aainstitutetext: Sede Esmeralda, Universidad de Tarapacá, Avenida Luis Emilio
Recabarren 2477, Iquique, Chilebbinstitutetext: Institute of Theoretical
Physics, Faculty of Mathematics and Physics, Charles University, V
Holešovičkách 2, 180 00 Prague 8, Czech Republicccinstitutetext: Departamento
de Física, Universidad de Concepción, Casilla, 160-C, Concepción,
Chileddinstitutetext: Laboratory of Theoretical Physics, Institute of Physics,
University of Tartu, W. Ostwaldi 1, 50411 Tartu, Estonia

# Mixing “Magnetic” and “Electric” Ehlers–Harrison transformations: The
Electromagnetic Swirling Spacetime and Novel Type I Backgrounds

José Barrientos a,b Adolfo Cisterna b Ivan Kolář c Keanu Müller c Marcelo
Oyarzo d and Konstantinos Pallikaris<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper, we obtain a complete list of stationary and axisymmetric
spacetimes, generated from a Minkowski spacetime using the Ernst technique. We
do so by operating on the associated seed potentials with a composition of
Ehlers and Harrison transformations. In particular, assigning an additional
“electric” or “magnetic” tag to the transformations, we investigate the new
spacetimes obtained either via a composition of magnetic Ehlers and Harrison
transformations (first part) or via a magnetic-electric combination (second
part). In the first part, the resulting type D spacetime, dubbed
electromagnetic swirling universe, features key properties, separately found
in swirling and (Bonnor–)Melvin spacetimes, the latter recovered in
appropriate limits. A detailed analysis of the geometry is included, and
subtle issues are addressed. A detailed proof that the spacetime belongs to
the Kundt family, is included, and a notable relation to the planar-Reissner-
Nordström-NUT black hole is also meticulously worked out. This relation is
further exploited to reverse-engineer the form of the solution in the presence
of a nontrivial cosmological constant. A Schwarzschild black hole embedded
into the new background is also discussed. In the second part, we present four
novel stationary and axisymmetric asymptotically nonflat type I spacetimes,
which are naively expected to be extensions of the Melvin or swirling solution
including a NUT parameter or electromagnetic charges. We actually find that
they are, under conditions, free of curvature and topological singularities,
with the physical meaning of the electric transformation parameters in these
backgrounds requiring further investigation.

##  1 Introduction

Since the advent of Einstein’s equations, the quest for exact solutions to
this set of coupled nonlinear partial differential equations has played an
important part in modern physics Stephani:2003tm . Exact solutions have
significantly aided us in understanding many classical and semiclassical
properties of gravity, and this is exactly what makes them integral to our
comprehension of diverse phenomena occurring at astrophysical and cosmological
scales. Furthermore, these solutions have set the stage for the theoretical
exploration of many groundbreaking concepts, including, but not limited to,
black hole thermodynamics Bekenstein:1972tm ; Bekenstein:1973ur ;
Hawking:1974rv ; Hawking:1975vcx , the information paradox Giddings:1995gd ,
and holography Susskind:1994vu .

Dealing with the field equations of General Relativity (GR) poses a nontrivial
challenge, which reduces to a tractable problem only when a high amount of
symmetry is imposed. In particular, exploiting the Lie point symmetries
inherent in Einstein’s equations, serves as a robust method for generating
exact solutions—solutions that would be practically impossible to integrate
with brute force. Two especially interesting Lie point symmetries of the
Einstein–Maxwell system, Ehlers Ehlers:1957zz ; Ehlers:1959aug  and Harrison
harrison1968new  transformations, suffice for the construction of novel
stationary and axially symmetric spacetimes. These two transformations are
part of a larger set of Lie point symmetries, which exist as hidden symmetries
of the Einstein–Maxwell system of field equations, and which are revealed only
when one formulates the theory in terms of the complex so-called Ernst
potentials Ernst:1967wx ; Ernst:1967by ; they are indeed potential-space
symmetries, the parameters of which comprise an eight-parameter isometry
group, a nonlinear representation of which was originally given in
NeugeKramer69 . The linear representation of this group and the apparent
isomorphism with SU(2,1), was only a few years later delivered by Kinnersley
Kinn1 , who extended the previous results of Geroch Geroch  to include
electromagnetism.

Recently, stationary and axisymmetric spacetimes have received a considerable
amount of attention within the framework of the Ernst description Ernst:1967wx
; Ernst:1967by  of gravity.111These equations were originally derived by Ernst
in Ernst:1967by  for stationary electrovac fields with the further assumption
of axisymmetry. Later, Israel and Wilson Israel:1972vx , as well as Harrison
harrison1968new , rederived them independently for general stationary
electrovac fields. In particular, the significant effect that Ehlers and
Harrison transformations have on accelerating spacetimes, has been explored in
detail Barrientos:2023tqb ; Astorino:2023elf ; Astorino:2023ifg ;
Barrientos:2023dlf . Using Ehlers or Harrison transformations, or a
combination of them for that matter, it has lately been demonstrated that
certain algebraically special accelerating spacetimes can be mapped to novel
algebraically general solutions. For example, some of us have successfully
constructed a complete hierarchy of type I spacetimes Barrientos:2023dlf
obtained via this generating technique, generalizing the well-known
Plebański–Demiański family, i.e., the most general family of type D solutions
in Einstein–Maxwell theory.

To better understand these developments, we shall briefly review the effects
of these transformations. For seed spacetimes cast into the electric form of
the Weyl–Lewis–Papapetrou (WLP) metric,222See the next section for definitions
of “electric” and “magnetic” forms. the standard lore is that Ehlers
transformations introduce an additional real parameter which in certain cases
can be associated with a NUT parameter in the target metric. On the other
hand, Harrison transformations introduce an additional complex parameter,
whose real and imaginary parts, can in certain cases be associated with
monopolic electromagnetic charges in the target configuration. When the seed
is a type D accelerating spacetime, it has been observed that, on top of the
above effects, the transformations also bring about a change in the algebraic
character of the generated solution, namely the target spacetimes are of type
I. An explanation for this peculiar effect seemingly lies in the way that the
transformation parameters enter the new metrics.

To get a good grasp on this, let us for a moment consider the Schwarzschild
black hole as a prototypical seed example. Casting it into the electric form
of the WLP metric and operating on the seed potentials with an Ehlers
transformation, it is known that the resulting spacetime is the Taub–NUT black
hole, modulo coordinate transformations, and parameter redefinitions. In other
words, the transformation—without changing the Petrov type Barrientos:2023tqb
; Astorino:2023elf —has introduced a new parameter, which now, together with
the mass, determines the location of the black hole horizon. In sharp contrast
to this, if we let our seed be the $C$-metric newman1961new ; robinson1962some
; witten1962gravitation , an Ehlers transformation will not only affect the
usual event horizon in the above sense, but also the Rindler one
Barrientos:2023dlf ; Astorino:2023ifg . As a byproduct of this, the resulting
spacetime turns out to be an accelerating Schwarzschild black hole with a NUT-
like parameter, which however is of type I, and as such, it cannot be found
within the Plebański–Demiański type D hierarchy. Similarly, a Harrison
transformation has its two real parameters entering both, black hole and
Rindler, horizons of the $C$-metric, thereby leading to a charged accelerating
black hole of type I and not in a Reissner–Nordström-$C$-metric of type D, as
one may have perhaps expected.

Important insight into the modified Rindler horizons in these type I
accelerating black holes can be obtained by viewing the new solutions as a
particular limit of black hole binariesAstorino:2023ifg . Recall that the
near-horizon geometry of a Schwarzschild black hole is described by the
Rindler metric, characteristic of an accelerating observer. A mathematically
equivalent way to “zoom in” on the event horizon is to take the infinite mass
limit of the solution. An accelerating Schwarzschild black hole can be
conceptualized as a binary system of two Schwarzschild black holes,
effectively described by the Bach–Weyl solution BachWeyl , where one of the
two grows infinitely large (becomes infinitely massive) while retaining a
finite distance from the other. The event horizon of the “big” black hole
appears then as an accelerating horizon to its small sibling. Consequently,
the Bach-Weyl solution ends up appearing as a $C$-metric in this limit.
Analogously, these type I accelerating black holes, featuring Rindler horizons
which depend on the transformation parameters among others, can be thought
about as a limit of the NUTty and/or charged extension of the Bach-Weyl
spacetime. A complete hierarchy of these novel type I spacetimes, including
also a seed angular momentum parameter, can be found in Barrientos:2023dlf
.333The solution with angular momentum has also later appeared in the fifth
revision of Astorino:2023ifg .

In light of these recent developments, this study aims to shed light on the
remaining spacetimes one can generate by composing Ehlers and Harrison
transformations. To further elaborate on our agenda, it is best if we use the
following terminology, which will be formally introduced in Sec. 2.1. An
electric (resp. magnetic) Ehlers transformation is an Ehlers transformation of
the Ernst potentials associated with a seed metric cast into the electric
(resp. magnetic) form of the WLP metric, and ditto for Harrison. To date, and
to the best of our knowledge, only the combination of electric transformations
has been investigated. In this work, we wish to fill the gaps, and we do so by
first combining a magnetic Ehlers transformation with a magnetic Harrison one
(first part), and then by taking all possible combinations of an electric and
a magnetic transformation (second part). It is a firmly established fact that
operating with magnetic Harrison and magnetic Ehlers transformations on the
seed potentials of Minkowski spacetime, one obtains two interesting
asymptotically nontrivial backgrounds, commonly known as Bonnor–Melvin
Bonnor_1954 ; Melvin1966  and swirling Astorino:2022aam ; Gibbons:2013yq
spacetimes, respectively. We will briefly review them later on. In the first
part of this work, we present a more general spacetime, dubbed electromagnetic
swirling universe, from which, as the name suggests, the aforementioned
solutions follow in appropriate limits. We study this geometry in detail, also
addressing a rather subtle issue concerning the uniqueness of timelike Killing
vectors, relevant also in the swirling case, which has been unfortunately
neglected so far. Moreover, we prove that the new metric is Kundt via an
intricate chain of coordinate reparametrizations and parameter redefinitions.
We also explicitly prove the existence of an intriguing relation between this
new background and a planar Reissner–Nordström–NUT spacetime. It is this
particular relation which we exploit to also analytically derive the
cosmological extension of the electromagnetic swirling universe, i.e., the
form of the solution in the presence of a cosmological constant. As a finale
to the first part, we embed a Schwarzschild black hole into the new
background, giving some emphasis on the dragging effect and the deformation of
the horizon surface.

In the second part, we give attention to electric-magnetic mixtures,
registering the complete list of novel spacetimes one can generate from
Minkowski spacetime by combining an electric with a magnetic transformation.
Four type I families are obtainable in this way, and we discuss the conditions
under which they can be legitimately called backgrounds. Due to the number of
solutions, the analysis will not be exhaustive. We show that all spacetimes in
the second part of this work, feature closed timelike/null curves (CTCs/CNCs).
These appear inside regions, the boundaries of which are also surfaces where
the frame-dragging angular velocity becomes singular. We then argue that one
may perhaps ascribe the occurrence of nonchronal regions to the intensity of
rotation building up very close to these singular surfaces; inertial frames
get dragged so strongly that the light cones end up being tilted in the
direction of the circumference. We remark that two of the backgrounds carry
everywhere finite electric and magnetic fields which decay in all directions
at infinity. Such behavior is in stark contrast to what happens in the
Bonnor–Melvin solution, where the fields are uniform in the vicinity of the
symmetry axis. Whether CTC-free parts of these two backgrounds could perhaps
be as suitable as the former spacetime, for example in describing
astrophysical black holes surrounded by strong magnetic fields, remains an
open question. The presence of closed timelike/null curves requires much
deeper scrutiny. Although they render the entirety of each solution
unrealistic for modeling (astro)physical phenomena, the fact that these
pathologies beset solutions to the Einstein–Maxwell theory, perhaps makes
these backgrounds interesting from a totally different perspective.

This paper is structured as follows. In Sec. 2.1 we communicate the very
basics of the Ernst formalism and briefly discuss the symmetry
transformations. We use this section to also derive the nomenclature, and we
further lay down the steps we follow to generate new solutions, giving an
algorithmic description of the generating technique we use in this work. In
Sec. 2.2, we review, in double-quick time, a highly convenient method for the
Petrov classification, which we will use throughout. In Sec. 3, we combine
magnetic Ehlers and Harrison transformations to obtain the electromagnetic
swirling universe, analyzing its geometric properties and making its relation
to a planar Reissner-Nordström-NUT spacetime manifest. We use the latter link
to derive the electromagnetic-swirling-(A)dS solution. Finally, a
Schwarzschild black hole is embedded into the new background, with its most
interesting features discussed in some depth. In Sec. 4, we direct our efforts
towards generating new spacetimes by combining electric and magnetic Ehlers
and Harrison transformations. We start with a Minkowski seed and work our way
up to the four different spacetimes one can obtain by combining a magnetic
transformation with an electric one. We investigate whether (and under which
conditions) these asymptotically nonflat geometries can be characterized as
backgrounds. Finally, in Sec. 5 we summarize our findings and conclude, also
suggesting new possible avenues for further research.

##  2 The essentials

###  2.1 The complex-potential formalism

In this section, we review the formulation of the Einstein–Maxwell field
equations in terms of two complex potentials, as presented by Ernst in his
seminal works Ernst:1967wx ; Ernst:1967by . Making a stationary and
axisymmetric ansatz, one can introduce two complex potentials, $\mathcal{E}$,
and $\Phi$, and write the Einstein–Maxwell field equations as a pair of
complex equations. Doing so, a set of Lie point symmetries is revealed, the
realization of which eludes one in the usual tensorial formalism. This set of
symmetry transformations can then be exploited to generate new solutions from
old ones. Let us briefly review the scheme.

The first step is to make a stationary and axisymmetric metric ansatz, the
Weyl–Lewis–Papapetrou (WLP) ansatz,

$ds^{2}=-f(dt-\omega\,d\phi)^{2}+f^{-1}[\mathrm{e}^{2\gamma}(d\rho^{2}+dz^{2})+\rho^{2}d\phi^{2}],$
(1)

where $f,\ \omega,$ and $\gamma$ are functions of $\rho$ and $z$. We further
take our gauge field to have the symmetry-compatible form

$A=A_{t}\,dt+A_{\phi}\,d\phi,$ (2)

where the scalar potentials are also functions of the Weyl coordinates $\rho$
and $z$. Defining

$\mathcal{E}:=f-\Phi\overline{\Phi}+i\chi,\quad\Phi:=A_{t}+i\widetilde{A}_{\phi},$
(3)

one may show, after some cumbersome algebra, that the Einstein–Maxwell field
equations for stationary axisymmetric fields are equivalent to the complex
Ernst equations

$\displaystyle(\operatorname{Re}\mathcal{E}+\Phi\overline{\Phi})\nabla^{2}\mathcal{E}$
$\displaystyle=$
$\displaystyle\boldsymbol{\nabla}\mathcal{E}\cdot(\boldsymbol{\nabla}\mathcal{E}+2\overline{\Phi}\boldsymbol{\nabla}\Phi),$
(4a)
$\displaystyle(\operatorname{Re}\mathcal{E}+\Phi\overline{\Phi})\nabla^{2}\Phi$
$\displaystyle=$
$\displaystyle\boldsymbol{\nabla}\Phi\cdot(\boldsymbol{\nabla}\mathcal{E}+2\overline{\Phi}\boldsymbol{\nabla}\Phi),$
(4b)

together with a pair of equations determining $\gamma$ via integration by
quadrature.444These will not bother us, for the symmetry transformations do
not transform $\gamma$ at all. The Laplacian and the gradient are understood
as operators in three-dimensional Euclidean space in cylindrical coordinates.
The potentials $\chi$ and $\widetilde{A}_{\phi}$ are twist potentials
satisfying the real equations

$\displaystyle\hat{\phi}\times\boldsymbol{\nabla}\chi$ $\displaystyle=$
$\displaystyle-\rho^{-1}f^{2}\boldsymbol{\nabla}\omega-2\hat{\phi}\times\operatorname{Im}(\overline{\Phi}\boldsymbol{\nabla}\Phi),$
(5a) $\displaystyle\hat{\phi}\times\boldsymbol{\nabla}\widetilde{A}_{\phi}$
$\displaystyle=$
$\displaystyle\rho^{-1}f(\boldsymbol{\nabla}A_{\phi}+\omega{\boldsymbol{\nabla}A_{t}}),$
(5b)

respectively, where $\hat{\phi}$ is the unit normal in the azimuthal
direction. Interestingly, if we define our complex potentials as

$\mathcal{E}:=-f-\Phi\overline{\Phi}-i\chi,\quad\Phi:=A_{\phi}+i\widetilde{A}_{t},$
(6)

and considering the metric ansatz

$ds^{2}=f(d\phi-\omega\,dt)^{2}+f^{-1}[\mathrm{e}^{2\gamma}(d\rho^{2}+dz^{2})-\rho^{2}dt^{2}],$
(7)

instead of (1), then for a gauge field as in eq. (2), we also arrive at eqs.
(4), with the twist potentials now satisfying the equations

