
it has been studied already in great detail in Ref. Podolsky:2020xkf  for the
bold case, also contains both type of topological structures. Its full
analysis, along with a proper study of the algebraic classification of the
solution (that here we have anticipated to be of the algebraically general
type I), will be presented in Ref. barrientosetall .

##  V Reissner-Nordström accelerating NUT black hole

In this section we tackle the construction of a NUT version of the Reissner-
Nordström C-metric. As roughly mentioned before, the presence of
electromagnetic charges restricts the usefulness of the standard electric
Ehlers transformation as a mechanism of endorsing NUT to a given spacetime. It
was proven in Ref. Astorino:2019ljy , by adding NUT onto the Kerr-Newman
solution, that the Ehlers map requires to be complemented with a duality
transformation affecting the electromagnetic potential. Specifically, the
Ehlers map rotates not only the mass, producing the appearance of the NUT
parameter, but also the gauge vector generating its misalignment. In order to
obtain a properly aligned gauge field, a further duality transformation, that
can be seen as a subcase of (15a), is required to act on the electromagnetic
potential. Then, the Kerr-Newman-NUT solution is obtained. An enhanced Ehlers
transformation, superposing the effect of the original Ehlers map and the
aforementioned duality transformation, was also provided and shown to
correctly add NUT to a nonaccelerating charged spacetime while simultaneously
maintaining the correct alignment of the electromagnetic potential. Here, we
shall see that complementing the usual Ehlers map with the aforementioned
duality transformation correctly produces an accelerating NUT Reissner-
Nordström solution with the correct alignment of the electromagnetic potential
in the corresponding limiting cases. To proceed, we start from the line
element (19) with metric functions

$\displaystyle
Q(R)=\left(1-A^{2}R^{2}\right)\left(1-\frac{2M}{R}+\frac{e^{2}+g^{2}}{R^{2}}\right),$
(70) $\displaystyle
P(\theta)=1+2AM\cos\theta+A^{2}\left(e^{2}+g^{2}\right)\cos^{2}\theta,$ (71)

and with the electromagnetic potential

$A=-\frac{e}{R}dt+g\cos\theta d\phi.$ (72)

We consider both electric and magnetic charges. Again, by comparison, we can
easily find the seed metric functions which take the same form as those of the
uncharged case (23). However, due to the presence of the electromagnetic
charges, we also need to identify the electromagnetic potentials composing the
seed potential $\Phi_{0}$. This, in turn, implies the computation of the seed
twisted potential $\tilde{A}_{\varphi 0}$. Using (13), we obtain

$\tilde{A}_{\varphi 0}=-\frac{g}{R}$ (73)

and identify the seed Ernst potentials

$\mathcal{E}_{0}=\frac{Q}{\Omega^{2}}-\frac{e^{2}+g^{2}}{R^{2}},\quad\Phi_{0}=-\frac{e+ig}{R},$
(74)

which by means of the corresponding Ehlers transformations deliver

$\mathcal{E}=\frac{\frac{Q}{\Omega^{2}}-\frac{e^{2}+g^{2}}{R^{2}}}{\Lambda},\quad\Phi=\frac{-\frac{e+ig}{R}}{\Lambda}.$
(75)

Here we have defined

$\Lambda(R,\theta)=1+ic\left(\frac{Q}{\Omega^{2}}-\frac{e^{2}+g^{2}}{R^{2}}\right).$
(76)

As usual, the new metric functions and the new electromagnetic potential can
be easily obtained by taking the real and imaginary parts of $\mathcal{E}$ and
$\Phi$. For the metric functions, we have

$f=\operatorname{Re}\mathcal{E}+|\Phi|^{2}=\frac{\frac{Q}{\Omega^{2}}}{|\Lambda|^{2}},$
(77)

while the imaginary part

$\chi=\operatorname{Im}\mathcal{E}=-c\frac{\left(\frac{Q}{\Omega^{2}}-\frac{e^{2}+g^{2}}{R^{2}}\right)^{2}}{|\Lambda|^{2}}$
(78)

provides, via (14), the corresponding function $\omega$. We obtain

$\omega(R,\theta)=2c\left(2M\cos\theta-A\left(e^{2}+g^{2}\right)\sin^{2}\theta+\frac{APR^{2}\sin^{2}\theta}{(1+AR\cos\theta)^{2}}\right).$
(79)

On the other hand, for the gauge potentials, we observe that

$\displaystyle
A_{t}(R,\theta)=\operatorname{Re}(\Phi)=-\frac{\frac{e}{R}}{|\Lambda|^{2}}-c\frac{\frac{g}{R}\left(\frac{Q}{\Omega^{2}}-\frac{e^{2}+g^{2}}{R^{2}}\right)}{|\Lambda|^{2}},$
(80)
$\displaystyle\tilde{A}_{\varphi}(R,\theta)=\operatorname{Im}(\Phi)=-\frac{\frac{g}{R}}{|\Lambda|^{2}}+c\frac{\frac{e}{R}\left(\frac{Q}{\Omega^{2}}-\frac{e^{2}+g^{2}}{R^{2}}\right)}{|\Lambda|^{2}}.$
(81)

