$\copyright$ Università degli Studi di Milano

The following thesis was deposited on 25 June 2024 and defended on 09 July
2024 as part of the requirements for the Master’s degree in Physics at the
Università degli Studi di Milano.

Title:
Charged and Rotating Black Holes in a Melvin-swirling Universe

Candidate: Andrea Di<EMAIL_ADDRESS><EMAIL_ADDRESS>
Supervisor: Prof. Silke Klemm
Cosupervisor: Dr. Adriano Viganò

UNIVERSITÀ DEGLI STUDI DI MILANO

FACOLTÀ DI SCIENZE E TECNOLOGIE

Corso di Laurea Magistrale in Fisica

CHARGED AND ROTATING

BLACK HOLES
IN A MELVIN-SWIRLING UNIVERSE

Relatrice: Candidato:
Prof.ssa Silke Klemm Andrea Di Pinto
Matricola 981352
Correlatore:
Dott. Adriano Viganò

Anno Accademico 2023/2024

To my parents

## Notations and conventions

Planck Natural Units:
Speed of light: $c=1$
Universal gravitational constant: $G=1$
Reduced Planck constant: $\hbar=1$
Boltzmann constant: $k_{B}=1$
Metric Signature: $\bigl{(}-,+,+,+\bigr{)}$
World Indices: Greek Letters ($\mu,\nu,\rho,\dots$)
Tangent Space Indices: Latin Letters ($a,b,c,\dots$)
Einstein Summation Convention:
Repeated indices are implicitly summed over (e.g. $\displaystyle
V^{\mu}V_{\mu}\coloneqq\sum_{\mu=0}^{\mu=3}V^{\mu}V_{\mu}$)
Symmetrization:
${T^{\sigma}}_{(\mu_{1}\dots\mu_{n})\rho}\coloneqq\frac{1}{n!}\bigl{(}{T^{\sigma}}_{\mu_{1}\dots\mu_{n}\rho}+\text{sum
over permutations of indices }\mu_{1}\dots\mu_{n}\bigr{)}$
Antisymmetrization:
${T^{\sigma}}_{[\mu_{1}\dots\mu_{n}]\rho}\coloneqq\frac{1}{n!}\bigl{(}{T^{\sigma}}_{\mu_{1}\dots\mu_{n}\rho}+\text{alternating
sum over permutations of indices }\mu_{1}\dots\mu_{n}\bigr{)}$
Gamma Matrices:
$\\{\gamma_{a},\gamma_{b}\\}=2\eta_{ab}\,\mathds{1}_{4}$
$\gamma_{a_{1}\dots a_{n}}\coloneqq\gamma_{[a_{1}}\dots\gamma_{a_{n}]}$

###### Contents

  1. Notations and conventions
  2. 1 Road to General Relativity
    1. 1.1 Basic Notions of (pseudo-)Riemannian Geometry
    2. 1.2 Einstein Field Equations
      1. 1.2.1 Trace-reversed Einstein Field Equations
    3. 1.3 General Relativity
  3. 2 Advanced Topics in General Relativity
    1. 2.1 Einstein-Hilbert Action
      1. 2.1.1 Einstein-Maxwell Equations
    2. 2.2 Killing Vectors
    3. 2.3 Spacetimes with Symmetries
      1. 2.3.1 Special Stationary and Axisymmetric Spacetimes
      2. 2.3.2 The Electric LWP Metric
      3. 2.3.3 The Magnetic LWP Metric
      4. 2.3.4 LWP Conjugation
    4. 2.4 Black Holes
      1. 2.4.1 Frame-dragging
      2. 2.4.2 Ergoregions
      3. 2.4.3 Black Holes Thermodynamics
    5. 2.5 Tetrad Formulation of General Relativity
      1. 2.5.1 Spinors in Curved Spacetime
  4. 3 The Ernst Formalism
    1. 3.1 The Electric Ansatz
    2. 3.2 Rewriting the Einstein-Maxwell Equations
    3. 3.3 Ernst Potentials and Ernst Equations
    4. 3.4 Symmetries of the Ernst Equations
      1. 3.4.1 Symmetries of the Ernst Equations in Pure Gravity
    5. 3.5 The Magnetic Ansatz
    6. 3.6 Magnetic Transformations on the Schwarzschild Black Hole
      1. 3.6.1 Magnetic Ehlers: Swirling
      2. 3.6.2 Magnetic Harrison: Electromagnetic Melvin
    7. 3.7 Electric Transformations on the Schwarzschild Black Hole
      1. 3.7.1 Electric Ehlers: Taub-NUT
      2. 3.7.2 Electric Harrison: Dyonic Reissner-Nordström
    8. 3.8 Physical Meaning of the Ehlers-Harrison Transformations
  5. 4 The Melvin-swirling Universe
    1. 4.1 The Electromagnetic-swirling Universe with a Cosmological Constant
  6. 5 New Charged and Rotating Black Holes in a Melvin-swirling Universe
    1. 5.1 Generating the New Solutions
    2. 5.2 Dyonic Kerr-Newman in a Melvin-swirling Universe
  7. 6 Analysis of the New Solutions
    1. 6.1 Coordinate Singularities
    2. 6.2 Conical Singularities
      1. 6.2.1 Sub-cases with Non-Removable Conical Singularities
      2. 6.2.2 Sub-cases with Removable Conical Singularities
    3. 6.3 Dirac Strings
      1. 6.3.1 Sub-cases with Non-Removable Dirac Strings
      2. 6.3.2 Sub-cases with Removable Dirac Strings
    4. 6.4 Closed Timelike Curves
    5. 6.5 Misner Strings
    6. 6.6 Curvature
      1. 6.6.1 Curvature Singularities at the “Center” of the Black Hole
      2. 6.6.2 Curvature Singularities Along the Symmetry Axis
    7. 6.7 Frame-dragging
      1. 6.7.1 Angular Velocity of the Event Horizon
    8. 6.8 Entropy and Temperature
    9. 6.9 Horizons and Ergoregions
    10. 6.10 Near-Horizon Extremal Geometry
  8. 7 Supersymmetric Extension
    1. 7.1 $\pazocal{N}=2$, $d=4$ Gauged Supergravity
      1. 7.1.1 Killing Spinors
    2. 7.2 Supersymmetric Extension of the Melvin-swirling Universe
  9. A Differential Operators in Cylindrical Coordinates
  10. B New Black Holes from Sub-cases of the New Solution
    1. B.1 Kerr in a Melvin-swirling Universe
    2. B.2 Dyonic Reissner-Nordström in a Melvin-swirling Universe
    3. B.3 Electric Kerr-Newman in a Melvin-swirling Universe
    4. B.4 Magnetic Kerr-Newman in a Melvin-swirling Universe
  11. C Other Useful Melvin-swirling Metrics
    1. C.1 Melvin-swirling Universe with a Cosmological Constant
    2. C.2 Melvin-swirling Universe
    3. C.3 Dyonic Reissner-Nordström in a Swirling Universe
  12. D Embedding of the Event Horizon in $\mathbb{E}^{3}$
  13. E Majorana Representation of the Gamma Matrices

## Introduction

> When I ask myself what are the great things, looking back in history, that
> we got from the era of the Renaissance, it’s the great art, the great music,
> the science insights of Leonardo da Vinci.
> Two hundred years from now, when you ask what are the great things that came
> from this era, I think it’s going to be an understanding of the universe
> around us.
>
> Kip Thorne

In 1915, Albert Einstein published the theory of General Relativity [1, 2],
which represents one of the most profound intellectual breakthroughs in the
history of physics and our current best description of gravity.

Before Einstein, the leading theory of gravity was Isaac Newton’s law of
universal gravitation. In this framework gravity is seen as a long-range
attractive force, acting between any two massive objects, directly
proportional to the product of their masses and inversely proportional to the
square of their distance.

On the other hand, in General Relativity, thanks to the sophisticated
mathematical formalism of differential geometry, developed by Bernhard Riemann
and other mathematicians in the 19th century [3], Einstein elevated the
concept of spacetime given in Special Relativity [4], in which it is seen as a
_flat_ and _immutable_ object, to a _dynamic_ and possibly _curved_
description, where gravity emerges not as a force but rather as a
manifestation of the curvature of spacetime, due to the presence of density
and flux of energy and momentum. Therefore, the main essence of this theory
can be eloquently captured in the words of the renowned physicist John
Archibald Wheeler: “ _Spacetime tells matter how to move; matter tells
spacetime how to curve_ ” [5, 6]. In other words, matter (or energy, or mass)
moves according to the geometry of spacetime where it is located, while
simultaneously, the presence of that matter (or energy, or mass) alters the
curvature of spacetime everywhere.

General Relativity has undergone extensive testing across various scales,
consistently demonstrating an excellent agreement with experimental
observations. From its first successes in the so-called “ _Three Classical
Tests of General Relativity_ ”, namely reproducing the perihelion precession
of Mercury’s orbit [7], predicting the deflection of light by the Sun [8], and
the gravitational redshift of light [9]; to the remarkable foresight of the
recently detected gravitational waves [10]. Among all predictions, black holes
[11] are arguably the most intriguing: they are regions of spacetime where the
gravitational field is so intense that nothing can escape from it, not even
light.

The primary object of this thesis is to analytically construct a new
particular type of black hole, which can be called the _dyonic Kerr-Newman
black hole in a Melvin-swirling universe_ , that corresponds to a rotating and
electromagnetically charged black hole, i.e. a (dyonic) Kerr-Newman black hole
[12, 13], embedded in a universe that itself is both rotating (_swirling_
[14]) and permeated by a uniform magnetic field (_Melvin_ [15]).

Starting from this new solution, we will also obtain other new black holes,
corresponding to its non-charged and non-rotating sub-cases, namely the _Kerr
black hole in a Melvin-swirling universe_ and the _dyonic Reissner-Nordström
black hole in a Melvin-swirling universe_.

More precisely, we will begin by introducing the basic concepts of General
Relativity in Sec. 1, and some advanced ones in Sec. 2, following the most
common textbooks [6, 16, 17].

Subsequently, Sec. 3 will consist in explaining the _Ernst formalism_ , a
solution-generating technique necessary to construct the desired black hole,
where we will follow the original Ernst’s papers [18, 19], as done in [20,
21]. This Ernst formalism is then applied in Sec. 5 in order to obtain the
aforementioned new solutions.

In Sec. 4 we will discuss the physical interpretation of the Melvin-swirling
universe, where we will also add a cosmological constant to this background.
Following that, in Sec. 6 we will study the physical and mathematical
properties of the new black hole and of all its sub-cases. Finally, Sec. 7 is
dedicated to discussing a possible supersymmetric extension of these
spacetimes in Supergravity, using [22, 23, 24, 25, 26] as references.

##  CHAPTER 1Road to General Relativity

### 1.1 Basic Notions of (pseudo-)Riemannian Geometry

The fundamental object in General Relativity is the metric tensor
$g_{\mu\nu}$, a $(0,2)$ symmetric tensor ($g_{\mu\nu}=g_{\nu\mu}$), taken to
be non-degenerate (the determinant $g\coloneqq|g_{\mu\nu}|$ never vanishes),
that represents a generalization of the dot product of ordinary Euclidean
space.

The metric captures all the causal and geometric structure of spacetime; some
of its main purposes consist of being utilized to lower and raise (with the
inverse metric $g^{\mu\nu}$) indices on tensors

$\displaystyle
g_{\mu\rho_{i}}T^{\rho_{1}\dots\rho_{i}\dots\rho_{n}}{}_{\sigma_{1}\dots\sigma_{m}}$
$\displaystyle=T^{\rho_{1}\dots}{}_{\mu}{}^{\dots\rho_{n}}{}_{\sigma_{1}\dots\sigma_{m}}\,,$
(1.1a) $\displaystyle
g^{\mu\sigma_{i}}T^{\rho_{1}\dots\rho_{n}}{}_{\sigma_{1}\dots\sigma_{i}\dots\sigma_{m}}$
$\displaystyle=T^{\rho_{1}\dots\rho_{n}}{}_{\sigma_{1}\dots}{}^{\mu}{}_{\dots\sigma_{m}}\,,$
(1.1b)

from which it is possible to define the norm of any vector field

$|V^{\mu}|^{2}=V^{\mu}V_{\mu}=g_{\mu\nu}V^{\mu}V^{\nu}\,,$ (1.2)

and thus to determine the line element, which is the squared distance $ds^{2}$
between two points infinitesimally close to each other

$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}\,,$ (1.3)

where $dx^{\mu}$ represents the dual basis of the coordinates $x^{\mu}$.

If the metric is not positive-definite, such as in General Relativity, the
norm of a vector may not be positive either, depending on the sign of this
number, vectors are:

$\displaystyle{timelike:\quad}$ $\displaystyle|V^{\mu}|^{2}<0\,,$ (1.4a)
$\displaystyle{lightlike:\quad}$ $\displaystyle|V^{\mu}|^{2}=0\,,$ (1.4b)
$\displaystyle{spacelike:\quad}$ $\displaystyle|V^{\mu}|^{2}>0\,.$ (1.4c)

The partial derivative $\partial$ is a good tensor operator only if the
spacetime is flat; in contrast, on curved spacetimes, it is necessary to use
the so-called covariant derivative (or connection) $\tilde{\nabla}$, whose
action on a generic $(n,m)$ tensor $T$ is:

$\displaystyle\begin{split}\tilde{\nabla}_{\mu}{T^{\rho_{1}\dots\rho_{n}}}_{\sigma_{1}\dots\sigma_{m}}=\partial_{\mu}{T^{\rho_{1}\dots\rho_{n}}}_{\sigma_{1}\dots\sigma_{m}}&+\tilde{\Gamma}{}^{\rho_{1}}{}_{\mu\lambda}\,{T^{\lambda\dots\rho_{n}}}_{\sigma_{1}\dots\sigma_{m}}+\,\dots\,+\tilde{\Gamma}{}^{\rho_{n}}{}_{\mu\lambda}\,{T^{\rho_{1}\dots\lambda}}_{\sigma_{1}\dots\sigma_{m}}\\\
&-\tilde{\Gamma}{}^{\lambda}{}_{\mu\sigma_{1}}{T^{\rho_{1}\dots\rho_{n}}}_{\lambda\dots\sigma_{m}}\,-\,\dots\,-\tilde{\Gamma}{}^{\lambda}{}_{\mu\sigma_{m}}{T^{\rho_{1}\dots\rho_{n}}}_{\sigma_{1}\dots\lambda}\,,\end{split}$
(1.5)

where $\tilde{\Gamma}{}^{\lambda}{}_{\mu\nu}$ are the connection coefficients,
which are _not_ the components of a tensor, but serve to correct the non-
tensorial behavior of the partial derivative in such a way that the result
becomes covariant.

Starting from the connection coefficients it is possible to construct two
important tensors: the torsion tensor

$T_{\mu\nu}{}^{\lambda}=\tilde{\Gamma}{}^{\lambda}{}_{\mu\nu}-\tilde{\Gamma}{}^{\lambda}{}_{\nu\mu}=2\,\tilde{\Gamma}{}^{\lambda}{}_{[\mu\nu]}\,,$
(1.6)

and the curvature tensor, also called the Riemann tensor,

${R^{\rho}}_{\sigma\mu\nu}=\partial_{\mu}\tilde{\Gamma}{}^{\rho}{}_{\nu\sigma}-\partial_{\nu}\tilde{\Gamma}{}^{\rho}{}_{\mu\sigma}+\tilde{\Gamma}{}^{\rho}{}_{\mu\lambda}\tilde{\Gamma}{}^{\lambda}{}_{\nu\sigma}-\tilde{\Gamma}{}^{\rho}{}_{\nu\lambda}\tilde{\Gamma}{}^{\lambda}{}_{\mu\sigma}\,,$
(1.7)

from which it can be verified that the commutator of two covariant derivatives
satisfies:

$\bigl{[}\tilde{\nabla}_{\mu},\tilde{\nabla}_{\nu}\bigr{]}V^{\rho}={R^{\rho}}_{\sigma\mu\nu}V^{\sigma}+{T_{\mu\nu}}^{\lambda}\tilde{\nabla}_{\lambda}V^{\rho}\,.$
(1.8)

The covariant derivative has another important application, indeed it can be
used to define the parallel transport, which is the generalization on a curved
spacetime of the concept _“keeping a vector constant as we move it along a
path”_. Given a $(m,n)$ tensor $T$ and a curve $x^{\mu}(\lambda)$, the
definition of the parallel transport of $T$ along $x^{\mu}(\lambda)$ is then
the requirement:

$\frac{dx^{\nu}}{d\lambda}\tilde{\nabla}_{\nu}{T^{\rho_{1}\dots\rho_{n}}}_{\sigma_{1}\dots\sigma_{m}}=0\,,$
(1.9)

also known as the equation of parallel transport. In particular, if $T$ is the
tangent vector $\frac{dx^{\mu}}{d\lambda}$ to the path $x^{\mu}(\lambda)$,
then Eq. (1.9) becomes the geodesic equation

$\frac{d^{2}x^{\mu}}{d^{2}\lambda}+\tilde{\Gamma}{}^{\mu}{}_{\rho\sigma}\frac{dx^{\rho}}{d\lambda}\frac{dx^{\sigma}}{d\lambda}=0\,,$
(1.10)

which describes the paths of the shortest distance between two points, called
geodesics.

Alternatively, by denoting $u^{\mu}=\frac{dx^{\mu}}{d\lambda}$ as the tangent
vector to the geodesic $x^{\mu}(\lambda)$, this equation is also typically
expressed as

$\tilde{\nabla}_{u}u^{\mu}=u^{\nu}\tilde{\nabla}_{\nu}u^{\mu}=0\,.$ (1.11)

With the concept of parallel transport, torsion can be interpreted as a
description of the twisting of tangent spaces upon parallel transport along a
curve, while curvature describes how tangent spaces roll along a path.
Finally, the commutator of two covariant derivatives (1.8) measures the
difference between parallel transporting a vector (tensor) first in one
direction and then the other, compared to the reverse transport order.

Moreover, it is always possible to define a unique connection on a manifold
with a metric. Indeed, the Fundamental Theorem of Riemannian Geometry states:

Let $\mathscr{M}$ be a differentiable manifold with a metric tensor
$g_{\mu\nu}$. Then there always exists a unique connection $\nabla$, called
the Levi-Civita connection, which satisfies the following conditions:
$\displaystyle\bullet\quad\text{Metric compatibility:
}\nabla_{\rho}g_{\mu\nu}=0\,,$ (1.12) $\displaystyle\bullet\quad\text{Torsion
free: }{T_{\mu\nu}}^{\lambda}=0\,.$ (1.13)

The metric compatibility condition (1.12) ensures the preservation, during
parallel transport, of the norm of vectors, the sense of orthogonality, and so
on. It also implies that a metric-compatible covariant derivative commutes
with the raising and lowering of indices

$V^{\mu}\nabla_{\rho}T_{\sigma\mu}=V_{\mu}\nabla_{\rho}{T_{\sigma}}^{\mu}\,.$
(1.14)

From the definition of the covariant derivative of a tensor (1.5) and the
metric compatibility condition (1.12), it is straightforward to verify that
the associated connection coefficients $\Gamma^{\nu}{}_{\mu\lambda}$ of the
Levi-Civita connection, called the Christoffel symbols, are entirely
determined by the metric tensor via the relation:

$\Gamma^{\nu}{}_{\mu\lambda}=\frac{1}{2}g^{\nu\rho}\Bigl{(}\partial_{\mu}g_{\rho\lambda}+\partial_{\lambda}g_{\rho\mu}-\partial_{\rho}g_{\mu\lambda}\Bigr{)}\,,$
(1.15)

whose symmetry
$\bigl{(}\Gamma^{\lambda}{}_{\mu\nu}=\Gamma^{\lambda}{}_{(\mu\nu)}\bigr{)}$ is
a direct consequence of the definition of the torsion tensor (1.6) and the
torsion-free requirement (1.13) of the connection.