$\displaystyle\hat{\phi}\times\boldsymbol{\nabla}\chi$ $\displaystyle=$
$\displaystyle-\rho^{-1}f^{2}\boldsymbol{\nabla}\omega+2\hat{\phi}\times\operatorname{Im}(\overline{\Phi}\boldsymbol{\nabla}\Phi),$
(8a) $\displaystyle\hat{\phi}\times\boldsymbol{\nabla}\widetilde{A}_{t}$
$\displaystyle=$
$\displaystyle\rho^{-1}f(\boldsymbol{\nabla}A_{t}+\omega{\boldsymbol{\nabla}A_{\phi}}).$
(8b)

As previously mentioned, eqs. (4) are invariant under a bunch of symmetry
transformations in potential space, whose domain and target (as maps) are
potentials associated with stationary and axisymmetric Einstein–Maxwell
fields. Their finite forms read NeugeKramer69

$\displaystyle\operatorname{G_{1}}[a]:(\mathcal{E}_{0},\Phi_{0})\mapsto(\mathcal{E},\Phi)$
$\displaystyle:=$ $\displaystyle(\mathcal{E}+ia,\Phi_{0}),$ (9a)
$\displaystyle\operatorname{G_{2}}[\alpha]:(\mathcal{E}_{0},\Phi_{0})\mapsto(\mathcal{E},\Phi)$
$\displaystyle:=$
$\displaystyle(\mathcal{E}_{0}-2\overline{\alpha}\Phi_{0}-\alpha\overline{\alpha},\Phi_{0}+\alpha),$
(9b)
$\displaystyle\operatorname{D}[\epsilon]:(\mathcal{E}_{0},\Phi_{0})\mapsto(\mathcal{E},\Phi)$
$\displaystyle:=$
$\displaystyle(\epsilon\overline{\epsilon}\mathcal{E}_{0},\epsilon\Phi_{0}),$
(9c)
$\displaystyle\operatorname{E}[c]:(\mathcal{E}_{0},\Phi_{0})\mapsto(\mathcal{E},\Phi)$
$\displaystyle:=$
$\displaystyle\frac{(\mathcal{E}_{0},\Phi_{0})}{1+ic\mathcal{E}_{0}},$ (9d)
$\displaystyle\operatorname{H}[\beta]:(\mathcal{E}_{0},\Phi_{0})\mapsto(\mathcal{E},\Phi)$
$\displaystyle:=$
$\displaystyle\frac{(\mathcal{E}_{0},\Phi_{0}+\beta\mathcal{E}_{0})}{1-2\overline{\beta}\Phi_{0}-\beta\overline{\beta}\mathcal{E}_{0}},$
(9e)

where $a,c$ are real parameters and $\alpha,\beta,\epsilon$ complex.
Transformations $\operatorname{G}_{1}$ and $\operatorname{G}_{2}$ are “gauge”
transformations which transform the potentials, but leave the metric and the
gauge field invariant, $\operatorname{D}$ is a duality-rescaling
transformation, $\operatorname{E}$ denotes the Ehlers transformation
Ehlers:1957zz , and $\operatorname{H}$ stands for the Harrison harrison1968new
one. Note that a composition of $\operatorname{G}_{1}$, $\operatorname{D}$ and
$\operatorname{E}$ gives the inversion map

$\operatorname{I}:(\mathcal{E}_{0},\Phi_{0})\mapsto(\mathcal{E},\Phi):=\frac{(1,\Phi_{0})}{\mathcal{E}_{0}},$
(10)

which thus is also a (discrete) symmetry of the Ernst equations. In
particular,

$\operatorname{I}=\operatorname{G}_{1}[a]\circ\operatorname{E}[1/a]\circ\operatorname{G}_{1}[a]\circ\operatorname{D}[ia],$
(11)

and we can easily verify that

$\operatorname{E}[a]=\operatorname{I}\circ\operatorname{G}_{1}[a]\circ\operatorname{I},\quad\operatorname{H}[\alpha]=\operatorname{I}\circ\operatorname{G}_{2}[\alpha]\circ\operatorname{I}.$
(12)

The transformations (9) are associated with eight Killing vectors (KVs)
locally generating an isometry group of the potential space whose linear
representation is a representation of SU(2,1) Kinn1 . As discussed in
Barrientos:2023dlf , Ehlers transformations form a one-dimensional subgroup,
i.e., $\operatorname{E}[a]\circ\operatorname{E}[b]=\operatorname{E}[a+b]$, and
they commute with Harrison transformations because
$[\operatorname{G}_{1},\operatorname{G}_{2}]=0$. On the other hand, two
Harrison transformations do not in general yield a Harrison, simply because
$[\operatorname{G}_{2},\operatorname{G}_{2}]\neq 0$. Actually,

$\operatorname{H}[\alpha]\circ\operatorname{H}[\beta]=\operatorname{E}[-2\operatorname{Im}(\alpha\overline{\beta})]\circ\operatorname{H}[\alpha+\beta],$
(13)

so, unless $\alpha\overline{\beta}=\beta\overline{\alpha}$, a “Harrison of
Harrison” gives an “Ehlers of Harrison” with the Ehlers parameter fixed in
terms of the Harrison parameters $\alpha$ and $\beta$. If we recall that the
Harrison map generates electrovac solutions from vacuum ones, then eq. (13)
implies that two Harrison transformations map static vacuum solutions to
stationary electrovac ones.

Let us also agree on the terminology to be used in this work. The metric (1)
will be called the electric form of the WLP, whereas (7) will be the magnetic
form. These names Vigano:2022hrg , which unfortunately are misleading, are
based on the simple observation that a Harrison transformation (with real
parameter) of the potentials (3) associated with a vacuum spacetime cast into
(1), introduces an electric charge, whereas the same transformation acting on
the potentials (6) associated with the very same vacuum spacetime cast into
(7), gives new potentials associated with a magnetized version of the seed.
Nevertheless, this way of naming things is convenient for the task at hand,
and we adhere to it henceforth. Therefore, the potentials (3) will be called
electric, as they are associated with a seed metric cast into the electric
WLP, while the potentials (6) will be dubbed magnetic by the same reasoning.
Finally, a symmetry transformation of electric potentials will be addressed as
electric, and ditto for the magnetic case.

Since this is a solution-generating technique, the process is purely
algorithmic. To the aid of the interested reader, we list the steps below for
the so-called electric case. The same steps ought to be followed in the
magnetic case using the relevant equations.

  1. 1.

Given a stationary and axisymmetric metric, identify $f,\ \omega,$ and
$\gamma$ in (1) via direct comparison.

  2. 2.

These should be fed to eq. (5b), together with the components of the seed
gauge field, which can then be integrated for the twist potential
$\widetilde{A}_{\phi}$. Results should in turn be fed to eq. (5a) to obtain
$\chi$.

  3. 3.

Substitute everything into the definitions (3) to get exact expressions for
the seed potentials. Next, do your favorite transformations and read off the
target functions $f,\ A_{t},\ \tilde{A}_{\phi},$ and $\chi$ again from (3).

  4. 4.

These should now be fed to eq. (5a), which can be integrated for $\omega$.
Having $\omega$, plug everything into eq. (5b) to get the azimuthal component
of the target gauge field.

  5. 5.

Substitute the target $f,\ \omega,\ A_{t},\ A_{\phi},$ and the seed $\gamma$
into the metric (1) and the gauge field (2), and voilà.

Although this procedure looks simple per se, the computational complexity
involved may become quite intractable.

###  2.2 Petrov classification

A fundamental way to distinguish gravitational fields, independent of the
coordinate system, is to classify them according to their Petrov type
Petrov2000TheCO . To do so, we need to study the algebraic structure of the
Weyl tensor, which in four dimensions has ten independent components, encoded
in the five complex Weyl–NP scalars $\Psi_{0},\ldots,\Psi_{4}$, within the
framework of the Newman–Penrose formalism. The Petrov classification is a
useful tool in our case, for one can directly prove the possible
nonequivalence between certain solutions appearing in this manuscript, by
simply looking at their Petrov type. This being the case, it is worth
including a double-quick review of the classification algorithm used in this
work.

First, we consider an arbitrary complex null tetrad (CNT)
$\mathfrak{e}=\\{k,l,m,\overline{m}\\}$ with $k,l$ real null vectors and
$m,\overline{m}$ complex conjugate null vectors, such that $k^{a}l_{a}=-1$,
$m^{a}\overline{m}_{a}=1$, and all other products zero. Latin indices are
lowered/raised with the use of the metric
$g_{ab}=g_{\mu\nu}\mathfrak{e}_{a}^{\mu}\mathfrak{e}_{b}^{\nu}$ and its
inverse. Having a complex null tetrad, we next consider the definition of the
five complex Weyl–NP scalar as given in Stephani:2003tm . The problem of
finding the Petrov type of a given spacetime will be attacked using the
d’Inverno and Russell-Clark method dInverno . Starting with an arbitrary null
basis, we wish to find the specific Lorentz transformation leading to a new
basis, in which the number of vanishing Weyl scalars is maximal. For
$\Psi_{4}\neq 0$, it is known that this is equivalent to the problem of
finding the roots of the complex quartic equation

$\Psi_{4}\epsilon^{4}-4\Psi_{3}\epsilon^{3}+6\Psi_{2}\epsilon^{2}-4\Psi_{1}\epsilon+\Psi_{0}=0.$
(14)

Based on the number and multiplicity of these roots, we can then determine the
Petrov type. To make things simpler, we will choose our CNT such that
$\Psi_{1}=0=\Psi_{3}$ and $\Psi_{4}\neq 0$. In App. A, we show that such a
choice is always possible for the general WLP metric, and we explicitly
suggest the way to construct it. Given this CNT, eq. (14) becomes a quadratic
for $z=\epsilon^{2}$, and dividing it by $\Psi_{4}$, we get

$z^{2}+\frac{6\Psi_{2}}{\Psi_{4}}z+\frac{\Psi_{0}}{\Psi_{4}}=0,$ (15)

with discriminant proportional to

$9\Psi_{2}^{2}-\Psi_{0}\Psi_{4}.$ (16)

The two roots of the quadratic are

$z_{\pm}=\frac{-3\Psi_{2}\pm\sqrt{9\Psi_{2}^{2}-\Psi_{0}\Psi_{4}}}{\Psi_{4}}.$
(17)

Thus, if $9\Psi_{2}^{2}\neq\Psi_{0}\Psi_{4}$, the quartic eq. (14) has four
simple roots $\\{\pm\sqrt{z_{+}},\pm\sqrt{z_{-}}\\}$, meaning that the Petrov
type is I. On the other hand, if the discriminant vanishes, the quadratic has
a double root $z_{0}=-3\Psi_{2}/\Psi_{4}$ which implies that the quartic eq.
(14) has two double roots $\pm\sqrt{z_{0}}$. This means that the Petrov type
is D in this case. Clearly, all spacetimes in the WLP family will be either O,
D, or I. Having completed the formalities, we shall now proceed with the
construction and study of a Schwarzschild black hole embedded into an
electromagnetic swirling universe, with enough emphasis given also on the
underlying background geometry.

##  3 The electromagnetic swirling universe

Since this section is dedicated to a sort of composite background, we shall
start it with a quick discussion about the separate building blocks of the
latter, the electromagnetic universe and the swirling spacetime. The magnetic
universe, also known as the Bonnor–Melvin solution, was first found by Bonnor
Bonnor_1954 , and it was only later rediscovered by Melvin Melvin1966 . It
describes a static and cylindrically symmetric magnetic field immersed in its
own gravitational field. In other words, it can be seen as describing a
magnetic flux tube held together by its own gravitational pull. The magnetic
field lines are parallel to the axis of symmetry, and the field can be treated
as a uniform one only near the vicinity of the axis. Since it contributes to
the stress tensor, and since stress-energy acts as a gravitating mass, its
intensity must be falling off far away from the symmetry axis to prevent a
collapse under its own gravity; and this is the case indeed.

Here, we shall present the solution with both, magnetic and electric, external
fields. We will refer to it as the electromagnetic universe. In cylindrical
coordinates, the metric describing it reads

$ds^{2}_{\mathsf{EM}}=\frac{\rho^{2}}{V^{2}}d\phi^{2}+V^{2}(-dt^{2}+d\rho^{2}+dz^{2}),$
(18)

where $V(\rho):=1+X\overline{X}\rho^{2}$ and $\overline{X}$ is the complex
conjugate of $X:=(E+iB)/2$, with $E$ (resp. $B$) controlling the intensity of
the electric (resp. magnetic) field. This metric is accompanied by the gauge
field

$A=-zE\,dt-\frac{B\rho^{2}}{2V}d\phi,$ (19)

and it belongs to the Kundt class of Petrov type D electrovac spacetimes.
Asymptotically, it approached the Levi-Civita spacetime555This is the Levi-
Civita metric with $\sigma=1=k$. See griffiths_podolsky_2009  for more
details.

$ds^{2}=\rho^{-2}d\phi^{2}+\rho^{4}(-dt^{2}+d\rho^{2}+dz^{2}).$ (20)

To see this, a rescaling of the noncompact coordinates is necessary.

Its motion group is locally generated by the four KVs

$T_{1}=\partial_{t},\quad T_{2}=\partial_{z},\quad T_{3}=\partial_{\phi},\quad
T_{4}=z\partial_{t}+t\partial_{z}.$ (21)

These KVs do not commute, thus forming a nonabelian Lie algebra $\mathfrak{g}$
with nonvanishing brackets

$[T_{1},T_{4}]=T_{2},\quad[T_{2},T_{4}]=T_{1}.$ (22)

The center of this Lie algebra is the one-dimensional subspace
$\operatorname{span}\\{T_{3}\\}$ which is obviously isomorphic to the real
line. This is a solvable algebra with its derived subalgebra being two-
dimensional abelian. One can actually observe that $\mathfrak{g}$ is a trivial
extension of $\mathfrak{e}(1,1)$, the latter being the Lie algebra of the
pseudo-Euclidean group $E(1,1)$ of rigid motions in Minkowski 2-space.

Another interesting feature of this solution is that, much like what happens
in anti-de Sitter space (AdS), timelike geodesics are forbidden to escape to
radial infinity due to the strong attraction towards the axis of symmetry.
Yet, here the source of this extreme gravitational pull is not a negative
cosmological constant, but rather the electromagnetic field itself (see
Melvin1966  for the study of geodesic motion in the magnetic universe).
Finally, let us remark that the electromagnetic universe can be obtained from
a Minkowski seed via a magnetic Harrison transformation with parameter
$i\overline{X}$.

Definitely, less has been said about the swirling spacetime. This is a
stationary vacuum solution of Einstein’s field equations,

$\displaystyle
ds^{2}_{\mathsf{S}}=\frac{\rho^{2}}{1+j^{2}\rho^{4}}(d\phi+4jz\,dt)^{2}+(1+j^{2}\rho^{4})(-dt^{2}+d\rho^{2}+dz^{2}),$
(23)

with the above expression first reported in Gibbons:2013yq , to the best of
our knowledge, as an analytic continuation of the Bianchi II cosmological
metric of Taub Taub:1950ez . This metric belongs to the Kundt family of Petrov
type D vacuum solutions,666Since type D vacuum solutions in Kundt’s class were
classified by Kinnersley in Kinnersley1969TYPEDV , this metric probably
appears therein, though definitely in a different chart. and its associated
isometry group is locally generated by the four KVs

$T_{1}=\partial_{t},\quad T_{2}=\partial_{\phi},\quad
T_{3}=z\partial_{t}+t\partial_{z}-2j(t^{2}+z^{2})\partial_{\phi},\quad
T_{4}=\partial_{z}-4jt\partial_{\phi}.$ (24)

Besides them, there is an additional irreducible rank-2 Killing tensor
obtained from the Killing–Yano 2-form

$-4\jmath\rho zdt\wedge
d\rho+\jmath\rho^{2}\left(1+\jmath^{2}\rho^{4}\right)dt\wedge dz+\rho
d\rho\wedge d\varphi.$ (25)

Since the KVs do not commute, their linear span ought to be a nonabelian Lie
algebra. To identify this algebra, it is best if we choose another set of
basis vectors, $\\{T_{+},T_{-},T,T_{3}\\}$, with $T_{\pm}=({T_{1}\pm
T_{4}})/\sqrt{2}$ and $T=4jT_{2}$. Concerning the new basis, the nonvanishing
Lie brackets read

$[T_{\pm},T_{3}]=\pm T_{\pm},\quad[T_{+},T_{-}]=T,$ (26)

and it is now easy to see that the derived subalgebra, spanned by
$\\{T_{\pm},T\\}$, is the three-dimensional Heisenberg algebra. It turns out
that the full Lie algebra, with center $\operatorname{span}\\{T\\}$, is
solvable and nondecomposable. If it bears a special name, then this name
unfortunately eludes us. It features as ${A}_{4,8}$ in the classification of
four-dimensional Lie algebras by Patera and Winternitz PateraWinternitz .

The limit of (23) to the Levi-Civita metric is discussed in Astorino:2022aam .
In the same work, a numeric treatment suggests that geodesic motion is vortex-
like.777Recently, a very detailed analysis of the geodesics in the swirling
background and in the exterior of the swirling black hole, has been carried
out in Capobianco:2023kse . The swirling spacetime is free of curvature
singularities, a Misner string, and nonchronal regions. Finally, it is
interesting to remark that the metric function $\omega$ grows infinitely large
as $|z|\to\infty$. Being linear in $z$, it is constant on fixed-$z$ planes and
zero on the equatorial plane $z=0$, where it changes sign. Do also note that
this solution can be obtained from a Minkowski seed via a magnetic Ehlers
transformation with parameter $j$.