Notice that here we have obtained $\tilde{A}_{\varphi}(R,\theta)$ and not the
real magnetic field. It is then necessary to use Eq. (13) to obtain
$A_{\varphi}$, for which it is first mandatory to know the explicit form of
the rotation function $\omega$. The magnetic field is found to be

$A_{\varphi}(R,\theta)=c\frac{eP\sin^{2}\theta}{(1+AR\cos\theta)^{2}\cos\theta}+g\cos\theta-c\frac{e}{\cos\theta}-\omega
A_{t}.$ (82)

At this stage, it is useful to proceed with the same reparametrization of the
mass parameter and change of coordinates performed in the uncharged case
provided in Sec. III. The line element is given by (45) with metric functions

$\displaystyle\Omega(r,\theta)$ $\displaystyle=1+A(r-r_{-})\cos\theta,$ (83)
$\displaystyle\mathcal{P}(\theta)$
$\displaystyle=1+A(r_{+}-r_{-})\cos\theta+A^{2}(e^{2}+g^{2})\cos^{2}\theta,$
(84) $\displaystyle\mathcal{Q}(r)$
$\displaystyle=(1-A^{2}(r-r_{-})^{2})(r^{2}-2mr-l^{2}+e^{2}+g^{2}),$ (85)
$\displaystyle\mathcal{T}(r,\theta)$
$\displaystyle=\frac{\mathcal{P}(\theta)(r-r_{-})^{2}}{(r_{+}-r_{-})\Omega(r,\theta)^{2}}-\frac{e^{2}+g^{2}}{r_{+}-r_{-}},$
(86) $\displaystyle\mathcal{R}^{2}(r,\theta)$
$\displaystyle=\frac{1}{r_{+}^{2}+l^{2}}\left[r_{+}^{2}(r-r_{-})^{2}+l^{2}\frac{\left(\mathcal{Q}(r)-(e^{2}+g^{2})\Omega(r,\theta)^{2}\right)^{2}}{\Omega(r,\theta)^{4}(r-r_{-})^{2}}\right],$
(87) $\displaystyle\omega(r,\theta)$
$\displaystyle=\frac{2l(r_{+}-r_{-})}{r_{+}}\left(\cos\theta+A\mathcal{T}(r,\theta)\sin^{2}\theta\right),$
(88)

while the corresponding electromagnetic potentials turn to be

$\displaystyle A_{\tau}(r,\theta)$
$\displaystyle=\frac{(r-r_{-})^{2}}{\mathcal{R}^{2}(r,\theta)}\left(-\frac{e}{(r-r_{-})}-gl\frac{(\mathcal{Q}(r)-(e^{2}+g^{2})\Omega(r,\theta)^{2})}{r_{+}\Omega(r,\theta)^{2}(r-r_{-})^{3}}\right),$
(89) $\displaystyle A_{\varphi}(r,\theta)$
$\displaystyle=l\frac{e\mathcal{P}(\theta)\sin^{2}\theta}{r_{+}\cos\theta\Omega(r,\theta)^{2}}+g\cos\theta-l\frac{e}{r_{+}\cos\theta}-\omega(r,\theta)A_{\tau}(r,\theta).$

It can be observed that from the line element it is direct to obtain the NUT
Reissner-Nordström metric in the limit of vanishing acceleration. However, as
we have pointed out before, the Ehlers transformation also rotates the vector
potential, and, as a consequence, the usual gauge field of the NUT Reissner-
Nordström configuration is not retrieved. It has been proposed in Ref.
Astorino:2019ljy  that an extra duality rotation acting on the vector
potential

$\Phi\rightarrow\bar{\Phi}=e^{i\beta}\Phi,$ (90)

with $\beta$ a constant, fixes the problem. This was proven for the case in
which the Kerr-Newman solution is affected by an Ehlers transformation in
order to obtain the Kerr-Newman-NUT spacetime. In addition, the usual Ehlers
and the previous duality transformations were combined to produce an enhanced
Ehlers transformation that automatically adds a NUT parameter onto a charged
spacetime with no extra rotation of the gauge potential. For the sake of
simplicity, here we consider $g=0$ and apply the duality transformation (90)
in order to align the gauge field in the corresponding subcases in which
either the acceleration or the NUT parameter vanishes. Before moving to the
$(\tau,r)$ coordinates and previous to the redefinition of the mass $M$, the
duality transformed electric and twisted magnetic potentials are given,
respectively, by