Starting from the Riemann tensor (1.7) it is possible to define several other
significant objects. The most notable among these are: the Ricci tensor

$R_{\mu\nu}={R^{\lambda}}_{\mu\lambda\nu}\,,$ (1.16)

the Ricci scalar (also know as the scalar curvature)

$R=g^{\mu\nu}R_{\mu\nu}\,,$ (1.17)

the Einstein tensor

$G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}R\,g_{\mu\nu}\,,$ (1.18)

and lastly, the Weyl tensor, which, for a spacetime of dimension $d$, takes
the form:

$C_{\mu\nu\rho\sigma}=R_{\mu\nu\rho\sigma}-\frac{2}{(d-2)}\bigl{(}g_{\mu[\rho}R_{\sigma]\nu}-g_{\nu[\rho}R_{\sigma]\mu}\bigr{)}+\frac{2}{(d-2)(d-1)}R\,g_{\mu[\rho}g_{\sigma]\nu}\,.$
(1.19)

The principal properties of these entities can be summarized as follows:

$\displaystyle R_{\mu[\nu\rho\sigma]}$ $\displaystyle=0\,,$ (1.20a)
$\displaystyle\nabla_{[\lambda}R_{\mu\nu]\rho\sigma}$ $\displaystyle=0\,,$
(1.20b) $\displaystyle R_{\mu\nu\rho\sigma}=R_{\rho\sigma\mu\nu}$
$\displaystyle=-R_{\nu\mu\rho\sigma}=-R_{\mu\nu\sigma\rho}\,,$ (1.20c)
$\displaystyle C_{\mu[\nu\rho\sigma]}$ $\displaystyle=0\,,$ (1.20d)
$\displaystyle C^{\mu}{}_{\nu\mu\sigma}$ $\displaystyle=0\,,$ (1.20e)
$\displaystyle C_{\mu\nu\rho\sigma}=C_{\rho\sigma\mu\nu}$
$\displaystyle=-C_{\nu\mu\rho\sigma}=-C_{\mu\nu\sigma\rho}\,,$ (1.20f)
$\displaystyle R_{\mu\nu}$ $\displaystyle=R_{\nu\mu}\,,$ (1.20g)
$\displaystyle\nabla_{\mu}R^{\mu}{}_{\nu}$
$\displaystyle=\frac{1}{2}\nabla_{\nu}R\,,$ (1.20h) $\displaystyle G_{\mu\nu}$
$\displaystyle=G_{\nu\mu}\,,$ (1.20i) $\displaystyle\nabla_{\mu}G^{\mu\nu}$
$\displaystyle=0,$ (1.20j)

of these, the first two are respectively referred to as the first and second
Bianchi identity.

Moreover, the same name is sometimes also used for those that are derived
starting from a Bianchi identity; in particular, the one involving the four-
divergence of the Einstein tensor (1.20j) is also often referred to as the
second Bianchi identity.

### 1.2 Einstein Field Equations

The most important equation of General Relativity is represented by the
Einstein field equations, or just Einstein equations, for which, given the
importance, the gravitation constant $G$ and the speed of light $c$ will just
be temporarily restored:

$G_{\mu\nu}+\Lambda g_{\mu\nu}=\frac{8\pi G}{c^{4}}T_{\mu\nu}\,,$ (1.21)

wherein $G_{\mu\nu}$ is Einstein tensor (1.18), $\Lambda$ is the cosmological
constant, and $T_{\mu\nu}$ is the energy-momentum tensor, which describes the
density and flux of energy and momentum within the spacetime.

Therefore, the core concept of General Relativity lies in the physical
interpretation of the Einstein field equations: the curvature of spacetime is
fundamentally tied to the distribution of energy and momentum.

These equations can be considered as second-order partial differential
equations (PDEs) for the metric $g_{\mu\nu}$. Indeed, the Ricci tensor (1.16)
and the scalar curvature (1.17) result from contractions of the Riemann tensor
(1.7), which involves derivatives and products of the Christoffel symbols
(1.15), which in turn are a combination of the inverse and derivatives of the
metric. Moreover, the energy-momentum tensor $T_{\mu\nu}$ typically also
involves the metric.

The Einstein equations are also non-linear, meaning that two known solutions
cannot be superposed to form a third. Nevertheless, there is a physical reason
for this, which is that, in General Relativity, the gravitational field
couples to itself.

Moreover, being second-order non-linear PDEs, finding a solution without
simplifying assumptions is usually quite a difficult task.

Since both sides of these equations are symmetric two-index tensors, there are
apparently only ten independent equations, which appear to correspond
precisely to the ten unknown components of the metric. However, the Bianchi
identity (1.20j) imposes four additional constraints, thus leaving only six
truly independent equations. This aligns well with the notion of covariance:
if a metric is a solution to the Einstein equations in one coordinate system
then it should also be a solution in any other coordinate system. Hence, there
exist four unphysical degrees of freedom in $g_{\mu\nu}$, corresponding to the
freedom to choose any coordinate system, and Einstein equations solely
constrain the six coordinate-independent degrees of freedom.

Furthermore, the Einstein equations automatically imply the energy-momentum
conservation $\nabla_{\mu}T^{\mu\nu}=0$, this is straightforwardly derived
from the requirement of metric compatibility (1.12) and the fact that the
four-diverge of the Einstein tensor is always zero (1.20j).

#### 1.2.1 Trace-reversed Einstein Field Equations

There is another useful way of expressing the Einstein equations (1.21).
Indeed, using the definition of the Einstein tensor (1.18) and taking the
trace with respect to the metric of both sides of these equations, one
obtains:

$\frac{\bigl{(}2-d\bigr{)}}{2}R+d\,\Lambda=8\pi T\,,$ (1.22)

where it has been used that the trace of the metric tensor is equal to the
spacetime dimension, $g^{\mu\nu}g_{\mu\nu}=d$, and where the trace of the
stress-energy tensor as has been defined as $T\coloneqq T^{\mu}{}_{\mu}$.
Solving for the Ricci scalar $R$ and substituting into the Einstein equation,
the following equivalent form of Einstein equations is obtained:

$R_{\mu\nu}-\frac{2\,\Lambda}{\bigl{(}d-2\bigr{)}}g_{\mu\nu}=8\pi\biggl{[}\,T_{\mu\nu}-\frac{T}{\bigl{(}d-2\bigr{)}}g_{\mu\nu}\biggr{]}\,,$
(1.23)

which are sometimes called the trace-reversed Einstein (field) equations.

In particular, for $d=4$, these reduce to:

$R_{\mu\nu}-\Lambda
g_{\mu\nu}=8\pi\Bigl{(}T_{\mu\nu}-\frac{1}{2}Tg_{\mu\nu}\Bigr{)}\,.$ (1.24)

### 1.3 General Relativity

With all the concepts introduced in this chapter, it is possible to ultimately
summarize the entire content of General Relativity as follows:  _Spacetime is
a four-dimensional differentiable manifold equipped with a Lorentzian metric
$\boldsymbol{g_{\mu\nu}}$.

The curvature of spacetime is fundamentally tied to the distribution of matter
by the Einstein field equations (1.21).
(Matter tells spacetime how to curve)

In the curved manifold, free particles move along geodesics (1.10).
(Spacetime tells matter how to move)_

##  CHAPTER 2Advanced Topics in General Relativity

### 2.1 Einstein-Hilbert Action

As understood by Hilbert [27], Einstein equations (1.21) can be completely
derived by means of an action principle, using the Einstein-Hilbert action

$\pazocal{I}_{E.H.}=\pazocal{I}_{matter}+\frac{1}{16\pi}\int
d^{4}x\sqrt{-g}\Bigl{(}R-2\Lambda\Bigr{)}\,.$ (2.1)

Indeed, to recover a physical law through the stationary-action principle, the
variation of this action with respect to the inverse metric must be zero, thus
yielding

$0=\delta\pazocal{I}_{E.H.}=\int
d^{4}x\frac{\delta\pazocal{I}_{matter}}{\delta g^{\mu\nu}}\delta
g^{\mu\nu}+\frac{1}{16\pi}\int d^{4}x\sqrt{-g}\Bigl{(}G_{\mu\nu}+\Lambda
g_{\mu\nu}\Bigr{)}\delta g^{\mu\nu}\,.$ (2.2)

Since this equation should hold for any variation $\delta g^{\mu\nu}$, it is
straightforward that it reduces precisely to the Einstein equations (1.21) if
the energy-momentum tensor is defined as:

$T_{\mu\nu}\coloneq-\frac{2}{\sqrt{-g}}\frac{\delta\pazocal{I}_{matter}}{\delta
g^{\mu\nu}}\,.$ (2.3)

Therefore, in principle, given the action that describes the theory, it is
always possible to recover the energy-momentum tensor (2.3). Thus, through the
integration of the Einstein equations, the metric tensor $g_{\mu\nu}$.

To be more precise, the matter action is also dependent on the matter fields
$\pazocal{I}_{matter}[g_{\mu\nu},\psi^{i}]$. This means that there are
additional field equations of motion, resulting from the variation of this
action with respect to the matter fields $\psi^{i}$.

#### 2.1.1 Einstein-Maxwell Equations

In relativity, the electric scalar potential and the magnetic vector potential
are combined into a single differential $1-$form, known as the electromagnetic
four-potential $A_{\mu}$, from which it is possible to construct a
differential $2-$form, and thus a totally antisymmetric $(0,2)$ tensor

$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}\,,$ (2.4)

referred to as the Faraday Tensor.

If no external currents are considered, the action of the electromagnetic
field is given by

$\pazocal{I}_{EM}=-\frac{1}{16\pi}\int d^{4}x\sqrt{-g}F_{\mu\nu}F^{\mu\nu}\,,$
(2.5)

therefore, by means of Eq. (2.3), the electromagnetic energy-momentum tensor
is

$T_{EM}^{\mu\nu}=\frac{1}{4\pi}\Bigl{(}F^{\mu\lambda}{F^{\nu}}_{\lambda}-\frac{1}{4}g^{\mu\nu}F_{\rho\sigma}F^{\rho\sigma}\Bigr{)}\,,$
(2.6)

and the Einstein equations (1.21) become

$G^{\mu\nu}+\Lambda g^{\mu\nu}=8\pi T_{EM}^{\mu\nu}\,,$ (2.7)

or equivalently, being the electromagnetic energy-momentum tensor (2.6)
traceless, $T_{EM}=0$, in the trace-reversed (1.24) form:

$R^{\mu\nu}-\Lambda g^{\mu\nu}=8\pi T_{EM}^{\mu\nu}\,.$ (2.8)

Analogously, the variation of the electromagnetic action (2.5), with respect
to the four-potential $A_{\mu}$, leads to the inhomogeneous Maxwell equations

$\nabla_{\mu}F^{\mu\nu}=0\,.$ (2.9)

A priori one should also consider the homogeneous Maxwell equations, which
take the form:

$\nabla_{[\mu}F_{\rho\sigma]}=0\,,$ (2.10)

however, these are always verified, given the definition of the Faraday tensor
(2.4) as a differential $2-$form.

### 2.2 Killing Vectors

Under an infinitesimal diffeomorphism $x^{\prime\mu}=x^{\mu}+\xi^{\mu}(x)$, it
can be verified that the variation of the metric is given by

$\delta
g_{\mu\nu}=-\bigl{(}\nabla_{\mu}\xi_{\nu}+\nabla_{\nu}\xi_{\mu}\bigr{)}\,.$
(2.11)

A four-vector $\xi^{\mu}$ that satisfies the condition $\delta g_{\mu\nu}=0$,
i.e.

$\nabla_{\mu}\xi_{\nu}+\nabla_{\nu}\xi_{\mu}=0\,,$ (2.12)

is called a Killing vector, while this last equation is known as the Killing
equation.

Thus, in general, a Killing vector generates a one-parameter family of
diffeomorphism, under which the metric remains invariant. Such a
diffeomorphism is called an isometry.

It can be proved that if a spacetime possesses a Killing vector $\xi^{\mu}$,
it is always possible to find a coordinate system whose metric is independent
of one of the coordinates $\tilde{x}$, corresponding to the integral curve
coordinate of that Killing vector, and for which
$\xi^{\mu}=\partial_{\tilde{x}}$.

Moreover, Killing vectors imply conserved quantities associated with the
motion of free particles. Indeed, if $x^{\mu}(\lambda)$ is a geodesic (1.10)
with tangent vector $u^{\mu}=\frac{dx^{\mu}}{d\lambda}$, and $\xi^{\mu}$ is a
Killing vector, then

$u^{\nu}\nabla_{\nu}(\xi_{\mu}u^{\mu})=u^{\nu}u^{\mu}\nabla_{\nu}\xi_{\mu}+\xi_{\mu}u^{\nu}\nabla_{\nu}u^{\mu}=0\,,$
(2.13)

where the first term is zero due to the antisymmetrical nature of the Killing
equation (2.12), while the second vanishes because of the assumption that
$x^{\mu}(\lambda)$ is a geodesic (1.11). Thus, the quantity $\xi_{\mu}u^{\mu}$
remains conserved along the particle’s worldline. Since, by definition, the
metric remains unchanged along the direction of a Killing vector, the physical
interpretation of this conservation is that a free particle will not
experience any “forces” in the direction given by a Killing vector, and
therefore the component of the particle’s momentum in that direction will be
conserved.

Furthermore, it can be proved that, for any Killing vector $\xi$, the
following useful relations hold:

$\displaystyle\nabla_{\rho}\nabla_{\mu}\xi^{\nu}$
$\displaystyle=R^{\nu}{}_{\mu\rho\sigma}\xi^{\sigma}\,,$ (2.14a)
$\displaystyle\nabla^{\nu}\nabla_{\nu}\xi^{\mu}$
$\displaystyle=R^{\mu\nu}\xi_{\nu}\,,$ (2.14b)
$\displaystyle\nabla_{\nu}\nabla_{\mu}\nabla^{\nu}\xi^{\mu}$
$\displaystyle=0\,.$ (2.14c)

### 2.3 Spacetimes with Symmetries

Since the presence of Killing vectors and symmetries simplifies the form of
the metric it is useful to state the following definitions:

$\bullet$ A spacetime is said to be stationary if there exists a one-parameter
group of isometries $\sigma_{t}$ whose orbits are timelike curves.
Equivalently, a stationary spacetime is one which possesses a timelike Killing
vector $k^{\mu}$. $\bullet$ A spacetime is said to be axisymmetric (axially
symmetric) if there exists a one-parameter group of isometries $\chi_{\phi}$
whose orbits are closed spacelike curves, which implies the existence of a
spacelike Killing vector $m^{\mu}$ whose integral curves are closed. $\bullet$
A spacetime is said to be stationary and axisymmetric if it possesses both
these symmetries and if, in addition, the action of $\sigma_{t}$ and
$\chi_{\phi}$ commutes:
$\sigma_{t}\circ\chi_{\phi}=\chi_{\phi}\circ\sigma_{t}\,.$ (2.15)
Equivalently, a stationary and axisymmetric is one which possesses both these
symmetries and if, in addition, the Killing vectors $k^{\mu}$ and $m^{\mu}$
commute: $[k,m]=0\,.$ (2.16) $\bullet$ A spacetime is said to be static if it
is stationary and if, in addition, there exists a spacelike hypersurface
$\Sigma$ that is orthogonal to the orbits of the isometry. $\bullet$ A
spacetime is said to be spherically symmetric if its isometry group contains a
subgroup isomorphic to the group SO$(3)$, and the orbits of this subgroup are
two-dimensional spheres.

It can be proved that, for a static spacetime, there always exists a
coordinate transformation such that, in the new coordinates, the timelike
Killing vector is $k=\partial_{t}$, with $t=x^{0}$ timelike coordinate, and
where

$ds^{2}_{static}=-f\bigr{(}x^{1},x^{2},x^{3}\bigr{)}dt^{2}+\sum_{i,j=1}^{3}h_{ij}\bigr{(}x^{1},x^{2},x^{3}\bigr{)}dx^{i}dx^{j}\,.$
(2.17)

For a spherically symmetric spacetime the metric induces a metric on each
orbit $2-$sphere, which, because of the rotational symmetry, must be a
multiple of the metric of a unit $2-$sphere

$ds^{2}_{unit-2-sphere}=\bigl{(}d\theta^{2}+\sin^{2}\theta
d\phi^{2}\bigr{)}\,,$ (2.18)

thus, if a spacetime is both static and spherically symmetric, and if the
static Killing vector $k=\partial_{t}$ is unique, then $k^{\mu}$ must be
orthogonal to the orbit $2-$spheres. Consequently, it is always possible to
choose a set of coordinates $x^{\mu}=(t,r,\theta,\phi)$ for which it holds

$ds^{2}_{static-spherically-
symmetric}=-f\bigr{(}r)dt^{2}+h(r)dr^{2}+r^{2}\bigl{(}d\theta^{2}+\sin^{2}\theta
d\phi^{2}\bigr{)}\,.$ (2.19)

Similarly, due to the definition of a stationary and axisymmetric spacetime,
and in particular for the commutativity of the two Killing vectors (2.16), it
holds that, for a spacetime of this type, it is always possible to choose a
set of coordinates $x^{\mu}=(t,x^{1},x^{2},\phi)$, in such a way that
$k=\partial_{t}$, $m=\partial_{\phi}$, and for which the metric has no
dependence on the two coordinates $t$ and $\phi$. Thus, in these coordinates,
the line element of a stationary and axisymmetric spacetime takes the form

$ds^{2}_{stationary-axisymmetric}=g_{\mu\nu}(x^{2},x^{3})dx^{\mu}dx^{\nu}\,.$
(2.20)

Moreover, due to Eq. (2.13), for a stationary and axisymmetric spacetime, it
holds that along a geodesic with tangent vector $u^{\mu}$ are both constant
the energy (per unit mass) $E$, associated with the Killing vector
$k=\partial_{t}$, and the angular momentum (per unit mass) $L$, associated
with $m=\partial_{\phi}$:

$\displaystyle E$ $\displaystyle=g_{\mu\nu}u^{\mu}k^{\nu}\,,$ (2.21a)
$\displaystyle L$ $\displaystyle=g_{\mu\nu}u^{\mu}m^{\nu}\,.$ (2.21b)

#### 2.3.1 Special Stationary and Axisymmetric Spacetimes

The line element of a stationary and axisymmetric spacetime (2.20) can be
further simplified if the following theorem is satisfied:

Theorem $\star$:
Let $k^{\mu}$ and $m^{\mu}$ be two commuting Killing fields such that:
($\ast$) $k_{[\mu}m_{\nu}\nabla_{\sigma}k_{\lambda]}$ and
$k_{[\mu}m_{\nu}\nabla_{\sigma}m_{\lambda]}$ each vanishes at, at least, one
point of the spacetime.
($\ast\ast$)
$k^{\mu}R_{\mu[\nu}k_{\sigma}m_{\lambda]}=m^{\mu}R_{\mu[\nu}k_{\sigma}m_{\lambda]}=0$.
Then the $2-$planes orthogonal to the Killing fields are integrable.

In particular, ($\ast$) is satisfied whenever the spacetime is asymptotically
flat since there must be a “rotation axis” on which $m^{\mu}$ vanishes;
similarly ($\ast\ast$) is trivially satisfied for vacuum spacetimes without a
cosmological constant, due to the trace-reversed Einstein equations (1.24),
which yield $R_{\mu\nu}=T_{\mu\nu}=0$. Moreover, ($\ast\ast$) can be proved to
hold for spacetimes with specific energy-momentum tensors $T_{\mu\nu}$, such
as that of a stationary and axisymmetric electromagnetic field [28], or that
of a perfect fluid with the four-velocity lying in the plane spanned by the
Killing vectors $k^{\mu}$ and $m^{\mu}$.

The meaning of Theorem $\star$ is that, at each point, the two-dimensional
subspaces of the tangent space spanned by the vectors orthogonal to $k^{\mu}$
and $m^{\mu}$ are integrable, thus tangent to two-dimensional surfaces. Hence,
it is possible to choose coordinates $(x^{1},x^{2})$ in one of the orthogonal
two-dimensional surfaces and then extend these coordinates to the rest of the
spacetime by “transporting” them along the integral curves of $k^{\mu}$ and
$m^{\mu}$.

Therefore, for a stationary and axisymmetric spacetime with commuting Killing
vectors $k=\partial_{t}$ and $m=\partial_{\phi}$, it is always possible to
choose a set of coordinates $(t,x^{1},x^{2},\phi)$ such that

$g_{\mu\nu}=\begin{pmatrix}-V&0&0&W\\\ 0&g_{11}&g_{12}&0\\\
0&g_{12}&g_{22}&0\\\ W&0&0&X\end{pmatrix}\,,$ (2.22)

where $V=-k_{\mu}k^{\mu}=-g_{tt}$, $W=k_{\mu}m^{\mu}=g_{t\phi}$, and
$X=m_{\mu}m^{\mu}=g_{\phi\phi}$, thus reducing the number of the non-vanishing
independent metric components from ten to six.