###  3.1 The geometry

Let us then construct a new spacetime which features both, an external
electromagnetic field and swirling rotation.888During the final stages of this
work, we have noticed the thesis Illy:2023iau . In there, an accelerated
Reissner–Nordström black hole was constructed in a magnetic swirling
background. We will create this from a Minkowski seed via a composition of
magnetic Ehlers and Harrison transformations, in particular

$\operatorname{E}[j]\circ\operatorname{H}[i\overline{X}],$ (27)

where the complex $X$ was defined directly below eq. (18). Since this is the
first solution we present in this work, we will execute the steps listed in
Sec. 2.1 one by one. Our seed metric is Minkowski in cylindrical coordinates,

$ds_{0}^{2}=-dt^{2}+d\rho^{2}+dz^{2}+\rho^{2}d\phi^{2}.$ (28)

This can be cast into the metric (7) with nonvanishing seed functions

$f_{0}=\rho^{2}=\mathrm{e}^{2\gamma}.$ (29)

Eqs. (8a) and (8b) then yield vanishing twist potentials up to the choice of
integration constants. The seed potentials are then the simplest possible,
$\mathcal{E}_{0}=-\rho^{2}$ and $\Phi_{0}=0$.

We act upon them with the transformation (27) to obtain the new potentials

$(\mathcal{E},\Phi)=\frac{(\mathcal{E}_{0},i\overline{X}\mathcal{E}_{0})}{1+(ij-|X|^{2})\mathcal{E}_{0}},$
(30)

from which we may read off

$\begin{split}f&=\frac{\rho^{2}}{V^{2}+j^{2}\rho^{4}},\\\
\chi&=jf\rho^{2},\end{split}\quad\begin{split}A_{\phi}&=-\frac{f}{2}(BV-
jE\rho^{2}),\\\ \widetilde{A}_{t}&=(A_{\phi})_{(E,B)\to(-B,E)},\end{split}$
(31)

where $V$ was defined directly below (18), and $(A_{\phi})_{(E,B)\to(-B,E)}$
denotes the value of $A_{\phi}$ with $B$ exchanged with $E$ and $E$ exchanged
with $-B$. We use $|w|=\sqrt{w\overline{w}}$ for the modulus of a complex
variable $w$. Plugging the above into eq. (8a), we obtain a pair of
differential equations, first-order in derivatives of $\omega$, which we can
integrate for

$\omega=-4jz.$ (32)

With $\omega$ available, everything shall be fed to eq. (8b) which now yields
a pair of differential equations, first-order in derivatives of $A_{t}$, the
solution of which reads

$A_{t}=-zf[EV^{2}\rho^{-2}+j(2BV-jE\rho^{2})].$ (33)

It follows that the metric describing the electromagnetic swirling universe
(EMS) is

$ds^{2}_{\mathsf{EMS}}=\frac{\rho^{2}}{V^{2}+j^{2}\rho^{4}}(d\phi+4jz\,dt)^{2}+(V^{2}+j^{2}\rho^{4})(-dt^{2}+d\rho^{2}+dz^{2}),$
(34)

accompanied by a gauge field

$A=\rho^{2}\frac{2z[EV^{2}\rho^{-2}+j(2BV-jE\rho^{2})]dt+(BV-
jE\rho^{2})d\phi}{2(V^{2}+j^{2}\rho^{4})}.$ (35)

It is straightforward to see that when the Harrison parameter vanishes, i.e.,
$X=0$ or equivalently, $E=0=B$, the gauge field vanishes and, taking into
account that $V=1$ in such a case, we recover the swirling metric (23). On the
other hand, when the Ehlers parameter vanishes, it is also easy to verify that
the resulting spacetime is the electromagnetic universe with metric (18) and
gauge field (19). The metric (34) admits a nonabelian group of motions
$G_{4}$, locally generated by the KVs in the swirling case, eq. (24). Of
course, equality at the level of the algebras does not in general imply a
group isomorphism (consider covering groups for example). In addition, we have
a different Killing–Yano 2-form,

$-4j^{2}z\rho\,dt\wedge
d\rho+(|X|^{2}V+j^{2}\rho^{2})(V^{2}+j^{2}\rho^{4})dt\wedge dz+j\rho
d\rho\wedge d\phi,$ (36)

which reduces to the Killing–Yano 2-form (25) when we switch off the Harrison
parameter (after a harmless overall division by $j$). On the other hand, if we
make $j$ vanish, we get a Killing tensor
$\propto-\partial_{t}\otimes\partial_{t}+\partial_{z}\otimes\partial_{z}$,
which is just a trivial Killing tensor in the case of the electromagnetic
universe.

Let us now have a closer look at the metric (34). First of all, observe that
the $\rho$ coordinate is not the so-called reduced circumference. The latter
reads

$\mathsf{R}:=\frac{\sqrt{g_{\phi\phi}}}{2\pi}=\frac{\rho}{\sqrt{V^{2}+j^{2}\rho^{4}}},$
(37)

which goes to zero both as $\rho\to 0$ and $\rho\to\infty$. In fact, its
maximum is at a $\rho_{\mathsf{max}}=(j^{2}+|X|^{4})^{-1/4}$ with

$0<\mathsf{R}\leq\mathsf{R}_{\mathsf{max}}:=\left(2|X|^{2}+2\sqrt{j^{2}+X^{4}}\right)^{-1/2}.$
(38)

Hence, this would make a very restricted coordinate, and this is why we will
stick to the use of the initial $\rho$ coordinate. There are only two metric
functions that change sign, $g_{tt}$ and $g_{t\phi}$. For the former, the
surface where the change of sign happens, that is the surface on which
$g_{tt}=0$, is given by the equation $\mathcal{S}=0$, where

$\mathcal{S}(\rho,z):=(V^{2}+j^{2}\rho^{4})^{2}-(4jz\rho)^{2},$ (39)

This actually defines two surfaces $\mathcal{S}_{\pm}=0$ (the + for positive
$z$ and the $-$ for negative), with

$\mathcal{S}_{\pm}:=V^{2}+j^{2}\rho^{4}\mp 4jz\rho,$ (40)

on which $\partial_{t}$ is null, and whose unit normals

$n_{\pm}^{\mu}=\frac{g^{\mu\nu}\partial_{\nu}\mathcal{S}_{\pm}}{\sqrt{g^{\lambda\sigma}(\partial_{\lambda}\mathcal{S}_{\pm})\partial_{\sigma}\mathcal{S}_{\pm}}},$
(41)

are spacelike, i.e.,
$n^{\mu}_{\pm}n^{\nu}_{\pm}g_{\mu\nu}\overset{\mathcal{S}_{\pm}}{=}1$. This
means that the surfaces are timelike.

Observe that $\mathcal{S}(\rho,-z)=\mathcal{S}(\rho,z)$, or, equivalently,
that $\mathcal{S}_{\pm}(\rho,-z)=\mathcal{S}_{\mp}(\rho,z)$, and that
$\mathcal{S}_{\pm}(\rho,0)\neq 0$. This implies that the equator, which is
also the timelike surface where $g_{t\phi}=0$, acts as a plane of reflection,
with $\mathcal{S}_{-}=0$ in the $z<0$ half-space being the mirror image of
$\mathcal{S}_{+}=0$ in the positive $z$ half-space. Since

$g_{tt}=-\frac{\mathcal{S}_{+}\mathcal{S}_{-}}{V^{2}+j^{2}\rho^{4}},$ (42)

it is clear that for $z>0$, $\mathcal{S}_{-}>0$, and that if

$z>\frac{V^{2}+j^{2}\rho^{4}}{4j\rho},$ (43)

then $g_{tt}>0$, viz. $\mathcal{S}_{+}<0$ gives a region in which
$\partial_{t}$ is spacelike. Similarly, for $z<0$, $\mathcal{S}_{+}>0$, and if

$z<-\frac{V^{2}+j^{2}\rho^{4}}{4j\rho},$ (44)

this provides another region, this time $\mathcal{S}_{-}<0$, in which
$\partial_{t}$ is again spacelike. These two regions pretty much fulfill the
criteria to be called ergoregions, with $\mathcal{S}_{\pm}=0$ giving the
ergosurfaces. However, there might be a caveat with this interpretation, which
requires further investigation and a deeper understanding.

In the familiar Kerr geometry, $\partial_{t}$ can be selected as the unique
timelike and normalized KV at infinity. Asymptotic flatness of a metric
guarantees that the ergoregion (if it exists) is confined. Here, the metric
exhibits a peculiar asymptotic behavior (it is basically asymptotically
swirling as we will soon see). In fact, it is not hard to see that regions,
where $\partial_{t}$ is spacelike, extend to infinity. Indeed, take $z$ to
grow faster than $\rho^{3}$, and notice that the second term in (40) becomes
dominant, yielding

$g_{tt}\sim\frac{(4j\rho z)^{2}}{V^{2}+j^{2}\rho^{4}}>0.$ (45)

On the other hand, if $z$ grows slower than $\rho^{3}$, it is the first term
in $\mathcal{S}_{\pm}$ that prevails, yielding

$g_{tt}\sim-(j^{2}+|X|^{4})\rho^{4}<0.$ (46)

Therefore, there are regions at infinity where $\partial_{t}$ is not timelike.
Actually, there is simply no such KV (or a linear combination of them) in our
case. To see this, consider the most general linear combination
$\xi=\sum_{i=1}^{4}C_{i}T_{i}$, where the $T_{i}$’s are given in eq. (24),
with the $C_{i}$’s being constant coefficients. Being a linear combination of
KVs with constant coefficients, this is obviously another KV. Fix a $t$, do
$(\rho,z)\to(\rho\epsilon^{-1},z\epsilon^{-4})$, and Taylor expand about
$\epsilon=0$. This ensures that we are probing a case where $z$ grows faster
than $\rho^{3}$, as fast as $\rho^{4}$ in particular. Doing so, one confronts
the following situation: it is impossible to choose the constants
$\\{C_{i}\\}$ in a way such that the leading-order term in the expansion is
negative! In other words, there is a region at asymptotic infinity where no KV
can be timelike. Therefore, the concept of $t$ as a time of distant observers
or a “time at infinity” seems to be somehow problematic, to say the least.

Well, even in Kerr spacetime, it is true that the interpretation of $t$ as a
“time”, universal in the entirety of “space” (as a time of distant observers),
is meaningful only outside the ergosphere Frolov:1998wf . Here, it just
happens that the ergoregion unfortunately extends to infinity. Of course, it
is always possible to find a KV which is timelike at $\rho$ infinity provided
that $z$ grows slower than $\rho^{3}$. Truly, $\partial_{t}$ is such a KV, but
so are other combinations $\xi$, e.g. $\xi=C_{1}T_{1}+C_{2}T_{2}+C_{4}T_{4}$
with $C_{1}^{2}>C_{4}^{2}$. Clearly, a normalization condition at infinity
cannot be used here to single out a unique combination, for there is no $\xi$
satisfying this at all; indeed,
$\xi\cdot\xi\sim-(C_{1}^{2}-C_{4}^{2})(j^{2}+|X|^{4})\rho^{4}$. One may
however demand that $\xi\cdot\xi\sim-1$ as $\rho\to 0$ (after all this is a
physical region). This forces $C_{1}^{2}=C_{4}^{2}+1$, but still leaves
$C_{2}$ completely arbitrary. Moreover, the condition that $\xi\cdot\xi$ is
time independent further fixes $C_{4}=0$, so we are left with
$\xi=\partial_{t}+C_{2}\partial_{\phi}$. There is no other condition, based on
limits, which we can use to somehow fix $C_{2}$. Note that one confronts the
same situation also in the swirling spacetime.

It is certainly tempting to consider the discrete symmetries of the metric as
a means to fix $C_{2}$. Recall that in a Kerr geometry, reflection of time
$t\to-t$ is not a symmetry unless it is accompanied by a change in the
direction of rotation, namely $\phi\to-\phi$, and vice versa. This is true
also for the EMS metric (34), except now we have additional ways to restore
time-reversal symmetry. As a matter of fact, time reversal here, if
accompanied by a transformation $z\to-z$ (which maps one semi-axis to the
other), is another (simultaneous) discrete symmetry transformation; it leaves
the metric invariant. Do also notice that a transformation $\phi\to-\phi$,
again accompanied by $z\to-z$, is yet another discrete symmetry. These
additional discrete symmetries are a key characteristic of the EMS and
swirling metrics. They do not exist for example in the Kerr case. It then
appears natural to ponder whether demanding that the norm of the KV candidate
is invariant under all the aforementioned discrete symmetries of the metric,
namely $(t,\phi)\to-(t,\phi)$, $(t,z)\to-(t,z)$, and $(z,\phi)\to-(z,\phi)$,
is a good condition (on top of the previous ones), one that does the job. This
indeed yields $C_{2}=0$, resulting in $\xi=\partial_{t}$. However, this does
not prove uniqueness, although it naively appears to do so.

To see this, consider the harmless coordinate transformation
$(t^{\prime},z^{\prime},\phi^{\prime})=(t,z-\alpha,\phi+4j\alpha t)$ where
$\alpha$ is some real constant. The transformation is (metric-)form
preserving, and therefore, the metric has the same discrete symmetries. Since
the inner product $\xi\cdot\xi$ is a coordinate scalar, we may directly write
it in terms of the prime coordinates. Now, $\partial_{t}\cdot\partial_{t}$ was
invariant under $(t,z)\to-(t,z)$, but it will not be invariant under
$(t^{\prime},z^{\prime})\to-(t^{\prime},z^{\prime})$. However, the norm of
another KV, namely
$\partial_{t^{\prime}}=\partial_{t}-4j\alpha\partial_{\phi}$, will. In fact,
$\partial_{t^{\prime}}$ also satisfies all the preceding criteria by default,
and we see that discrete symmetries cannot help us single out a candidate KV
after all. Our naively “unique” $\partial_{t}$ is as good as any other member
of the family $\partial_{t}-4j\alpha\partial_{\phi}$. Therefore, in the
absence of a robust selection mechanism, we argue that one should indeed use
the whole family $\xi=\partial_{t}+C\partial_{\phi}$ as the timelike KV, where
$C$ is an arbitrary real constant. It then follows that ergosurfaces would be
understood as (the timelike) surfaces where $\xi$ is null, ergoregions as
regions where $\xi$ is spacelike, and the frame-dragging angular velocity
would be given by

$\Omega:=-\frac{\xi\cdot\partial_{\phi}}{g_{\phi\phi}}.$ (47)

The latter deserves a few more comments. In particular, consider for a moment
the general WLP metric in the form (7), for which we have the convenient
equation

$\Omega=-C+\omega.$ (48)

It becomes evident, that using $\Omega$ to measure the value of the angular
speed at each point is not really a meaningful practice, for $C$ is completely
arbitrary. Instead, the meaningful quantity to look at is
$\Delta\Omega(\rho,z):=\Omega(\rho,z)-\Omega_{0}=\omega(\rho,z)-\omega_{0}$.
Remarkably, it then seems that for such spacetimes, in which the timelike KV
is the one-parameter family $\xi$, rotation can only be understood in a
relative manner. For example, in the EMS case we currently study,
$\Omega=-C-4jz$. It is clear that, taking $\Omega_{0}$ to be the value of
$\Omega$ on an arbitrary $z$ slice, this slice will be the surface where the
difference $\Delta\Omega$ changes sign.999This will also act as the plane of
reflection for the ergoregions. This ultimately implies the presence of
counter-rotating regions, though the exact localization of these regions is
obviously observer-dependent. After this elucidating aside, let us now, once
and for all, choose coordinates adapted to a $C=0$ observer (our timelike KV
being $\xi=\partial_{t}$), and let us proceed with discussing further features
of the solution.

Going to a rectangular coordinate system, one can easily show that the metric
has no coordinate singularities. There is no event horizon, and the absence of
a conical singularity can be verified by the fact that

$\lim\limits_{\epsilon\to
0}\frac{\int_{0}^{2\pi}\sqrt{g_{\phi\phi}}|_{\rho=\epsilon}\,d\phi}{\int_{0}^{\epsilon}\sqrt{g_{\rho\rho}}\,d\rho}=2\pi.$
(49)

The absence of a Misner string is also evident since the norm of the azimuthal
KV vanishes as $\sim\rho^{2}$ near the symmetry axis. The electromagnetic
swirling universe is also free of Closed Timelike Curves (CTCs), for
$\partial_{\phi}$ is everywhere spacelike. Probing the spacetime for curvature
singularities, we shall have a look at $R^{\mathbf{ab}}{}_{\mathbf{cd}}$,
i.e., (the components of) the Riemann tensor in the orthonormal basis
$\\{e_{\mathbf{a}}\\}$, defined in App. A. If this tensor is regular near a
coordinate singularity, then the singularity is just due to a poor choice of
chart. Indeed, all curvature invariants up to arbitrary polynomial order can
be constructed using this particular tensor. Thus, if the tensor itself is
regular, the regularity of the invariants follows. On the other hand, if
$R^{\mathbf{ab}}{}_{\mathbf{cd}}$ is singular near a locus of interest, this
does not automatically imply the existence of a curvature singularity, for the
poles appearing in the tensor components could, in theory, not appear when
taking traces to form curvature invariants. We find that
$R^{\mathbf{ab}}{}_{\mathbf{cd}}$ depends solely on $\rho$, and that it is
regular everywhere since the denominator of all components is just
$(V^{2}+j^{2}\rho^{4})^{3}$.101010Recall that this goes to 1 when $\rho\to 0$.
We also verify that it falls off quite fast as $\rho\to\infty$, which
reassures us that tidal forces are diminishing as one moves far away from the
axis. To give an example, we mention that the Kretschmann scalar goes as
$64(5|X|^{4}-3j^{2})+\mathcal{O}(\rho)$ in the neighborhood of the symmetry
axis, whereas it falls off as $\sim\rho^{-12}$ when $\rho\to\infty$.