$\displaystyle\bar{A}_{t}(R,\theta)$
$\displaystyle=\operatorname{Re}(\bar{\Phi})=-\frac{\frac{e}{R}}{|\Lambda|^{2}}\cos\beta-c\frac{\frac{e}{R}\left(\frac{Q}{\Omega^{2}}-\frac{e^{2}}{R^{2}}\right)}{|\Lambda|^{2}}\sin\beta,$
(91) $\displaystyle\bar{\tilde{A}}_{\varphi}(R,\theta)$
$\displaystyle=\operatorname{Im}(\bar{\Phi})=c\frac{\frac{e}{R}\left(\frac{Q}{\Omega^{2}}-\frac{e^{2}}{R^{2}}\right)}{|\Lambda|^{2}}\cos\beta-\frac{\frac{e}{R}}{|\Lambda|^{2}}\sin\beta,$

from where the dipole magnetic potential is extracted:

$\displaystyle\bar{A}_{\varphi}$
$\displaystyle=\frac{ce}{\Omega^{2}}\left[e^{2}A^{2}\sin^{2}\theta\cos\theta+2AM\sin^{2}\theta-(\Omega+1)AR-\cos\theta\right]\cos\beta+e\cos\theta\sin\beta-\omega\bar{A}_{t}.$
(92)

In the $A\rightarrow 0$ limit, we get that

$\displaystyle\bar{A}_{\varphi}$
$\displaystyle=-ce\cos\theta\cos\beta+e\cos\theta\sin\beta-\omega\bar{A}_{t},$
(93)

an expression that can be easily rearranged into the fashion
$\bar{A}_{\varphi}=-\omega\bar{A}_{t}$, by suitably selecting the constant
parameter $\beta$. Passing to $\tau$, $r$, and $m$ and rescaling the
coordinates $\tau$ and $r$ and the parameters $m$, $l$, and $e$ by the
constant factor $\sqrt{\frac{r_{+}}{r_{+}-r_{-}}}$, the electric potential
reduces to

$\bar{A}_{\tau}=-\frac{er}{r^{2}+l^{2}},$ (94)

namely, to the standard NUT Reissner-Nordström gauge field.888An accelerating
version of the NUT Reissner-Nordström black has been also constructed in Ref.
Astorino:2023elf . A hierarchy of solutions of this kind is depicted in FIG.
2.

Figure 2: Hierarchy of solutions contained in the Reissner-Nordström
accelerating NUT spacetime.

##  VI Outlook

Our study leverages Ernst’s insightful description of axially symmetric and
stationary spacetimes to devise a novel and more computationally efficient
approach for constructing accelerating NUT black holes. Specifically, we
employed an electric Ehlers transformation on the conventional C-metric to
rederive Chng, Mann, and Stelea’s solution Chng:2006gh  while also presenting
for the first time its scalar conformally dressed generalization. In addition,
we provide the Reissner-Nordström accelerating NUT black hole of Einstein-
Maxwell theory, providing the charged extension of the Chng, Mann, and Stelea
solution written in Podolský and Vrátný coordinates. A duality transformation
has been applied on top of the usual Ehlers map in order to restore the
alignment of the gauge field in the corresponding nonaccelerating limit. As a
matter of fact, the outcome of this procedure is the obtention of the proper
Reissner-Nordström NUT black hole configuration in the vanishing acceleration
case. Indeed, this solution is of type I algebraic nature. In our
constructions, we have found the coordinate system introduced by Podolský and
Vrátný Podolsky:2020xkf  to be particularly useful, as it allows for the
transparent identification of the NUT charge. Moreover, it is the appropriate
change of coordinates, no matter the presence of electromagnetic charges, that
allows for clean nonaccelerating limits. Our metric in the conformally coupled
case possesses clear limits to well-known subcases, including the conformally
dressed C-metric Charmousis:2009cm ; Anabalon:2009qt  and NUT
Bhattacharya:2013hvm ; Bardoux:2013swa  black holes, which are obtained by
taking the limits $l\rightarrow 0$ and $A\rightarrow 0$ on Eq. (60),
respectively. These conformally dressed spacetimes, which depart from the
Plebanśki-Demiański family, could not have been obtained from the wide family
of metrics presented in Ref. Anabalon:2009qt .