In order to define one of the two-dimensional surface coordinates it is useful
to introduce the quantity

$\rho^{2}=VX+W^{2}\,,$ (2.23)

which corresponds to (minus) the determinant of the $t-\phi$ part of the
metric. Under the assumption $\nabla_{\mu}\rho\neq 0$ it is possible to choose
$x^{1}$ as $\rho$. Similarly, $x^{2}$ is chosen as a certain $z$ such that
$\bigl{(}\nabla_{\mu}z\bigr{)}\bigl{(}\nabla^{\mu}\rho\bigr{)}=0$, where this
requirement of orthogonality is accomplished by setting $z$ as constant along
the integral curves of $\nabla^{\mu}\rho$, requirement which uniquely
determines $z$ up to $z\rightarrow z\textquoteright=f(z)$.

These coordinates completely determine the metric of a stationary and
axisymmetric spacetime that satisfies the hypothesis of Theorem $\star$, whose
line element is then

$ds^{2}=\frac{\rho^{2}}{V}d\phi^{2}+\Omega^{2}\bigl{(}d\rho^{2}+\Sigma\,dz^{2}\bigr{)}-V\biggl{[}dt-\frac{W}{V}d\phi\biggr{]}^{2}\,,$
(2.24)

and for which there are only four unknown components of the metric,
corresponding to $V$, $W$, $\Sigma$, and $\Omega$, that are functions only of
the two variables $\rho$ and $z$.

#### 2.3.2 The Electric LWP Metric

The form of the metric given by Eq. (2.24) can be furthermore simplified for
spacetimes in a vacuum, $R_{\mu\nu}=0$. Indeed, using Eq. (2.14b) it is
possible to compute the components of $R_{\mu\nu}$ in the plane spanned by the
Killing vectors $k^{\mu}$ and $m^{\mu}$. Therefore, the equation

$0=R^{tt}+R^{\phi\phi}=R^{\mu\nu}k_{\nu}\nabla_{\mu}t+R^{\mu\nu}m_{\nu}\nabla_{\mu}\phi\,,$
(2.25)

yields

$D_{\mu}D^{\mu}\rho=0\,,$ (2.26)

where $D$ is the covariant derivative on the two-dimensional surface spanned
by $\rho$ and $z$.

This equation has the consequence that $\Sigma$ is a function of only the
coordinate $z$. Hence, given the remaining degree of freedom on $z$ that was
discussed above, it is possible to transform
$z\rightarrow\int\Sigma^{\frac{1}{2}}dz$ and thereby set $\Sigma=1$. Finally,
defining $\gamma=\frac{1}{2}\log\bigl{(}V\,\Omega^{2}\bigr{)}$, $f=V$ and
$\omega=\frac{W}{V}$, the metric takes the remarkably simpler form

$ds^{2}_{electric-
LWP}=\frac{1}{f}\Bigl{[}\rho^{2}d\phi^{2}+e^{2\gamma}\bigl{(}d\rho^{2}+dz^{2}\bigr{)}\Bigr{]}-f\bigl{(}dt-\omega
d\phi\bigr{)}^{2}\,,$ (2.27)

known as the electric Lewis-Weyl-Papapetrou (LWP) metric [29, 30, 31, 32].
Thus, the electric LWP metric (2.27) represents the most general metric for
stationary and axisymmetric spacetimes in a vacuum without a cosmological
constant.

Actually, the condition $R_{\mu\nu}=0$ is sufficient but not necessary for the
electric LWP metric to be the most general for a stationary and axisymmetric
spacetime. Indeed, this metric is also the most general for a spacetime of
this type equipped with a stationary and axisymmetric electromagnetic field
[33].

It can be noted that a flat spacetime corresponds to $V=1$, $\omega=\gamma=0$,
for which the electric LWP metric reduces to the Minkowski metric, where the
coordinates $(\rho,z,\phi)$ are the ordinary cylindrical coordinates. For this
reason, the coordinates $(t,\rho,z,\phi)$ are called Weyl, or cylindrical,
coordinates.

#### 2.3.3 The Magnetic LWP Metric

Historically the electric LWP metric (2.27) has always been called just as the
LWP metric, without the _electric_ adjective. On the other hand, in recent
years, it has been widely used another rewriting of this metric, dubbed as
_magnetic_ [20, 14], which is particularly useful in the Ernst formalism, that
will be explained in Sec. 3 where it will also be given another interpretation
on the use of the terms “magnetic” and “electric” for these metrics.

At this stage, it is still possible to explain the terminology by drawing a
parallel with the dualism between electric and magnetic charges. Indeed, it is
possible to apply a “duality” transformation on the electric LWP metric (2.27)
by an analytical continuation, or discrete double-Wick rotation:

$DW\coloneqq\Bigl{\\{}t\to i\psi\,,\quad\quad\\\ \phi\to i\tau\Bigr{\\}}\,,$
(2.28)

which leads to the magnetic Lewis-Weyl-Papapetrou (LWP) metric

$ds^{2}_{magnetic-
LWP}=\frac{1}{f}\Bigl{[}-\rho^{2}d\tau^{2}+e^{2\gamma}\bigl{(}d\rho^{2}+dz^{2}\bigr{)}\Bigr{]}+f\bigl{(}d\psi-\omega
d\tau\bigr{)}^{2}\,,$ (2.29)

that, given the construction of this metric, is another independent solution
of the Einstein equations (1.21) if the same double-Wick rotation (2.28) is
also applied to all the other possible fields of the theory.

#### 2.3.4 LWP Conjugation

Moreover, without using any transformation, it is still always possible to
rewrite the _same_ solution from an electric to a magnetic form, and vice-
versa. Indeed, the electric metric (2.27) can be rewritten as

$ds^{2}_{e\rightarrow
m}=-\frac{1}{xf}\Bigl{[}-\rho^{2}dt^{2}-xe^{2\gamma}\bigl{(}d\rho^{2}+dz^{2}\bigr{)}\Bigr{]}-xf\Bigl{[}d\phi-\frac{\omega}{x}dt\Bigr{]}^{2}\,,$
(2.30)

where $\displaystyle x=\frac{f^{2}\omega^{2}-\rho^{2}}{f^{2}}$ , which leads
to the magnetic form (2.29) if the following redefinitions are applied:

$\tilde{f}=-xf\,,\quad\tilde{\omega}=\frac{\omega}{x}\,,\quad\tilde{\gamma}=\gamma+\frac{1}{2}\log\bigl{(}-x\bigr{)}\,.$
(2.31)

In the same way, the magnetic metric (2.29) can be rewritten as

$ds^{2}_{m\rightarrow
e}=-\frac{1}{xf}\Bigl{[}\rho^{2}d\psi^{2}-xe^{2\gamma}\bigl{(}d\rho^{2}+dz^{2}\bigr{)}\Bigr{]}+xf\Bigl{[}d\tau-\frac{\omega}{x}d\psi\Bigr{]}^{2}\,,$
(2.32)

which results in the electric form (2.27) by using the same redefinitions
(2.31).

The reason why the same redefinitions (2.31) are applied in both cases relies
upon the fact that the double-Wick rotation (2.28) can be seen as the
transformation

$W\coloneqq\biggl{\\{}f\to-\biggl{[}\frac{f^{2}\omega^{2}-\rho^{2}}{f^{2}}\biggr{]}f\,,\,\,\,\omega\to\biggl{[}\frac{f^{2}}{f^{2}\omega^{2}-\rho^{2}}\biggr{]}\omega\,,\,\,\,e^{2\gamma}\to-\biggl{[}\frac{f^{2}\omega^{2}-\rho^{2}}{f^{2}}\biggr{]}e^{2\gamma}\biggr{\\}}\,,$
(2.33)

which is commonly referred to as conjugation [34], and that acts as an
involution operator, i.e. $W\circ W=\mathds{1}$. A widely known example of the
use of the conjugation is that it maps the Schwarzschild black hole (2.34)
into the Witten bubble of nothing [35].

### 2.4 Black Holes

The spacetime geometry in the region surrounding an object of mass $M$, in a
vacuum and without a cosmological constant, is described by the static and
spherically symmetric Schwarzschild metric [11]

$ds^{2}=-\biggl{(}1-\frac{2M}{r}\biggr{)}dt^{2}+\frac{dr^{2}}{\Bigl{(}1-\frac{2M}{r}\Bigr{)}}+r^{2}\bigl{(}d\theta^{2}+\sin^{2}\theta\,d\phi^{2}\bigr{)}\,.$
(2.34)

It is straightforward to verify that this spacetime is singular whenever $r=0$
or $r=2M$.

Spacetime’s singularities can be classified as coordinate singularities, if at
that point the coordinate system breaks down while maintaining a finite
curvature, thus being removable by choosing a different coordinate system, or
as curvature singularities, if the curvature at that point diverges.

To distinguish between curvature and coordinate singularities one could
naïvely use the components of the Riemann tensor $R^{\mu}{}_{\nu\rho\sigma}$.
On the other hand, since these are the components of a tensor, they are
actually coordinate-dependent. Thus, the correct choice relies on constructing
scalar quantities starting from $R^{\mu}{}_{\nu\rho\sigma}$, which are always
coordinate-independent. The scalar commonly used for this purpose is the
Kretschmann scalar

$K=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\,.$ (2.35)

In the case that the radius of the object described by the Schwarzschild
metric (2.34) becomes less than two times the mass $r<2M$, the object
undergoes gravitational collapse, forming what is known as a black hole, a
region of spacetime where gravity is so strong that nothing, not even light,
can escape from it. This region is delimited by a sphere of radius $r=2M$,
which is called an event horizon, thus representing a surface that, once
crossed, can never be traveled back. The singularity at $r=2M$, i.e. where the
event horizon is located, is actually just a _coordinate_ singularity, which
can thus be transformed away using alternative coordinate systems, such as the
Kruskal coordinates. Instead, at the center of the black hole, $r=0$, lies a
_curvature_ singularity, representing the point where all the mass of the
starting object concentrates after the gravitational collapse.

The generalization to an electromagnetically charged and rotating black hole
is given by the dyonic Kerr-Newman black hole [12, 13], where the term
_dyonic_ signifies the presence of both the electric and magnetic charge. The
spacetime of this black hole is stationary and axisymmetric, and can be
expressed in the canonical (ADM) form as:

$\displaystyle ds^{2}$
$\displaystyle=R^{2}\biggl{[}-\frac{\Delta}{\Sigma}\,dt^{2}+\frac{dr^{2}}{\Delta}+d\theta^{2}\biggr{]}+\frac{\Sigma\sin^{2}\theta}{R^{2}}\biggl{[}d\phi-\frac{a\,\lambda}{\Sigma}\,dt\biggr{]}^{2}\,,$
(2.36a) $\displaystyle A$
$\displaystyle=\frac{A_{0}}{\Sigma}\,dt+\frac{A_{3}}{R^{2}}\biggl{[}d\phi-\frac{a\,\lambda}{\Sigma}\,dt\biggr{]}\,,$
(2.36b)

where

$\displaystyle Z^{2}$ $\displaystyle=Q^{2}+H^{2}\,,$ (2.37a)
$\displaystyle\Delta$ $\displaystyle=r^{2}-2Mr+Z^{2}+a^{2}\,,$ (2.37b)
$\displaystyle\lambda$ $\displaystyle=r^{2}+a^{2}-\Delta=2Mr-Z^{2}\,,$ (2.37c)
$\displaystyle\Sigma$ $\displaystyle=\bigl{(}r^{2}+a^{2}\bigr{)}^{2}-\Delta
a^{2}\sin^{2}\theta\,,$ (2.37d) $\displaystyle R^{2}$
$\displaystyle=r^{2}+a^{2}\cos^{2}\theta\,,$ (2.37e) $\displaystyle A_{0}$
$\displaystyle=aH\Delta\cos\theta-Qr\bigl{(}r^{2}+a^{2}\bigr{)}\,,$ (2.37f)
$\displaystyle A_{3}$
$\displaystyle=aQr\sin^{2}\theta-H\bigl{(}r^{2}+a^{2}\bigr{)}\cos\theta\,,$
(2.37g)

$M$ is the mass, $a$ is the angular momentum per unit mass, while $Q$ and $H$
are respectively the electric and the magnetic charge.

It is straightforward to verify that the only curvature singularity of this
spacetime is located at the “center” $R^{2}=0$, i.e. for $r=\cos\theta=0$,
which is called a _ring singularity_ due to its topology. Conversely, the
other singularities at $\theta=0$, $\Delta=0$, and $\Sigma=0$, are all
coordinate singularities.

In particular, and in contrast to the Schwarzschild case, the presence of at
least the rotation or one of the charges gives rise to two horizons, located
at:

$\Delta=0\quad\Rightarrow\quad r_{\pm}=M\pm\sqrt{M^{2}-a^{2}-Z^{2}}\,,$ (2.38)

that may return to a single event horizon at $r=M$, resulting in what is
called an extremal black hole, if the mass satisfies the condition:

$M^{2}=a^{2}+Z^{2}\quad\Rightarrow\quad r_{+}=r_{-}\,.$ (2.39)

Additionally, Eq. (2.38) also implies an upper limit for the charges and the
angular momentum, indeed, if $M^{2}<a^{2}+Z^{2}$ the black hole becomes what
is known as a naked singularity, i.e. a curvature singularity not hidden
behind an event horizon.

#### 2.4.1 Frame-dragging

An important property of rotating black holes is the frame-dragging, which is
an effect where observers with zero angular momentum actually rotate with
respect to infinity. Indeed, for observers with four-velocity

$u=\sqrt{\frac{g_{\phi\phi}}{g_{t\phi}^{2}\,-\,g_{tt}g_{\phi\phi}}}\biggl{[}\partial_{t}-\frac{g_{t\phi}}{g_{\phi\phi}}\partial_{\phi}\biggr{]}\,,$
(2.40)

it is straightforward to verify that
$u\cdot\partial_{r}=u\cdot\partial_{\theta}=0$, meaning that these observers
do not move with respect to the surfaces of constant time. Furthermore, since
$u\cdot\partial_{\phi}=0$, they also have zero angular momentum (2.21b).

On the other hand, they _do_ rotate with respect to infinity, with angular
velocity:

$\frac{d\phi}{dt}=\frac{u^{\phi}}{u^{t}}=-\frac{g_{t\phi}}{g_{\phi\phi}}\,,$
(2.41)

which implies that, despite being locally non-rotating observers due to their
zero angular momentum, they actually rotate with respect to infinity with
angular velocity $\omega=-\frac{g_{t\phi}}{g_{\phi\phi}}$.

In this sense, the quantity (2.41) represents the frame-dragging, or
gravitational dragging, of the whole spacetime.

This angular velocity increases as the observer approaches the black hole, and
it goes in the same direction as the black hole rotation. In particular, this
implies that the angular velocity of the event horizons is

$\omega^{H}_{\pm}=-\biggr{(}\frac{g_{t\phi}}{g_{\phi\phi}}\biggr{)}\bigg{\rvert}_{r=r_{\pm}}\\!=\frac{a}{r_{\pm}^{2}+a^{2}}\\\
\,.$ (2.42)

#### 2.4.2 Ergoregions

Another peculiar property of rotating black holes is that the Killing vector
$k=\partial_{t}$ is not always timelike outside the outer event horizon.
Taking, for simplicity, the non-charged sub-case $Q=H=0$ of the Kerr-Newman
black hole (2.36), which is called the Kerr black hole and is described by the
line element

$\displaystyle ds^{2}$
$\displaystyle=R^{2}\biggl{[}-\frac{\Delta}{\Sigma}\,dt^{2}+\frac{dr^{2}}{\Delta}+d\theta^{2}\biggr{]}+\frac{\Sigma\sin^{2}\theta}{R^{2}}\biggl{[}d\varphi-\frac{2\,aMr}{\Sigma}\,dt\biggr{]}^{2}\,,$
(2.43a) $\displaystyle\Delta$ $\displaystyle=r^{2}-2Mr+a^{2}\,,$ (2.43b)
$\displaystyle\Sigma$ $\displaystyle=\bigl{(}r^{2}+a^{2}\bigr{)}^{2}-\Delta
a^{2}\sin^{2}\theta\,,$ (2.43c) $\displaystyle R^{2}$
$\displaystyle=r^{2}+a^{2}\cos^{2}\theta\,.$ (2.43d)

It is straightforward to verify that this black hole has the horizons (2.38)
located at:

$r_{\pm}=M\pm\sqrt{M^{2}-a^{2}}\,.$ (2.44)

On the other hand,

$k^{2}=g_{tt}=-\frac{\Delta-a^{2}\sin^{2}\theta}{R^{2}}=-\biggl{(}1-\frac{2Mr}{r^{2}+a^{2}\cos^{2}\theta}\biggr{)}\,,$
(2.45)

is not always negative for $r>r_{+}$. In fact, $k$ becomes timelike for
$r>r_{+}$ only outside the surface described by

$r_{erg}(\theta)=m+\sqrt{m^{2}-a^{2}\cos^{2}\theta}\,,$ (2.46)

which is known as the ergosphere, while the region between the outer event
horizon $r_{+}$ and the ergosphere $r_{erg}(\theta)$ is called the ergoregion.
It is interesting to note that the ergosphere intersects the outer event
horizon only for $\theta=0,\pi$. Thus, except for these two angles, observers
can freely enter and exit the ergoregion, as doing so does not require
crossing the event horizon.

$r=0$$r_{-}$$r_{+}$ErgoregionErgosphere Figure 2.1: Graphical representation
of the Kerr black hole. $r_{\pm}$ are the inner and outer horizons, $r=0$ is
the ring singularity, while the ergoregion is the region between the
ergosphere (2.46) and the outer event horizon $r_{+}$.

An interesting theoretical phenomenon due to ergoregions is the Penrose
process, which offers a method for extracting energy from a rotating black
hole. Indeed, if a particle with four-momentum $p^{\mu}$ enters the ergoregion
along a geodesic (1.11), from Eq. (2.21a), there is a constant of motion
associated with the Killing vector $k=\partial_{t}$, which corresponds to the
energy, $E=-g_{\mu\nu}p^{\mu}k^{\nu}$. If this particle decays into two other
particles, one of which falls into the black hole with energy
$E_{1}=-p_{1}\cdot k$, while the other exits the ergoregion and escapes to
infinity, then, conservation of energy dictates that the escaping particle
will have energy $E_{2}=E-E_{1}$. On the other hand, since $k$ is spacelike
inside the ergoregion, it is possible to obtain $E_{2}>E$, thereby extracting
energy from a rotating black hole.

Furthermore, the ergosphere is also referred to as the static limit surface.
This is because once observers enter the region inside the ergosphere, it is
impossible for them to remain _static_ , in the sense that they cannot
maintain a four-velocity equal to $u=\sqrt{-\frac{1}{g_{tt}}}\partial_{t}$.

Similarly, the outer event horizon $r_{+}$ is also the stationary limit
surface, because inside this horizon it is impossible for observers to remain
_stationary_ , in the sense that a stationary observer is one that does not
perceive any time variation in the black hole’s gravitational field, due to
having the four-velocity equal to
$u=\sqrt{\frac{g_{\phi\phi}}{g_{t\phi}^{2}\,-\,g_{tt}g_{\phi\phi}}}\biggl{[}\partial_{t}-\frac{g_{t\phi}}{g_{\phi\phi}}\partial_{\phi}\biggr{]}$
(2.40).

Finally, it can be proved that the ergoregion is also the region where the
black hole’s frame dragging becomes so extreme that, even if an arbitrarily
large force is applied, all observers must orbit with a non-null angular
velocity whose direction matches that of the black hole’s rotation.

#### 2.4.3 Black Holes Thermodynamics

A hypersurface $\pazocal{N}$ whose normal vectorial fields are lightlike is
called a Killing horizon of a Killing vector $\xi$ if, on $\pazocal{N}$, this
Killing vector $\xi$ is normal to $\pazocal{N}$, and thus lightlike.

As an example, the two horizons $r_{\pm}$ (2.38) of the Kerr-Newman black hole
are indeed Killing horizons of the Killing vectors

$\xi^{KN}_{\pm}=\partial_{t}+\omega^{H}_{\pm}\partial_{\phi}\,,\\\ $ (2.47)

where $\omega^{H}_{\pm}$ are the angular velocities of the horizons (2.42).

Additionally, for a Killing horizon $\pazocal{N}$ of the Killing vector $\xi$,
it is possible to define the surface gravity

$\kappa=\sqrt{-\frac{1}{2}\nabla_{\mu}\xi_{\nu}\nabla^{\mu}\xi^{\nu}}\bigg{\rvert}_{\pazocal{N}}\,.$
(2.48)

Classically, a black hole should have zero temperature, however, in 1974,
Hawking [36] proposed that due to the quantum effect known as Hawking
radiation, an event horizon should radiate like a black body with a
temperature

$T=\frac{\kappa}{2\pi}\,,$ (2.49)

in this sense, the surface gravity (2.48) actually represents the black hole
temperature.