The new metric is asymptotic to

$ds^{2}_{\mathsf{EMS}}\underset{\rho\to\infty}{\sim}\frac{1}{(|X|^{4}+j^{2})\rho^{2}}(d\phi+4jz\,dt)^{2}+(|X|^{4}+j^{2})\rho^{4}(-dt^{2}+d\rho^{2}+dz^{2}),$
(50)

provided that $z$ grows slower than $\rho^{3}$. A coordinate rescaling

$\\{t,\rho,z\\}=\frac{|j|}{\sqrt{j^{2}+|X|^{4}}}\\{t^{\prime},\rho^{\prime},z^{\prime}\\},\quad\phi=\frac{j^{2}+|X|^{4}}{j^{2}}\phi^{\prime},$
(51)

brings the above to a form asymptotic to a swirling metric with parameter

$j^{\prime}=\frac{(j^{2}+|X|^{4})^{2}}{j^{3}}.$ (52)

However, the gauge field strength 2-form does not vanish as
$\rho^{\prime}\to\infty$. In fact,

$F\underset{\rho^{\prime}\to\infty}{\sim}\frac{j^{2}[(j^{2}-|X|^{4})E-2jB|X|^{2}]}{(j^{2}+|X|^{4})^{2}}dz^{\prime}\wedge
dt^{\prime}$ (53)

Therefore, the complete solution is not asymptotic to the swirling spacetime
because the latter is a vacuum solution. Note that $\ast(F\wedge\ast F)\to 0$
as $\rho^{\prime}\to\infty$.

Concerning the Petrov type of the EMS spacetime, it is straightforward to
conclude that it is D, for we find that $9\Psi_{2}^{2}=\Psi_{0}\Psi_{4}$ (see
the reasoning and other details in Sec. 2.2). On the contrary, it is not
trivial at all to prove that (34) actually belongs to Kundt’s class. Solutions
in the Kundt family admit a shearfree, nonexpanding, and nontwisting null
geodesic congruence, with the general metric being Kundt1961 ;
Kundt1962ExactSO ; Stephani:2003tm ; griffiths_podolsky_2009

$ds^{2}=2P^{-2}d\zeta\,d\overline{\zeta}-2du(dv+W\,d\zeta+\overline{W}\,d\overline{\zeta}+H\,du),$
(54)

where $P,H$ are real functions, and $W$ is complex. Now, consider the specific
functions

$P^{2}=\frac{\widetilde{z}^{2}+\gamma^{2}}{\widetilde{z}+k},\quad
W=-\frac{\sqrt{2}v}{(\widetilde{z}+i\gamma)P^{2}},\quad
H=\frac{\widetilde{z}^{2}+\gamma^{2}}{2}-\frac{2\gamma^{2}v^{2}}{(\widetilde{z}^{2}+\gamma^{2})^{2}P^{2}},$
(55)

where

$\gamma=\frac{j}{2^{2/3}(j^{2}+|X|^{4})^{2/3}},\quad k=-\frac{\gamma
X\overline{X}}{j},$ (56)

and where $\widetilde{z}$ is supposed to be given in terms of
$\operatorname{Re}\zeta$ via
$\sqrt{2}P^{2}d\widetilde{z}=d\zeta+d\overline{\zeta}$. Let us then perform
the coordinate redefinitions

$\begin{split}\sqrt{2}\zeta&=\frac{\widetilde{z}(\widetilde{z}-2k)}{2}+(\gamma^{2}+k^{2})\ln\frac{\widetilde{z}^{2}+\gamma^{2}}{P^{2}}+i(\psi+\gamma
q^{2}),\\\ v&=q(\widetilde{z}^{2}+\gamma^{2}),\quad u=\tau-q,\end{split}$ (57)

in order to express (54) in the coordinate system
$\\{\tau,\widetilde{z},q,\psi\\}$. We get

$ds^{2}=(\widetilde{z}^{2}+\gamma^{2})(-d\tau^{2}+dq^{2})+P^{-2}(d\psi+2\gamma
q\,d\tau)^{2}+P^{2}\,d\widetilde{z}^{2},$ (58)

which we readily recognize as a member of the general family of nonexpanding
type D solutions (see (16.27) in griffiths_podolsky_2009 ). At this stage, yet
another coordinate transformation with

$\begin{split}\tau&=t(\gamma^{2}+k^{2})^{-1/2},\\\
\widetilde{z}&=-k+\rho^{2}(\gamma^{2}+k^{2})^{-1}/4,\end{split}\quad\begin{split}q&=z(\gamma^{2}+k^{2})^{-1/2},\\\
\psi&=2\phi(\gamma^{2}+k^{2}),\end{split}$ (59)

finally brings us to the metric (34), and that is all. Therefore, we conclude
that the EMS universe is also Kundt, as a combination of two Kundt spacetimes,
the swirling one ($k=0$ via $X=0$) and the EM universe ($\gamma=0$ via $j=0$).

At the same time, there is a gauge field that we completely neglected so far.
To find its form in coordinates $\\{u,v,\zeta,\overline{\zeta}\\}$, we start
the other way around. Let

$j=\frac{\gamma}{4(\gamma^{2}+k^{2})^{2}},\quad
X=\frac{\widetilde{E}+i\widetilde{B}}{2(\gamma^{2}+k^{2})},$ (60)

where $k=-\widetilde{E}^{2}-\widetilde{B}^{2}$, and perform the coordinate
transformations

$\begin{split}t&=\tau\sqrt{\gamma^{2}+k^{2}},\\\
\rho&=2\sqrt{\gamma^{2}+k^{2}}\sqrt{\widetilde{z}+k},\end{split}\quad\begin{split}z&=q\sqrt{\gamma^{2}+k^{2}},\\\
\phi&=\psi(\gamma^{2}+k^{2})^{-1}/2,\end{split}$ (61)

to arrive at

$\displaystyle A$ $\displaystyle=$ $\displaystyle
q\frac{2\gamma\widetilde{B}(\widetilde{z}+k)(k\widetilde{z}-\gamma^{2})-\widetilde{E}[k^{2}\widetilde{z}^{2}-(k^{2}+4k\widetilde{z}+\widetilde{z}^{2})\gamma^{2}+\gamma^{4}]}{(\gamma^{2}+k^{2})(\widetilde{z}^{2}+\gamma^{2})}d\tau$
(62)
$\displaystyle+(\widetilde{z}+k)\frac{\gamma\widetilde{E}(\widetilde{z}+k)+\widetilde{B}(k\widetilde{z}-\gamma^{2})}{(\gamma^{2}+k^{2})(\widetilde{z}^{2}+\gamma^{2})}d\psi,$

which is the form of the gauge field (35) in the coordinate system
$\\{\tau,\widetilde{z},q,\psi\\}$ with the spacetime metric being (58). At
this stage, we shall consider

$\begin{split}\tau&=u+v/(\widetilde{z}^{2}+\gamma^{2}),\\\
q&=v/(\widetilde{z}^{2}+\gamma^{2}),\end{split}\quad\begin{split}\psi&=\sqrt{2}\operatorname{Im}\zeta-\gamma
v^{2}/(\widetilde{z}^{2}+\gamma^{2})^{2},\\\
d\widetilde{z}&=(\widetilde{z}+k)(d\zeta+d\overline{\zeta})(\widetilde{z}^{2}+\gamma^{2})^{-1}/\sqrt{2},\end{split}$
(63)

which bring us to

$\displaystyle A$ $\displaystyle=$ $\displaystyle
v\frac{2\gamma\widetilde{B}(\widetilde{z}+k)(k\widetilde{z}-\gamma^{2})-\widetilde{E}[k^{2}\widetilde{z}^{2}-(k^{2}+4k\widetilde{z}+\widetilde{z}^{2})\gamma^{2}+\gamma^{4}]}{(\gamma^{2}+k^{2})(\widetilde{z}^{2}+\gamma^{2})}du$
(64)
$\displaystyle-\frac{v\widetilde{E}}{(\widetilde{z}^{2}+\gamma^{2})^{2}}dv+A_{\zeta}\,d\zeta+\overline{A}_{\zeta}\,d\overline{\zeta},$

where

$\displaystyle A_{\zeta}$ $\displaystyle=$
$\displaystyle-i\widetilde{B}\frac{(\widetilde{z}+k)(k\widetilde{z}-\gamma^{2})}{\sqrt{2}(\gamma^{2}+k^{2})(\widetilde{z}^{2}+\gamma^{2})}+\widetilde{E}(\widetilde{z}+k)\frac{2\widetilde{z}v^{2}(\gamma^{2}+k^{2})-i\gamma(\widetilde{z}+k)(\widetilde{z}^{2}+\gamma^{2})^{3}}{\sqrt{2}(\gamma^{2}+k^{2})(\widetilde{z}^{2}+\gamma^{2})^{4}},$

and where $\widetilde{z}$ is implicitly given in terms of
$\operatorname{Re}\zeta$ via the second equation in the right column of (63).
From here, one can check that using the backwards coordinate transformations
(57) and (59), together with (56) and

$\widetilde{E}=E(\gamma^{2}+k^{2}),\quad\widetilde{B}=B(\gamma^{2}+k^{2}),$
(66)

one indeed reaches eq. (35).

Finally, let us have a look at the electric and magnetic fields in the EMS
universe. Following the procedure presented in App. B, we find that

$\mathbf{E}=-\frac{2jBV\rho^{2}+E(V^{2}-j^{2}\rho^{4})}{(V^{2}+j^{2}\rho^{4})^{2}}\hat{z},\quad\mathbf{B}=(\mathbf{E})_{(E,B)\to(-B,E)}.$
(67)

Both depend only on $\rho$ with field lines parallel to the axis of symmetry,
in the vicinity of which they acquire a constant profile, $-E\hat{z}$ and
$B\hat{z}$, respectively. Moreover, they fall off as $\sim\rho^{-4}$ when
$\rho$ grows large, with the extrema of their magnitude given by the positive
real roots of a hexic polynomial in $\rho$.

###  3.2 Double Wick rotation and the planar Reissner–Nordström–NUT spacetime

It is a well-established result that the Bonnor-Melvin solution can be mapped
to a planar Reissner–Nordström (RN) spacetime Gibbons:2001sx . It is also
known that the swirling solution can be mapped to a planar Taub–NUT spacetime
Astorino:2022aam . These mappings are in general achieved by employing a
double Wick rotation, coordinate transformations, and parameter redefinitions.
Therefore, one may reasonably expect to be able, in a similar fashion, to map
the electromagnetic swirling universe, which utterly is a combination of the
above, to a planar RN–NUT spacetime. Indeed, we prove that this is the case.

First, let us show how to “planarize” the standard RN–NUT metric via a
limiting process. Here, the word “standard” refers to the metric

$ds^{2}_{\mathsf{RNN}}=-f(d\tau-2lx\,d\varphi)^{2}+f^{-1}{dr^{2}}+(r^{2}+l^{2})[dx^{2}{(1-x^{2})^{-1}}+(1-x^{2})d\varphi^{2}],$
(68)

and the gauge field

$A_{\mathsf{RNN}}=\frac{gl-
er}{r^{2}+l^{2}}d\tau+x\frac{2elr+g(r^{2}-l^{2})}{r^{2}+l^{2}}d\varphi,$ (69)

where

$f(r)=\frac{r^{2}-2mr-l^{2}+q^{2}}{r^{2}+l^{2}},\quad q^{2}=e^{2}+g^{2},$ (70)

with $e$ and $g$ being the electric and magnetic charge parameters,
respectively, $l$ denoting the NUT parameter, and $m$ standing for the mass
parameter.

We can “planarize” this solution, i.e., “flatten” the $S^{2}$ into
$\mathbb{R}^{2}$, by performing the rescalings

${y}=\lambda\widetilde{y},\quad{r}=\widetilde{r}/\lambda,\quad{m}=\widetilde{m}/\lambda^{3},\quad
e=\widetilde{e}/\lambda^{2},\quad g=\widetilde{g}/\lambda^{2},\quad
l=\widetilde{l}/\lambda,$ (71)

and sending $\lambda\to 0$ with $m,e,g,l\to\infty$ while $\widetilde{m},\
\widetilde{e},\ \widetilde{g}$, and $\widetilde{l}$ are kept fixed. Here,
$y=\\{\tau,x,\varphi\\}$. This procedure results in the metric

$\displaystyle
ds^{2}_{\mathsf{pRNN}}=-{f}(d{\tau}-2{l}x\,d\varphi)^{2}+f^{-1}{dr^{2}}+(r^{2}+l^{2})(dx^{2}+d\varphi^{2}),$
(72)

with

${f}=\frac{-2mr+q^{2}}{r^{2}+l^{2}},$ (73)

and in a gauge field equal to (69). We have dropped the use of tilde accents
for convenience. Notice that the gauge field is invariant under this process.
Of course, the result is not necessarily guaranteed to be a solution; this is
something that has to be checked. Nevertheless, one can verify that the full
solution, comprised of the metric (72) and the gauge field (69), does indeed
satisfy the Einstein–Maxwell field equations. Note that if $m>0$ and $r>0$,
there is a Killing horizon at $r=q^{2}/(2m)$ (same as in the case of the
planar RN solution), separating the inner region $0<r<q^{2}/(2m)$ from the
outer one, where $t$ is timelike and $r$ is spacelike in the former and the
other way around in the latter. On the other hand, if $m<0$ and $r>0$, and in
contrast to the situation in the planar RN spacetime, the planar RN-NUT does
not suffer from a naked singularity exactly due to the presence of the NUT
parameter; the curvature scalars are everywhere regular. Of course, the planar
RN–NUT solution is also plagued with a Misner string, for the symmetry axis
cannot be well-behaved both at $x=1$ and $x=-1$.

Now, let us perform a double Wick rotation
$(t,\phi)=i(\widetilde{\phi},\widetilde{t}\,)$ of (34) and (35), also doing
$E=i\widetilde{E}$ and $B=i\widetilde{B}$ such that
$\widetilde{X}:=\widetilde{E}+i\widetilde{B}=iX$. After the coordinate
transformations

$\begin{split}\widetilde{t}&={4|m|^{3}}(4m^{2}l^{2}+q^{4})^{-1}\tau,\\\
\rho&=\sqrt{4m^{2}l^{2}+q^{4}}\sqrt{-2mr+q^{2}}/(2m^{2}),\end{split}\quad\begin{split}z&=x\sqrt{4m^{2}l^{2}+q^{4}}/(2|m|),\\\
\widetilde{\phi}&=\varphi\sqrt{4m^{2}l^{2}+q^{4}}/(2|m|),\end{split}$ (74)

and the parameter redefinitions

$j=-8l|m|^{5}(4m^{2}l^{2}+q^{4})^{-2},\quad|\widetilde{X}|^{2}=4m^{4}q^{2}(4m^{2}l^{2}+q^{4})^{-2},$
(75)

we find out that the resulting metric is exactly the planar RNN metric (72).
However, to bring the resulting gauge field into the form (69), an additional
gauge transformation is necessary, the purpose of which is to shift the
temporal component by the specific constant $2m(2glm-
eq^{2})(4m^{2}l^{2}+q^{4})^{-1}$. Only then, the further parameter
redefinitions

$\widetilde{E}=4m^{2}\frac{4elmq^{2}-g(4m^{2}l^{2}-q^{4})}{(4m^{2}l^{2}+q^{4})^{2}},\quad\widetilde{B}=(|m|/m)(\widetilde{E})_{(e,g)\to(g,-e)},$
(76)

which satisfies the right equation in (75), leading to the desired result,
namely the mapping of the gauge field in the EMS universe to (69).
Consequently, we conclude that the full solution can be consistently mapped to
a planar RN–NUT spacetime via the above sequence of operations. The various
limits are then clear. Killing $l$ is tantamount to switching off $j$, and
vice versa; this provides the (bijective) mapping of the electromagnetic
universe to the planar RN spacetime Gibbons:2001sx . Killing $e,g$ is
tantamount to switching off $X$, and the other way around; this gives the
mapping of the swirling solution to the planar Taub–NUT spacetime
Astorino:2022aam .

###  3.3 Adding a cosmological constant

The previous result motivates one to use the extension of the RN–NUT spacetime
for a nonvanishing cosmological constant $\Lambda$, to derive a generalization
of the EMS solution which includes a $\Lambda\neq 0$. In general, when a
cosmological constant is included, the system of field equations is no longer
integrable. Equations that were homogeneous in the absence of $\Lambda$,
become inhomogeneous in its presence. In particular, the WLP metric itself is
not suitable for stationary axisymmetric fields in the presence of a
cosmological constant,111111See Charmousis:2006fx  for a generalized metric
which, given a certain harmonic condition, can be reduced to the WLP metric.
ergo the machinery used so far is not applicable. This is why the task of
extending stationary and axisymmetric solutions of the Ernst equations to
account for the presence of $\Lambda$, is a highly nontrivial one.