It is known that the source of acceleration in these spacetimes is given by
the presence of cosmic strings or struts, which is an unavoidable consequence
of a nontrivial function $P(\theta)$ causing a deficit or excess angle on the
metric. To construct accelerating spacetimes with no conical singularities, it
is necessary to provide an alternative mechanism for the acceleration.
Therefore, it would be interesting to explore immersing these types of
solutions in external magnetic fields Astorino:2013xc  or in rotating
backgrounds as has recently been done for the standard C-metric
Astorino:2022prj . This can be easily performed by using magnetic Harrison and
Ehlers transformations. There are several other avenues that can be explored
to continue studying these geometries. In the context of our charged and hairy
accelerating NUT black hole, it would be desirable to study its Euclidean
version and its Eguchi-Hanson formulation Eguchi:1978xp ; Eguchi:1978gw . This
would allow us to explore charged and hairy accelerating gravitational
solitons, an avenue that has thus far been explored only for the
nonaccelerating case Barrientos:2022yoz . Additionally, the thermodynamic
analysis of these solutions offers a highly nontrivial arena for exploration.
As is known, the C-metric and NUT solutions in GR already offer interesting
subtleties regarding the computation of their conserved charges and
thermodynamic behavior BallonBordo:2021pzm ; Clement:2019ghi ;
Ciambelli:2020qny ; Anabalon:2018qfv . Therefore, accelerating NUT geometries
provide an interesting class of spacetimes to test standard methods for
computing charges and black hole thermodynamic laws. On the contrary,
investigating the geodesic motion of accelerating NUT black holes presents a
promising avenue for exploration. Recent findings have demonstrated that
geodesic observers in NUT spacetimes can avoid encountering the Misner string
or experiencing any causality violations, provided that the constant parameter
resulting from the coordinate transformation that determines the position of
the Misner string is appropriately restricted Clement:2015cxa . This allows
for reinterpretation of NUT black holes, in some cases as traversable wormhole
geometries with no energy violation whatsoever Clement:2015aka ;
Clement:2022pjr . Then, it would be interesting to seek for accelerating NUT
traversable wormholes, especially in the charged case scenario.

Last but not least, a major challenge will be the construction of a rotating
(Kerr-like) extension of these (charged) accelerating NUT spacetimes. As
pointed out in Ref. vratny , an educated guess indicates that adding rotation
to these spacetimes would guide us to a different solution than the metric
contained in the Plebanśki-Demiański family. This is expectable from the fact
that, contrary to the spinning C-metric with NUT charge contained in the
Plebanśki-Demiański line element, here our accelerating NUT spacetimes are not
centered with respect to $r=0$ but with respect to the position of the inner
horizon $r=r_{-}$. This can be observed from the $\left(r-r_{-}\right)$
dependence of the conformal factor and accelerating horizons. In addition, the
nonrotating solution already departs from the wide Plebanśki-Demiański class,
and rotation is not expected to restore the special algebraic character of
this spacetime. An analytic construction of such a spacetime would be hard to
imagine without the utilization of sophisticated solution generating
techniques; we expect that some light might be shed on this problem by
considering the inverse scattering method Belinski:2001ph .

##  VII Acknowledgments

We thank Professor Jiři Podolský for his valuable comments and suggestions.
The work of J. B. is supported by the Academy of Sciences of the Czech
Republic (RVO 67985840), Project No. L100192101 and by FONDECYT Grant No.
3230596. A. C.’s work is funded by FONDECYT Regular Grant No. 1210500. Both
authors appreciate the technical support provided by Marcel Yáñez when dealing
with heavy computational calculations.

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# Instability of social network dynamics with stubborn links

S. Sheykhali
Department of Physics
University of Zanjan (ZNU)
Zanjan, Iran
<EMAIL_ADDRESS>
&A.H. Darooneh
Department of Physics
University of Zanjan (ZNU)
Zanjan, Iran
&G.R. Jafari
Department of Physics
Shahid Beheshti University
Tehran, Iran
Corresponding AuthorDepartment of Network and Data Science, Central European
University, Budapest, Hungary

###### Abstract

This paper studies the signed networks in the presence of stubborn links,
based on the structural balance theory. Each agent in the network has a
mixture of positive and negative links represent friendly and antagonistic
interactions, and his stubbornness about interactions. Structural balance
theory affirms that in signed social networks with simultaneous
friendly/hostile interactions, there is a general tendency of evolving over
time to reduce the tensions. From this perspective, individuals iteratively
invert their own sentiments to reduce the felt tensions induced by imbalance.
In this paper, we investigate the consequences of the agents’ stubbornness on
their interactions. We define stubbornness as an extreme antagonistic
interaction which is resistant to change. In the current paper we investigated
if the presence of stubborn links renders an impact on the balance state of
the network and whether or not the degree of balance in a signed network
depends on the location of stubborn links. Our results show that a poorly
balanced configuration consists of multiple antagonistic groups. Both
analytical and simulation results demonstrate that the global level of balance
of the network is more influenced by the locations of stubborn links in the
resulting network topology than by the fraction of stubborn links. This means
that even with a large fraction of stubborn links the network would evolve
towards a balanced state. On the other hand, if a small fraction of stubborn
links are clustered in five stubborn communities, the network evolves to an
unbalanced state.