Moreover, as proposed by Bekenstein and Hawking [37, 38], black holes should
also have an entropy, proportional to the area of the event horizon,

$S=\frac{A}{4}\,.$ (2.50)

These considerations have solidified the idea that black holes are
thermodynamic objects, since a few years earlier it had already been
formulated what are called The Four Laws of Black Hole Mechanics [39], which
already bore a strong resemblance to the four laws of thermodynamics:

The Four Laws of Black Hole Mechanics
---
|  Thermodynamics |  Black Holes
Zeroth |  The temperature $T$ is constant throughout a system in thermal equilibrium. |  The surface gravity $\kappa$ is constant over the entire horizon of a stationary black hole.
First |  $dE=T\,dS\,\,+$ work terms, where $E$ is the energy, $T$ is the temperature and $S$ is the entropy. |  $dM=\frac{\kappa}{8\pi}\,dA\,+\,\Omega_{H}\,dJ\,+\,\Phi\,dQ$, where $M$ is the black hole mass, $\kappa$ is the surface gravity, $A$ is the horizon area, $\Omega_{H}$ is the angular velocity of the horizon, $J$ is the black hole angular momentum, $\Phi$ is the electrostatic potential and $Q$ is the black hole electric charge.
Second |  In any process, the variation of the entropy $S$ is always greater than or equal to zero $\delta S\geq 0$. |  In any process, the variation of the black hole area $A$ is always greater than or equal to zero $\delta A\geq 0$.
Third |  It is impossible to achieve zero temperature $T=0$ in any physical process. |  It is impossible to achieve zero surface gravity $\kappa=0$ in any physical process.
Table 2.1: The Four Laws of Black Hole Mechanics, expressed in Planck’s units,
alongside their counterparts in Classical Thermodynamics.

However, it must be noted that the second law fails in the presence of
magnetic charges, such as in the case of the dyonic Kerr-Newman black hole.
Additionally, the third law excludes the possibility of extremal black holes
(2.39), since for these black holes the surface gravity is always zero.
Furthermore, one might argue that Hawking’s discovery that black holes radiate
supersedes the second law, as this radiation leads, over time, to a decrease
in the black hole mass, and consequently a reduction in the area of the event
horizon.

In particular, the surface gravity and the area of the Kerr-Newman black hole
are respectively:

$\displaystyle\kappa^{KN}_{\pm}$
$\displaystyle=\sqrt{-\frac{1}{2}\nabla_{\mu}\xi_{\nu}\nabla^{\mu}\xi^{\nu}}\,\bigg{\rvert}_{r=r_{\pm}}\\!=\frac{r_{\pm}-r_{\mp}}{2\bigl{(}r_{\pm}^{2}+a^{2}\bigr{)}}\,,$
(2.51) $\displaystyle\pazocal{A}^{KN}_{\pm}$
$\displaystyle=\int_{0}^{2\pi}\\!d\phi\int_{0}^{\pi}\\!d\theta\,\sqrt{g_{\theta\theta}g_{\varphi\varphi}}\,\bigg{\rvert}_{r=r_{\pm}}\\!=4\pi\bigl{(}r_{\pm}^{2}+a^{2}\bigr{)}\,,$
(2.52)

from which the temperature and the entropy of this black hole, i.e. those of
the outer horizon, are given by:

$\displaystyle T^{KN}$
$\displaystyle=\frac{\kappa_{+}}{2\pi}=\frac{r_{+}-r_{-}}{4\pi\bigl{(}r_{+}^{2}+a^{2}\bigr{)}}\,,$
(2.53) $\displaystyle S^{KN}$
$\displaystyle=\frac{\pazocal{A}_{+}}{4}=\pi\bigl{(}r_{+}^{2}+a^{2}\bigr{)}\,.$
(2.54)

### 2.5 Tetrad Formulation of General Relativity

Up until now, at each point in the manifold, it has been used a natural basis
for tangent spaces given by the partial derivatives with respect to the
coordinates at that point, $e_{\mu}=\partial_{\mu}$, which led to a basis for
the cotangent spaces given by the gradients of the coordinate functions,
$\theta^{\mu}=dx^{\mu}$. However, a coordinate basis $\partial_{\mu}$ is in
general _not_ orthonormal, except of course for the trivial case of the flat
spacetime in Cartesian coordinates. Therefore, at each point in the manifold,
it can be introduced a basis $e_{a}$ not related to any coordinate
system111The index is a Latin letter, instead of a Greek one, as a reminder
that this is a basis not related to any coordinate system. as a linear
combination of the coordinate basis:

$e_{a}=e^{\mu}_{a}e_{\mu}\,,$ (2.55)

and also required to be orthonormal, which traduces in the following
conditions:

$g_{\mu\nu}e^{\mu}_{a}e^{\nu}_{b}=\eta_{ab}\,,$ (2.56)

where $\eta_{ab}=\text{diag}(-1,+1,+1,+1)$ is the Minkowski metric.

Thus, $e^{\mu}_{a}$ represents the coefficients matrix to pass from a non-
orthonormal coordinate system to an orthonormal non-coordinate one. Denoting
their inverse as $e^{a}_{\mu}$, which then satisfy the conditions

$\displaystyle e^{\mu}_{a}e^{a}_{\nu}=\delta^{\mu}_{\nu}\,,$ (2.57a)
$\displaystyle e^{a}_{\mu}e^{\mu}_{b}=\delta^{a}_{b}\,,$ (2.57b)

it is possible to define an orthonormal basis for the cotangent space as

$e^{a}=e_{\mu}^{a}\theta^{\mu}\,,$ (2.58)

which is compatible with the basis vector $e_{a}$, in the sense that

$e^{a}(e_{b})=\delta^{a}_{b}\,.$ (2.59)

The set of vectors $e_{a}$ comprising the orthonormal basis is known as a
tetrad222From the Greek _tetras_ , “a group of four”., or sometimes as a
vielbein333From the German for “many legs”. However, in different numbers of
dimensions, it occasionally becomes a zweibein (two), dreibein (three),
vierbein (four), and so on..

Furthermore, the requirement of ortogonality (2.56) also results in the
following useful relation between the determinant of the tetrad
$e\coloneqq|e^{a}_{\mu}|$ and the determinant of the metric $g$:

$e=\sqrt{-g}\,.$ (2.60)

With these definitions, for any tensor, it is then possible to convert _world_
Greek indices into _tangent space_ Latin indices, and vice-versa, as:

$\displaystyle
e^{a}_{\rho_{i}}\,T^{\rho_{1}\dots\rho_{i}\dots\rho_{n}}{}_{\sigma_{1}\dots\sigma_{m}}$
$\displaystyle=T^{\rho_{1}\dots
a\dots\rho_{n}}{}_{\sigma_{1}\dots\sigma_{m}}\,,$ (2.61a) $\displaystyle
e^{\sigma_{i}}_{a}\,T^{\rho_{1}\dots\rho_{n}}{}_{\sigma_{1}\dots\sigma_{i}\dots\sigma_{m}}$
$\displaystyle=T^{\rho_{1}\dots\rho_{n}}{}_{\sigma_{1}\dots
a\dots\sigma_{m}}\,,$ (2.61b) $\displaystyle e^{\mu}_{c_{i}}\,T^{c_{1}\dots
c_{i}\dots c_{n}}{}_{d_{1}\dots d_{m}}$ $\displaystyle=T^{c_{1}\dots\mu\dots
c_{n}}{}_{d_{1}\dots d_{m}}\,,$ (2.61c) $\displaystyle
e^{d_{i}}_{\mu}\,T^{c_{1}\dots c_{n}}{}_{d_{1}\dots d_{i}\dots d_{m}}$
$\displaystyle=T^{c_{1}\dots c_{n}}{}_{d_{1}\dots\mu\dots d_{m}}\,.$ (2.61d)

Being a non-coordinate basis, the tetrad can be changed independently of the
coordinates, with the only restriction of preserving the orthonormality
(2.56). Therefore, since the transformations that preserve the flat metric
$\eta_{ab}$ are the Lorentz transformations, it holds that, at any point in
spacetime $x$, the tetrad can be transformed as

$e^{\prime}_{a}=\Lambda_{a}{}^{b}(x)e_{b}\,,$ (2.62)

where $\Lambda(x)$ are now _local_ Lorentz transformations, in the sense that
$\Lambda(x)$ represents a position-dependent Lorentz transformation.

Moreover, there is still the usual freedom to perform a general coordinate
transformations, i.e. a diffeomorphism. Thus, the transformation law for
tensors with mixed indices is in general given by:

$T^{\prime a\mu}{}_{c\nu}=\Lambda^{a}{}_{b}\,\frac{\partial
x^{\prime\mu}}{\partial x^{\rho}}\,\Lambda_{d}{}^{c}\,\frac{\partial
x^{\sigma}}{\partial x^{\prime\nu}}\,T^{b\rho}{}_{d\sigma}\,.$ (2.63)

For this reason, it is necessary to define the action of the covariant
derivative upon tensors with Latin indices, in such a way that it has the
correct transformation law under local Lorentz transformations. This requires
new connection coefficients, known as the spin connection coefficients
$\omega_{\mu}{}^{a}{}_{b}$, by means of which the covariant derivative on a
tensor with Latin indices is defined as:

$\displaystyle\begin{split}\nabla_{\mu}{T^{c_{1}\dots c_{n}}}_{d_{1}\dots
d_{m}}=\partial_{\mu}{T^{c_{1}\dots d_{n}}}_{c_{1}\dots
d_{m}}&+\omega_{\mu}{}^{c_{1}}{}_{a}\,{T^{a\dots c_{n}}}_{d_{1}\dots
d_{m}}+\,\dots\,+\omega_{\mu}{}^{c_{n}}{}_{a}\,{T^{c_{1}\dots a}}_{d_{1}\dots
d_{m}}\\\ &-\omega_{\mu}{}^{a}{}_{d_{1}}{T^{c_{1}\dots c_{n}}}_{a\dots
d_{m}}\,-\,\dots\,-\omega_{\mu}{}^{a}{}_{d_{m}}{T^{c_{1}\dots
c_{n}}}_{d_{1}\dots a}\,,\end{split}$ (2.64)

while the action of the covariant derivative on Greek indices is the same as
before, using the Christoffel symbols (1.5) as the connection coefficients.
These new spin connection coefficients are related to the Christoffel symbols
via the so-called tetrad postulate

$\nabla_{\mu}e^{a}_{\nu}=0\,,$ (2.65)

which is _always_ true, and not only for the Levi-Civita connection, and
gives:

$\omega_{\mu}{}^{a}{}_{b}=e^{a}_{\nu}e^{\lambda}_{b}\Gamma^{\nu}{}_{\mu\lambda}-e^{\lambda}_{b}\partial_{\mu}e^{a}_{\lambda}\,.$
(2.66)

On the other hand, for the Levi-Civita connection (1.12), the metric
compatibility requirement (1.12), which now reads $\nabla\eta_{ab}=0$, yields
that the spin connection coefficients are antisymmetric in the Latin indices:

$\omega_{\mu ab}=-\omega_{\mu ba}\,.$ (2.67)

#### 2.5.1 Spinors in Curved Spacetime

The connection $\omega_{\mu}{}^{a}{}_{b}$ is dubbed with the adjective _spin_
because it can be used to construct the Lorentz-covariant derivate on spinors
$\pazocal{D}_{\mu}$, which on a spinor field $\psi$ act as:

$\pazocal{D}_{\mu}\psi=\Bigl{(}\partial_{\mu}+\frac{1}{4}\omega_{\mu}{}^{ab}\gamma_{ab}\Bigr{)}\psi\,.$
(2.68)

Indeed, as the name suggests, under a local Lorentz transformation
$\Lambda(x)$ for which spinors transform as

$\psi^{\prime}=\rho\bigl{(}\Lambda(x)\bigr{)}\psi\,,$ (2.69)

it holds that $\pazocal{D}_{\mu}\psi$ transforms covariantly:

$\pazocal{D}_{\mu}\psi^{\prime}=\pazocal{D}_{\mu}\Bigl{[}\rho\bigl{(}\Lambda(x)\bigr{)}\psi\Bigr{]}=\rho\bigl{(}\Lambda(x)\bigr{)}\pazocal{D}_{\mu}\psi\,.$
(2.70)

Moreover, for a spin $3/2$ field $\psi_{\nu}$, the Lorentz-covariant
derivative satisfies the following relation with the Levi-Civita connection:

$\nabla_{[\mu}\psi_{\nu]}=\pazocal{D}_{[\mu}\psi_{\nu]}\,,$ (2.71)

which is an equality that holds because the extra term in the Levi-Civita
connection, $\Gamma^{\lambda}{}_{[\mu\nu]}\psi_{\lambda}$, is zero due to the
requirement of the connection being torsion free (1.13).

It is interesting to note that the commutator of two Lorentz-covariant
derivatives is related to the curvature, indeed it can be proved that

$\bigl{[}\pazocal{D}_{\mu},\pazocal{D}_{\nu}]\psi=\frac{1}{4}\Bigl{(}R_{\mu\nu}{}^{ab}\gamma_{ab}\Bigr{)}\psi\,,$
(2.72)

which, in a certain sense, is similar to what happens for a connection with no
torsion (1.8).

Finally, it is possible to define the gamma matrices with a curved index as

$\gamma_{\mu}\coloneqq e^{a}_{\mu}\gamma_{a}\,.$ (2.73)

##  CHAPTER 3The Ernst Formalism

As pointed out in Sec. 1.2, Einstein equations are a set of highly non-linear
PDEs of the second order. For this reason, finding new exact analytical
solutions just resting on ansätze is usually extremely challenging, a problem
that led to the development of numerous methods and techniques with the aim of
constructing new exact solutions _without_ the need to perform a direct
integration of the equations of motion. The fundamental idea behind these
methods consists of starting from a known solution, often referred to as a
seed, and then applying a transformation or a map to generate a new, usually
non-equivalent, solution of the field equations, thereby circumventing the
necessity of directly integrating the equations of motion.

One of these methods is due to Ernst [18, 19], which is applicable to
stationary and axisymmetric spacetimes whose metric can be expressed in the
electric LWP form (2.27). The core concept of this method involves utilizing
what are called the complex Ernst potentials, which are defined from the
metric and the electromagnetic potential, in order to rewrite the Einstein-
Maxwell equations as a pair of new complex equations, known as the Ernst
equations, due to the fact these new equations unveil certain symmetries whose
associated transformations can map one solution of the field equations to
another.

In general, not all symmetry maps of the Ernst equations actually generate new
independent solutions; in fact, some are merely gauge transformations.
However, there are two maps, namely the _Ehlers transformations_ [40] and the
_Harrison transformations_ [41], which always act non-trivially on a seed
black hole solution. It is worth noting that these maps actually encompass
four distinct transformations. Indeed, as noted in Sec. 2.3.4, the electric
LWP metric (2.27) can be always double-Wick rotated (2.28) in the magnetic LWP
(2.29) and subsequently reshuffled again in the electric form. This means that
the same transformation can generate another independent solution when applied
to the same seed expressed in the magnetic form, provided that the same
double-Wick rotation is also applied to the potentials of the Ernst method.

### 3.1 The Electric Ansatz

The action of the electromagnetic theory without a cosmological constant is
given by the Einstein-Maxwell action

$\pazocal{I}_{E.M.}=\frac{1}{16\pi}\int
d^{4}x\sqrt{-g}\biggl{(}R-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\biggr{)}\,,$ (3.1)

which, as seen in Sec. 2.1.1, leads to the following Einstein-Maxwell field
equations

$\displaystyle\nabla_{\mu}F^{\mu\nu}$ $\displaystyle=0\,,$ (3.2a)
$\displaystyle G_{\mu\nu}$ $\displaystyle=8\pi T_{\mu\nu}\,,$ (3.2b)

with the electromagnetic energy-momentum tensor $T_{\mu\nu}$ given by Eq.
(2.6).

As discussed in Sec. 2.3.2, the most general metric for a stationary and
axisymmetric spacetime, eventually in the presence of a stationary and
axisymmetric electromagnetic field, is the electric Lewis-Weyl-Papapetrou
metric (2.27). Thus, the first step in the Ernst formalism consists of
choosing an ansatz, in cylindrical coordinates, for the metric and the
electromagnetic field, which is called the electric ansatz and given
by444Actually, as explained in [33], this is not the most general form for a
stationary and axisymmetric four-potential. Nonetheless, this is the
electromagnetic potential that is used for the Ernst formalism.

$\displaystyle ds_{e}^{2}$
$\displaystyle=\frac{1}{f}\Bigl{[}\rho^{2}d\phi^{2}+e^{2\gamma}\bigl{(}d\rho^{2}+dz^{2}\bigr{)}\Bigr{]}-f\bigl{(}dt-\omega
d\phi\bigr{)}^{2}\,,$ (3.3a) $\displaystyle A$
$\displaystyle=A_{t}dt+A_{\phi}d\phi\,,$ (3.3b)

where the dependence on the coordinates of the functions $f$, $\omega$,
$\gamma$, $A_{t}$ and $A_{\phi}$ is only on $(\rho,z)$.

Since the formalism is formulated in cylindrical coordinates $(\rho,z,\phi)$,
and the functions depend only on $(\rho,z)$, it will be convenient to express
the equations in a three-dimensional Euclidean space notation. This involves
introducing quantities like the axis unit vectors
$(\vec{e}_{\rho},\vec{e}_{z},\vec{e}_{\phi})$, the gradient, the curl, the
divergence, and the Laplacian in these coordinates, which are reported in
Appendix A.