Having said that, let us now attack the problem of generalizing the EMS
solution. It can be straightforwardly checked that the form of the planar
RN–NUT spacetime in the presence of $\Lambda$, is given by (72) and (69), with

$f=\frac{-2mr+q^{2}+\Lambda l^{4}-\Lambda
r^{2}(r^{2}+6l^{2})/3}{r^{2}+l^{2}}.$ (77)

Considering the inverse forms of the coordinate transformations (74), together
with the parameter redefinitions

$l=-\frac{j|m|^{1/3}}{2^{1/3}(j^{2}+|\widetilde{X}|^{4})^{2/3}},\quad
q^{2}=\frac{(2m^{2})^{2/3}|\widetilde{X}|^{2}}{(j^{2}+|\widetilde{X}|^{4})^{2/3}},$
(78)

and doing a double Wick rotation $(\widetilde{t},\widetilde{\phi})=i(\phi,t)$
of the resulting spacetime, also setting $\widetilde{X}=iX$, we obtain the
generalized metric

$ds^{2}_{\mathsf{EMS\Lambda}}=\frac{\mathcal{R}}{V^{2}+j^{2}\rho^{4}}(d\phi+4jz\,dt)^{2}+(V^{2}+j^{2}\rho^{4})\left(-dt^{2}+\frac{\rho^{2}d\rho^{2}}{\mathcal{R}}+dz^{2}\right),$
(79)

where

$\displaystyle\mathcal{R}(\rho)$ $\displaystyle=$
$\displaystyle\Lambda\frac{12j^{4}-(3j^{2}+|{X}|^{4})^{2}}{12(j^{2}+|{X}|^{4})^{3}}+\left(1-\frac{\Lambda(3j^{2}+|{X}|^{4})|{X}|^{2}}{3(j^{2}+|X|^{4})^{2}}\right)\rho^{2}$
(80)
$\displaystyle-\frac{\Lambda\rho^{4}}{2}\left(1+\frac{2|X|^{2}}{3}\rho^{2}+\frac{j^{2}+|X|^{4}}{6}\rho^{4}\right).$

Up to gauge transformations, the resulting gauge field is (35) as expected. Of
course, there is no guarantee that the metric (79) and the gauge field (35)
solve the Einstein–Maxwell-$\Lambda$ field equations, but we verify that this
is the case indeed. Therefore, we have successfully constructed the
cosmological extension of the EMS spacetime, which also is of Petrov type D,
as well as a member of the Kundt class.121212Applying the parameter
redefinitions (60) and the coordinate transformations (61), one should be
readily convinced that the transformed metric belongs to the general family of
nonexpanding type D solutions. From there, reaching the Kundt form is more or
less straightforward (see griffiths_podolsky_2009 ).

Observe that for $\partial_{\phi}$ to remain spacelike at infinity, we must
consider $\Lambda<0$. We also see that there is a spinning string at $\rho=0$,
for

$\lim\limits_{\rho\to
0}g_{\phi\phi}=\Lambda\frac{3j^{4}-|X|^{4}(|X|^{4}+6j^{2})}{12(j^{2}+|X|^{4})^{3}},$
(81)

Let us then do $(t,\phi)=(\widetilde{t}+a\widetilde{\phi},\widetilde{\phi})$,
with $a$ being a constant, together with a reidentification of the new
coordinates such that $\widetilde{\phi}$ is $2\pi$ periodic, to see whether we
can obtain a new spacetime free of it. It is not difficult to check that
$g_{\widetilde{\phi}\widetilde{\phi}}$ has a $z$ dependence, and that the axis
can be made regular only at a single $z$ (by fixing $a$), meaning that the
spinning string will persist. Therefore, we deduce that it is not artificial
(it cannot be removed everywhere along the $z$ axis); it rather corresponds to
an actual Misner string! Fortunately, we can remove it by tuning the Ehlers
and Harrison parameters as

$j^{2}={\frac{3+2\sqrt{3}}{3}}|X|^{4},$ (82)

before any regluing. Doing so, we then observe that the induced metric with
${t}=\mathrm{cte}=z$, after the optional rescaling
$\rho=\widetilde{\rho}\sqrt{4|X|^{2}-\Lambda}/(2|X|)$, assumes the form

$d{s}^{2}_{\mathsf{EMS\Lambda}}|_{t,z}\underset{\widetilde{\rho}\to
0}{\sim}d\widetilde{\rho}^{2}+\frac{(4|X|^{2}-\Lambda)^{2}}{16|X|^{4}}\widetilde{\rho}^{2}d{\phi}^{2},$
(83)

close to the symmetry axis.131313If we do not rescale $\rho$, then eq. (83)
will be the same up to multiplication by a constant factor. This will not
change the value of the deficit, since it does not alter the ratio between the
proper length of a circumference and the radius. The above expression suggests
the presence of an infinite strut with negative mass per unit length

$\mu=\delta/4=\Lambda/(4|X|^{2}),$ (84)

where $2\pi\delta$ is the excess angle of the line source. Thankfully, this
can be made to vanish via the rescaling

$\phi=\frac{4|X|^{2}}{4|X|^{2}-\Lambda}\widetilde{\phi},$ (85)

if we reidentify $\widetilde{\phi}$ as our new azimuthal coordinate with
period $2\pi$. Consequently, it is always possible to obtain a regular
spacetime with a negative cosmological constant, free of a Misner string,
conical singularities, and CTCs, with the caveat of having the particular
relation (82) between $j$ and $X$.

After imposing the tuning (82) by replacing $j$, and with $\widetilde{\phi}$
being our new azimuthal coordinate, the $\mathcal{R}$ function in the metric
(79) acquires the form

$\mathcal{R}=\rho^{2}\frac{-9\Lambda+2|X|^{2}[18-\Lambda\rho^{2}(9+6|X|^{2}\rho^{2}+(3+\sqrt{3})|X|^{4}\rho^{4}]}{36|X|^{2}}.$
(86)

This is a positive function, with the reduced circumference being (dropping
the tilde accent)

$\mathsf{R}:=\frac{\int_{0}^{2\pi}\sqrt{g_{{\phi}{\phi}}}\,d{\phi}}{2\pi}=\frac{4|X|^{2}\sqrt{\mathcal{R}}}{(4|X|^{2}-\Lambda)\sqrt{V^{2}+{\frac{3+2\sqrt{3}}{3}}|X|^{4}\rho^{4}}}.$
(87)

In contrast to $\mathsf{R}$ in the case $\Lambda=0$, here the reduced
circumference, being a monotonically increasing function of $\rho$, has the
same range as the $\rho$ coordinate. Moreover, the presence of a negative
cosmological constant has a significant impact on the full extent of the
ergoregions as we see in the instructive Fig. 1.

Figure 1: Cross-sections of the ergoregions in the EMS (stippled dark gray
areas) and EMS-$\Lambda$ (stippled light gray areas) spacetimes. Plots are for
$|X|=0.35$, and are given by eq. (82). The cosmological constant in the EMS
solution has been set to $-3$. In panel [a] we display a ${\mathrm{y}}=0$
cross-section with rectangular coordinates
$\mathrm{(x,y)}=\rho(\sin\phi,\cos\phi)$. In panel [c] we display a $|z|=3$
cross-section. Similarly, in panel [b], a $u_{\mathrm{y}}=0$ slice is
displayed using coordinates
$(u_{\mathrm{x}},u_{\mathrm{y}})=(2/\pi)(\operatorname{atan}\rho)(\sin\phi,\cos\phi)$
and $v=(2/\pi)\operatorname{atan}z$, where $0\leq
u_{\mathrm{x}}^{2}+u_{\mathrm{y}}^{2}<1$. Finally, in panel [d] we show a
$|v|=(2/\pi)\operatorname{atan}3$ plane section. Coordinates
$\\{u_{\mathrm{x}},u_{\mathrm{y}},v\\}$ are particularly convenient, since
radial infinity is approached via $u_{\mathrm{x}}^{2}+u_{\mathrm{y}}^{2}\to 1$
and $|z|$ infinity via $|v|\to 1$, with the symmetry axis located at
$u_{\mathrm{x}}=0=u_{\mathrm{y}}$.

Indeed, if we let both $\rho$ and $|z|$ approach infinity, we find that

$g_{tt}\underset{\rho,|z|\to\infty}{\sim}-\frac{4j^{2}\Lambda
z^{2}\rho^{4}}{3}>0,$ (88)

where $j,X$ are understood as being related via eq. (82). In fact, as
$\rho\to\infty$, it suffices that $|z|\gtrapprox 1.05/\sqrt{-\Lambda}$ for
$\partial_{t}$ to be spacelike. Close to the axis, that is as $\rho\to 0$, we
have

$g_{tt}\overset{|z|\in\mathcal{O}(\rho^{-2})}{\underset{\rho,|z|^{-1}\to
0}{\sim}}\propto(4|X|^{2}-\Lambda)\rho^{2}z^{2}>0,$ (89)

where we let $z$ approach infinity at least as fast as $\rho^{-2}$. As in the
EMS case, the ergoregions do not touch the $z$ axis.

Finally, for completeness, let us mention that the metric (79) with
$\Lambda>0$, after taking care of the Misner string by imposing eq. (82),
features a cosmological horizon located at the largest positive real root of a
polynomial hexic in $\rho$. This can be utterly written as a polynomial cubic
in $\rho^{2}$ which, using Descartes’ rule of signs, appears to have a single
positive real root if and only if $X^{2}>\Lambda/4$. This is then a double
root of the hexic equation. Alas, the whole situation (with positive
$\Lambda$) gets more complicated, if we notice that as we approach this root,
call it $\rho_{+}$, the reduced circumference vanishes, meaning that
$\rho=\rho_{+}$ behaves as a sort of “axis” besides $\rho=0$. This observation
is rather expected, simply because the root of $g^{\rho\rho}$ is necessarily a
root of $g_{\phi\phi}$, as can be seen from the line element (79).

Let us also briefly go through the various limits. When $X=0$, the gauge field
vanishes, and the cosmological EMS metric (79) reduces to the cosmological
extension of the swirling solution presented in Astorino:2022aam . The latter
is free of curvature singularities and free of any horizons if $\Lambda<0$.
Notwithstanding these good features, a fact completely neglected in
Astorino:2022aam , is that the cosmological swirling solution actually
features a cosmic spinning string, evident from

$\lim\limits_{\rho\to 0}g_{\phi\phi}=\frac{\Lambda}{4j^{2}},$ (90)

which proves to be irremovable through coordinate transformations and
regluing. As such, it shall again be understood as a Misner string. Since we
no longer have the freedom to tune parameters in order to remove it (as we did
previously), one comes to the unfortunate conclusion that the
swirling-$\Lambda$ solution of Astorino:2022aam  is in general plagued with a
Misner string and the CTCs accompanying it.

When $j=0$, the metric (79) acquires the static form

$ds^{2}=\frac{\rho^{2}-\Lambda|X|^{-4}V^{4}/12}{V^{2}}d\phi^{2}+V^{2}\left(-dt^{2}+\frac{\rho^{2}d\rho^{2}}{\rho^{2}-\Lambda|X|^{-4}V^{4}/12}+dz^{2}\right),$
(91)

and the gauge field becomes (19). We remark that this is not the metric
presented in the perhaps pertinent cases Zofka:2019yfa ; Astorino:2012zm . The
above spacetime seems to have a spinning string, for

$\lim\limits_{\rho\to 0}g_{\phi\phi}=-\Lambda|X|^{-4}/12.$ (92)

If $\Lambda>0$, it is impossible to remove this string by any means. If on the
other hand $\Lambda<0$, we can do

$(t,\phi)=\left(\widetilde{t}+\frac{\sqrt{-\Lambda}}{2\sqrt{3}X\overline{X}}\widetilde{\phi},\widetilde{\phi}\right)$
(93)

and reglue the spacetime to get rid of it. The new spacetime is then also free
of conical singularities. An expansion of the induced metric with
$\widetilde{t}=\mathrm{cte}=z$ near the symmetry axis attests to that.
Moreover, $\partial_{\widetilde{\phi}}$ is clearly spacelike everywhere if we
adhere to the use of a negative cosmological constant, while we also remark
that there are no curvature singularities, nor are any ergoregions present.

###  3.4 Embedding a Schwarzschild black hole

Having analyzed the background, it is time to discuss the EMS black hole which
is just the Schwarzschild black hole embedded in the previously discussed
spacetime. For this reason, we will refer to it also as Schwarzschild–EMS.
This embedding shall be understood as a composition of magnetic Ehlers and
Harrison transformations acting on the potentials associated with a
Schwarzschild spacetime. We will work with the spherical-like coordinates
$\\{t,r,x=\cos\theta,\phi\\}$,141414Here, the coordinate $x$ ought not to be
confused with the usual Cartesian coordinate $\mathrm{x}$ used in other parts
of this work. since these prove to be the most convenient for integration.

It is quite straightforward to carry out the first steps which result in
identifying the Schwarzschild metric

$ds^{2}_{\mathcal{S}}=-(1-2M/r)dt^{2}+(1-2M/r)^{-1}dr^{2}+r^{2}(1-x^{2})^{-1}dx^{2}+r^{2}(1-x^{2})d\phi^{2},$
(94)

with the magnetic WLP metric (7), the nonvanishing metric functions being

$f_{0}=r^{2}(1-x^{2}),\quad\mathrm{e}^{2\gamma}=\frac{r^{4}(1-x^{2})}{(r-M)^{2}-M^{2}x^{2}},$
(95)

with Weyl’s coordinates $\rho,z$ given in terms of $r,x$ via

$\rho=\sqrt{r(r-2M)(1-x^{2})},\quad z=(r-M)x.$ (96)

Since this is a static vacuum solution, it is evident that $\Phi_{0}=0$ and
$\mathcal{E}_{0}=-f_{0}$. Observe that the mass does not appear in the
potentials, but rather in the function $\gamma$ and the coordinate
transformations (96).

Having retrieved the seed data, we now act with the transformation (30) on the
seed potentials to obtain—after integrating the twist equations—a new
spacetime, comprised of the metric

$ds^{2}_{\mathsf{\mathcal{S}EMS}}=f(d\phi-\omega\,dt)^{2}+(1-x^{2})f^{-1}r^{2}ds^{2}_{\mathcal{S}}|_{d\phi=0},$
(97)

with functions

$\begin{split}f&=\frac{r^{2}(1-x^{2})}{V^{2}+j^{2}r^{4}(1-x^{2})^{2}},\\\
\omega&=-4j(r-2M)x,\end{split}$ (98)

together with the gauge field

$\displaystyle A$ $\displaystyle=$
$\displaystyle-(r-2M)x\frac{E[V^{2}-j^{2}r^{4}(1-x^{2})^{2}]+2jBr^{2}(1-x^{2})V}{V^{2}+j^{2}r^{4}(1-x^{2})^{2}}dt$
(99)
$\displaystyle+\frac{r^{2}(1-x^{2})}{2}\frac{jEr^{2}(1-x^{2})-BV}{V^{2}+j^{2}r^{4}(1-x^{2})^{2}}d\phi,$

where the function $V$ is now defined as

$V(r,x):=1+|X|^{2}r^{2}(1-x^{2}).$ (100)

This spacetime describes the exterior of a black hole with a Killing horizon,
located at $r_{+}=2M$, dressing a singularity at the radial origin. Indeed,
$g^{rr}(r_{+},x)=0$, and the Kretschmann scalar blows up as $\sim M^{2}r^{-6}$
only when $r\to 0$. The surface area of the horizon is that of a 2-sphere of
radius $r_{+}$, namely $4\pi r_{+}^{2}$. However, its circumference at the
equator $x=0$ is not $2\pi r_{+}$ in the presence of the transformation
parameters, i.e.,

$\int_{0}^{2\pi}\sqrt{g_{\phi\phi}(r=r_{+},x=0)}\,d\phi=\frac{2\pi
r_{+}}{\sqrt{(1+|X|^{2}r_{+}^{2})^{2}+j^{2}r_{+}^{4}}}.$ (101)

This suggests that the horizon surface, its embedding in particular, can be
visualized as a 2-sphere of area $4\pi r_{+}^{2}$, which is deformed when we
switch on $j$ and/or $X$. In particular, the deformation caused by these
parameters can be visualized as squeezing the 2-sphere on the equatorial
plane, progressively obtaining an egg-shaped surface which is deformed into a
peanut-like one as we squeeze stronger. Moreover, the solution is stationary;
the black hole gets dragged due to the swirling property of the background. It
also got “electromagnetized”, in the sense that it is no longer a vacuum
solution, but rather an electrovac one, a fact ascribed to the presence of the
external electric and magnetic fields in the background geometry. Note that
(97) enjoys the good features of the EMS background, i.e., it is free of
topological singularities and nonchronal regions,151515The reader may convince
herself/himself by quickly performing the pertinent checks we previously did
in the case of the background. while it also shares the same asymptotic
behavior with the latter.

Concerning $\omega$, we display various isolines over a heatmap in Fig. 2 (see
caption for details). We do so in coordinates
$\\{u_{\mathrm{x}},u_{\mathrm{y}},v\\}$, introduced previously.

Figure 2: Heatmap of
$\widetilde{\omega}:=(2/\pi)\operatorname{atan}|\omega(u_{\mathrm{x}},v)|$ on
the $u_{\mathrm{x}}v$-plane for $j=0.5$ and $M=1$. Lightest hue corresponds to
$\widetilde{\omega}=1$, whereas darkest hue (pure black) to
$\widetilde{\omega}=0$. The darker the hue, the smaller $\widetilde{\omega}$
is. The horizon should be understood as the $[-1/2,1/2]$ part of the $v$ axis.
Note that $\widetilde{\omega}$ vanishes only on the equatorial plane and in
the horizon limit. We also display the following isolines:
$\widetilde{\omega}=10^{-3}$ (solid), $\widetilde{\omega}=10^{-2}$ (dashed),
and $\widetilde{\omega}=0.1$ (dotted).

In fact, we first switch to coordinates (96) via the inverse transformations

$\begin{split}r&=M+\sqrt{\frac{\rho^{2}+z^{2}+M^{2}+\sqrt{(\rho^{2}+z^{2}+M^{2})^{2}-4M^{2}z^{2}}}{2}},\\\
x&=\frac{\sqrt{2}z}{\sqrt{\rho^{2}+z^{2}+M^{2}+\sqrt{(\rho^{2}+z^{2}+M^{2})^{2}-4M^{2}z^{2}}}}.\end{split}$
(102)

In this chart, the horizon surface is understood as a closed line segment of
length $2M$ on the $z$ axis (with center at $z=0$). We can formally approach
it for $|z|\leq M$ by taking $\rho\to 0$. Then, we once again employ the
convenient coordinates

$u_{\mathrm{x}}=(2/\pi)(\operatorname{atan}\rho)\sin\phi,\quad
u_{\mathrm{y}}=(2/\pi)(\operatorname{atan}\rho)\cos\phi,\quad
v=(2/\pi)\operatorname{atan}z$ (103)

in which $|z|$ infinity sits at $|v|=1$ and $\rho$ infinity at
$u_{\mathrm{x}}^{2}+u_{\mathrm{y}}^{2}=1$. The horizon is again understood as
a line segment, now of length $(4/\pi)\operatorname{atan}M$, on the $v$ axis
(with center at $v=0$).

In Fig. 2 we (indirectly) see that $|\omega|$ grows infinitely large as
$|z|\to\infty$. Since it describes the frame-dragging angular velocity, the
fact that it is linear in $z$, implies that the two $z$ half-spaces (positive
and negative) counterrotate. We also observe that it gets smaller and smaller
as we approach the equator, where it vanishes. It also decreases as we
approach the horizon where it also vanishes in the respective limit. Note
that, for $j>0$, $\omega<0$ in the half-space defined by $z>0$ (or $v>0$),
positive otherwise; the exact opposite holds true when $j<0$. Regarding the
ergoregions, these prove to be more or less insensitive to the introduction of
a mass. Actually, their behavior at asymptotic infinity is exactly the same as
that in the case of the EMS solution we previously discussed, and there is
nothing, really worth reporting, going on close to the horizon.

Attacking the Petrov classification next, and following the reasoning
presented in Sec. (2.2), we can deduce that the general Petrov type is I, for
we have that $9\Psi_{2}^{2}\neq\Psi_{0}\Psi_{4}$. In particular,

$\displaystyle 9\Psi_{2}^{2}-\Psi_{0}\Psi_{4}$ $\displaystyle=$ $\displaystyle
18M(r-2M)(1-x^{2})(j-i|X|^{2})$ (104)
$\displaystyle\times\frac{\\{jr^{2}(1-x^{2})+i[1-|X|^{2}r^{2}(1-x^{2})]\\}^{2}[jr^{2}(1-x^{2})+iV]^{3}}{r^{4}[V^{2}+j^{2}r^{4}(1-x^{2})^{2}]^{5}}.$

This becomes zero when $M=0$, or $j=0=X$, or at the poles $x=1$ and $x=-1$, or
at the horizon $r=2M$. In the limit of vanishing mass, we recover the EMS
universe (discussed in the previous sections), albeit in the spherical-like
coordinate system $\\{t,r,x,\phi\\}$. When $j=0=X$, we obtain the
Schwarzschild seed which is type D everywhere. At the poles, we observe that
$\Psi_{4}\neq 0$, meaning that the Petrov type is D on the axis (at least in
the exterior). On the horizon surface, the spacetime is algebraically special.
Since $r=2M$ turns out to be a pole of $\Psi_{4}$, it would be erroneous to
make a definite claim that the Petrov type is D; we can only argue (as we just
did) that the solution is algebraically special, because
$9\Psi_{2}^{2}-\Psi_{0}\Psi_{4}$,161616Note that
$9\Psi_{2}^{2}-\Psi_{0}\Psi_{4}=\sqrt{(I^{3}-27J^{2})/(\Psi_{0}\Psi_{4})}$
with $I,J$ as defined in Stephani:2003tm . which vanishes at $r=2M$, utterly
corresponds to a combination of curvature invariants and thus, it is
independent of the reference frame. Note that our choice (148) of the
orthonormal tetrad, from which we constructed the CNT (see App. A for
details), corresponds to a zero angular momentum observer in circular motion;
it is not suitable for studying the horizon limit. To determine the actual
Petrov type at $r=2M$, one must rather consider a falling observer
Pravda:2005uv . Finally, when $X=0$, we obtain the swirling black hole, the
Petrov type of which is also I, whereas for $j=0$ we recover the static type I
Schwarzschild–Melvin spacetime ErnstSchwMel , which describes a Schwarzschild
black hole embedded into an (electro)magnetic universe. The latter solution is
of Petrov type II on the horizon surface (again located at $r=2M$), and it
also features another locus of interest given by

$|X|\sqrt{1-x^{2}}=r^{-1},\quad x\neq 1,-1,$ (105)

where all Weyl scalars vanish and the Petrov type is, therefore, O
Pravda:2005uv .

##  4 Mixing electric and magnetic transformations

In this section, we wish to scrutinize another possible route. Instead of
composing transformations of the same “kind”, i.e., either electric or
magnetic, we shall explore their mixture. Once again, we will consider only
Ehlers and Harrison transformations. In general, there are eight possible
mixed compositions,

$U^{\alpha}{}_{\beta}\circ U^{\gamma}{}_{\delta},$ (106)

with $\beta\neq\delta$, where $\alpha,\ldots,\delta=1,2$ and

$(U^{\alpha}{}_{\beta})=\begin{pmatrix}\operatorname{E}_{e}&\operatorname{E}_{m}\\\
\operatorname{H}_{e}&\operatorname{H}_{m}\end{pmatrix}.$ (107)

However, if the seed is Minkowski, the number of available compositions
reduces to four, namely

$U^{\alpha}{}_{1}\circ U^{\beta}{}_{2}.$ (108)

This happens because

$U^{\alpha}{}_{2}\circ U^{\beta}{}_{1}\sim U^{\alpha}{}_{2},$ (109)

where $\sim$ denotes a rough equivalence relation here, in the sense that
electric transformations of the seed potentials associated with Minkowski
space, result in a metric which is also Minkowski modulo coordinate
transformations.

Our first course of action is to study the novel spacetimes obtained via these
mixed compositions in the case of a Minkowski seed. For starters, we wish to
see if these geometries, presented here for the first time, turn out to be
backgrounds, i.e., that they are first and foremost free of curvature
singularities, topological defects, and other sorts of pathologies. Since we
know that $U^{1}{}_{\alpha}\circ U^{2}{}_{\alpha}=U^{2}{}_{\alpha}\circ
U^{1}{}_{\alpha}$, we could directly operate on the seed potentials with
$\operatorname{E}_{e}[c]\circ\operatorname{H}_{e}[Q]\circ\operatorname{E}_{m}[j]\circ\operatorname{H}_{m}[i\overline{X}]$,
where $j,c$ are real parameters, $X$ was previously introduced in the case of
the electromagnetic universe, namely $X=(E+iB)/2$, and we also define a new
complex parameter $Q:=(q_{e}+iq_{m})/2$. This would be tantamount to casting
the EMS metric in the form (1) and acting on the associated electric
potentials with $\operatorname{E}_{e}[c]\circ\operatorname{H}_{e}[Q]$. We
could then consider all possible limits, in which we remain with two layers of
transformations, one magnetic and one electric. Nevertheless, computationally
speaking, this would not be a wise strategy to pursue. For this reason, we
will generate each case separately. We will probe the new geometries only for
curvature singularities, spinning strings, conical singularities, and
nonchronal regions (regions with CTCs). We will call them backgrounds if they
are free of curvature and topological singularities, proper backgrounds if
they are also free of CTCs/CNCs.

###  4.1 Electromagnetic universe and electric Harrison transformations

Here, we operate on the magnetic potentials $\mathcal{E}_{0}=-\rho^{2}$ and
$\Phi_{0}=0$, associated with Minkowski spacetime, with
$\operatorname{H}_{e}[Q]\circ\operatorname{H}_{m}[i\overline{X}]$.171717See p.
3 of this manuscript for nomenclature. Of course, we do not have to truly do
the composition; we may directly act on the electric potentials, associated
with the electromagnetic universe, with $\operatorname{H}_{e}[Q]$. Hence, our
seed potentials are

$\mathcal{E}_{0}=V^{2}-4|X|^{2}z^{2},\quad\Phi_{0}=-2\overline{X}z,$ (110)

where we recall that $V=1+|X|^{2}\rho^{2}$. We thus follow the prescription
presented at the end of Sec. 2.1, skipping the integration details to get the
target metric (1), with

$\begin{split}f&=\frac{V^{2}}{\mathcal{V}^{2}+2\mathcal{V}(q_{e}E-q_{m}B)z+16|QX|^{2}z^{2}},\\\
\omega&=C_{\omega}-(q_{e}B+q_{m}E)\frac{1+|Q|^{2}(V^{2}+4|X|^{4}\rho^{2}z^{2})}{|X|^{2}V},\\\
\mathrm{e}^{2\gamma}&=V^{4},\end{split}$ (111)

where we further defined $\mathcal{V}(\rho,z):=1-|Q|^{2}\mathcal{E}_{0}$ for
convenience. Soon, we will also display the gauge field; before that, let us
study this new metric which has Petrov type I.

The first thing that one observes, is that on the equatorial plane $z=0$, $f$
has a double (and single) pole at $\rho=\rho_{*}$, where $\rho_{*}$ denotes
the positive real root of $\mathcal{V}=0$, or $|Q|^{2}V^{2}=1$ equivalently.
We find that

$\rho_{*}=\frac{\sqrt{1-|Q|}}{|X|\sqrt{|Q|}},$ (112)

which is clearly real if-f $|Q|\leq 1$. Saturating the bound, this coordinate
singularity is put exactly at $\rho=0=z$. For $Q<1$, the locus is a ring. For
$q_{e}B\neq-q_{m}E$, this is the only pole of $f$. In any case, if this is a
true curvature singularity (be it a point or a ring), and since $f\neq 0$ for
all $\rho,z$, indicating the absence of a horizon, it must be that it is a
naked one, with the general consensus being that such configurations are
unphysical. Note that the metric does not have any other singular limits
besides the one we just reported, which unfortunately happens to be a
curvature singularity, for we find that the Kretschmann scalar at the equator
blows up as $\sim(\rho-\rho_{*})^{-8}$ in the limit $\rho\to\rho_{*}$.
Parameter tuning cannot be a remedy to this; there are simply not enough
parameters to tune in order to get rid of all poles up to octic order in the
pertinent Taylor expansion. The situation is worse when $|Q|=1$, in which case
$R^{\mu\nu}{}_{\rho\sigma}R^{\rho\sigma}{}_{\mu\nu}\sim\rho^{-16}$ when
$\rho\to 0$. Consequently, it is mandatory to restrict $|Q|>1$ in order to
expel $\rho_{*}$ from the physical range of $\rho$. Still, regularity (of
curvature invariants) is guaranteed only if the components of the Riemann
tensor in the orthonormal basis of App. A are everywhere regular. Fortunately,
we find that their denominators are proportional to $f^{-1}$, with the
proportionality factors being nonvanishing functions of $\rho$. Since
$f^{-1}=0$ has no positive real solutions for $|Q|>1$, it follows that the
spacetime is indeed everywhere regular.

Having successfully tackled this important issue, it is time to address
another subtlety. Observe that at fixed $z$,

$\lim\limits_{\rho\to
0}g_{\phi\phi}=-\left(C_{\omega}-\frac{(1+|Q|^{2})(q_{e}B+q_{m}E)}{|X|^{2}}\right)^{2}\lim\limits_{\rho\to
0}f,$ (113)

where the limit of $f$ as $\rho\to 0$ is a nonvanishing expression that
involves the parameters and $z$, appearing as a denominator of the leading
term in the above. Once again, we are confronted with a spinning string, which
we can remove by fixing the integration constant $C_{\omega}$ as

$C_{\omega}=\frac{(1+|Q|^{2})(q_{e}B+q_{m}E)}{|X|^{2}}.$ (114)

Of course, keeping the integration constant free in this case, only to fix it
now, was after all a proactive action. Had we set it to zero, we would again
have the string removed via a coordinate transformation, followed by a
regluing of spacetime. Nevertheless, the final expression becomes

$\omega=\frac{(q_{e}B+q_{m}E)[1-|Q|^{2}(V+4|X|^{2}z^{2})]\rho^{2}}{V},$ (115)

and we now also have well-defined limits $X\to 0$ (Minkowski), or $Q\to 0$
(electromagnetic universe).181818Observe that, if $C_{\omega}$ is not fixed as
in eq. (114), $X\to 0$ is a singular limit of $\omega$ in (111). Additionally,
we remark that after taking care of the string, the induced metric with
$t=\mathrm{cte}=z$ becomes $\sim C(d\rho^{2}+\rho^{2}d\phi^{2})$ in the
vicinity of the symmetry axis, where $C$ denotes a proportionality factor
depending on the parameters and the chosen value of $z$. This tells us that
there are no conical singularities to be bothered with. Henceforth, we
consider only the family with $|Q|>1$ and $C_{\omega}$ as in eq. (114).

For $|Q|>1$, the solution is everywhere stationary (not just static) provided
that $q_{e}B\neq-q_{m}E$. If $q_{e}B+q_{m}E>0$, we have that $\omega<0$. If
the former is negative, the latter is positive. It is also clear that as
$\rho\to 0$, it holds that $\omega\to 0$. However, the metric function
$\omega$ is no longer the object of interest here, because the frame-dragging
velocity (in coordinates adapted to $\partial_{t}$) is actually given by

$\widetilde{\omega}=-\frac{\omega f^{2}}{\rho^{2}-(\omega f)^{2}}.$ (116)

This function is too lengthy to write it down explicitly. However, it is
obvious that its zeroes are the zeroes of $\omega$ (given that $f$ has no
poles), its poles the zeroes of

$g_{\phi\phi}=\frac{\rho^{2}-(f\omega)^{2}}{f},$ (117)

i.e., the surface where the reduced circumference shrinks to zero. Since
$\omega$ has a fixed sign depending on the choice of $q_{e}B+q_{m}E$, and
since $f>0$ everywhere, it follows that the sign of $\widetilde{\omega}$ may
change if and only if $g_{\phi\phi}$ changes sign, namely if there exist CTCs
in this spacetime. The boundary of these nonchronal regions, that is the
surface on which the norm of $\partial_{\phi}$ vanishes, will then be a
surface of infinite $|\widetilde{\omega}|$. These observations are manifest in
Fig. 3.

Figure 3: In panel [a] we plot a heatmap of
$(2/\pi)\operatorname{atan}|\widetilde{\omega}|$ on the
$u_{\mathrm{x}v}$-plane for $q_{e}=1.05$, $q_{m}=2.06$, $E=1$, and $B=-1$. We
do so in coordinates (103). The lighter the hue, the greater
$|\widetilde{\omega}|$ is, with white denoting the singular surface. Regions
enclosed by the white curves are regions of negative $\widetilde{\omega}$. In
panel [b] we display a $u_{\mathrm{y}}=0$ cross-section of the
$g_{\phi\phi}=0$ surface for $q_{e}=2.07$, $q_{m}=2.33$, $E=-0.31$, $B=0.06$
(solid), $q_{e}=1.05$, $q_{m}=2.06$, $E=1$, $B=-1$ (dashed), $q_{e}=0.01$,
$q_{m}=-3.14$, $E=-0.07$, $B=0.92$ (dotted). Regions bounded by the curves are
nonchronal regions (filled with CTCs). Keep in mind that the full picture is
obtained via a complete revolution of these profiles about the $v$ axis,
whereby infinity is depicted as a cylinder of unit height and radius.

Indeed, observe that the white curves in panel [a] are exactly the dashed ones
in panel [b]. Note that appearances can be deceiving here, for it looks like
the nonchronal regions extend to infinity. This is not true; by doing
$\rho\to\rho/\lambda$, $z\to z/\lambda^{\alpha}$ with $\alpha$ real positive,
and expanding about $\lambda=0$, one may check that the leading term is always
positive, whichever the value of $\alpha$ is. Interestingly, the fact that the
chronology horizons SHchrono  (the surfaces where the nonchronal regions meet
the chronal ones) coincide with the surfaces of singular frame-dragging
angular velocity, perhaps admits a physical interpretation. Rotation becomes
very rapid very close to these singular surfaces, thereby dragging inertial
frames so strongly that the light cones are completely tilted in the direction
of the circumference! Such a situation is not unfamiliar generally speaking. A
similar interpretation, roughly speaking, appears, for example, in the case of
the Van Stockum solution vanStockum:1937zz ; BonnorDust1 .