_K_ eywords Structural balance theory, Signed network, Stubborn links, Link
communities

##  1 Introduction

The problem of minimizing social tension or frustration among a group of
agents contributes remarkably to the development of interactions, hence
shaping communities, alliances, and political groups. Structural balance
theory (SBT), proposed by Heider [1], was the first attempt to explain the
structure and origin of human tensions in terms of friendship and hostility
relationships. This yields that social systems with simultaneous
friendly/hostile interactions tend to evolve towards a more balanced situation
with minimum stress. Signed networks representing the relation (positive or
negative, i.e. friendly or hostile) between interacting agents can provide us
with significant insights into unveiling the global properties from local
interactions [2, 3, 4, 5]. Cartwright and Harary [6] formerly modeled SBT to
analyze the dynamics and construction of signed networks. Structural balance
theory studies the formation, the dynamic features of pairwise interactions
among individuals or communities and finally the system’s final configuration
of interactions.

A generalization of Heider’s original hypothesis has been extensively applied
to many interesting real-world systems, including social systems [7, 8, 9,
10], the entities within economical activities [11], ecologic communities [12,
13] and biological neural networks[14]. There has been a large number of
previous works studying structural balance in signed networks [15, 16], using
methods motivated by spin glasses [17, 18], the dynamic evolution of
structural balance [19, 20, 21, 22, 23], reverse transformation of balanced
structure [24], time-varying relationship strengths [25] and networks with
non-active edges [26].

The SBT states that the network is either balanced or moves towards a balanced
state. For the latter case, a measure of balance is needed to trace movement
towards balance. Energy function (as specifically defined in Sec. 2), stated
in terms of cycles is defined to measure the balance degree of the signed
network. If a network becomes balanced, it achieves the global minimum of the
energy landscape. However, the energy landscape also contains many metastable
states, i.e., local minima, named as jammed states [19]. The concept of such
energy landscapes has been studied in a wide range of systems from structural
glasses [27] to brain activity [28] and social networks [29], to explain how
the system gets trapped into local minima as it moves down the energy
landscape. Following the SBT, balance dynamics act as the driving mechanism
bringing the network to its minimum energy state. In its dynamical evolution,
the network may fall into a jammed state in which no single sign change can
decrease the energy of the system. Marvel et al. [29] observed that the
maximum energy allowed for a jammed state is usually found in the lower half
of the energy spectrum and it is hard for any system to achieve this upper
bound. However, the transformation cost of changing an edge from -1 to 1 is
ignored during the dynamical or optimization process. The findings reveal some
factors affecting the relationship outside of balance theory in the real
world. There is a growing literature studying opinion dynamics in social
systems with stubborn agents (whose opinions are kept constant throughout
time) [30, 31, 32, 33, 34]. Therefore, complete balance is almost never
achieved in real systems as a consequence of the complexity of interactions
[33]. As such, the decisions that people make and the beliefs that people hold
have profound consequences on the structure of the network. Here we consider
social systems in the presence of "stubborn links", which basically do not
follow the network dynamics of attitude change, which is likely to generate
tension. We study the stubbornness of negative links, since in accordance with
SBT they play a key role in both the structure and dynamics of the network.
How does the degree of balance of any signed network depend on the locations
of stubborn links? We analyze under which conditions we can avoid certain
conflicting situations.

The SBT at most predicts partitioning nodes into one or two strong subsets
(bipolar) at the macro-level. This strict condition makes it quite unlikely
for a signed network to achieve complete balance in practice. Human social
networks are rarely found to be in general strongly balanced ([35, 36]). As
most empirical structures are not balanced [37], Doreian and Mrvar have
proposed a method for the composition of the network to plus-sets (more than
two) as close to balance as possible [38].

In this study, the community formation of stubborn links is observed. Such
structures can result in preventing the network from falling into a socially
balanced absorbing state. A link-clustering approach to detect communities in
terms of closely interrelated links was proposed by Ahn et al. [39]. Here, we
discuss how the so-called stubborn negative links presence results in the
system achieving high energy of a jammed state. By means of the energy
function and the signed network topology, we address the relationship between
them in the presence of such stubborn links. Moreover, we study how many
negative subgraphs exist on a network and how does this number depend on the
number of nodes. Through simulations, we show that a network can non-trivially
tolerate a large population of stubborn links depending on their distribution.
Still, the network with a large number of stubborn links does not have a high
probability to reach a balanced state. Finally we study how these links affect
the energy of the network in terms of imbalanced triads clusters and
distribution of community sizes.

##  2 Structural Balance with stubborn links

Following structural balance theory, the two signed triadic configurations (+
- -) and (+ + +) are considered as balanced and (+ + -) and (- - -) signed
triadic configurations are considered as unbalanced. The two unbalanced
triangles are identified as a source of inherent tension which may drive them
into more balanced configurations (Figure. 1).

Figure 1: Different balanced and unbalanced configurations of a triangle.
Solid lines with ’+’ label represent positive edges and dashed lines with ’–’
label represent negative edges.