### 3.2 Rewriting the Einstein-Maxwell Equations

By defining the Maxwell equations as

$M^{\nu}\coloneq\nabla_{\mu}F^{\mu\nu}\,,$ (3.4)

it can be shown that the $\phi-$component of these equations can be written as

$\begin{split}M^{\phi}&=\frac{e^{-2\gamma}f}{\rho^{2}}\biggl{[}\,\vec{\nabla}f\cdot\bigl{(}\vec{\nabla}A_{\phi}+\omega\vec{\nabla}A_{t}\bigr{)}-\frac{2f}{\rho}\biggl{(}\frac{\partial
A_{\phi}}{\partial\rho}+\omega\frac{\partial A_{t}}{\partial\rho}\biggr{)}\\\
&\quad+f\Bigl{(}\nabla^{2}A_{\phi}+\omega\nabla^{2}A_{t}+\vec{\nabla}\omega\cdot\vec{\nabla}A_{t}\Bigr{)}\biggr{]}\equiv
0\,,\end{split}$ (3.5)

that can be proved to be equivalent to

$\vec{\nabla}\cdot\biggl{[}\,\frac{f}{\rho^{2}}\bigl{(}\vec{\nabla}A_{\phi}+\omega\vec{\nabla}A_{t}\bigr{)}\biggr{]}=0\,.$
(3.6)

Similarly, it is possible to manipulate the $t-$component of the Maxwell
equations as

$\begin{split}M^{t}&=e^{-2\gamma}\biggl{[}\frac{\omega
f^{2}}{\rho^{2}}\Bigl{(}\vec{\nabla}f\cdot\bigl{(}\vec{\nabla}A_{\phi}+\omega\vec{\nabla}A_{t}\bigr{)}+\nabla^{2}A_{\phi}+\omega\nabla^{2}A_{t}\Bigr{)}+\frac{1}{f}\vec{\nabla}f\cdot\vec{\nabla}A_{t}\\\
&\quad+\frac{f^{2}}{\rho^{2}}\vec{\nabla}\omega\cdot\bigl{(}\vec{\nabla}A_{\phi}+2\,\omega\vec{\nabla}A_{t}\bigr{)}-\frac{2\,\omega
f^{2}}{\rho^{3}}\biggl{(}\frac{\partial
A_{\phi}}{\partial\rho}+\omega\frac{\partial
A_{t}}{\partial\rho}\biggr{)}-\nabla^{2}A_{t}\biggr{]}\equiv 0\,,\end{split}$
(3.7)

which is equivalent to

$\vec{\nabla}\cdot\biggl{[}\,\frac{1}{f}\vec{\nabla}A_{t}-\frac{\omega
f}{\rho^{2}}\bigl{(}\vec{\nabla}A_{\phi}+\omega\vec{\nabla}A_{t}\bigr{)}\biggr{]}=0\,.$
(3.8)

Analogously, the Einstein equations in the trace-reversed form (2.8) can be
defined as

$E_{\mu\nu}\coloneq R_{\mu\nu}-8\pi T_{\mu\nu}\,.$ (3.9)

Hence, the $tt-$component of these equations is then given by

$\begin{split}E_{tt}&=-\frac{e^{-2\gamma}}{2\rho^{2}}\biggl{[}\,2f\bigl{(}\vec{\nabla}A_{t}\bigr{)}^{2}+\frac{2f^{3}}{\rho^{2}}\Bigl{(}\omega^{2}\bigl{(}\vec{\nabla}A_{t}\bigr{)}^{2}+\bigl{(}\vec{\nabla}A_{\phi}\bigr{)}^{2}+2\
\omega\vec{\nabla}A_{\phi}\cdot\vec{\nabla}A_{t}\Bigr{)}\\\
&\quad+\bigl{(}\vec{\nabla}f\bigr{)}^{2}-\frac{f^{4}}{\rho^{2}}\bigl{(}\vec{\nabla}\omega\bigr{)}^{2}-f\nabla^{2}f\biggr{]}\equiv
0\,,\end{split}$ (3.10)

that can also be written as

$f\nabla^{2}f=\bigl{(}\vec{\nabla}f\bigr{)}^{2}-\frac{f^{4}}{\rho^{2}}\bigl{(}\vec{\nabla}\omega\bigr{)}^{2}+2f\bigl{(}\vec{\nabla}A_{t}\bigr{)}^{2}+\frac{2f^{3}}{\rho^{2}}\bigl{(}\vec{\nabla}A_{\phi}+\omega\vec{\nabla}A_{t}\bigr{)}^{2}\,.$
(3.11)

Likewise, the $t\phi-$component gives

$\begin{split}E_{t\phi}&=-\frac{e^{-2\gamma}}{2\rho^{2}}\biggl{[}\rho^{2}\omega\bigl{(}\vec{\nabla}f\bigr{)}^{2}+2\,\omega
f^{3}\Bigl{(}\omega^{2}\bigl{(}\vec{\nabla}A_{t}\bigr{)}^{2}+\bigl{(}\vec{\nabla}A_{\phi}\bigr{)}^{2}+2\,\omega\vec{\nabla}A_{\phi}\cdot\vec{\nabla}A_{t}\Bigr{)}\\\
&\quad-2\rho^{2}f\Bigl{(}\vec{\nabla}f\cdot\vec{\nabla}\omega+2\vec{\nabla}A_{\phi}\cdot\vec{\nabla}A_{t}+\omega(\vec{\nabla}A_{t})^{2}+\frac{\omega}{2}\nabla^{2}f\Bigr{)}\\\
&\quad-\rho
f^{2}\biggl{(}\rho\nabla^{2}\omega-2\frac{\partial\omega}{\partial\rho}\biggl{)}-\omega
f^{4}\bigl{(}\vec{\nabla}\omega\bigr{)}^{2}\biggr{]}\equiv 0\,.\end{split}$
(3.12)

As a result, the combination of Eq. (3.12) and Eq. (3.12) as

$\begin{split}E_{t\phi}+\omega
E_{tt}&=-\frac{fe^{-2\gamma}}{2\rho^{4}}\biggl{[}4\vec{\nabla}A_{\phi}\cdot\vec{\nabla}A_{t}+4\,\omega\bigl{(}\vec{\nabla}A_{t}\bigr{)}^{2}\\\
&\quad+2\vec{\nabla}f\cdot\vec{\nabla}\omega+f\vec{\nabla}^{2}\omega-\frac{2f}{\rho}\frac{\partial\omega}{\partial\rho}\biggr{]}\equiv
0\,,\end{split}$ (3.13)

can be shown to be equivalent to

$\vec{\nabla}\cdot\biggl{[}\
\frac{f^{2}}{\rho^{2}}\vec{\nabla}\omega+\frac{4f}{\rho^{2}}\bigl{(}\vec{\nabla}A_{\phi}+\omega\vec{\nabla}A_{t}\bigr{)}A_{t}\biggl{]}=0\,.$
(3.14)

The other non-trivial Einstein equations define $\gamma$ by quadratures.
Indeed, the equations for $E_{\rho z}$ and $E_{\rho\rho}-E_{zz}$ result in

$\displaystyle\begin{split}&\rho\,\frac{\partial\gamma}{\partial
z}+\frac{\rho^{2}}{f}\biggl{(}\frac{\partial A_{t}}{\partial
z}\biggr{)}\biggl{(}\frac{\partial
A_{t}}{\partial\rho}\biggr{)}-2f\biggl{(}\omega\frac{\partial A_{t}}{\partial
z}-\frac{\partial A_{\phi}}{\partial z}\biggr{)}\biggl{(}\omega\frac{\partial
A_{t}}{\partial\rho}-\frac{\partial A_{\phi}}{\partial\rho}\biggr{)}\\\
&\quad-\frac{\rho^{2}}{2f^{2}}\biggl{(}\frac{\partial f}{\partial
z}\biggr{)}\biggl{(}\frac{\partial
f}{\partial\rho}\biggr{)}+\frac{f^{2}}{2}\biggl{(}\frac{\partial\omega}{\partial
z}\biggr{)}\biggl{(}\frac{\partial\omega}{\partial\rho}\biggr{)}=0\,,\end{split}$
(3.15a)
$\displaystyle\begin{split}&\rho\,\frac{\partial\gamma}{\partial\rho}+\frac{\rho^{2}}{f}\biggl{(}\frac{\partial
A_{t}}{\partial\rho}\biggr{)}^{2}\\!\\!-\biggl{(}\frac{\partial
A_{t}}{\partial
z}\biggr{)}^{2}\\!\\!+\frac{\rho^{2}}{f^{2}}\biggl{[}\biggl{(}\frac{\partial
f}{\partial z}\biggr{)}^{2}\\!\\!-\biggl{(}\frac{\partial
f}{\partial\rho}\biggr{)}^{2}\,\biggr{]}+f^{2}\biggl{[}\biggl{(}\frac{\partial\omega}{\partial
z}\biggr{)}^{2}\\!\\!-\biggl{(}\frac{\partial\omega}{\partial\rho}\biggr{)}^{2}\,\biggr{]}\\\
&\quad+f\biggl{[}\biggl{(}\frac{\partial A_{\phi}}{\partial
z}\biggr{)}^{2}\\!\\!-\biggl{(}\frac{\partial
A_{\phi}}{\partial\rho}\biggr{)}^{2}\,\biggl{]}+\,\omega^{2}\biggl{[}\biggl{(}\frac{\partial
A_{t}}{\partial z}\biggr{)}^{2}\\!\\!-\biggl{(}\frac{\partial
A_{t}}{\partial\rho}\biggr{)}^{2}\,\biggr{]}\\\
&\quad+2\,\omega\biggl{[}\biggl{(}\frac{\partial
A_{t}}{\partial\rho}\biggr{)}\biggl{(}\frac{\partial
A_{\phi}}{\partial\rho}\biggr{)}-\biggl{(}\frac{\partial A_{t}}{\partial
z}\biggr{)}\biggl{(}\frac{\partial A_{\phi}}{\partial
z}\biggr{)}\biggr{]}=0\,.\end{split}$ (3.15b)

Thus, $\gamma$ can be found by integrating these latter equations (3.15) once
the functions present in the metric are known.

### 3.3 Ernst Potentials and Ernst Equations

The rewriting of the Maxwell equation for the $\phi-$component (3.6) results
in a total divergence. Hence, for this equation, it is possible to introduce a
potential for which the divergence of the corresponding vector is equal to
zero. Indeed, it can be proved that for any function “sufficiently well-
behaved” $\chi(\rho,z)$, it holds

$\vec{\nabla}\cdot\biggl{[}\,\frac{1}{\rho}\bigl{(}\vec{e}_{\phi}\times\vec{\nabla}\chi\bigr{)}\biggr{]}=0\,.$
(3.16)

Thus, after defining the so-called electromagnetic twisted potential
$\tilde{A}_{\phi}$ through

$\vec{e}_{\phi}\times\vec{\nabla}\tilde{A}_{\phi}\coloneqq\frac{f}{\rho}\bigl{(}\vec{\nabla}A_{\phi}+\omega\vec{\nabla}A_{t}\bigr{)}\,,$
(3.17)

it is straightforward, thanks to (3.16), that the equation (3.6) resulting
from the $\phi-$component of the Maxwell equations, can be rewritten as

$\vec{\nabla}\cdot\biggl{[}\,\frac{1}{\rho}\bigl{(}\vec{e}_{\phi}\times\vec{\nabla}\tilde{A}_{\phi}\bigr{)}\biggr{]}=0\,.$
(3.18)

Moreover, applying $\displaystyle\vec{e}_{\phi}\,\times$ to the definition of
the electromagnetic twisted potential (3.17) yields

$\frac{1}{\rho}\bigl{(}\vec{e}_{\phi}\times\vec{\nabla}A_{\phi}\bigr{)}=-\frac{1}{f}\vec{\nabla}\tilde{A}_{\phi}-\frac{\omega}{\rho}\bigl{(}\vec{e}_{\phi}\times\vec{\nabla}A_{t}\bigr{)}\,,$
(3.19)

which, when composed with the operator $\displaystyle\vec{\nabla}\,\cdot$,
results in

$\vec{\nabla}\cdot\biggl{[}\,\frac{1}{f}\vec{\nabla}\tilde{A}_{\phi}+\frac{\omega}{\rho}\bigl{(}\vec{e}_{\phi}\times\vec{\nabla}A_{t}\bigr{)}\biggr{]}=-\vec{\nabla}\cdot\biggl{[}\,\frac{1}{\rho}\bigl{(}\vec{e}_{\phi}\times\vec{\nabla}A_{\phi}\bigr{)}\biggr{]}\overset{\eqref{potential-
regular}}{=}0\,,$ (3.20)

that substitutes Eq. (3.6) as an equation of motion.

In a similar manner, the equation (3.8) arising from the $t-$component of the
Maxwell equations can be written as

$\vec{\nabla}\cdot\biggl{[}\,\frac{1}{f}\vec{\nabla}A_{t}-\frac{\omega}{\rho}\bigl{(}\vec{e}_{\phi}\times\vec{\nabla}\tilde{A}_{\phi}\bigr{)}\biggr{]}=0\,.$
(3.21)

In summary, after implementing the definition of the potential
$\tilde{A}_{\phi}$ (3.17), the Maxwell equations are equivalent to the
following system:

$\displaystyle\vec{\nabla}\cdot\biggl{[}\,\frac{1}{f}\vec{\nabla}\tilde{A}_{\phi}+\frac{\omega}{\rho}\bigl{(}\vec{e}_{\phi}\times\vec{\nabla}A_{t}\bigr{)}\biggr{]}$
$\displaystyle=0\,,$ (3.22a)
$\displaystyle\vec{\nabla}\cdot\biggl{[}\,\frac{1}{f}\vec{\nabla}A_{t}-\frac{\omega}{\rho}\bigl{(}\vec{e}_{\phi}\times\vec{\nabla}\tilde{A}_{\phi}\bigr{)}\biggr{]}$
$\displaystyle=0\,.$ (3.22b)

Furthermore, the two potentials $A_{t}$ and $\tilde{A}_{\phi}$ can be
effectively packed in what is known as the electromagnetic complex Ernst
potential

$\Phi\coloneqq A_{t}+i\tilde{A}_{\phi}\,,$ (3.23)

so that Eqs. (3.22) can be expressed as the following single complex equation

$\vec{\nabla}\cdot\biggr{[}\,\frac{1}{f}\vec{\nabla}\Phi+\frac{i\,\omega}{\rho}\bigl{(}\vec{e}_{\phi}\times\vec{\nabla}\Phi\bigr{)}\biggr{]}=0\,.$
(3.24)

It is useful to note that Eq. (3.16), applied to the product of the two
functions $\chi=A_{t}\tilde{A}_{\phi}$, gives

$\vec{\nabla}\cdot\biggl{[}\,\frac{1}{\rho}\bigl{(}\vec{e}_{\phi}\times
A_{t}\vec{\nabla}\tilde{A}_{\phi}\bigr{)}\biggr{]}=-\vec{\nabla}\cdot\biggl{[}\frac{1}{\rho}\bigl{(}\vec{e}_{\phi}\times\tilde{A}_{\phi}\vec{\nabla}A_{t}\bigr{)}\biggr{]}\,.$
(3.25)

With this result, the Einstein equation Eq. (3.14) can be written as

$\vec{\nabla}\cdot\biggl{[}\,\frac{f^{2}}{\rho^{2}}\vec{\nabla}\omega+\frac{2}{\rho}\Bigl{(}\vec{e}_{\phi}\times\operatorname{Im}\bigl{(}\Phi^{*}\vec{\nabla}\Phi\bigr{)}\Bigr{)}\biggr{]}=0\,,$
(3.26)

as can be explicitly verified by using the expansion of the product

$\operatorname{Im}\bigl{(}\Phi^{*}\vec{\nabla}\Phi\bigr{)}=A_{t}\vec{\nabla}\tilde{A}_{\phi}-\tilde{A}_{\phi}\vec{\nabla}A_{t}\,.$
(3.27)

Consequently, as done for the electromagnetic twisted potential
$\tilde{A}_{\phi}$ (3.17), it is also possible to introduce a twisted
gravitational potential $h$ from Eq. (3.26), in such a way that

$\vec{e}_{\phi}\times\vec{\nabla}h\coloneqq-\frac{f^{2}}{\rho}\vec{\nabla}\omega-2\,\vec{e}_{\phi}\times\operatorname{Im}\bigl{(}\Phi^{*}\vec{\nabla}\Phi\bigr{)}\,.$
(3.28)

Hence, applying $\displaystyle\vec{e}_{\phi}\,\times$ to the gravitational
twisted potential (3.28) results in

$\frac{1}{\rho}\bigl{(}\vec{e}_{\phi}\times\vec{\nabla}\omega\bigr{)}=\frac{1}{f^{2}}\vec{\nabla}h+\frac{2}{f^{2}}\,\operatorname{Im}\bigl{(}\Phi^{*}\vec{\nabla}\Phi\bigr{)}\,,$
(3.29)

which, when composed with the operator $\displaystyle\vec{\nabla}\,\cdot$,
yields

$\vec{\nabla}\cdot\biggl{[}\,\frac{1}{f^{2}}\vec{\nabla}h+\frac{2}{f^{2}}\operatorname{Im}\bigl{(}\Phi^{*}\vec{\nabla}\Phi\bigr{)}\biggr{]}=-\vec{\nabla}\cdot\biggl{[}\,\frac{1}{\rho}\bigl{(}\vec{e}_{\phi}\times\vec{\nabla}\omega\bigr{)}\biggr{]}\overset{\eqref{potential-
regular}}{=}0\,,$ (3.30)

that substitutes Eq. (3.14).

Finally, the last Einstein equation (3.11) with the implementation of the
twisted potential $h$ becomes

$f\nabla^{2}f=(\vec{\nabla}f)^{2}-\Bigl{[}\vec{\nabla}h+2\operatorname{Im}\bigl{(}\Phi^{*}\vec{\nabla}\Phi\bigr{)}\Bigr{]}^{2}+2f\bigl{(}\vec{\nabla}\Phi^{*}\cdot\vec{\nabla}\Phi\bigr{)}\,.$
(3.31)

Summarising, the Einstein equations provided by Eq. (3.11) and Eq. (3.14) are
replaced by the equations

$\displaystyle\vec{\nabla}\cdot\biggl{[}\,\frac{1}{f^{2}}\vec{\nabla}h+\frac{2}{f^{2}}\operatorname{Im}\bigl{(}\Phi^{*}\vec{\nabla}\Phi\bigr{)}\biggr{]}=0\,,$
(3.32a) $\displaystyle
f\nabla^{2}f=(\vec{\nabla}f)^{2}-\Bigl{[}\vec{\nabla}h+2\operatorname{Im}\bigl{(}\Phi^{*}\vec{\nabla}\Phi\bigr{)}\Bigr{]}^{2}+2f\bigl{(}\vec{\nabla}\Phi^{*}\cdot\vec{\nabla}\Phi\bigr{)}\,.$
(3.32b)

By defining the gravitational complex Ernst potential as

$\pazocal{E}\coloneqq f-\Phi^{*}\Phi+ih\,,$ (3.33)

together with the electromagnetic complex Ernst potential $\Phi$ (3.23), the
equations of motion represented by the Eqs. (3.22) and Eqs. (3.32) can be
expressed as the following two complex equations:

$\displaystyle\bigl{(}\operatorname{Re}\pazocal{E}+\Phi^{*}\Phi\bigr{)}\nabla^{2}\pazocal{E}$
$\displaystyle=\vec{\nabla}\pazocal{E}\cdot\bigl{(}\vec{\nabla}\pazocal{E}+2\Phi^{*}\vec{\nabla}\Phi\bigr{)}\,,$
(3.34a)
$\displaystyle\bigl{(}\operatorname{Re}\pazocal{E}+\Phi^{*}\Phi\bigr{)}\nabla^{2}\Phi$
$\displaystyle=\vec{\nabla}\Phi\cdot\bigl{(}\vec{\nabla}\pazocal{E}+2\Phi^{*}\vec{\nabla}\Phi\bigr{)}\,,$
(3.34b)

called the Ernst equations, which thus formally reduces the problem of solving
the Einstein-Maxwell equations for a stationary and axisymmetric spacetime to
the problem of solving these equations.

Similarly, the equations for $\gamma$, in terms of the Ernst potentials $\Phi$
and $\pazocal{E}$, are given by

$\displaystyle\begin{split}\partial_{\rho}\gamma&=\frac{\rho}{4\bigl{(}\operatorname{Re}\pazocal{E}+\Phi^{*}\Phi\bigr{)}^{2}}\biggl{[}\bigl{(}\partial_{\rho}\pazocal{E}+2\Phi^{*}\partial_{\rho}\Phi\bigr{)}\bigl{(}\partial_{\rho}\pazocal{E}^{*}+2\Phi\partial_{\rho}\Phi^{*}\bigr{)}\\\
&\quad-\bigl{(}\partial_{z}\pazocal{E}+2\Phi^{*}\partial_{z}\Phi\bigr{)}\big{(}\partial_{z}\pazocal{E}^{*}+2\Phi\partial_{z}\Phi^{*}\bigr{)}\biggr{]}\\\
&\quad-\frac{\rho}{\bigl{(}\operatorname{Re}\pazocal{E}+\Phi^{*}\Phi\bigr{)}}\big{(}\partial_{\rho}\Phi^{*}\partial_{\rho}\Phi-\partial_{z}\Phi^{*}\partial_{z}\Phi\big{)}\,,\end{split}$
(3.35a)
$\displaystyle\begin{split}\partial_{z}\gamma&=\frac{\rho}{4\bigl{(}\operatorname{Re}\pazocal{E}+\Phi^{*}\Phi\bigr{)}^{2}}\biggl{[}\big{(}\partial_{\rho}\pazocal{E}+2\Phi^{*}\partial_{\rho}\Phi\big{)}\big{(}\partial_{z}\pazocal{E}^{*}+2\Phi\partial_{z}\Phi^{*}\big{)}\\\
&\quad+\big{(}\partial_{z}\pazocal{E}+2\Phi^{*}\partial_{z}\Phi\big{)}\big{(}\partial_{\rho}\pazocal{E}^{*}+2\Phi\partial_{\rho}\Phi^{*}\big{)}\biggr{]}\\\
&\quad-\frac{\rho}{\bigl{(}\operatorname{Re}\pazocal{E}+\Phi^{*}\Phi\bigr{)}}\big{(}\partial_{\rho}\Phi^{*}\partial_{z}\Phi+\partial_{z}\Phi^{*}\partial_{\rho}\Phi\big{)}\,,\end{split}$
(3.35b)

that, as already mentioned, can be solved by quadratures.

### 3.4 Symmetries of the Ernst Equations

It can be noted that Ernst equations (3.34) represents an effective three-
dimensional problem, thus allowing one to overlook the four-dimensional origin
of the problem. Furthermore, these equations also enable an efficient way to
study the symmetries of the Einstein-Maxwell equations and subsequently make
use of such symmetries to construct new solutions from an already known one.
Indeed, it can be proved that the Ernst equations (3.34) can be derived from
the effective three-dimensional action

$S=\int
d^{3}x\,\biggl{[}\frac{\bigl{(}\vec{\nabla}\pazocal{E}+2\Phi^{*}\vec{\nabla}\Phi\bigr{)}\cdot\bigl{(}\vec{\nabla}\pazocal{E}^{*}+2\Phi\vec{\nabla}\Phi^{*}\bigr{)}}{\bigl{(}\pazocal{E}+\pazocal{E}^{*}+2\Phi^{*}\Phi\bigr{)}^{2}}-\frac{2\vec{\nabla}\Phi^{*}\cdot\vec{\nabla}\Phi}{\bigl{(}\pazocal{E}+\pazocal{E}^{*}+2\Phi^{*}\Phi\bigr{)}}\biggr{]}\,.$
(3.36)

Therefore, the symmetries of the Ernst equations (3.34) can be found just by
studying the symmetries of the Ernst action (3.36)555To be more precise, not
all the symmetries of the equations of motion correspond to the symmetries of
the action. Indeed, the equations of motion might possess more symmetries than
the action does. However, the symmetries of the action are always symmetries
of the equations of motion, and for the Ernst equations they also coincide..