Now, observe that if we tune our transformation parameters such that
$q_{m}E=-q_{e}B$, $\omega$ vanishes and thus we obtain a static metric, which
also is of Petrov type I. The new metric is now free of CTCs, because
$g_{\phi\phi}=\rho^{2}/f$ with $f>0$ everywhere; there is no rotation taking
place any longer to tilt the light cones. However, our claim that the
spacetime is regular is not valid under the particular tuning. Indeed, the
assumptions for that were $q_{m}E\neq-q_{e}B$ and $|Q|>1$. Unfortunately, when
$q_{m}E=-q_{e}B$, there are two spatial surfaces, one for $z<0$ and the other
for positive $z$, on which the curvature invariants become singular. The
explicit surface equations are found by requesting the vanishing of the
denominator of $f$. Therefore, although we got rid of rotation and the CTCs,
we ended up with a far worse situation, namely a naked singularity with a
weird disconnected geometry. It goes without saying that there is no reason to
further discuss this scenario, and one should stick to the previous
assumptions which at least guarantee regularity.

Let us finally have a look at the gauge field. Its components read

$\begin{split}A_{t}&=f\frac{[\mathcal{V}+(q_{e}E-q_{m}B)z](-Ez+{q_{e}}\mathcal{E}_{0}/2)-(Bz+{q_{m}}\mathcal{E}_{0}/2)(q_{e}B+q_{m}E)z}{V^{2}},\\\
A_{\phi}&=\rho^{2}\frac{4|X|^{2}q_{m}z+(V+4|X|^{2}z^{2})[({3q_{e}^{2}}-{q_{m}^{2}})B/4+q_{e}q_{m}E]-B}{2V}-\omega
A_{t},\end{split}$ (118)

modulo gauge transformations. A visual of the field lines (and more) can be
found in Fig. 4.

Figure 4: Field lines plotted over the heatmap of (a function of) the norm for
the electric field (left panel) and the magnetic field (right panel). The size
of the arrows is proportional to the magnitude of the vector at each sampled
point. The heatmaps show the values of $\operatorname{atan}|\mathbf{E}|$
(left) and $\operatorname{atan}|\mathbf{B}|$ (right). The lighter the color,
the larger the number. The norms are everywhere finite, and the
$\operatorname{atan}$ function is used for display purposes. Plots are for
$q_{e}=-2$, $q_{m}=131/151$, $E=-2/17$, and $B=-131/2567$. These values
satisfy eq. (LABEL:eq:ProperTune).

We mention that the electric and magnetic fields vanish at asymptotic infinity
in all directions. Contrasting this with the behavior of the fields in the
electromagnetic universe, in which they are uniform close to the axis for all
$z$, we can argue that the electric and magnetic fields in this spacetime are
better-behaved, at least in terms of asymptotic behavior.

###  4.2 Electromagnetic universe and electric Ehlers transformations

Next, let us discuss an alternative possibility that generates a new type I
axisymmetric stationary electrovac field, starting again with the
electromagnetic universe as our seed. This scenario involves acting with
$\operatorname{E}[c]$, where $c$ is a real parameter, upon the seed potentials
(110). Skipping the integration details (we just follow the algorithmic
process outlined at the end of Sec. 2.1), we obtain the target metric (1),
with functions

$\begin{split}f&=\frac{V^{2}}{1+c^{2}(V^{2}-4|X|^{2}z^{2})^{2}},\\\
\omega&=8c\frac{|X|^{2}\rho^{2}z}{V},\\\
\mathrm{e}^{2\gamma}&=V^{4},\end{split}$ (119)

together with a gauge field, whose components read

$\begin{split}A_{t}&=\frac{-E+cB(V^{2}-4|X|^{2}z^{2})}{1+c^{2}(V^{2}-4|X|^{2}z^{2})^{2}}z,\\\
A_{\phi}&=-\frac{B+cE(V+4|X|^{2}z^{2})}{2V}\rho^{2}-\omega A_{t}.\end{split}$
(120)

It is easy to see that if $c\neq 0$, the Petrov type is I. Of course, for
$c=0$ the solution reduces to the electromagnetic universe which has Petrov
type D. The metric does not exhibit any coordinate singularities, and the axis
is not plagued with a spinning string. In particular, the induced metric with
$t=\mathrm{cte}=z$ behaves as

$ds^{2}\underset{\rho\to
0}{\sim}[1+c^{2}(1-4|X|^{2}z^{2})^{2}](d\rho^{2}+\rho^{2}d\phi^{2}),$ (121)

near the symmetry axis, which also proves that there is no conical singularity
there. The components $R^{\mathbf{ab}}{}_{\mathbf{cd}}$ of the Riemann tensor
in the pertinent orthonormal basis of App. A, have a denominator of the
general form $V^{n}f^{-k}$, where $n$ and $k$ are irrelevant positive
integers. Since this denominator cannot be made to vanish, we conclude that
curvature invariants up to arbitrary polynomial order will be everywhere
regular. Moreover, $\lim_{\rho\to\infty}R^{\mathbf{ab}}{}_{\mathbf{cd}}=0$
(ditto for $z\to\infty$) further ensures that tidal forces vanish as we move
far away from the $z$ axis and/or the equator. We may then call this geometry
a background, although not a proper one.

Note that $f$ is strictly positive continuous, meaning that there are no
ergoregions in this stationary spacetime. However, there are surfaces where
the reduced circumference shrinks to zero. These are, once again, exactly the
surfaces where the frame-dragging angular velocity (116) blows up. Actually,
there are two separate $g_{\phi\phi}=0$ surfaces, one in each $z$ half-space,
which moreover behave as chronology horizons, in the sense that they separate
CTC-free regions from CTC-full ones. Therefore, the interpretation we gave in
the previous section applies also here.191919One can check that the nonchronal
regions do not extend to infinity and that they exist for arbitrary values of
$X$ and $c$. Their structure is more or less similar to the one in the
previous case, although now the equator functions as a plane of reflection,
simply because $g_{\phi\phi}$ is invariant under $z\to-z$. For this reason, we
do not bother plotting them.

Now, regarding the electric and magnetic fields, the electric Ehlers
transformation has—besides modifying the $z$ component—generated a nontrivial
component in the $\hat{\rho}$ direction which, as expected, is proportional to
$c$. Interestingly, there are surfaces where the radial component of the
electric field (with respect to the unit basis) vanishes. These are given by

$z^{2}=\frac{c(-E+cBV^{2})\pm 2|c||X|}{4Bc^{2}|X|^{2}},$ (122)

provided that the right hand side is positive. On these surfaces,

$\mathbf{E}=\frac{B^{2}(2|c||X|\pm
cE)}{2V^{2}[4E|c||X|-c(B^{2}+2E^{2})]}\hat{z}=:CV^{-2}\hat{z},$ (123)

which is exactly the form of the electric (or magnetic) field in the
electromagnetic universe! Note that both, electric and magnetic, fields vanish
as $\rho\to\infty$. This was also the case in the seed spacetime. Remarkably,
however, here they also vanish far away in the $\hat{z}$ direction, which was
not at all the case in the Bonnor–Melvin solution, where the fields did not
depend on $z$. In particular, we have that

$\mathbf{E}+i\mathbf{B}\underset{z\to\pm\infty}{\sim}\frac{1}{4c|X|^{2}V}\left(\frac{1}{z^{2}V}\hat{z}\mp\frac{\rho}{z^{3}}\hat{\rho}\right)(B+iE).$
(124)

###  4.3 Swirling universe and electric Ehlers transformations

Having explored the scenarios with an electromagnetic universe as our seed, we
shall now discuss our options when considering a swirling seed. Let us
initiate this discussion with the following composition. In theory, we start
with Minkowski space and operate with
$\operatorname{E}_{e}[c]\circ\operatorname{E}_{m}[j]$ on the associated seed
potentials. In practice, we will just act with $\operatorname{E}[c]$ upon the
seed potentials associated with the swirling spacetime cast into the electric
WLP form. Consequently, it is necessary to identify the seed quantities anew;
the swirling spacetime can be described by a metric (1) with functions

$\displaystyle f_{0}$ $\displaystyle=\frac{S^{2}-(4j\rho z)^{2}}{S},$ (125)
$\displaystyle\omega_{0}$ $\displaystyle=\frac{4j\rho^{2}z}{S^{2}-(4j\rho
z)^{2}},$ (126) $\displaystyle e^{2\gamma}$ $\displaystyle=S^{2}-(4j\rho
z)^{2},$ (127)

where we have defined $S(\rho):=1+j^{2}\rho^{4}$. Given the fact that the seed
is stationary, the solution to (5a) is a nontrivial seed potential

$\chi_{0}=-2j\left(\rho^{2}+6z^{2}-\frac{8z^{2}}{S}\right).$ (128)

With the previous quantities at hand, the only nonvanishing seed Ernst
potential is found to be

$\displaystyle\mathcal{E}_{0}$ $\displaystyle=f_{0}+i\chi_{0}.$ (129)

After acting with $\operatorname{E}[c]$, we arrive at the metric (1) with
functions

$\begin{split}f&=\frac{f_{0}}{(1-c\chi_{0})^{2}+c^{2}f_{0}^{2}},\\\
\omega&=\omega_{0}\left\\{1+2jc[(S+2)\rho^{2}-4z^{2}]-c^{2}[4-8j^{2}z^{2}\boldsymbol{(}2z^{2}+(S-2)\rho^{2}\boldsymbol{)}-3S^{2}]\right\\}.\end{split}$
(130)

The former is directly read off from the target potential $\mathcal{E}$ since
it corresponds to $\operatorname{Re}\mathcal{E}$ in the absence of $\Phi$. The
latter requires integrating eq. (5a). Concerning the Petrov type, here we can
explicitly write down the relevant expression because it is fairly short,

$9\Psi_{2}^{2}-\Psi_{0}\Psi_{4}=-\frac{144cj^{3}\rho^{2}}{\left\\{1+cj[(S+2)\rho^{2}-4z^{2}]+i[c-j\rho^{2}-cj^{2}\rho^{2}(12z^{2}+\rho^{2})]\right\\}^{5}}$
(131)

in particular. Therefore, the Petrov type—based on the analysis we did in Sec.
2.2—of the solution is I. The swirling universe is obtained in the limit $c\to
0$. In this case, expression (131) vanishes, but $\Psi_{4}\neq 0$. On the
other hand, in the limit $j\to 0$, all five complex Weyl-NP scalars become
zero, indicating a Petrov type O. Indeed, the solution reduces to flat
spacetime. This totally agrees with the fact that electric transformations,
when applied to the potentials of Minkowski spacetime, give rise to a target
spacetime which is again Minkowski modulo coordinate rescalings. For the case
at hand, these rescalings read

$t\rightarrow\frac{t}{\sqrt{1+c^{2}}},\quad(\rho,z)\rightarrow\sqrt{1+c^{2}}(\rho,z).$
(132)

The new metric is free of a spinning string, for $\lim_{\rho\to
0}g_{\phi\phi}=0$ at fixed arbitrary $z$. Moreover, it is also free of conical
singularities; the induced metric with $t=\mathrm{cte}=z$ behaves as $\sim
C(\rho^{2}+\rho^{2}d\phi^{2})$ near the symmetry axis, where $C$ is a constant
depending on the parameters and the fixed value of $z$. Note that there are no
coordinate singularities evident. However, the components of the Riemann
tensor in the orthonormal basis, i.e.,
$R^{\mathbf{a}\mathbf{b}}{}_{\mathbf{c}\mathbf{d}}$, come with a denominator
of the general form $[S^{2}-(4j\rho
z)^{2}]^{m}[(1-c\chi_{0})^{2}+c^{2}f_{0}^{2}]^{n}S^{n}$, where the exact
values of the integers $m\geq 0<n$ are utterly unimportant at this stage. When
$m=0$ (and for some components it is), the denominator can never vanish in the
admissible coordinate range. On the other hand, when $m\neq 0$ (true for some
components), there are potential poles on the surfaces $\mathcal{S}_{\pm}=0$
(the ergosurfaces as we will soon see), with $\mathcal{S}_{\pm}$ given in eq.
(40) for $X=0$. Therefore, one cannot argue that the spacetime is everywhere
regular, at least not in the fashion we previously did; one needs to compute
curvature invariants explicitly. Such behavior is solely due to the swirling
nature of the target spacetime, for $\mathcal{S}_{\pm}$ is independent of $c$.
Nevertheless, calculating the Kretschmann scalar, one finds that the
denominator of the latter has $m=0$ and $n=6$, meaning that it is everywhere
regular. Cubic polynomials also have $m=0$ ($n$=9). Consequently, at least up
to third-order curvature polynomials (which are coordinate scalars), the
absence of singularities is verified. The appearance of poles in higher order
polynomials is highly unlikely then, although regularity is not guaranteed in
the robust sense of having an everywhere regular
$R^{\mathbf{ab}}{}_{\mathbf{cd}}$. Therefore, we may consider this as a
background geometry, although we will immediately see that it cannot be
proper.

Note that the denominator of $f$ is everywhere positive, meaning that the sign
of $g_{tt}=-f$ only depends on the numerator. Its vanishing happens on loci
satisfying the surface equation

$\mathcal{S}_{+}\mathcal{S}_{-}=(1+j^{2}\rho^{4})^{2}-(4jz\rho)^{2}=0,$ (133)

which gives the ergosurfaces. Consequently, the ergosurfaces in this spacetime
are exactly the same as the ones in the swirling universe. Finally, let us
once again probe for CTCs. By now, the narrative should be clear. For the
metric (1), the frame-dragging angular velocity is given by (116). Rotation
becomes infinitely rapid on the surfaces where the denominator vanishes,
provided that the zeroes of the denominator are not zeroes of the numerator,
or if they are, that the denominator grows faster than $\omega f^{2}$ close to
the surface. Then, these surfaces are necessarily zeroes of $g_{\phi\phi}$,
and if $g_{\phi\phi}$ changes sign there, these act as chronology horizons,
separating the chronal from the nonchronal parts of spacetime. In the previous
examples of this section, the denominator of $g_{\phi\phi}$, namely the
function $f$, was strictly positive. Thus, the sign of $g_{\phi\phi}$ solely
depended on the sign of the numerator $\rho^{2}-(\omega f)^{2}$. Here, because
there are ergoregions, both the numerator and the denominator are allowed to
change sign. In fact, we expect that the ergoregions and the nonchronal
regions containing CTCs, partially overlap, namely that chronology horizons
and ergosurfaces cross each other. Indeed, this can be seen in Fig. 5.

Figure 5: Left panel. Showing $u_{\mathrm{y}}=0$ cross-section of the
ergoregions (light gray) and the nonchronal regions (dark gray). Right panel.
3D Illustration of the nonchronal regions (regions containing CTCs) in
rectangular coordinates
$\\{\mathrm{x}=\rho\sin\phi,\mathrm{y}=\rho\cos\phi,z\\}$. Plots are for
$j=0.5$ and $c=-0.75$.

Quite interestingly, it turns out that in the spacetime under study, there can
actually be up to four disconnected regions filled with CTCs for certain
parameter ranges, two toroidal regions with finite volume, and two other
regions that extend to infinity, as can be seen from

$g_{\phi\phi}\underset{\rho,|z|\to\infty}{\sim}-\frac{1024(cj\rho^{2}z^{3})^{2}}{144z^{4}+j^{2}\rho^{8}}<0,$
(134)

where we let $z$ grow exactly as fast as $\rho^{2}$. Do note that although it
seems that the nonchronal “tori” closer to the equator comes into contact with
the ergosurfaces, this is not the case. There are only two rings where the two
surfaces intersect. Looking at the left panel of Fig. 5, these would be at
$|v|\approx 0.47$ with radius $|u_{\mathrm{x}}|\approx 0.66$.

###  4.4 Swirling universe and electric Harrison transformations

The remaining spacetime to consider involves the action of the composition
$\operatorname{H}_{e}[Q]\circ\operatorname{E}_{m}[j]$ on the seed potentials
of Minkowski spacetime. In practice, we just act with $\operatorname{H}[Q]$ on
the seed potentials (129). The integration details are “left as an exercise”;
the whole process is already described in the introduction of this work. We
obtain the target metric (1) with functions

$\begin{split}f&=\frac{f_{0}}{(1-|Q|^{2}f_{0})^{2}+|Q|^{4}\chi_{0}^{2}},\\\
\omega&=\omega_{0}\left[1-|Q|^{4}\left(4-8j^{2}z^{2}(2z^{2}+(S-2)\rho^{2})-3S^{2}\right)\right].\end{split}$
(135)

By doing this transformation, we have further excited a gauge field with
nonvanishing components

$\begin{split}A_{t}&=\frac{f}{2f_{0}}\left\\{q_{e}[f_{0}(1-|Q|^{2}f_{0})-|Q|^{2}\chi_{0}^{2}]-q_{m}\chi_{0}\right\\},\\\
A_{\phi}&=\frac{\omega_{0}}{2}\left\\{q_{m}j[(S+2)\rho^{2}-4z^{2}]+[1-(\omega/\omega_{0})(q_{e}/|Q|^{2})]-\omega
A_{t}\right\\}.\end{split}$ (136)

One can immediately check that $9\Psi_{2}^{2}\neq\Psi_{0}\Psi_{4}$ with
$\Psi_{4}\neq 0$ for $j,Q\neq 0$. Therefore, the Petrov type of the solution
is I. The swirling universe solution is recovered in the limit $Q\to 0$
(recall that this implies $q_{e},q_{m}\to 0$). Indeed, also
$9\Psi_{2}^{2}=\Psi_{0}\Psi_{4}$ in this case, with $\Psi_{4}\neq 0$, which
gives the Petrov type we expect, that is D. Minkowski spacetime (up to
irrelevant coordinate rescalings) is obtained in the limit $j\to 0$. In this
limit, all Weyl–NP scalars vanish and the type is O as expected. This
spacetime is also free of a spinning string. The fact that $g_{\phi\phi}$
vanishes as $\sim\rho^{2}$ near the symmetry axis, proves the claim. It is
also free of conical singularities, for the induced metric with
$t=\mathrm{cte}=z$ behaves as $\sim C(d\rho^{2}+\rho^{2}d\phi^{2})$ in the
vicinity of the axis, with $C$ being a constant depending on the parameters
and the fixed value of $z$. What about coordinate singularities?