We begin by modeling a fully connected network as a signed complete graph
$G(N;S)$, with $N$ agents and $S$ edges. Each positive or negative edge
$S_{ij}$ representing a feeling of trust/mistrust, like/dislike, or
friendship/enmity between the node $i$ and $j$ [40, 17]. Following the SBT, a
potential energy function to quantify the degree of balance of any signed
social network is defined as[29]:

$\displaystyle E=\frac{-1}{\binom{N}{3}}\sum S_{ij}S_{jk}S_{ki}.$ (1)

where $S_{ij}$, $S_{jk}$, and $S_{ki}$ are signed links forming a triad and
the sum is overall triangles ${i,j,k}$ of the network. The energy ranges
between $-1$ and $+1$ corresponding to a balanced and an imbalanced structure
respectively.

Structural balance theory provides the basic components of a dynamic model by
specifying sign changes of links in the unbalanced triples to be balanced—a
graph is balanced if all of its triples are balanced. Two internally friendly
cliques with mutual antagonism between them, is the only sign configuration in
global minima (The paradise configuration is considered as the extreme in
which one clique is empty).

###  2.1 Stubborn links

There is no guarantee, however, that all of the links in the network will
cooperate in balance dynamics. Some links may refuse to switch and lead the
network to an imbalanced final state (stubborn links). In this term, a network
may fall into a final state containing imbalanced triads (Figure. 1). Changing
the sign of any such negative links would thereby render all the triads
imbalanced. To create the network, we first generated a matrix of $N$ nodes.
The signs "+" or "-" are randomly assigned to the links. The negative stubborn
links ($n_{s}=1:N(N-1)/2$) are randomly distributed. Generally, we observe
that the presence of stubborn links affects the final balance state, and the
deviation depends not only on the fraction but specifically on the location of
the stubborn links in the network structure.

###  2.2 Stubborn Communities

In a balanced system, the individuals can be partitioned into two or more
subgroups [41] or balanced cliques to minimize the structural conflicts. It
has been shown natural modular structure for networks in jammed states [29].
The states with high-energy local minima must be structurally more complex
than low-energy ones. The communities formed by nodes with only negative or
very few positive links have been studied before [42]. We define stubborn
communities as those cliques made of stubborn links only and having positive
links with the rest of the network. Our study shows that when stubborn
communities are formed (Figure. 2), a few number of stubborn links can
substantially make the energy positive.

Figure 2: Schematic pattern of a network with stubborn links (red dash lines)
clustered in stubborn communities (blue dashed circles); they provoke a
stubborn-community, i.e. stubborn links forming a cluster.

It can be safely said that a tendency to form community structure (community
by stubborn links) is present. This can be quantified with the clustering
coefficient, $C$ Which is defined as the probability that two incident edges
are completed by a third edge to form a triangle number of closed triangles
[43]:

$C=\frac{\\{number\;of\;closed\;traingles\\}}{\\{number\;of\;all\;open\;and\;closed\;triplets\\}}$
(2)

In order to measure the density of stubborn communities, we use the clustering
coefficient measure for negative sign link configurations. Figure. 3.B shows
the scatter plot of energy against the clustering coefficient. What we observe
in this Figure is that the formation of stubborn communities moderately grows
with energy, while it remains fairly constant in $E=0$ for $0.3<C<0.7$ and
again grows to $C=1$ as unbalanced communities grow in numbers.

Figure 3: (A) Energy as a function of the fraction of stubborn links for a
network with $N=100$ nodes. Each point represents the final state of a single
realization. The red points represent the energy reached by networks with a
random displacement of stubborn links in the network. The blue points
represent the extreme values of the energy of the final state when stubborn
links form communities. The blue solid lines are related to a network with
approximately same size stubborn communities, the upper line represents the
communities formed by the minimum fraction of stubborn links and the lower
line correspond to the maximum fraction of stubborn links. (B) Energy as a
function of the clustering coefficient (C) for stubborn links in networks in
the final state of every simulation.

Here, the optimization problem is to minimize the fraction of stubborn links
necessary to reach an imbalanced state of a network. Starting from an initial
random sign configuration, we select and switch single link signs until the
network falls into a local minimum. In order to have a statistic of the final
states, in every 1000 independent realizations, we embed stubborn links
randomly in a fully connected sign networks, with fraction varying from 0 to
1, and calculated the energy. The red points in Figure. 3.A, show the energy
of the final state versus the fraction of stubborn links when they are
randomly displaced. Stubborn links clustering is crucial for the definition of
the final state network energy, as shown in Figure. 3.B. Although, a high
fraction of such links does not imply a high clustering coefficient. There is
a minimum fraction of such links able to move the network to a positive energy
final state. In this case, we define the energy upper bound as the maximum
energy reachable by the system with this configuration. On the other side
there is also a maximum number of stubborn links able to keep the system into
negative energy final states. We define the range of variation of the energy
function with respect to these extremes and the energy lower bound as the
minimum energy reachable by this configuration.