A smart way to study the symmetries of the Ernst action (3.36) is to analyze
the quadratic form associated with this action, also called the associated
metric [42]. Hence, after considering $\pazocal{E}$ and $\Phi$ as complex
coordinates with real coordinates $(x,y,u,v)$ as

$\displaystyle\pazocal{E}$ $\displaystyle=x+i\,y\,,$ (3.37a)
$\displaystyle\Phi$ $\displaystyle=u+i\,v\,,$ (3.37b)

the metric associated with the action (3.36) is given by

$\displaystyle ds^{2}$
$\displaystyle=\frac{\bigl{(}d\pazocal{E}+2\Phi^{*}d\Phi\bigr{)}\bigl{(}d\pazocal{E}^{*}+2\Phi
d\Phi^{*}\bigr{)}}{\bigl{(}\pazocal{E}+\pazocal{E}^{*}+2\Phi^{*}\Phi\bigr{)}^{2}}-\frac{2\,d\Phi^{*}d\Phi}{\bigl{(}\pazocal{E}+\pazocal{E}^{*}+2\Phi^{*}\Phi\bigr{)}}\,$
(3.38)
$\displaystyle=\frac{dx^{2}+dy^{2}-4\,x\bigl{(}du^{2}+dv^{2}\bigr{)}+4\,dy\bigl{(}\,u\,dv-v\,du\bigr{)}+4\,dx\,\bigl{(}v\,dv+u\,du\bigr{)}}{4\,\bigl{(}x+u^{2}+v^{2}\bigr{)}^{2}}\,.$
(3.39)

Solving the Killing equation (2.12) for this four-dimensional metric yields a
total of eight Killing vectors

$\displaystyle\xi_{1}$
$\displaystyle=4\,x\,y\,\partial_{x}+2\bigl{(}y^{2}-x^{2}\bigr{)}\partial_{y}+2\bigl{(}x\,v+y\,u\bigr{)}\partial_{u}+2\bigl{(}y\,v-x\,u\bigr{)}\partial_{v}\,,$
(3.40a) $\displaystyle\xi_{2}$
$\displaystyle=2\bigl{(}x\,v+y\,u\bigr{)}\partial_{x}+2\bigl{(}y\,v-x\,u\bigr{)}\partial_{y}+\bigl{(}4\,u\,v-y\bigr{)}\partial_{u}+\bigl{(}2\,v^{2}-2\,u^{2}+x\bigr{)}\partial_{v}\,,$
(3.40b) $\displaystyle\xi_{3}$
$\displaystyle=2\bigl{(}x\,u-y\,v\bigr{)}\partial_{x}+2\bigl{(}x\,v+y\,u\bigr{)}\partial_{y}+\bigl{(}2\,u^{2}-2\,v^{2}+x\bigr{)}\partial_{u}+\bigl{(}4\,u\,v+y\bigr{)}\partial_{v}\,,$
(3.40c) $\displaystyle\xi_{4}$
$\displaystyle=4\,x\,\partial_{x}+4\,y\,\partial_{y}+2\,u\,\partial_{u}+2\,v\,\partial_{v}\,,$
(3.40d) $\displaystyle\xi_{5}$
$\displaystyle=-v\,\partial_{u}+u\,\partial_{v}\,,$ (3.40e)
$\displaystyle\xi_{6}$
$\displaystyle=-2\,v\,\partial_{x}+2\,u\,\partial_{y}+\partial_{v}\,,$ (3.40f)
$\displaystyle\xi_{7}$
$\displaystyle=2\,u\,\partial_{x}+2\,v\,\partial_{y}-\partial_{u}\,,$ (3.40g)
$\displaystyle\xi_{8}$ $\displaystyle=4\,\partial_{u}\,,$ (3.40h)

which are then equivalent to the eight infinitesimal generators of the
symmetries of the Ernst action (3.36).

In order to find the finite transformations generated by these Killing vectors
(3.40), it is necessary to integrate the flow generated by such vectors. The
equations that define the flow are:

$\frac{\partial x^{i}}{\partial\epsilon}=\xi^{i}\bigl{(}x^{i}\bigr{)}\,,$
(3.41)

where $x^{i}=(x,y,u,v)$ are the coordinates, and $\epsilon$ is the flow
parameter of the finite transformation.

Integrating the infinitesimal transformations provided by Eqs. (3.40) yields
the so-called finite Kinnersley transformations [42]:

$\displaystyle\pazocal{E}^{\prime}$
$\displaystyle=\lambda^{*}\lambda\,\pazocal{E}\,,\qquad\qquad\qquad\quad\;\,\hskip
0.75pt\Phi^{\prime}=\lambda\Phi\,,$ (3.42a)
$\displaystyle\pazocal{E}^{\prime}$
$\displaystyle=\pazocal{E}+i\,b\,,\qquad\qquad\quad\quad\;\;\;\;\Phi^{\prime}=\Phi\,,$
(3.42b) $\displaystyle\pazocal{E}^{\prime}$
$\displaystyle=\frac{\pazocal{E}}{1+i\,\jmath\,\pazocal{E}}\,,\qquad\qquad\qquad\;\Phi^{\prime}=\frac{\Phi}{1+i\,\jmath\,\pazocal{E}}\,,$
(3.42c) $\displaystyle\pazocal{E}^{\prime}$
$\displaystyle=\pazocal{E}-2\beta^{*}\Phi-\beta^{*}\beta\,,\qquad\quad\\!\hskip
0.35pt\Phi^{\prime}=\Phi+\beta\,,$ (3.42d) $\displaystyle\pazocal{E}^{\prime}$
$\displaystyle=\frac{\pazocal{E}}{1-2\,\alpha^{*}\Phi-\alpha^{*}\alpha\,\pazocal{E}}\,,\quad\;\;\,\Phi^{\prime}=\frac{\alpha\,\pazocal{E}+\Phi}{1-2\,\alpha^{*}\Phi-\alpha^{*}\alpha\,\pazocal{E}}\,,$
(3.42e)

where $\alpha$, $\beta$ and $\lambda$ are complex parameters, while $b$ and
$\jmath$ are real.

It can be proved [42, 21] that the transformations given by Eq. (3.42a), Eq.
(3.42b) and Eq. (3.42d) are actually gauge transformations, which thus do not
produce any new solutions, in the sense that these transformations are
diffeomorphisms which do not alter the nature of the solutions. For this
reason, these maps are not particularly interesting. In contrast, the
transformations provided by Eq. (3.42c) and Eq. (3.42e), respectively known as
Ehlers transformations [40] and Harrison transformations [41], usually act
non-trivially on a seed solution of the Einstein-Maxwell equations, therefore
generating a new and independent solution666This is always true for black
holes, however, an Ehlers or Harrison transformation always results in a
change of coordinates when applied to the Minkowski background in the electric
ansatz.. Moreover, the Kinnersley transformations (3.42) form a SU$(2,1)$
group [43, 44], which is consistent with the fact that SU$(2,1)$ is also the
symmetry group of the Einstein-Maxwell equations.

#### 3.4.1 Symmetries of the Ernst Equations in Pure Gravity

In the vacuum case, $\Phi=0$, the Ernst equations reduce to the single complex
equation

$\bigl{(}\operatorname{Re}\pazocal{E}\bigr{)}\nabla^{2}\pazocal{E}=\vec{\nabla}\pazocal{E}\cdot\vec{\nabla}\pazocal{E}\,,\\\
$ (3.43)

whose corresponding action is then

$S=\int
d^{3}x\,\frac{\vec{\nabla}\pazocal{E}^{*}\\!\cdot\vec{\nabla}\pazocal{E}}{\bigl{(}\pazocal{E}+\pazocal{E}^{*}\bigr{)}^{2}}\,,$
(3.44)

which, after considering the gravitational Ernst potential $\pazocal{E}$ as a
complex coordinate

$\pazocal{E}=x+i\,y\,,$ (3.45)

gives rise to the following associated metric

$ds^{2}=\frac{d\pazocal{E}^{*}d\pazocal{E}}{\bigl{(}\pazocal{E}+\pazocal{E}^{*}\bigr{)}^{2}}=\frac{dx^{2}+dy^{2}}{4\,x^{2}}\,.$
(3.46)

As previously seen, the number of Killing vectors in the electrovacuum case is
eight in total (3.40). On the other hand, the associated metric when $\Phi=0$
(3.46) has only the following three Killing vectors

$\displaystyle\xi_{1}$
$\displaystyle=4\,x\,y\,\partial_{x}+2\bigl{(}y^{2}-x^{2}\bigr{)}\partial_{y}\,,$
(3.47a) $\displaystyle\xi_{2}$
$\displaystyle=4\,x\,\partial_{x}+4\,y\,\partial_{y}\,,$ (3.47b)
$\displaystyle\xi_{3}$ $\displaystyle=2\,\partial_{y}\,,$ (3.47c)

which, after integration, results in the generalized Ehlers transformations
[45]

$\pazocal{E}^{\prime}=\frac{a\bigl{(}a\pazocal{E}+i\,b\bigr{)}}{\bigl{(}1+b\,\jmath+i\,a\,\jmath\bigr{)}}\,,$
(3.48)

where $a$, $b$ and $\jmath$ are real parameters.

As expected, the generalized Ehlers transformations (3.48) have the same
symmetry group SL$(2,\mathbb{R})$ (or SU$(1,1)$) as General Relativity.
Moreover, these transformations are equivalent to an Ehlers transformation
(3.42c) composed with the two gauge transformations provided by Eq. (3.42a)
and Eq. (3.42b). Therefore, in contrast to the Ehlers transformations, it
holds that the Harrison transformations (3.42e) are additional independent
transformations only for the Einstein-Maxwell theory.

### 3.5 The Magnetic Ansatz

As mentioned in Sec. 2.3.4, it is always possible to perform a double-Wick
rotation (2.28) on a metric written in the electric LWP form (2.27) in order
to obtain a new solution expressed in the magnetic LWP (2.29), which can
subsequently be recast in the electric one just by performing a reshuffling of
the terms and a redefinition of the functions.

Therefore, the Ernst formalism can also be applied to what is called the
magnetic ansatz

$\displaystyle ds^{2}_{m}$
$\displaystyle=\frac{1}{f}\Bigl{[}-\rho^{2}dt^{2}+e^{2\gamma}\bigl{(}d\rho^{2}+dz^{2}\bigr{)}\Bigr{]}+f\bigl{(}d\phi-\omega
dt\bigr{)}^{2}\,,$ (3.49a) $\displaystyle A$
$\displaystyle=A_{t}dt+A_{\phi}d\phi\,,$ (3.49b)

by using the following “magnetic” Ernst potentials

$\displaystyle\pazocal{E}$ $\displaystyle\coloneqq-f-\Phi^{*}\Phi+ih\,,$
(3.50a) $\displaystyle\Phi$ $\displaystyle\coloneqq\tilde{A}_{t}-iA_{\phi}\,,$
(3.50b)

and “magnetic” twisted potentials

$\displaystyle\vec{e}_{\phi}\times\vec{\nabla}\tilde{A}_{t}$
$\displaystyle=-\frac{f}{\rho}\Bigl{(}\vec{\nabla}A_{t}+\omega\vec{\nabla}A_{\phi}\Bigr{)}\,,$
(3.51a) $\displaystyle\vec{e}_{\phi}\times\vec{\nabla}h$
$\displaystyle=-\frac{f^{2}}{\rho}\vec{\nabla}\omega-2\,\vec{e}_{\phi}\times\operatorname{Im}\bigl{(}\Phi^{*}\vec{\nabla}\Phi\bigr{)}\,,$
(3.51b)

where then

$\operatorname{Im}\bigl{(}\Phi^{*}\vec{\nabla}\Phi\bigr{)}=A_{\phi}\vec{\nabla}\tilde{A}_{t}-\tilde{A}_{t}\vec{\nabla}A_{\phi}\,.$
(3.52)

As can be seen, the roles of the time and the azimuthal coordinates are
exchanged, as expected from the application of the double-Wick rotation
(2.28), which also introduces a correction in the signs of the functions
defining the “magnetic” Ernst potentials $\pazocal{E}$ (3.50) and $\Phi$
(3.51).

Moreover, the magnetic ansatz has the same symmetries (3.42) as the electric
one, with the Ehlers (3.42c) and Harrison transformations (3.42e) being the
only possible non-trivial transformations.

Therefore, this new ansatz implies that the same Ehlers or Harrison
transformation can generate another independent solution depending on which
ansatz the seed solution is expressed in, increasing the number of independent
and meaningful transformations from two to four. Hence, depending on which
ansatz the transformation is applied to, it will be referred to either as an
electric Ehlers-Harrison transformation or as a magnetic Ehlers-Harrison
transformation. The terminology “electric” and “magnetic” has gained
popularity only in recent years [20, 21] [46, 14, 47]. Nevertheless, the
meaning should be straightforward given the dual nature of the double-Wick
rotation. In general, these terms are also employed due to the way the
Harrison transformation works with the different seeds when the transformation
parameter is real. Indeed, a real Harrison map adds an electric charge when
applied to the electric ansatz (3.3), while it adds an external magnetic field
in the case of the magnetic ansatz (3.49).

### 3.6 Magnetic Transformations on the Schwarzschild Black Hole

The Schwarzschild metric (2.34), which is reported here for convenience:

$ds^{2}=-\biggl{(}1-\frac{2M}{r}\biggr{)}dt^{2}+\frac{dr^{2}}{\Bigl{(}1-\frac{2M}{r}\Bigr{)}}+r^{2}\bigl{(}d\theta^{2}+\sin^{2}\theta\,d\phi^{2}\bigr{)}\,,$

can be expressed in the magnetic ansatz (3.49) as

$\displaystyle ds^{2}_{m}$
$\displaystyle=\frac{1}{f}\Bigl{[}-\rho^{2}dt^{2}+e^{2\gamma}\bigl{(}d\rho^{2}+dz^{2}\bigr{)}\Bigr{]}+f\bigl{(}d\phi-\omega
dt\bigr{)}^{2}\,,$ $\displaystyle A$ $\displaystyle=A_{t}dt+A_{\phi}d\phi\,,$

by means of the following transformation from cylindrical coordinates

$\displaystyle\rho$ $\displaystyle=\sqrt{r^{2}-2Mr}\sin\theta\,,$ (3.53a)
$\displaystyle z$ $\displaystyle=\bigl{(}r-M\bigr{)}\cos\theta\,,$ (3.53b)

from which the functions present in the magnetic ansatz (3.49) are

$\displaystyle f$ $\displaystyle=r^{2}\sin^{2}\theta\,,$ (3.54a)
$\displaystyle\gamma$
$\displaystyle=\frac{1}{2}\log\biggl{[}\frac{r^{4}\sin^{2}\theta}{r^{2}-2Mr+M^{2}\sin^{2}\theta}\biggr{]}\,,$
(3.54b) $\displaystyle\omega$ $\displaystyle=A_{t}=A_{\phi}=0\,.$ (3.54c)

It has to be noted that the value of $\gamma$ is not fundamental because it is
invariant under the Ehlers and Harrison transformations, however, it has been
made explicit for completeness. With these functions, it is possible to obtain
the magnetic twisted potentials by solving the differential equations that
define them (3.51), which results in these twisted potentials being zero up to
an integration constant

$h=\tilde{A}_{t}=0\,.$ (3.55)

The following step will be applying an Ehlers or Harrison transformation to
the Ernst potentials of the magnetic ansatz, that were defined as (3.50)

$\displaystyle\pazocal{E}$ $\displaystyle\coloneqq-f-\Phi^{*}\Phi+ih\,,$
$\displaystyle\Phi$ $\displaystyle\coloneqq\tilde{A}_{t}-iA_{\phi}\,,$

which, using Eqs. (3.54a) (3.55), in this case are

$\displaystyle\pazocal{E}$ $\displaystyle=-r^{2}\sin^{2}\theta\,,$ (3.56a)
$\displaystyle\Phi$ $\displaystyle=0\,.$ (3.56b)

#### 3.6.1 Magnetic Ehlers: Swirling

An Ehlers transformation (3.42c) with real parameter $\jmath$ :

$\displaystyle\pazocal{E}^{\prime}$
$\displaystyle=\frac{\pazocal{E}}{1+i\,\jmath\,\pazocal{E}}\,,$
$\displaystyle\Phi^{\prime}$
$\displaystyle=\frac{\Phi}{1+i\,\jmath\,\pazocal{E}}\,,$

when applied to the Ernst potentials under consideration (3.56), yields

$\displaystyle\pazocal{E}^{\prime}$
$\displaystyle=-\frac{r^{2}\sin^{2}\theta}{1-i\,\jmath\,r^{2}\sin^{2}\theta}\,,$
(3.57a) $\displaystyle\Phi^{\prime}$ $\displaystyle=0\,,$ (3.57b)

which, using Eqs. (3.50), result in

$\displaystyle f^{\prime}$
$\displaystyle=-\operatorname{Re}\pazocal{E}^{\prime}-\Phi^{\prime*}\Phi^{\prime}=\frac{r^{2}\sin^{2}\theta}{1+\jmath^{2}r^{4}\sin^{4}\theta}\,,$
(3.58a) $\displaystyle h^{\prime}$
$\displaystyle=\operatorname{Im}\pazocal{E}^{\prime}=-\frac{\jmath\,r^{4}\sin^{4}\theta}{1+\jmath^{2}r^{4}\sin^{4}\theta}\,,$
(3.58b) $\displaystyle\gamma^{\prime}$ $\displaystyle=\gamma\,,$ (3.58c)
$\displaystyle A^{\prime}_{t}$ $\displaystyle=A^{\prime}_{\phi}=0\,,$ (3.58d)

while $\omega^{\prime}$ has to be found from the definition of the twisted
potential $h^{\prime}$ (3.51b), which gives

$\omega^{\prime}=-4\jmath\bigl{(}r-2M\bigr{)}\cos\theta+\omega_{0}\,,$ (3.59)

where $\omega_{0}$ is an integration constant that can be reabsorbed being
related to the frame of reference.

Consequently, the result for the metric is

${ds}^{\prime}\mkern
1.0mu{}^{2}=F\biggl{[}-\biggl{(}1-\frac{2M}{r}\biggr{)}dt^{2}+\frac{dr^{2}}{\Bigl{(}1-\frac{2M}{r}\Bigr{)}}+r^{2}d\theta^{2}\biggr{]}+\frac{r^{2}\sin^{2}\theta}{F}\biggl{[}d\varphi\,+\,4\,\jmath\bigl{(}r-2M\bigr{)}\cos\theta
dt\biggr{]}^{2},$ (3.60)

where the function $F$ has been defined as

$F\coloneqq 1+\jmath^{2}r^{4}\sin^{4}\theta\,,$ (3.61)

and the electromagnetic potential remains null as stated in Eq. (3.58d),
meaning that a magnetic Ehlers transformation maps a vacuum solution into a
vacuum solution. Obviously, Eq. (3.60) reduces again to the seed Schwarzschild
metric for $\jmath=0$.

This solution thus represents a Schwarzschild black hole (2.34) with mass $M$,
embedded in what is called the swirling universe [14], which is a sort of
rotating universe, in the sense that, by setting to zero the mass, $M=0$, it
is possible to remove the black hole and recover the background swirling
universe:

${ds}^{\prime}\mkern
1.0mu{}^{2}=F\biggl{[}-dt^{2}+dr^{2}+r^{2}d\theta^{2}\biggr{]}+\frac{r^{2}\sin^{2}\theta}{F}\biggl{[}d\varphi+4\,\jmath\,r\cos\theta
dt\biggr{]}^{2}\,,$ (3.62)

corresponding to a universe with a rotation due to a frame-dragging effect
(2.41) equal to

$\Omega=\frac{d\phi}{dt}=-4\,\jmath\,r\cos\theta\,.$ (3.63)

Moreover, this rotation is also modified by the black hole presence. Indeed,
in the massive case, this parameter has an additional term proportional to the
mass777As we will see in Sec. 5 the rotation is in general modified by all the
parameters of the theory.:

$\Omega=-4\jmath\bigl{(}r-2M\bigr{)}\cos\theta\,.$ (3.64)

In particular, using the cylindrical coordinates (3.53a), this can be
understood as a rotation around the symmetry $z-$axis, with the two
hemispheres rotating in the opposite direction from each other, indeed, in
these coordinates the rotation becomes:

$\Omega=-4\,\jmath\,z\,,$ (3.65)

which is an expression that holds both for the background and also for the
Schwarzschild black hole embedded in this universe.