Let us focus on $f$ and probe it for poles. The function is expected to blow
up at solutions to the equation $(1-|Q|^{2}f_{0})^{2}+|Q|^{4}\chi_{0}^{2}=0$.
This is an equation quadratic in $z^{2}$, which does not have any real
solution unless $z=0$. On the equatorial plane, the form

$(1-|Q|^{2})^{2}+2j^{2}|Q|^{2}(3|Q|^{2}-1)\hat{\rho}+j^{4}|Q|^{4}\hat{\rho}^{2}=0,$
(137)

is assumed, where $\hat{\rho}:=\rho^{1/4}$. It is clear that the above admits
only a single positive real solution if-f $|Q|=1$, the solution being
$\rho=0$. This localizes the coordinate singularity to a single point
$\rho=0=z$ (the origin). We then need to check whether if this is an honest
curvature singularity, or just due to a poor choice of coordinates. Once
again, we turn to $R^{\mathbf{ab}}{}_{\mathbf{cd}}$ in order to draw
conclusions. In the case under study, the components have a denominator of the
general form $[S^{2}-(4j\rho
z)^{2}]^{m}[(1-|Q|^{2}f_{0})^{2}+|Q|^{4}\chi_{0}^{2}]^{n}S^{n}$.202020Again,
the exact values of $m\geq 0<n$ are not needed to make the argument. Clearly,
regularity cannot be directly deduced from $R^{\mathbf{ab}}{}_{\mathbf{cd}}$,
because this guy simply blows up on surfaces (133) and at the origin
$\rho,z=0$. This is not problematic per se, as long as the invariants are
regular. Upon examining the behavior of the Kretschmann scalar for example, it
becomes evident that the thing can be singular only at the origin, which it
approaches as $\sim\rho^{-12}$, and only when $|Q|=1$. Consequently, if
$|Q|=1$, the singularity sitting at $\rho=0=z$ constitutes a true curvature
singularity. However, there is absolutely no null hypersurface to act as a
sort of event horizon here, and one thus fails to comply with the censorship
hypothesis; the singularity is naked. Moreover, the zeroes of the function
$f$, i.e., the surfaces (133), which represent the ergosurfaces in this
spacetime, are completely independent of $Q$ (they coincide with the
ergosurfaces in the swirling universe), meaning that they will also exist for
$|Q|=1$. A naked singularity with ergoregions sounds indeed like things really
took the wrong turn, and they most probably did. However, such a situation is
not that unfamiliar, with the case of the hyperextreme Kerr black hole first
coming to mind griffiths_podolsky_2009 . As previously mentioned, naked
singularities are generally regarded as unphysical. Fortunately, all we have
to do to avoid this worrying issue, is to simply exclude the value $|Q|=1$
from the pool of allowed parameter values. Doing so, we can then argue that
spacetime is regular, of course in the less robust sense of the foregoing
section, namely that curvature invariants at least up to some low order are
regular everywhere. Then, since we have a geometry free of all sorts of
“physical” singularities, we can again call this a background.

It turns out that this background cannot get the “proper” attribute, because,
once again, we are confronted with CTCs. The nonchronal regions in this
spacetime are more or less similar to the ones in the previous case (see Fig.
5); there can be up to four disjoint regions, two in the $z>0$ half-space and
their mirror images in the negative half-space, with the equator acting as the
plane of reflection. As before, closer to the equator we have compact toroidal
regions full of CTCs which exist if $|Q|>1$ regardless of the value of $j$. As
$|z|$ grows larger, one enters the other nonchronal regions which again extend
to infinity, as can be seen from

$g_{\phi\phi}\underset{\rho,|z|\to\infty}{\sim}-\frac{1024(j|Q|^{2}\rho^{2}z^{3})^{2}}{144z^{4}+j^{2}\rho^{8}}<0,$
(138)

where we take $z$ to grow exactly as fast as $\rho^{2}$. The two regions in
each half-space merge exactly when $|Q|=1$, which was previously excluded to
avoid the curvature singularity at the origin.

Concerning the electric and magnetic fields in this electrovac solution, their
expressions are quite lengthy, and we prefer to plot them instead. We do so in
Fig. 6.

Figure 6: Field lines plotted over the heatmap of (a function of) the norm for
the electric field (left panel) and the magnetic field (right panel). The
heatmaps show the values of $\operatorname{atan}|\mathbf{E}|$ (left) and
$\operatorname{atan}|\mathbf{B}|$ (right). The lighter the color, the larger
the number. Pure white color denotes an infinite vector magnitude. Plots are
for $q_{e}=-3$, $q_{m}=-1/2$, and $j=-1/2$.

There, we see that the fields decay at infinity and that their magnitude is
everywhere finite besides on the ergosurfaces, the latter indeed appearing as
poles in the expressions of the norms. We mention here that the graphic
inspection is certainly not sufficient, and that the behavior of the electric
and magnetic fields in this spacetime deserves further investigation which we
will unfortunately not include in this work.

##  5 Conclusions

This paper focused on obtaining the complete list of stationary and
axisymmetric spacetimes, generated from Minkowski spacetime by operating on
the associated seed potentials with a composition of Ehlers and Harrison
transformations. Adopting the terminology laid down in Sec. 2.1, it is clear
that the metric functions in the electric and magnetic forms of the WLP metric
are completely different functions, though ultimately related via function
redefinitions. This implies that the Ernst potentials in the two cases are
also going to differ, and thus, any symmetry transformation in potential space
is going to provide us also with different target potentials, and ultimately
different spacetimes. This is especially clarified in the case of a Minkowski
seed, where Ehlers or Harrison transformations of the electric potentials
eventually lead us again to seed spacetime (in cylindrical coordinates),
modulo rescalings of the noncompact coordinates. On the contrary, an Ehlers or
Harrison transformation of the magnetic potentials gives rise to the swirling
or electromagnetic universe, respectively. Hence, the form of the WLP metric,
electric or magnetic, with which we identify our seed metric, crucially
determines the output of the generating process, a deep-seated fact already.

Consequently, taking into account that there are two ways to identify a seed
metric and two kinds of transformations we are interested in, we should a
priori expect sixteen different spacetimes. However, our seed is Minkowski and
thus, this number quickly reduces to ten because a composition of electric
transformations does nothing, and $U_{m}\circ U_{e}\sim U_{m}$ where
$U=\\{\operatorname{E},\operatorname{H}\\}$. Recalling that
$\operatorname{E}_{m}\circ\operatorname{E}_{m}\sim\operatorname{E}_{m}$,
$\operatorname{H}_{m}\circ\operatorname{E}_{m}\sim\operatorname{E}_{m}\circ\operatorname{H}_{m}$,
and that $\operatorname{H}_{m}\circ\operatorname{H}_{m}$ is a particular case
of $\operatorname{E}_{m}\circ\operatorname{H}_{m}$, while also excluding cases
where the resulting spacetime is either the swirling or the electromagnetic
universe, we should be expecting at most five distinct target spacetimes.
Indeed, one of them was extensively studied in Sec. 3.1, and the other four in
Sec. 4.1.

In Sec. 3, we started by reviewing the electromagnetic universe (the extension
of the Bonnor–Melvin solution including an additional external electric field)
and the swirling solution. These can be obtained from a Minkowski spacetime by
operating on its associated magnetic seed potentials with a Harrison and an
Ehlers transformation, respectively. Since our first goal was to combine these
two, it was reasonable to expect that the resulting spacetime correctly
reduces to its building blocks in the appropriate limits, and that it also
inherits properties from both. Indeed, applying a magnetic “Ehlers of a
Harrison” map, we were led to the electromagnetic swirling universe, which met
the above expectations. We first probed for surfaces where the metric
functions change sign. This revealed the presence of a timelike surface, on
which $\partial_{t}$ is null. Had we been able to single out $\partial_{t}$ as
a unique timelike Killing vector, we would call this the ergosurface. However,
we pointed out that the usual selection mechanism one uses in Kerr geometry,
for example, is not applicable here; there is simply no KV, neither timelike
nor normalized, everywhere at infinity. Again, this was not necessarily
problematic, since also in the case of Kerr, the definition of $t$ as a “time”
is meaningful outside the ergosphere; it just happens that here the
ergoregions eventually extend to infinity in particular directions.

We then pondered whether—excluding these regions at infinity where no KV can
be timelike—there are other sensible criteria, specific to our case, which can
be used to advocate for the existence of a single special timelike Killing
vector, to which we should adapt our coordinates. Without being able to
rigorously prove that there is no such set of conditions, we nevertheless were
not able to find one. The best we could do is to restrict the list of
candidates to the one-parameter family $\partial_{t}+C\partial_{\phi}$, with
$C$ being an arbitrary real parameter. It was then natural to argue that
notions such as ergosurfaces, ergoregions, and frame-dragging angular
velocity, should be understood with respect to the complete one-parameter
family and not just $\partial_{t}$; the latter is as special as any other
member of the former. Moreover, taking the above into consideration, we
emphasized that rotation in these spacetimes can only be perceived in a
relative manner. The frame-dragging angular velocity $\Omega$, defined with
respect to the timelike KV $\partial_{t}+C\partial_{\phi}$ in eq. (47), cannot
provide an absolute measure, for there is a free parameter $C$ roaming around.
However, we remarked that differences (in the value of $\Omega$ on two
different $(\rho,z)$ surfaces) are independent of $C$, and indeed, one may use
these differences to (i) prove the existence of rotating regions in the first
place, and (ii) to further probe for counterrotating regions, with their exact
localization being observer-dependent. Notwithstanding this interesting
observation, relevant also in the swirling case (though not addressed so far),
we eventually decided to adapt our coordinates to a $C=0$ observer for clarity
and simplicity. However, the particular issue definitely requires further
investigation, a task we plan to undertake in future work.

Next, we showed that the EMS spacetime is free of curvature singularities, a
Misner string, and conical singularities. We also verified, by checking the
components of the Riemann tensor in the suitable orthonormal basis of App. A,
that tidal forces diminish at infinity in all directions. Actually, the EMS
metric is asymptotic to a swirling metric, given that the growth rate of the
ratio $\rho/z$ satisfies a certain inequality. However, the complete solution
is not, for the gauge field strength does not in general vanish. We applied
the method described in Sec. 2.2 to conclude that the Petrov type of the
solution is D, and then we gave a detailed sequence of coordinate
transformations and parameter redefinitions, which ultimately proves that the
EMS spacetime belongs to the Kundt family of solutions admitting a shearfree,
nonexpanding, and nontwisting null geodesic congruence. The electric and
magnetic fields in this spacetime were found to decay far away from the
symmetry axis, but, exactly as in the case of the electromagnetic universe,
they were uniform in the proximity of the latter, meaning that they do not
vanish far away from the equator near the axis.

Motivated by the well-established relations of the Bonnor–Melvin solution to a
planar Reissner–Nordström spacetime, and of the swirling solution to a planar
Taub–NUT spacetime, we managed to successfully demonstrate the relation of our
EMS solution to a planar Reissner–Nordström–NUT spacetime; we did so by
performing a double Wick rotation, and by appropriately redefining our
coordinates and parameters. This relation, interesting in its own right,
proves to be even more intriguing, if one notices that it can be exploited to
directly derive the cosmological extension of the solution. The presence of
$\Lambda$ renders the field equations into a set of inhomogeneous equations,
and the system is no longer integrable. The generating technique no longer
applies, for the potential-space symmetries are lost. Thus, the only course of
action practically is direct integration, which can be a daunting task.
Remarkably, starting from a planar Reissner–Nordström–NUT spacetime with a
cosmological constant, and applying the previous transformations and
redefinitions in reverse order, we arrive at a new spacetime, which for
$\Lambda=0$ reduces to the EMS spacetime, and which also is a solution to the
field equations of the theory. We study this spacetime in some detail,
highlighting the presence of a Misner string which can fortunately be
exorcised if we properly tune our parameters. Note that the string exists also
in the swirling-$\Lambda$ solution of Astorino:2022aam , despite having been
overlooked therein. The crucial difference is that in the latter solution,
which can be recovered from the EMS-$\Lambda$ solution we presented here in
the limit $X\to 0$, there is absolutely no way to “banish” it.

Next, we immersed a Schwarzschild black hole in the EMS background. The
presence of a mass source in the seed brought in a singularity and an event
horizon also in the target spacetime, their locations not modified. We made it
clear that whilst the surface area of the horizon does not change, the surface
itself is deformed, with the deformation controlled by the transformation
parameters. The structure of the ergoregions was qualitatively the same, while
no topological singularities or closed timelike curves were present. The
asymptotic behavior of the black hole was that of the background. The solution
was found to have Petrov type I almost everywhere besides the axis, where the
type was D, and the horizon surface, where we proved that it is algebraically
special. We remarked that in order to determine the exact Petrov type on the
horizon, a frame attached to a freely falling observer is needed.

In the second part of this manuscript, we extracted all the spacetimes one can
obtain obtain from Minkowski spacetime, by operating on its associated seed
potentials with one electric and one magnetic transformation, Ehlers or
Harrison. We showed that the order of operations matters. We excluded cases
with $U_{m}\circ U_{e}$ for reasons previously explained, though we remark
that one must consider these when the seed spacetime is other than Minkowski.
For example, given a Schwarzschild seed, a combination
$\operatorname{U}_{m}\circ\operatorname{U}_{e}$ would lead to a completely
different spacetime. We registered four novel type I asymptotically nontrivial
spacetimes:

  1. I.

A four-parameter family containing the real parameters $E,B$, found in the
electromagnetic universe, and two additional real parameters $q_{e},q_{m}$
introduced via the electric Harrison transformation.

  2. II.

A three-parameter family containing $E,B$ and one additional real parameter
$c$ entering via the electric Ehlers transformation.

  3. III.

A two-parameter family containing $j$ and one additional parameter $c$
introduced via the electric Ehlers transformation.

  4. IV.

A three-parameter family containing $j$, found in the swirling spacetime, and
two additional parameters $q_{e},q_{m}$ brought in by the electric Harrison
transformation.

As expected, the new spacetimes are complicated modifications of either the
electromagnetic universe or the swirling solution. However, these are neither
NUTty nor charged, extensions of the latter. To be precise, there is at least
no direct evidence to support the association of $c$ with a NUT parameter, or
the association of $q_{e},q_{m}$ with monopolic charges, and the “physical”
meaning of these parameters deserves further scrutiny.

Starting with the electrovac spacetime I, here the interaction of
$q_{e},q_{m}$ with the parameters $E,B$ controlling the magnitude of the
external electric and magnetic fields in the electromagnetic universe,
remarkably produced a stationary spacetime without ergoregions. We argued that
for $q_{e}^{2}+q_{m}^{2}\leq 4$, the spacetime describes a naked singularity
(a ring singularity on the equatorial plane which can be contracted to a point
at $\rho=0=z$ when the bound is saturated), otherwise it is regular. We then
showcased the presence of a spinning string on the axis which was removable,
and once we took care of that, we showed that there are no further topological
singularities to be bothered with. Limits of the new geometry to Minkowski
spacetime and the electromagnetic universe were checked. By looking at the
frame-dragging angular velocity (in coordinates adapted to $\partial_{t}$), we
deduced the presence of counterrotating regions and of surfaces where the it
cannot even be defined. These last surfaces were then identified with the so-
called chronology horizons which bound the nonchronal regions appearing in
this spacetime (regions filled with CTCs). Indeed, we claimed that such
regions are expected in the configurations we study if the angular velocity is
singular on some surface, and a physical interpretation was given, namely that
very close to these surfaces, light cones are completely tipped in the
direction of angular motion due to the extreme intensity of rotation building
up there. The electromagnetic fields in the target spacetime, finite
everywhere, were found to decay in all directions at infinity, a behavior
contrasting the one in the electromagnetic universe, where the fields appear
uniformly close to the axis.

Electrovac spacetime II can be thought of as another modification of the
electromagnetic universe. We showed that this is yet another stationary
background without ergoregions, but with nonchronal regions which exist for
arbitrary values of the involved parameters. The electric Ehlers parameter $c$
has nontrivially modified the gauge field of the electromagnetic universe,
with the target electric and magnetic fields acquiring an additional component
in the $\hat{\rho}$ direction. They again were everywhere nonsingular, and
they vanished in all directions at infinity. We remarked that there exists a
$(\rho,z)$ surface, on which the fields behave exactly like in the
electromagnetic universe. Next, we extracted vacuum spacetime III, practically
a modification of the swirling universe. We showed that this stationary
spacetime inherited its ergoregion structure from the swirling solution and
that it was regular, at least in the sense of regular curvature invariants up
to some low order. The absence of topological singularities was verified. As
in the preceding backgrounds, this one also featured nonchronal regions, with