Without stubborn links, the network eventually reaches a balanced state. By
imposing a subset of stubborn links the energy changes significantly. A small
fraction of stubborn links has more chance to locate randomly in the network
in such a way that they do not link to any other stubborn link. With few
stubborn links (less than $20\%$ of all the links) the chance of reaching a
balanced state is still high (Figure. 3). However, by increasing the number of
stubborn links many imbalanced triads cannot reach a balance, resulting in a
frozen state. In this case, the only way to reduce the energy of the system
would be by flipping a stubborn link, which does not occur by definition. We
aim to determine the optimal placement of the minimum number of stubborn links
with the maximum impact on the energy.

The spatial distance apart stubborn links allowing more triangles to be
unbalanced. This feature cause more (- + +) unbalance triangles and also
increases energy. Progressively increasing the fraction of stubborn links,
they less and less likely locate separately and the probability of connecting
each other increases. Which the number of (- - +) balance triangles increases
and also energy can decreases. On the other hand, more stubborn links, more
probability of clustering, more probability of reaching high energy (red
points in 3.A). When stubborn links assemble in negative communities, the
number of (- + +) unbalance triangles increases causing a raise of energy
(upper blue points in 3.A). The lower blue points in 3.A show the extremely
opposite state with positive links inside communities and stubborn negative
links between them. We quantify these observations by measuring the clustering
coefficient ($C$) of the negative stubborn links in the network, which shows
how they progressively assemble into communities (3.B).

Figure 4: (A) Distribution of energy of the final state. (B) Distribution of
number of triangles for any of the four types. For networks with $N=100$ nodes
and $1000$ realizations.

For energy reduction, the distribution of balanced and unbalanced triangles
should be taken explicitly into account. In order to visually compare how the
energy and the number of balanced and unbalanced triangles change as the
fraction of stubborn links increases, we plot in Figure. 4 the distribution of
triangles and the distribution of energy. As expected, energy increases and
the range of energy decreases due to stubborn links higher density. This is
related to the fact that increasing the fraction of stubborn links causes more
unbalanced (- - -) triangles to appear.

The tendency of a network over time to become more balanced leads to a
reduction in the number of imbalanced triangles, which in this case is
followed by a reduction in the number of negative links. On the other hand, to
find the minimum number of negative links to allow a positive energy final
state we should look at the configurations with more positive links. It
ultimately could turn the system into an unstable state in which there is no
propensity to reduce the unbalanced triangles in favor of balance [44]. Our
results show that stubborn links forming communities may lead the system to
fall into a local minimum with positive energy. Even a small fraction of
stubborn links may evolve into stubborn communities.

##  3 OPTIMUM NUMBER OF Stubborn-Communities

Marvel et al. [29] showed that jammed states may be derived from the
undirected Paley graphs [45]. Our first result is that local minima of
clustered networks with stubborn links can have energies above zero (defined
in Eq. 1). To see this, note that in a balanced state, there are at least as
many unbalanced triangles as balanced triangles. A triad is balanced if any
three nodes from any three distinct stubborn-communities are selected (+ + +)
or two nodes from one community and the third one from another community (+ -
-). On the other hand, three nodes from the same community form an unbalanced
triangle. There are therefore $\sum_{i}\binom{m_{i}}{2}(N-m_{i})+\binom{m}{3}$
balanced triangles and $(N-m_{i})\binom{m_{i}}{3}$ unbalanced triangles, where
$i=1,...,k$ is the number of cliques and $m_{i}$ is the number of vertices in
the clique i.

Thus, summing over all triads yields the upper bound on the energy function:

$\displaystyle
E\geqslant-\sum_{i}\frac{m_{i}\binom{k}{3}-\frac{m_{i}(m_{i}-1)}{2}k(N-m_{i})+\binom{m_{i}}{3}}{\binom{N}{3}}.$
(3)

Our results show that the number of stubborn communities formed in a network
with similar size is limited (Figure 3). Consider a fixed number of stubborn
cliques m, in a network of size N. If all N nodes in the network are uniformly
distributed among the $m\sim N/k$ cliques, we have:

$\displaystyle
E\geqslant-\frac{\frac{N}{k}\binom{k}{3}-\frac{N}{2k}(\frac{N}{k}-1)k(N-\frac{N}{k})+\binom{\frac{N}{k}}{3}}{\binom{N}{3}}.$
(4)

In the extreme case $k\rightarrow\infty$, the condition implies that the
number of stubborn-communities will be $m\sim 5$. This is consistent with the
obtained numerical results, which point at the existence of bounds for the
stubborn-community size. Figure. 5 shows the numerical solution of energy as a
function of the number of clusters. As presented in this figure, the number of
these stubborn-communities is limited to five communities.

Figure 5: Numerical solution of energy (E) as a function of number of clusters
(m), for a network of N=100 nodes.