#### 3.6.2 Magnetic Harrison: Electromagnetic Melvin

Similarly, a Harrison transformation with a complex parameter $\alpha$ :

$\displaystyle\pazocal{E}^{\prime}$
$\displaystyle=\frac{\pazocal{E}}{1-2\,\alpha^{*}\Phi-\alpha^{*}\alpha\,\pazocal{E}}\,,$
$\displaystyle\Phi^{\prime}$
$\displaystyle=\frac{\alpha\,\pazocal{E}+\Phi}{1-2\,\alpha^{*}\Phi-\alpha^{*}\alpha\,\pazocal{E}}\,,$

applied to the Ernst potentials given by Eqs. (3.56) results in

$\displaystyle\pazocal{E}^{\prime}$
$\displaystyle=-\frac{r^{2}\sin^{2}\theta}{1+\alpha^{*}\alpha\,r^{2}\sin^{2}\theta}\,,$
(3.66a) $\displaystyle\Phi^{\prime}$
$\displaystyle=-\frac{\alpha\,r^{2}\sin^{2}\theta}{1+\alpha^{*}\alpha\,r^{2}\sin^{2}\theta}\,,$
(3.66b)

and therefore

$\displaystyle f^{\prime}$
$\displaystyle=\frac{r^{2}\sin^{2}\theta}{1+\alpha^{*}\alpha\,r^{2}\sin^{2}\theta}\,,$
(3.67a) $\displaystyle h^{\prime}$ $\displaystyle=\omega^{\prime}=0\,,$
(3.67b) $\displaystyle\gamma^{\prime}$ $\displaystyle=\gamma\,.$ (3.67c)

Additionally, the magnetic Harrison transformations also generate an
electromagnetic field, which can be obtained by using the definition of the
Ernst potential $\Phi^{\prime}=\tilde{A}^{\prime}_{t}-iA^{\prime}_{\phi}$
(3.50b) and by integrating the equations for the electromagnetic twisted
potential $\tilde{A}^{\prime}_{t}$ (3.51a). Hence, after defining the complex
Harrison parameter as $\alpha\coloneqq\frac{1}{2}\bigl{(}E-i\,B\bigr{)}$, the
metric and the electromagnetic potential are given by

$\displaystyle{ds}^{\prime}\mkern 1.0mu{}^{2}$
$\displaystyle=F^{2}\biggl{[}-\biggl{(}1-\frac{2M}{r}\biggr{)}dt^{2}+\frac{dr^{2}}{\Bigl{(}1-\frac{2M}{r}\Bigr{)}}+r^{2}d\theta^{2}\biggr{]}+\biggl{[}\frac{r^{2}\sin^{2}\theta}{F^{2}}\biggr{]}d\phi^{2}\,,$
(3.68a) $\displaystyle A^{\prime}$
$\displaystyle=E\bigl{(}r-2M\bigr{)}\cos\theta
dt+\biggl{[}\frac{B\,r^{2}\sin^{2}\theta}{2\,F}\biggr{]}d\phi\,,$ (3.68b)

where

$F\coloneqq 1+\biggl{[}\frac{E^{2}+B^{2}}{4}\biggr{]}r^{2}\sin^{2}\theta\,.$
(3.69)

The background is again recovered by setting $M=0$:

$\displaystyle{ds}^{\prime}\mkern 1.0mu{}^{2}$
$\displaystyle=F^{2}\bigl{(}-dt^{2}+dr^{2}+r^{2}d\theta^{2}\bigr{)}+\biggl{[}\frac{r^{2}\sin^{2}\theta}{F^{2}}\biggr{]}d\phi^{2}\,,$
(3.70) $\displaystyle A^{\prime}$ $\displaystyle=E\,r\cos\theta
dt+\biggl{[}\frac{B\,r^{2}\sin^{2}\theta}{2\,F}\biggr{]}d\phi\,.$ (3.71)

This background (3.70) thus corresponds to a universe filled with a “uniform”
electric field $E$ and magnetic field $B$, called the electromagnetic
universe, and that represents a generalization of the purely magnetic sub-
case, $E=0$, known as the Melvin universe [15]. For this reason, Eq. (3.68)
represents a Schwarzschild black hole embedded in an electromagnetic universe
[48].

In particular, the purely magnetic Melvin universe is

$\displaystyle{ds}^{\prime}\mkern 1.0mu{}^{2}$
$\displaystyle=\bigl{(}1+\frac{1}{4}B^{2}\,r^{2}\sin^{2}\theta\bigr{)}^{2}\biggl{[}-dt^{2}+dr^{2}+r^{2}d\theta^{2}\biggr{]}+\frac{r^{2}\sin^{2}\theta}{\bigl{(}1+\frac{1}{4}B^{2}\,r^{2}\sin^{2}\theta\bigr{)}^{2}}d\phi^{2}\,,$
(3.72a) $\displaystyle A^{\prime}$
$\displaystyle=\frac{B\,r^{2}\sin^{2}\theta}{2\bigl{(}1+\frac{1}{4}B^{2}\,r^{2}\sin^{2}\theta\bigr{)}}\
d\phi\,.$ (3.72b)

### 3.7 Electric Transformations on the Schwarzschild Black Hole

In a similar way to what was done for the magnetic ansatz in Sec. 3.6, it
holds that the Schwarzschild spacetime (2.34)

$ds^{2}=-\biggl{(}1-\frac{2M}{r}\biggr{)}dt^{2}+\frac{dr^{2}}{\Bigl{(}1-\frac{2M}{r}\Bigr{)}}+r^{2}\bigl{(}d\theta^{2}+\sin^{2}\theta
d\phi^{2}\bigr{)}\,,$

can be written in the electric ansatz (3.3)

$\displaystyle ds_{e}^{2}$
$\displaystyle=\frac{1}{f}\Bigl{[}\rho^{2}d\phi^{2}+e^{2\gamma}\bigl{(}d\rho^{2}+dz^{2}\bigr{)}\Bigr{]}-f\bigl{(}dt-\omega
d\phi\bigr{)}^{2}\,,$ $\displaystyle A$
$\displaystyle=A_{t}dt+A_{\phi}d\phi\,,$

using the same cylindrical coordinates as the magnetic ansatz

$\displaystyle\rho=\sqrt{r^{2}-2Mr}\sin\theta\,,$ (3.73a) $\displaystyle
z=\bigl{(}r-M\bigr{)}\cos\theta\,,$ (3.73b)

from which the functions present in the electric ansatz (3.3) correspond to

$\displaystyle f$ $\displaystyle=1-\frac{2M}{r}\,,$ (3.74a)
$\displaystyle\gamma$
$\displaystyle=\frac{1}{2}\log\biggl{[}\frac{r^{2}-2Mr}{r^{2}-2Mr+M^{2}\sin^{2}\theta}\biggr{]}\,,$
(3.74b) $\displaystyle\omega$ $\displaystyle=A_{t}=A_{\phi}=0\,.$ (3.74c)

With these functions, it is possible to obtain the twisted potentials, using
their definitions as Eqs. (3.17) (3.28), which can be proved to be zero up to
an integration constant,

$h=\tilde{A}_{\phi}=0\,,$ (3.75)

while the Ernst potentials of the electric ansatz (3.33) (3.23)

$\displaystyle\pazocal{E}$ $\displaystyle=f-\Phi^{*}\Phi+i\,h\,,$
$\displaystyle\Phi$ $\displaystyle=A_{t}+i\tilde{A}_{\phi}\,,$

in this case, are given by

$\displaystyle\pazocal{E}$ $\displaystyle=1-\frac{2M}{r}\,,$ (3.76a)
$\displaystyle\Phi$ $\displaystyle=0\,.$ (3.76b)

#### 3.7.1 Electric Ehlers: Taub-NUT

Therefore, an Ehlers transformation (3.42c)

$\displaystyle\pazocal{E}^{\prime}$
$\displaystyle=\frac{\pazocal{E}}{1+i\,\jmath\,\pazocal{E}}\,,$
$\displaystyle\Phi^{\prime}$
$\displaystyle=\frac{\Phi}{1+i\,\jmath\,\pazocal{E}}\,,$

applied to the seed Ernst potentials (3.76), results in

$\displaystyle\pazocal{E}^{\prime}$
$\displaystyle=\frac{r^{2}-2Mr}{r^{2}+\jmath^{2}\bigl{(}r-2M\bigr{)}^{2}}-i\,\frac{\jmath\bigl{(}r-2M\bigr{)}^{2}}{r^{2}+\jmath^{2}\bigl{(}r-2M\bigr{)}^{2}}\,,$
(3.77a) $\displaystyle\Phi^{\prime}$ $\displaystyle=0\,.$ (3.77b)

This implies that an electric Ehlers transformation does not add an
electromagnetic field to a seed solution that does not possess one from the
beginning, thus mapping a vacuum solution into a vacuum solution, as was the
case for the magnetic Ehlers transformation.

Consequently, the functions needed to construct the new metric are

$\displaystyle f^{\prime}$
$\displaystyle=\frac{r^{2}-2Mr}{r^{2}+\jmath^{2}\bigl{(}r-2M\bigr{)}^{2}}\,,$
(3.78a) $\displaystyle h^{\prime}$
$\displaystyle=-\frac{\jmath\bigl{(}r-2M\bigr{)}^{2}}{r^{2}+\jmath^{2}\bigl{(}r-2M\bigr{)}^{2}}\,,$
(3.78b) $\displaystyle\omega^{\prime}$
$\displaystyle=4\,\jmath\,M\cos\theta\,.$ (3.78c)

Therefore, the new metric after the electric Ehlers transformation is:

$\begin{split}{ds}^{\prime}\mkern
1.0mu{}^{2}&=-\frac{r^{2}-2Mr}{r^{2}+\jmath^{2}\bigl{(}r-2M\bigr{)}^{2}}\bigl{(}dt-4\jmath
M\cos\theta
d\phi\bigr{)}^{2}+\frac{r^{2}+\jmath^{2}\bigl{(}r-2M\bigr{)}^{2}}{r^{2}-2Mr}dr^{2}\\\
&\quad+\Bigl{[}r^{2}+\jmath^{2}\bigl{(}r-2M\bigr{)}^{2}\Bigr{]}\bigl{(}d\theta^{2}+\sin^{2}\theta
d\phi^{2}\bigr{)}\,.\end{split}$ (3.79)

In order to identify the physical meaning of this metric (3.79), it is useful
to perform the following change of coordinates888The transformation of the
radial coordinate is suggested by the $S^{2}$ element in the metric.:

$\displaystyle R$
$\displaystyle=\frac{r+\jmath^{2}\bigl{(}r-2M\bigr{)}}{\sqrt{1+\jmath^{2}}}\,,$
(3.80a) $\displaystyle T$ $\displaystyle=\frac{t}{\sqrt{1+\jmath^{2}}}\,,$
(3.80b)

that, together with a redefinition of the parameters as

$\displaystyle M$ $\displaystyle=-\frac{n}{2\jmath}\sqrt{1+\jmath^{2}}\,,$
(3.81a) $\displaystyle\jmath$
$\displaystyle=\frac{m-\sqrt{m^{2}+n^{2}}}{n}\,,$ (3.81b)

allows rewriting the obtained solution (3.79) as the well-known Taub-NUT
metric [49, 50]:

$\begin{split}{ds}^{\prime}\mkern
1.0mu{}^{2}&=-\frac{R^{2}-2mR-n^{2}}{R^{2}+n^{2}}\bigl{(}dT-2n\cos\theta
d\phi\bigr{)}^{2}+\frac{R^{2}+n^{2}}{R^{2}-2mR-n^{2}}dR^{2}\\\
&\quad+\bigl{(}R^{2}+n^{2}\bigr{)}\bigl{(}d\theta^{2}+\sin^{2}\theta
d\phi^{2}\bigr{)}\,,\end{split}$ (3.82)

where $m$ is the mass and $n$ is the so-called NUT parameter.

An important consideration is that if the seed is the flat Minkowski
spacetime, i.e. the massless case, it holds that, after an electric Ehlers
transformation, the result is again the Minkowski spacetime, thus without the
addition of any new parameter. Indeed, for $M=0$ the metric in Eq. (3.79)
reduces to

${ds}^{\prime}\mkern
1.0mu{}^{2}=-\frac{dt^{2}}{\bigl{(}1+\jmath^{2}\bigr{)}}+\bigl{(}1+\jmath^{2}\bigr{)}dr^{2}+r^{2}\bigl{(}1+\jmath^{2}\bigr{)}\bigl{(}d\theta^{2}+\sin^{2}\theta
d\phi^{2}\bigr{)}\,,$ (3.83)

that is exactly the Minkowski background modulo a rescaling of the
coordinates.

#### 3.7.2 Electric Harrison: Dyonic Reissner-Nordström

Finally, applying a Harrison transformation

$\displaystyle\pazocal{E}^{\prime}$
$\displaystyle=\frac{\pazocal{E}}{1-2\,\alpha^{*}\Phi-\alpha^{*}\alpha\,\pazocal{E}}\,,$
$\displaystyle\Phi^{\prime}$
$\displaystyle=\frac{\alpha\,\pazocal{E}+\Phi}{1-2\,\alpha^{*}\Phi-\alpha^{*}\alpha\,\pazocal{E}}\,,$

to the considered seed (3.76), leads to the following Ernst potentials

$\displaystyle\pazocal{E}^{\prime}$
$\displaystyle=\frac{r-2M}{r-\alpha^{*}\alpha\bigl{(}r-2M\bigr{)}}\,,$ (3.84a)
$\displaystyle\Phi^{\prime}$
$\displaystyle=\frac{\alpha\bigl{(}r-2M\bigr{)}}{r-\alpha^{*}\alpha\bigl{(}r-2M\bigr{)}}\,,$
(3.84b)

from which

$\displaystyle f^{\prime}$
$\displaystyle=\frac{r\bigl{(}r-2M\bigr{)}}{\Bigl{[}\,r-\alpha^{*}\alpha\bigl{(}r-2M\bigr{)}\Bigr{]}^{2}}\,,$
(3.85a) $\displaystyle h^{\prime}$ $\displaystyle=\omega^{\prime}=0\,,$
(3.85b) $\displaystyle\gamma^{\prime}$ $\displaystyle=\gamma\,,$ (3.85c)
$\displaystyle A^{\prime}$
$\displaystyle=\frac{\alpha_{r}\bigl{(}r-2M\bigr{)}}{r-\alpha^{*}\alpha\bigl{(}r-2M\bigr{)}}dt+2\,\alpha_{I}\,M\cos\theta
d\phi\,,$ (3.85d)

where the electromagnetic twisted potential $\tilde{A}_{\phi}$ has been
obtained using its definition as in Eq. (3.17), and the complex parameter of
the Harrison transformation has been defined as
$\alpha\coloneqq\alpha_{R}+i\,\alpha_{I}$.

Thus, using the functions obtained after performing the Harrison
transformation (3.85a), the resulting metric is

$\begin{split}{ds}^{\prime}\mkern
1.0mu{}^{2}&=-\frac{r\bigl{(}r-2M\bigr{)}}{\Bigl{[}\,r-\alpha^{*}\alpha\bigl{(}r-2M\bigr{)}\Bigr{]}^{2}}dt^{2}+\frac{\Bigl{[}\,r-\alpha^{*}\alpha\bigl{(}r-2M\bigr{)}\Bigr{]}^{2}}{r\bigl{(}r-2M\bigr{)}}dr^{2}\\\
&\quad+\Bigl{[}\,r-\alpha^{*}\alpha\bigl{(}r-2M\bigr{)}\Bigr{]}^{2}\bigl{(}d\theta^{2}+\sin^{2}\theta
d\phi^{2}\bigr{)}\,,\end{split}$ (3.86)

which, after performing the following change of coordinates

$\displaystyle R$
$\displaystyle=r\bigl{(}1-\alpha^{*}\alpha\bigr{)}+2M\alpha^{*}\alpha\,,$
(3.87a) $\displaystyle T$ $\displaystyle=\frac{t}{1-\alpha^{*}\alpha}\,,$
(3.87b)

together with the reparametrizations

$\displaystyle p$ $\displaystyle=2\,\alpha_{I}M\,,$ (3.88a) $\displaystyle e$
$\displaystyle=-\frac{2\,\alpha_{R}M}{1-\alpha^{*}\alpha}\,,$ (3.88b)
$\displaystyle m$ $\displaystyle=M\bigl{(}1+\alpha^{*}\alpha\bigr{)}\,,$
(3.88c)

and by defining the dyon $q^{2}\coloneqq e^{2}+p^{2}$, it holds that the final
result for the electromagnetic potential and the metric is

$\displaystyle{ds}^{\prime}\mkern 1.0mu{}^{2}$
$\displaystyle=-\biggl{(}1-\frac{2m}{R}+\frac{q^{2}}{R^{2}}\biggr{)}dt^{2}+\frac{dR^{2}}{\Bigl{(}1-\frac{2m}{R}+\frac{q^{2}}{R^{2}}\Bigr{)}}+R^{2}\bigl{(}d\theta^{2}+\sin^{2}\theta
d\phi^{2}\bigr{)}\,,$ (3.89) $\displaystyle A^{\prime}$
$\displaystyle=\frac{e}{R}dt+p\cos\theta d\phi\,,$ (3.90)

known as the dyonic Reissner-Nordström black hole [51, 52], i.e. the non-
rotating, $a=0$, sub-case of the dyonic Kerr-Newman black hole (2.36), thus
representing an electromagnetically charged black hole, with mass $m$,
electric charge $e$ and magnetic charge $p$.

As in the case of the electric Ehlers transformation, an electric Harrison
transformation also does _not_ introduce any new parameters when applied to
the Minkowski spacetime. Indeed, the metric given by Eq. (3.79) also reduces
to the Minkowski background modulo a rescaling of the coordinates for $M=0$:

${ds}^{\prime}\mkern
1.0mu{}^{2}=-\frac{dt^{2}}{\bigl{(}1+\alpha^{*}\alpha)}+\bigl{(}1+\alpha^{*}\alpha)dr^{2}+r^{2}\bigl{(}1+\alpha^{*}\alpha\bigr{)}\bigl{(}d\theta^{2}+\sin^{2}\theta
d\phi^{2}\bigr{)}\,.$ (3.91)

### 3.8 Physical Meaning of the Ehlers-Harrison Transformations

The physical meaning of the parameters added by the electric and magnetic
Ehlers-Harrison transformations can therefore be understood from the metrics
obtained in Sec. 3.6 and Sec. 3.7, through the application of these
transformations to the simple case of a Schwarzschild black hole (2.34).

Summarizing, a Harrison transformation applied to an electric ansatz has the
effect of adding a dyon, i.e. it adds both an electric and a magnetic charge.
Thus generating the so-called dyonic Reissner-Nordström black hole when
starting from the Schwarzschild one. Similarly, on the magnetic ansatz, the
Harrison transformation embeds the seed spacetime in the electromagnetic
universe, which is a spacetime filled with a “uniform” electromagnetic field.
In particular, when the parameter of the Harrison transformation is real and
the mass is set to zero, the resulting background is a universe filled just
with a _uniform magnetic field_ , known as the Melvin universe.

On the other hand, an Ehlers transformation applied to an electric ansatz has
the effect of adding the NUT parameter [53], therefore generating what is
referred to as the Taub-NUT black hole when the seed is the Schwarzschild
black hole. Analogously, on the magnetic ansatz, an Ehlers transformation
embeds the given spacetime in the swirling universe, which is a sort of
_rotating universe_ [14], as explained in Sec. 3.6.1.

Ehlers-Harrison on the Schwarzschild Black Hole
---
|  Harrison |  Ehlers
Electric Transformation |  Reissner-Nordström BH |  Taub-NUT BH
Magnetic Transformation |  Schwarzschild BH in an |  Schwarzschild BH in a
|  electromagnetic universe |  swirling universe
Table 3.1: Table summarizing the effect of an electric or magnetic Ehlers-
Harrison transformation applied to the Schwarzschild black hole.

Moreover, an Ehlers transformation never generates an electromagnetic field if
not already present in the seed solution, thus always mapping a vacuum
solution into a vacuum solution. In contrast, a Harrison transformation always
generates an additional electromagnetic field, except when applied to the
Minkowski spacetime in the electric ansatz.