One possible way to find the sign configuration that achieves this upper bound
on E is through computational searches: From an initial state in which all the
edges are negative, one edge is randomly selected and the sign switches
following SBT. This process continues until the network reaches a local
minimum of E. For a small network in a zero-energy local minimum,
configurations on nodes with several stubborn links are complex. Finding such
states on larger networks is more computationally expensive. We test whether
such communities are significantly likely to appear in the networks:

$\displaystyle P_{m,k}\propto\frac{1}{\binom{N}{k}\frac{k(k-1)}{2}k(N-k)}$ (5)

where $P_{m,k}$ is the probability that $k$ nonempty $m$-vertices stubborn-
communities form in $G$. Which $P_{m,k}$ is calculated as the product of the
probability that no vertex inside the subgraph is connected to any outside
vertex ($m(N-m)$) and the probability that the vertices are fully connected,
for the same size communities $m=\frac{N}{k}$. With large numbers of nodes,
($N\rightarrow\infty$), $P_{m,k}\rightarrow 0$. Which is consistent with
findings of Antal et al. [19], who showed that probability to reach jammed
states using such local search methods decreases to zero as a function of the
network size. Such a probability increases in the presence of stubborn links.
According to the computational difficulty in finding such local minima via
algorithmic search, we now show how the sign configuration in positive-energy
can be seen through a direct construction which is motivated by computational
searching for small examples.

Consider a fixed number of stubborn cliques k in which the nodes are
equidistributed among the stubborn communities. As the fewest number of edges
are in the cliques, hence the least edges are available for inclusion in
unbalanced triangles. The simulations show that the stubborn-communities with
an equal number of nodes, move the network towards positive energy with a
lower stubborn link ratio in comparison with randomly distributed nodes(Figure
3B). We now show that this clustered signed complete graph has positive
energy. Consider a network with $N$ nodes and $m$ non-overlapping mutually
antagonistic cliques of equal sizes $k=N/m$. In the extreme state, all
negative links are stubborn and flipping positive links leads to increase
unbalanced triads, so energy is increased by $\delta E$:

$\displaystyle\delta E=\binom{m}{2}(N-k)+\binom{m}{2}[N-2(k)-2\binom{k}{2}k].$
(6)

According to the dynamic models for structural balance, the overall number of
imbalanced triads cannot increase in an update event, which implies an upper
bound on the energy ($E$). Hence, the network evolves into sub-communities
with uniformly distributed negative links having the maximum energy and all
adjacent sign configurations having lower energies. This lower bound is
related to the constraints $\sum_{m}c_{m}=n$, where $c_{m}$ is the number of
nodes in the $m$th stubborn-community. If all the nodes are distributed in $5$
roughly equal-sized groups, then the proportion of unbalanced triangles (and
consequently energy) would be maximized. This configuration is the most
stressful configuration with the minimum number of stubborn links.

Figure 6: S-S distribution of stubborn degree for a network with $N=20$ nodes.
The symmetric graphics represent the cases of the minimum number of stubborn
links with positive energy.

The correlation between nodes degree based on their number of stubborn links
also shows interesting behavior (S in Figure 6). Nodes with the same
stubbornness degree (defined as the number of stubborn links normalized by the
node’s degree) are strongly connected to each other. In other words, stubborn
nodes tend to connect to those who are similar to them, a condition essential
in peer to peer multi-agent competition and cooperation dynamics [46]. The
graphics represent the case of the minimum number of stubborn links with
positive energy. This correlation supports the idea that emerging stubborn-
communities formed by stubborn links lead to positive energies. We argue that
if networks with stubborn links are divided into stubborn-communities, the
division alone can produce both degree-degree correlations and clustering.
This evidence is compatible with the results regarding the stubborn-community
forming.

##  4 Conclusions

While structural balance theory shows the tendency of locally interacting
agents to avoid conflictual situations, this dynamic changes upon the
introduction of stubborn links into social systems. We investigated how
instability can emerge as a result of such local stubborn interactions. By
means of the structural balance theory, we study the fraction of stubborn
links and where to place them in order to maximize the energy. Our results
show that these links which are typically few in numbers can alter the
equilibrium of the balance dynamics in a fully connected signed network. We
observed the tendency to form stubborn communities, such that there is a kind
of stubbornness assortativity emerging from the balance dynamics.
Specifically, the final state energy of networks does not directly depend on
the fraction of stubborn links, it depends on their position in the network,
i.e. on the way they topologically assemble into the system, e.g. stubborn
communities. To understand how much the fraction of stubborn links could
affect the network energy, we numerically found upper and lower bounds to the
energy function depending on the number of stubborn links present in the
network. We computed the maximum positive energy reachable by a minimum
fraction of such links and the minimum negative energy reachable by a maximum
fraction of stubborn links. Our analytical results show that in the infinite
network size limit, the energy upper bound is reachable with only 5 stubborn
communities. This results provide new insights on SBT dynamics in presence of
stubborn relations among people, parties or countries.

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