Indeed, as pointed out in Sec. 3.7, while any _magnetic_ transformation always
generates a new independent solution when applied to the Minkowski background,
this is not the case for an _electric_ transformation, since the application
of any electric transformation to the Minkowski spacetimes always results in a
mere change of coordinates. The reason behind this result relies upon the fact
that the magnetic transformations add new parameters characterizing the
background universe, whereas the electric transformations add new parameters
to the black hole, which the Minkowski spacetime obviously does _not_ contain.

Another important consideration is that the Elhers and Harrison
transformations are Lie point symmetry of the Ernst equations, which basically
means that, after one of these transformations, the result will always be in
the same ansatz as the seed solution, thus implying that it is always possible
to compose an arbitrary number of these transformations. Moreover, these
transformations also commute, hence, for example, a Melvin and swirling
universe can be obtained either through the application of a magnetic Ehlers
transformation followed by a magnetic Harrison, or equivalently using the
opposite order of compositions. Since it will be relevant in the following
sections, the explicit metric and electromagnetic potential of the Melvin-
swirling universe are reported in Appendix C.2.

Furthermore, In Sec. 5, the magnetic transformations will be used to obtain,
for the first time, the dyonic Kerr-Newman black hole (2.36) embedded in a
Melvin-swirling universe.

##  CHAPTER 4The Melvin-swirling Universe

There is a profound relation between the parameters that can be added using
the Ernst formalism. Indeed, the swirling background (3.62) can also be
obtained in a particular way starting from the Taub-NUT spacetime with a flat
base manifold [14], which in turn can be generated via an Ehlers
transformation (3.42c), from the Schwarzschild metric (2.34), previously
composed with a double-Wick rotation (2.28)999While this is true for a generic
sign of the constant curvature of the seed base manifold, only the metrics
with positive curvature can be interpreted as black holes in General
Relativity..

In fact, the flat Taub-NUT spacetime, with mass $m$ and NUT parameter $n$ is
given by the metric

$ds^{2}=-\frac{2mr}{r^{2}+n^{2}}\bigl{(}d\tau-2n\theta
d\varphi\bigr{)}^{2}+\frac{r^{2}+n^{2}}{2mr}dr^{2}+\bigl{(}r^{2}+n^{2}\bigr{)}\bigl{(}d\theta^{2}+d\varphi^{2}\bigr{)}\,,$
(4.1)

which, after the following double-Wick rotation and reparametrizations

$\displaystyle n$
$\displaystyle=\Bigl{(}\,\frac{m}{2\jmath}\,\Bigr{)}^{\frac{1}{3}}\,,$ (4.2a)
$\displaystyle\varphi$
$\displaystyle=-\frac{i\,t}{n}=-i\Bigl{(}\,\frac{2\jmath}{m}\,\Bigr{)}^{\frac{1}{3}}t\,,$
(4.2b) $\displaystyle\tau$
$\displaystyle=-\frac{i\,n^{2}}{m}\phi=-i\Bigl{(}\,\frac{m}{2\jmath}\,\Bigr{)}^{\frac{2}{3}}\frac{\phi}{m}\,,$
(4.2c) $\displaystyle\theta$
$\displaystyle=-\frac{z}{n}=-\Bigl{(}\,\frac{2\jmath}{m}\,\Bigr{)}^{\frac{1}{3}}z\,,$
(4.2d) $\displaystyle r$
$\displaystyle=\frac{m}{2n^{2}}\rho^{2}=\frac{m}{2}\Bigl{(}\,\frac{2\jmath}{m}\,\Bigr{)}^{\frac{2}{3}}\rho^{2}\,,$
(4.2e)

yields

$ds^{2}=\bigl{(}1+\jmath^{2}\rho^{4}\bigr{)}\bigl{(}-dt^{2}+d\rho^{2}+dz^{2}\bigr{)}+\frac{\rho^{2}}{\bigl{(}1+\jmath^{2}\rho^{4}\bigr{)}}\bigl{(}d\phi+4\jmath
z\,dt\bigr{)}^{2}\,,\\\ $ (4.3)

that is exactly the swirling background, with swirling parament $\jmath$, in
cylindrical coordinates

$\displaystyle\rho$ $\displaystyle=r\sin\theta\,,$ (4.4a) $\displaystyle z$
$\displaystyle=r\cos\theta\,\,.$ (4.4b)

Analogously, the Melvin background can be also obtained starting from the
magnetic Reissner-Nordström spacetime with a flat base manifold [54, 55],
which is given by

$\displaystyle ds^{2}$
$\displaystyle=-\frac{2mr+p^{2}}{r^{2}}dt^{2}+\frac{r^{2}}{2mr+p^{2}}dr^{2}+r^{2}\bigl{(}d\theta^{2}+d\varphi^{2}\bigr{)}\,,$
(4.5a) $\displaystyle A$ $\displaystyle=p\,\theta\,d\varphi\,,$ (4.5b)

where $m$ is the mass and $p$ is the magnetic charge. On the other hand, this
magnetic Reissner-Nordström with a flat base manifold can also be generated
from the Schwarzschild metric via a Harrison transformation (3.42e) previously
composed with a double-Wick rotation.

Moreover, both the Melvin universe and the swirling universe are obtained
respectively by applying a Harrison transformation and an Ehlers
transformation on a seed expressed in the magnetic ansatz. Thus, it is
possible to summarize the correspondence between these backgrounds by the
following proportion:

$\boxed{\text{\small Melvin Universe : Harrison transformations = Swirling
Universe : Ehlers transformations}}$ (4.6)

Furthermore, the Melvin universe can be demonstrated to be equivalent to the
spacetime generated by two magnetic Reissner-Nordström black holes with
opposite charges that are infinitely distant along the symmetry axis [56].
Similarly, as explained in [14], the swirling background should have an
analogous interpretation, with the Reissner-Nordström black holes replaced by
two Taub-NUT black holes with opposite NUT parameters. Although this
interpretation has not been yet demonstrated, it seems to be a valid
hypothesis given the many analogies between the Melvin and the swirling
universes, where, in particular, we have that the swirling universe can be
obtained as a double-Wick rotation of the flat Taub-NUT black hole in the same
way as the Melvin universe can be obtained as a double-Wick rotation of the
flat magnetic Reissner-Nordström black hole. Moreover, the swirling background
can also be thought of, with a certain degree of approximation, as the
gravitational setting generated by the interplay of two sources counter-
rotating around the same axis, such as two counter-rotating galaxies or black
holes [57].

Combining these interpretations, the Melvin-swirling universe101010The metric
and the electromagnetic potential for this solution are reported in Sec. C.2.

, which is obtained by applying a magnetic Ehlers transformation to the Melvin
universe, or equivalently, by applying a real magnetic Harrison transformation
to the swirling universe, can be thought of as the spacetime generated by two
magnetic Reissner-Nordström-Taub-NUT black holes pushed at infinity along the
symmetry axis and with opposite magnetic and NUT charges, as sketched in
Figure 4.1. Therefore, performing a double-Wick rotation on the flat magnetic
Reissner-Nordström-Taub-NUT black hole should reproduce the Melvin-swirling
background.

Black hole at infinity with magnetic
charge p and NUT parameter n Coordinate origin of the
Melvin-swirling spacetime Black hole at infinity with magnetic
charge -p and NUT parameter -n Figure 4.1: Schematic representation of the
Melvin-swirling background as two infinitely distant, along the symmetry axis,
magnetic Reissner-Nordström-Taub-NUT black holes with opposite charges.

In addition, as we will prove in Sec. 6.6, the Melvin-swirling universe is a
spacetime locally asymptotically flat

$R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\overset{r\to\infty}{\approx}\frac{49152}{\bigl{[}B^{4}+16\jmath^{2}\bigr{]}^{2}\sin^{12}\theta\,r^{12}}\,,$
(4.7)

with a constant curvature on the symmetry axis

$R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\big{\rvert}_{\theta=0,\pi}=4\bigl{(}5B^{4}-48\jmath^{2}\bigl{)}\,,$
(4.8)

where we recall that $B$ and $\jmath$ are, respectively, the Melvin and
swirling parameters.

### 4.1 The Electromagnetic-swirling Universe with a Cosmological Constant

There are two other considerations to address before testing the claim that
the Melvin-swirling background should result from a double-Wick rotation on
the flat magnetic Reissner-Nordström-Taub-NUT black hole.

The first consideration is that it is also possible to add a cosmological
constant with the method aforementioned. Indeed, the flat Taub-Nut black hole
with a cosmological constant is given by

$ds^{2}=-\frac{\pazocal{R}}{r^{2}+n^{2}}\bigl{(}d\tau-2n\theta
d\varphi\bigr{)}^{2}+\frac{r^{2}+n^{2}}{\pazocal{R}}dr^{2}+\bigl{(}r^{2}+n^{2}\bigr{)}\bigl{(}d\theta^{2}+d\varphi^{2}\bigr{)}\,,$
(4.9)

where

$\pazocal{R}=2mr+\Lambda\,n^{4}-2\,\Lambda\,n^{2}r^{2}-\frac{\Lambda}{3}r^{4}\,,$
(4.10)

$m$ is the mass, $n$ is the nut parameter, and $\Lambda$ is the cosmological
constant.

Thus, using the double-Wick rotation and the reparametrizations given by Eqs.
(4.2), this solution becomes:

$ds^{2}=\bigl{(}1+\jmath^{2}\rho^{4}\bigr{)}\biggl{[}-dt^{2}+\frac{\rho^{2}d\rho^{2}}{\pazocal{S}}+dz^{2}\biggr{]}+\frac{{\pazocal{S}}}{\bigl{(}1+\jmath^{2}\rho^{4}\bigr{)}}\bigl{(}d\phi+4\jmath
z\,dt\bigr{)}^{2}\,,\\\ $ (4.11)

where

$\pazocal{S}=\rho^{2}+\frac{\Lambda}{4\jmath^{2}}-\frac{\Lambda}{2}\rho^{4}-\frac{\jmath^{2}\Lambda}{12}\rho^{8}\,.$
(4.12)

As can be verified, the metric provided by Eq. (4.11), is a solution of the
Einstein equations with a cosmological constant. Therefore, it is the correct
generalization, in cylindrical coordinates (4.4), of the swirling universe
with a cosmological constant. Moreover, the same can be done for the Melvin
universe, starting from the flat magnetic Reissner-Nordström black hole with a
cosmological constant [45].

The second consideration is that the electromagnetic generalization of the
Melvin universe, potentially with a cosmological constant, can also be
obtained in the same way starting from the flat _dyonic_ Reissner-Nordström
black hole.

Collecting all these results, we will now obtain the electromagnetic-swirling
universe with a cosmological constant, starting from the flat dyonic Reissner-
Nordström-Taub-NUT black hole with a cosmological constant, whose metric and
potential are

$\displaystyle ds^{2}$
$\displaystyle=-\frac{\pazocal{R}}{r^{2}+n^{2}}\bigl{(}d\tau-\,2n\theta
d\varphi\bigr{)}^{2}+\frac{r^{2}+n^{2}}{\pazocal{R}}dr^{2}+\bigl{(}r^{2}+n^{2}\bigr{)}\bigl{(}d\theta^{2}+d\varphi^{2}\bigr{)}\,,$
(4.13a) $\displaystyle A$
$\displaystyle=-\frac{e\,r}{\bigl{(}r^{2}+n^{2}\bigr{)}}\bigl{(}d\tau-2n\theta
d\varphi\bigr{)}+\frac{p}{r^{2}+n^{2}}\Bigl{[}n\,d\tau+\bigl{(}r^{2}-n^{2}\bigr{)}\theta
d\varphi\Bigr{]}\,,$ (4.13b)

where

$\pazocal{R}=2mr+e^{2}+p^{2}+\Lambda\,n^{4}-2\,\Lambda\,n^{2}r^{2}-\frac{\Lambda}{3}r^{4}\,,\\\
$ (4.14)

$m$ is the mass, $e$ is the electric charge, $p$ is the magnetic charge, $n$
is the NUT parameter and $\Lambda$ is the cosmological constant. In addition
to the double-Wick rotation and reparametrizations given by Eqs. (4.2), which
were used to obtain the swirling spacetime with a cosmological constant, we
can also define a reparametrization for the charges as:

$\displaystyle e$
$\displaystyle=\frac{i\,m}{n}E=i\bigl{(}2m^{2}\jmath\bigr{)}^{\frac{1}{3}}E\,,$
(4.15a) $\displaystyle p$
$\displaystyle=\frac{i\,m}{n}B=i\bigl{(}2m^{2}\jmath\bigr{)}^{\frac{1}{3}}B\,.$
(4.15b)

Therefore, by means of Eqs. (4.2) and Eqs. (4.15), the flat dyonic Reissner-
Nordström-Taub-NUT black hole with a cosmological constant (4.13a) becomes:

$\displaystyle ds^{2}$
$\displaystyle=\bigl{(}1+\jmath^{2}\rho^{4}\bigr{)}\biggl{[}-dt^{2}+\frac{\rho^{2}d\rho^{2}}{\pazocal{S}}+dz^{2}\biggr{]}+\frac{{\pazocal{S}}}{\bigl{(}1+\jmath^{2}\rho^{4}\bigr{)}}\bigl{(}d\phi+4\jmath
zdt\bigr{)}^{2}\,,$ (4.16a) $\displaystyle A$
$\displaystyle=\biggl{[}\frac{1}{1+\jmath^{2}\rho^{4}}\biggr{]}\biggl{[}\bigl{(}B-\jmath
E\rho^{2})d\phi+2\jmath z\bigl{(}B-2\jmath
E\rho^{2}-\jmath^{2}B\rho^{4}\bigr{)}dt\biggr{]}\,,$ (4.16b)
$\displaystyle\pazocal{S}$
$\displaystyle=\rho^{2}-\bigl{(}E^{2}+B^{2}\bigr{)}+\frac{\Lambda}{4\jmath^{2}}-\frac{\Lambda}{2}\rho^{4}-\frac{\jmath^{2}\Lambda}{12}\rho^{8}\,.$
(4.16c)

However, this new solution (4.16) presents some odd behaviors. Indeed, if the
cosmological constant is zero, $\Lambda=0$, there is a singularity for
$\rho^{2}=\bigl{(}E^{2}+B^{2}\bigr{)}$, nonetheless, this is a _coordinate_
singularity, that can be fixed by the following coordinate transformation

${\rho}^{\prime}\mkern 1.0mu{}^{2}=\rho^{2}+\bigl{(}E^{2}+B^{2}\bigr{)}\,.$
(4.17)

In order to better understand the coordinate nature of this singularity, it
can be useful to install the same coordinate singularity in the Minkowski
metric expressed in cylindrical coordinates. Indeed, after the opposite shift,
${\rho}^{\prime}\mkern 1.0mu{}^{2}=\rho^{2}-c^{2}$, we have that the Minkowski
metric results in

$ds^{2}=-dt^{2}+\frac{{\rho}^{\prime}\mkern 1.0mu{}^{2}d{\rho}^{\prime}\mkern
1.0mu{}^{2}}{{\rho}^{\prime}\mkern
1.0mu{}^{2}-c^{2}}+dz^{2}+\bigl{(}{\rho}^{\prime}\mkern
1.0mu{}^{2}-c^{2}\bigr{)}d\phi^{2}\,.$ (4.18)

Moreover, there is also the problem that in the non-swirling case, $\jmath=0$,
the electromagnetic potential (4.16a) becomes zero, up to a gauge
transformation, meaning that setting the swirling parameter to zero also
removes the electromagnetic background fields. Problem that can be solved by
performing a rescaling of the electric and magnetic fields,
$E^{\prime}=\frac{E}{\jmath}$, $B^{\prime}=\frac{B}{\jmath}$.

When we reached this point, during the mid-stages of this thesis, an article
[47] was published where they obtained all the possible backgrounds
constructed by composing two different Ehlers-Harrison transformations. In
particular, they also found the electromagnetic Melvin-swirling universe with
a cosmological constant, using the same double-Wick rotation as we were doing.
For this reason, we did not proceed any further in searching for a better
possible coordinate transformation.

Nevertheless, we can express this result in the following form:

$\displaystyle ds^{2}$
$\displaystyle=F\biggl{[}-dt^{2}+\frac{\rho^{2}d\rho^{2}}{\pazocal{S}}+dz^{2}\biggr{]}+\frac{{\pazocal{S}}}{F}\bigl{(}d\phi-\Omega\,dt\bigr{)}^{2}\,,$
(4.19a) $\displaystyle A$
$\displaystyle=A_{0}\,dt+\frac{A_{3}}{F}\bigl{(}d\phi-\Omega\,dt\bigr{)}\,,$
(4.19b)

where

$\displaystyle A_{0}$ $\displaystyle=Ez\,,$ (4.20a) $\displaystyle A_{3}$
$\displaystyle=\frac{1}{2}\Bigl{[}B\rho^{2}-\jmath
E\rho^{4}+\frac{BX^{2}}{4}\rho^{4}\Bigr{]}\,,$ (4.20b) $\displaystyle\Omega$
$\displaystyle=-4\jmath z\,,$ (4.20c) $\displaystyle F$
$\displaystyle=1+\frac{X^{2}}{2}\rho^{2}+\biggl{[}\jmath^{2}+\frac{X^{4}}{16}\biggr{]}\rho^{4}\,,$
(4.20d)
$\displaystyle\begin{split}\pazocal{S}&=-\frac{4\Bigl{[}\bigr{(}48\jmath^{2}+X^{4}\bigr{)}^{2}-3072\jmath^{4}\Bigr{]}\Lambda}{3\bigl{(}16\jmath^{2}+X^{4}\bigr{)}^{3}}+\biggl{[}1-\frac{4X^{2}\bigl{(}48\jmath^{2}+X^{4}\bigr{)}\Lambda}{3\bigl{(}16\jmath^{2}+X^{4}\bigr{)}^{2}}\biggr{]}\rho^{2}\\\
&\quad-\frac{\Lambda}{2}\rho^{4}-\frac{X^{2}\Lambda}{12}\rho^{6}-\frac{\bigl{(}16\jmath^{2}+X^{4}\bigr{)}\Lambda}{192}\rho^{8}\,,\end{split}$
(4.20e) $\displaystyle X^{2}$ $\displaystyle=B^{2}+E^{2}\,,$ (4.20f)

with $\jmath$ the swirling parameter, $\Lambda$ the cosmological constant, $E$
the uniform electric field and $B$ the uniform magnetic field. This solution
(4.19) thus represents an electromagnetic-Melvin-swirling universe with a
cosmological constant.

In particular, in the case of a negative cosmological constant
$\Lambda=-\frac{3}{\ell^{2}}$, we find that the function $\pazocal{S}$
becomes:

$\displaystyle\begin{split}\pazocal{S}&=\frac{4\Bigl{[}\bigr{(}48\jmath^{2}+X^{4}\bigr{)}^{2}-3072\jmath^{4}\Bigr{]}}{\ell^{2}\bigl{(}16\jmath^{2}+X^{4}\bigr{)}^{3}}+\biggl{[}1+\frac{4X^{2}\bigl{(}48\jmath^{2}+X^{4}\bigr{)}}{\ell^{2}\bigl{(}16\jmath^{2}+X^{4}\bigr{)}^{2}}\biggr{]}\rho^{2}\\\
&\quad+\frac{3}{2\ell^{2}}\rho^{4}+\frac{X^{2}}{4\ell^{2}}\rho^{6}+\frac{\bigl{(}16\jmath^{2}+X^{4}\bigr{)}}{64\ell^{2}}\rho^{8}\,.\end{split}$
(4.21)

Similarly, if the cosmological constant is set to zero, $\Lambda=0$, we
recover the electromagnetic-Melvin-swirling universe, whose function
$\pazocal{S}$ is simply

$\pazocal{S}=\rho^{2}\,.$ (4.22)

Therefore, as sketched in Figure 4.2, we can conclude that the electromagnetic
Melvin-swirling background can be thought of as the spacetime generated by two
dyonic Reissner-Nordström-Taub-NUT black holes pushed at infinity along the
symmetry axis and with opposite charges.

Black hole at infinity with electric charge e, magnetic charge p and NUT
parameter n Coordinate origin of the electromagnetic Melvin-swirling spacetime
Black hole at infinity with electric charge -e, magnetic charge -p and NUT
parameter -n Figure 4.2: Schematic representation of the electromagnetic
Melvin-swirling background as two infinitely distant, along the symmetry axis,
dyonic Reissner-Nordström-Taub-NUT black holes with opposite charges.

Moreover, similarly to the magnetic Melvin-swirling universe, we have that, by
means of the following coordinate transformation: