### List of Publications

Marco Astorino and Adriano Viganò
_Binary black hole system at equilibrium_
Physics Letters B 820, 136506 (2021)

Marco Astorino and Adriano Viganò
_Many accelerating distorted black holes_
The European Physical Journal C 81, 891 (2021)

Marco Astorino, Roberto Emparan and Adriano Viganò
_Bubbles of nothing in binary black holes and black rings, and viceversa_
Journal of High Energy Physics 07, 007 (2022)

Marco Astorino, Riccardo Martelli and Adriano Viganò
_Black holes in a swirling universe_
Physical Review D 106, 064014 (2022)

Marco Astorino and Adriano Viganò
_Charged and rotating multi-black holes in an external gravitational field_
The European Physical Journal C 82, 829 (2022)

###### Contents

  1. List of Publications
  2. 1 Solution generating techniques
    1. 1.1 Ernst equations
      1. 1.1.1 Einstein–Maxwell equations
      2. 1.1.2 Ernst potentials
      3. 1.1.3 Symmetries of the Ernst equations
      4. 1.1.4 Ehlers transformation
      5. 1.1.5 Harrison transformation
    2. 1.2 Inverse scattering method in vacuum
      1. 1.2.1 Integrable ansatz
      2. 1.2.2 Integration scheme
      3. 1.2.3 General $n$-soliton solution
      4. 1.2.4 Kerr–NUT spacetime
      5. 1.2.5 Rotating C-metric
      6. 1.2.6 Multi-black hole solutions
  3. 2 Black holes in an external gravitational field
    1. 2.1 Background gravitational field
      1. 2.1.1 Background field multipoles
      2. 2.1.2 Seed for the inverse scattering construction
    2. 2.2 Array of static black holes
      1. 2.2.1 Conical singularities and regularisation
      2. 2.2.2 Smarr law
      3. 2.2.3 Schwarzschild black hole
    3. 2.3 Binary system: a case study
      1. 2.3.1 Near-horizon limit
      2. 2.3.2 Thermodynamics
      3. 2.3.3 Binary charged system
      4. 2.3.4 Binary Kerr–NUT system
    4. 2.4 Array of accelerating black holes
      1. 2.4.1 Accelerating multipolar gravitational background
      2. 2.4.2 Array of static accelerating black holes
      3. 2.4.3 Accelerating Schwarzschild black hole
      4. 2.4.4 Accelerating binary system
      5. 2.4.5 Accelerating particles
    5. 2.5 Outlook
  4. 3 Black holes in an expanding bubble of nothing
    1. 3.1 Black holes, black rings and Kaluza–Klein bubbles
      1. 3.1.1 Expanding bubbles of nothing from black hole binaries and black rings
      2. 3.1.2 Black hole binaries and black rings inside expanding bubbles
    2. 3.2 Bubbles as limits of black hole binaries and black rings
      1. 3.2.1 From black ring to bubble
      2. 3.2.2 Weyl metrics and rod structures
      3. 3.2.3 From binary black hole to bubble
      4. 3.2.4 From black ring to bubble, Weyl style
      5. 3.2.5 Five-dimensional black hole binaries and bubbles
    3. 3.3 Static black hole binaries and black rings in expanding bubbles
      1. 3.3.1 4D black hole binary in equilibrium inside the expanding bubble
      2. 3.3.2 Black ring in equilibrium inside the expanding bubble
    4. 3.4 Other configurations
      1. 3.4.1 Four dimensions
      2. 3.4.2 Five dimensions
    5. 3.5 Outlook
  5. 4 Black holes in a swirling universe
    1. 4.1 Generation of the solution
    2. 4.2 Analysis of the background spacetime
      1. 4.2.1 Geodesics in the background spacetime
    3. 4.3 Schwarzschild black hole in a swirling universe
      1. 4.3.1 Physical properties
      2. 4.3.2 Ergoregions
      3. 4.3.3 Petrov type
      4. 4.3.4 Geodesics
      5. 4.3.5 Charges and thermodynamics
    4. 4.4 Kerr black hole in a swirling universe
    5. 4.5 Double-Wick rotation of the background: flat Taub–NUT spacetime
    6. 4.6 Outlook
  6. Conclusions
  7. A Conical singularities and energy conditions
  8. B Harrison and Kramer–Neugebauer charging trasformations
  9. C Geodesics of the swirling spacetime
    1. C.1 Background geodesics
    2. C.2 Black hole geodesics
  10. D Zipoy–Voorhees spacetime embedded in the swirling universe

### Introduction

> Oh leave the Wise our measures to collate
> One thing at least is certain, light has weight
> One thing is certain, and the rest debate–
> Light rays, when near the Sun, do not go straight.
>
> Arthur Eddington

General Relativity [1, 2] represents one of the most astonishing achievements
of humankind, and one the most successful theories in the history of physics.
From its early successes in reproducing the perihelion precession of Mercury
and the deflection of light [3, 4, 5], to the remarkable prediction of the
existence of gravitational waves, that have been recently detected [6], the
theory has been tested on different scales and an excellent agreement with the
experiments has been established [7].

Among the predictions of General Relativity, black holes [8] perhaps represent
the most fascinating one: they are regions of spacetime where the
gravitational field is so intense that nothing can escape from it, not even
light. Such objects are quite difficult to observe, and indeed many years
passed before the existence of black holes was accepted by the physics
community. In particular, the hypothesis that a supermassive black hole,
dubbed Sagittarius A*, is located in the centre of our Galaxy [9, 10, 11] has
recently received a spectacular confirmation, with the first image of such a
black hole shot by the Event Horizon Telescope. Moreover, there are
observations of gravitational lensing around Messier 87* [12, 13, 14, 15, 16,
17], a proposed black hole, that are in good agreement with the Kerr metric
[18, 19].

Binary black hole systems are of utmost relevance from the astrophysical
perspective: their merging processes are the major source of gravitational
waves [6, 20, 21, 22] (together with binary neutron star systems [23, 24, 24]
and, on the theoretical side, they disclose the non-linear nature of General
Relativity, and represent an important playground in which testing the laws of
black hole mechanics and of quantum gravity. More generally, an analytical
description of multi-black hole spacetimes is of fundamental importance both
for the interpretation of the measurements and for the investigation of the
structure of the gravitational theory.

Of course one of the main obstacle in modelling a stationary multi-
gravitational source system is the necessity of a mechanism to balance the
gravitational attraction of the bodies: otherwise the system naturally tends
to collapse. Usually the equilibrium is granted by the introduction of
cosmological struts which prop up the gravitating bodies, but these one-
dimensional objects must be constituted by matter that violates physically
reasonable energy conditions. Alternatively, in some cases cosmic strings of
infinite length can be used to support the gravitational collapse. In both
cases these objects are symptomatic for conical singularities which plague the
spacetime.

It is known that in Newtonian mechanics the (uncharged) $N$-body
configurations cannot be in equilibrium for $N>1$. Indeed, since the Newtonian
gravitational force is always attractive, it is clear that separated bodies
cannot be in balance. However, the situation might be different in General
Relativity: if one considers rotating objects, then the effect of spin-spin
repulsion might compensate for the gravitational attraction. Therefore, in
order to better understand the nature of the gravitational interaction in
General Relativity, it is an important question as to whether an equilibrium
of (physically reasonable) rotating bodies is possible [25].

The simplest instance of such a configuration is provided by two aligned black
holes in vacuum, i.e. a double Kerr–NUT solution [26, 27]. Many authors, over
the years, addressed the problem of the equilibrium configuration for such a
system [28, 29, 30, 31], by refining the parametrisation in order to simplify
the resulting metric and to clarify the physical significance of the
parameters [32]. However, it turned out that the double Kerr–NUT solution
describes an equilibrium configuration when one of the two black holes
violates an inequality between the the horizon area and the angular momentum
[33] or, equivalently, when one of the two horizons disappears, i.e. we are
left with a naked singularity. Thus, regular stationary double black holes
configurations in vacuum do not exist: the spin-spin repulsion is not strong
enough to compensate for the gravitational attraction.

The situation is slightly different in the electrovacuum case: in Newtonian
physics, we know that a collection of $N$ charged bodies can reach the
equilibrium, provided that the charges are sufficiently large to balance the
gravitational attraction. In the relativistic case, however, there exist upper
limits on the charges and angular momenta, such that the black hole horizons
are not broken. This implies that the existence of equilibrium configurations
due to charge-charge (and spin-spin interaction) is not guaranteed _a priori_.

The most known example of a static equilibrium configuration is provided by
the Majumdar–Papapetrou solution [34, 35], which describes a collection of an
arbitrary number of extremal Reissner–Nordström black holes: the interesting
feature of this solution is that the black holes are not constrained on an
axis, but can be located in any position in the plane. The Majumdar–Papapetrou
spacetime was extended to the presence of a positive cosmological constant in
[36], and then to the general case (which also includes a negative
cosmological constant) in [37]. These solutions, despite being very
interesting, contain extremal black holes (i.e. vanishing surface gravity),
which represent limiting configurations that cannot be realised in Nature, as
shown by Thorne [38]. Thus, one may wonder if it is possible to construct
_non-extremal_ and stable multi-black hole configurations in the electrovacuum
case.

In the case of static, non-extremal multi-black hole configurations, the
answer is negative: it was shown [39] that a static multi-black hole solution
must reduce to the Majumdar–Papapetrou metric. The stationary case is more
complicated, and a definitive answer is not known: some recent papers that
tackle this problem are [25, 40, 41].

Turning back to the vacuum case, we showed that it is not possible to
construct asymptotically flat multi-black hole configurations that are at
equilibrium. Thus, it is natural to wonder if it is possible to relax the
boundary conditions and to consider more general backgrounds (different than
Minkowski), such that the embedding of multi-black hole systems can reach an
equilibrium configuration. This is the main goal of this Thesis: we will see,
in the next Chapters, that there exist non-trivial backgrounds (given by
external multipolar gravitational fields and by Kaluza–Klein bubbles) that
allow for a multi-black hole configuration that can achieve an equilibrium
state, provided that the parameters of the background (and of the holes) are
fine tuned in order to remove the conical singularities.

However, as one can imagine, the construction of a multi-black hole solution
is a highly non-trivial task: because of the non-linearity of the Einstein
equations, one can not simply superimpose many black hole solutions in order
to obtain a new solution. Because of these difficulties, we will rely on the
application of some powerful _solution generating techniques_ , i.e.
techniques that have been developed to construct exact solutions of the
Einstein equations without the direct integration of the equations of motion.
Since the new solutions that we will present in this Thesis are based on such
techniques, we will dedicate some space to the explanation of the methods.

More precisely we begin, in Chapter 1, by introducing the solution generating
techniques: after an overview (and a history) of the many techniques that have
been developed over the years, we explain in some detail the two techniques
that allow us to construct the desired multi-black hole solutions, i.e. the
Ernst formalism and the inverse scattering method. The inverse scattering
method will be our main tool in Chapters 2 and 3, while we will take advantage
of the Ernst transformations in Chapter 4.

Subsequently, Chapter 2 deals with multi-black holes embedded in an external
gravitational field: such a field is given by a multipolar expansion and,
thus, represents a generic static and axisymmetric gravitational field. We
will see that, by choosing appropriately the multipole parameters of the
field, it is possible to obtain an equilibrium configuration in many
interesting cases, like a collection of collinear static black holes or a
chain of accelerating black holes.

Chapter 3 considers another interesting example of a regularising background:
the expanding Kaluza–Klein bubbles, also known as “bubbles of nothing”. These
are interesting solutions of vacuum General Relativity, that are also
connected to the vacuum stability, that resemble the expanding de Sitter
spacetime. Such an expanding behaviour provides the force necessary to balance
the gravitational attraction among the black holes, and hence to reach the
equilibrium.

Finally, Chapter 4 takes a different perspective: it is not focused on a
multi-black hole solution, but rather it explores the action of a symmetry
transformation of the Ernst equations (unexplored until now), which produces a
new solution. Such a solution represents a black hole embedded in a rotating
(or “swirling”, as we dubbed it) universe, which possesses many interesting
features. Even if we limit ourselves to constructing single-black hole
solutions in Chapter 4, we discuss the possibility of implementing the
swirling background in order to enforce the spin-spin configuration explained
above, and reach an equilibrium configuration in a double-Kerr spacetime.

###  Chapter 1 Solution generating techniques

Einstein equations represent a set of highly non-linear partial differential
equations: thus, it is extremely difficult to find exact analytical solutions
just resting on ansätze. For this reason, many people have developed, over the
years, methods and techniques that allow one to construct exact solution
_without_ the direct integration of the equations of motion. The basic idea
behind these methods is to take a known solution, that is usually called a
_seed_ , and then to apply to it a transformation or a map in order to produce
a new, non-equivalent solution of the equations of motion, by overcoming the
integration of the equations themselves. We begin this chapter by presenting a
short review of the history of the solution generating techniques, in such a
way to provide to the reader a perspective of the many methods that have been
developed over the years, and to not limit ourselves only to the ones that we
will use in this Thesis. The historical account we will present is largely
based on the one given by Alekseev in [42].

The history of solution generating techniques began in the mid-60s, when a
class of non-linear partial differential equations, which were called
completely integrable, were discovered [43]: that class was represented by the
famous Korteweg–de Vries equation. Such equations admit various solution
generating methods for the explicit construction of infinite hierarchies of
solutions with an arbitrary number of parameters and for the non-linear
superposition of fields. This class is extremely important, because many
fundamental equations of mathematics and physics were found, upon two-
dimensional reduction, to belong to it111In fact, the integrability we are
talking about was found to hold only for the symmetry reduced Einstein
equations, i.e. for spacetimes which admit $D-2$ commuting Killing vectors in
$D$ dimensions..

The discoveries in the field of integrability led to the hope that Einstein
equations could be integrable as well. One of the very first examples was
provided by Ernst [44, 45], who showed, both for the vacuum and the
electrovacuum case, that the equations of motion can be written in a very
compact and elegant way, such that some symmetry transformations that leave
the theory invariant can be unraveled. Subsequently, Geroch conjectured [46]
that four-dimensional vacuum Einstein equations with two commuting Killing
vectors admit an infinite-dimensional group of internal symmetries which
allows one to obtain any solution starting from Minkowski spacetime.

Later, Kinnersley and Chitre [47, 48, 49, 50] inspected the internal
symmetries of stationary and axisymmetric Einstein–Maxwell equations, and
indeed they discovered an infinite-dimensional algebra by constructing its
representation in terms of potentials which characterise any solution. Shortly
after, Maison [51] found that vacuum gravity can be related to a linear
eigenvalue problem in the form of Lax, i.e. that it is possible to reduce the
solutions of non-linear Einstein equations to a sequence of linear problems.

The period of conjectures about integrability of Einstein gravity came to end
in 1978, when Belinski and Zakharov [52], by means of their _inverse
scattering method_ , constructed an over-determined linear system with a
complex spectral parameter, whose integrability conditions are equivalent to
the two-dimensional reduced vacuum Einstein equations. Thus, the original non-
linear problem was reformulated in terms of an equivalent spectral problem.
This also implied the discovery of gravitational solitons and the formulation
of the spectral problem in the language of a classical matrix Riemann problem.
The procedure was then completed in [53], where the authors found an
expression for the conformal factor. The electrovacuum version of the inverse
scattering method does not admit a straightforward generalisation: Alekseev
[54] proved that, in the presence of the electromagnetic field, the solitons
must possess complex (but not real) poles.

In the same year, Harrison [55] constructed the so-called _Bäcklund
transformations_ for vacuum Einstein equations, by using an approach due to
Estabrook and Wahlquist known as “prolongation structures” for non-linear
equations [56]. Almost simultaneously, Neugebauer [57] obtained the Bäcklund
transformations following a different approach, that was further developed in
[58, 27]. Finally, Julia [59, 60] proved that the infinite-dimensional
symmetries found by Geroch and Kinnersley are nothing but Kac–Moody
symmetries, together with a wide treatment of the symmetries of generalised
$\sigma$-models and supergravities. In [61] the corresponding infinite-
dimensional Geroch group was studied in detail.

These last developments were followed by a series of papers by Hauser and
Ernst [62, 63, 64, 65], who proposed the integral equation method in order to
exponentiate the infinitesimal symmetries of Kinnersley and Chitre and to
obtain the finite solutions generating transformations, that culminated in the
proof of the Geroch conjecture mentioned above [66]. A modification of the
Hauser and Ernst formalism was then proposed by Sibgatullin [67, 68] in order
to make its applicability simpler, and to make use only of the seed data on
the axis of symmetry.

The monodromy transform approach proposed by Alekseev [69, 70, 71, 72] started
with the formulation of a spectral problem, like the inverse scattering
method, but it also defined a monodromy data as a set of coordinate-
independent functions of a complex parameter that characterises any local
solution of vacuum or electrovacuum gravity.

All the developments we have described up to now are related to vacuum or
electrovacuum gravity; however, there are situations in which the presence of
matter fields still makes the reduced Einstein equations integrable.

A very simple example is given by gravity coupled to a perfect fluid with
stiff matter equation of state, where the inverse scattering method still
works in the very same way as the vacuum case [73]. This case is equivalent to
gravity plus a minimally coupled scalar field, which is particularly
interesting because, as shown by Bekenstein [74], it is possible to map the
minimally coupled theory into the conformally coupled one: thus, the
generation methods can be used in the conformally coupled theory, where black
holes and wormholes are known to exist [75, 76].

The Einstein–Maxwel-dilaton system, which can be obtained from five-
dimensional pure gravity upon Kaluza–Klein reduction, enjoys the symmetries of
the vacuum case [53]. Symmetries are present even in the case of a dilatonic
coupling different from the Kaluza–Klein one [77, 78]. Regarding scalar
fields, also the dilaton-axion systems are known to possess an integrable
structure [79, 80, 81, 82].

Higher-dimensional gravity benefits of generating techniques as well, at least
in some particular cases. An extension of the inverse scattering method to
higher dimensions in vacuum, which manages the problem of the metric
normalisation, has been proposed in [83, 84], and it was lately applied to
construct many interesting solutions which generalise the Myers–Perry black
holes [85], black rings [86, 87, 88, 89, 90, 91, 92], black Saturn [93], black
di-rings [94, 95], bycicling black rings [96] and others [97, 98, 99]. The
Einstein–Maxwell reduced system represents a $\sigma$-model under a special
ansatz, as shown in [100].

Supergravity theories, in the ungauged case, are known to admit the
construction of a Belinski–Zakharov spectral problem, as shown in the simple
case of the dilaton-axion system in [101, 102]. In the case of more
complicated theories, part of the symmetry reduced equations takes the form of
a $\sigma$-model. The existence of a $\sigma$-model is a strong evidence of
the integrability properties of a theory, and indeed it is usually possible to
construct an associated linear spectral problem. However, the existence of a
spectral problem does not lead itself to a method for generating new
solutions, because the space of solutions of the linear system is typically
larger than the space of solutions of the original non-linear equations: some
reduction constraints must be imposed in order to provide the equivalence
between the two systems. An example is provided by 5D supergravities, whose
integrability was established in [103, 104, 105, 106], and for which the
inverse scattering method was constructed in [107]. The inverse scattering
procedure was also applied to some supergravity models in [108, 109, 110].
Studies on the integrability of the heterotic string theory were pursued in
[111, 112].

One can observe, as a consequence of the historical perspective that we have
presented, that many generation techniques exist, and thus there are many
choices that can be made when dealing with the construction of exact
solutions. Each of the methods mentioned above comes with pros and cons, and
their usefulness depends on the specific system that one is studying. However,
they are all restricted to the existence of a certain number of Killing
vectors in the class of spacetime under consideration, more precisely $D-2$,
(where $D$ is the spacetime dimension): for our purposes, the class of metrics
that we will be interested in are the _stationary_ and _axisymmetric_
spacetimes222Let us notice that, for $D>4$, a slightly different definition of
stationary and axisymmetric spacetime than the $D=4$ case, is used [113].. In
this Thesis, we will focus on the Ernst formalism (described in Sec. 1.1) and
on the inverse scattering method (described in Sec. 1.2).

The Ernst formalism [44, 45] is based on a clever rewriting of the
Einstein–Maxwell equations, which is suitable to make explicit the symmetries
of the equations themselves: the system is studied in an effective three-
dimensional space, thus the Ernst construction operates a sort of dimensional
reduction from four to three dimensions. The Ernst equations, which represent
the aforementioned rewriting, are a couple of complex equations written in
terms of two complex potentials, dubbed $\mathcal{E}$ and $\Phi$. Such
potentials are defined in terms of the relevant components of the metric and
of the Maxwell field, and thus they are in one-to-one correspondence with the
fields of the theory: given the metric and the Maxwell 1-form, the Ernst
potentials are uniquely determined, and viceversa.

The Ernst equations allow us to reveal (a part of) the symmetries of the
Einstein–Maxwell system: such symmetries are encoded into some transformations
that the Ernst potentials undergo, which leave the theory unchanged. Thus,
such transformations map a solution into another one, without changing the
theory. We will see that most of the symmetry maps of the Ernst equations are
nothing but gauge transformations; however, there are two maps, the _Ehlers
transformation_ [114] and the _Harrison transformation_ [115] that act non-
trivially on a seed solution. We should specify that these maps actually
represent four different transformations, basically because it is possible to
Wick-rotate them in order to obtain non-equivalent behaviours.

More precisely, the Ehlers map allows one to add the NUT parameter to a given
seed solution [116], or, after a Wick-rotation, to add a rotation parameter
that we will study extensively in Chapter 4. On the other hand, the Harrison
map adds a dyonic charge to a seed or, in its Wick-rotated version, an
external electromagnetic field of Melvin type. The second usage, in
particular, was popularised by Ernst himself, who made use of it in order to
regularise the charged C-metric [117].

The inverse scattering method [52], that we will study in its original vacuum
version, is based on the integrability properties of the Einstein equations.
It works, in contrast with the Ernst formalism, on an effective two-
dimensional system, and it is thus a dimensional reduction from four to two
dimensions. The key intuition of Belinski and Zakharov is to rewrite the the
gravitational equations as first order matrix equations, and then to recognise
them as the integrability conditions of an over-determined system of matrix
equations related to an eigenvalue problem for some linear differential
operator. The eigenvalue problem is linear, and for this reason it is easier
to solve than the original non-linear problem represented by the Einstein
equations: the strategy of the approach is to solve the eigenvalue problem
associated to a seed solution, and then use it to algebraically build a new
solution, without the direct integration of the equations of motion. The
fundamental building blocks needed in the construction of a new solution are
the _solitons_ , that we will define later in Sec. 1.2.

Because of the reduction to two dimensions, the inverse scattering method
unravels more symmetries than the Ernst equations, and indeed it allows to add
not only the NUT parameter to a seed, but also mass, angular momentum and
acceleration. Furthermore, and this is our main motivation in using it, the
inverse scattering procedure allows us to easily construct multi-black hole
spacetimes: once one recognises that a pair of the aforementioned solitons
gives rise to a black hole, then it suffices to add many pairs of solitons to
obtain many black holes. This modular structure is extremely powerful, since
it allows us to keep under control the parametrisation of the solution under
consideration and to construct a multi-source solution by using a non-linear
superposition principle.

The rest of the Chapter is devoted to an accurate description of the Ernst
formalism and of the inverse scattering method, since they will play a central
role in the construction of the new solutions that we will present in the
subsequent Chapters.

#### 1.1 Ernst equations

In this section we will derive the Ernst equations for the Einsten–Maxwell
theory and unravel their symmetries, in order to find non-trivial
transformations that map a solution of the theory into another one. We will
follow the original Ernst’s papers [44, 45].

We begin by considering the Einstein–Maxwell action

$S=\int d^{4}x\sqrt{-g}\biggl{(}R-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\biggr{)}\,,$
(1.1)

which gives rise to the Maxwell equations

$\nabla_{\mu}F^{\mu\nu}=0\,,$ (1.2)

and to the Einstein equations with electromagnetic source

$G_{\mu\nu}=8\pi T_{\mu\nu}\,,$ (1.3)

that can also purposefully rewritten, by taking the trace, as

$R_{\mu\nu}=8\pi\biggl{(}T_{\mu\nu}-\frac{1}{2}Tg_{\mu\nu}\biggr{)}\,.$ (1.4)

We have defined the electromagnetic stress-energy tensor as

$T_{\mu\nu}=\frac{1}{4\pi}\biggl{(}F_{\mu\rho}F_{\nu}^{\;\;\rho}-\frac{1}{4}F_{\rho\sigma}F^{\rho\sigma}g_{\mu\nu}\biggr{)}\,.$
(1.5)

Being the stress-energy tensor traceless, $T=0$, the Einstein equations are
just given by

$R_{\mu\nu}=8\pi T_{\mu\nu}\,.$ (1.6)

We have to consider an ansatz for the electromagnetic field and for the
metric. We are interested in stationary and axisymmetric fields [118]: we
recall that a spacetime is _stationary_ when it admits a nowhere vanishing
timelike Killing field, and it is _axisymmetric_ when it is invariant under
the action of the 1-parameter group $SO(2)$ and the fixed point set under the
group is non-empty.

One can prove [118] that a metric with such symmetries is given by the
Lewis–Weyl–Papapetrou ansatz333Actually, such a form of the metric is legit
when the trace of the Ricci tensor over the manifold orthogonal to the Killing
fields is zero [118].

${ds}^{2}=-f(dt-\omega
d\phi)^{2}+f^{-1}\bigl{[}e^{2\gamma}\bigl{(}{d\rho}^{2}+{dz}^{2}\bigr{)}+\rho^{2}{d\phi}^{2}\bigr{]}\,,$
(1.7)

where the functions $f$, $\omega$, $\gamma$ depend only on $(\rho,z)$. The
coordinates $(t,\phi,\rho,z)$ are called _Weyl_ or _cylindrical_ coordinates.
We will see that they will prove very useful to express the equations of
motion in a suitable way. We also choose an ansatz for the Maxwell
potential444The ansatz is coherent with the Lewis–Weyl–Papapetrou symmetries,
however it is not the most general stationary and axisymmetric choice. A
discussion of the constraints on the Maxwell field can be found in [118].

$A=A_{t}dt+A_{\phi}d\phi\,,$ (1.8)

where $A_{t}$ and $A_{\phi}$ depend only on $(\rho,z)$.

Since we are working in cylindrical coordinates $(\rho,\phi,z)$, and our
functions depend on $(\rho,z)$ only, it will be convenient to express our
equations in three-dimensional Euclidean space notation, thus introducing
quantities like the axis unit vectors $(\hat{\rho},\hat{\phi},\hat{z})$, and
the gradient, divergence, curl and Laplacian, that we report here for
convenience

$\displaystyle\nabla f$ $\displaystyle=\frac{\partial
f}{\partial\rho}\hat{\rho}+\frac{1}{\rho}\frac{\partial
f}{\partial\phi}\hat{\phi}+\frac{\partial f}{\partial z}\hat{z}\,,$ (1.9a)
$\displaystyle\nabla\cdot\vec{V}$
$\displaystyle=\frac{1}{\rho}\frac{\partial(\rho
V_{\rho})}{\partial\rho}+\frac{1}{\rho}\frac{\partial
V_{\phi}}{\partial\phi}+\frac{\partial V_{z}}{\partial z}\,,$ (1.9b)
$\displaystyle\nabla\times\vec{V}$
$\displaystyle=\biggl{(}\frac{1}{\rho}\frac{\partial
V_{z}}{\partial\phi}-\frac{\partial V_{\phi}}{\partial
z}\biggr{)}\hat{\rho}+\biggl{(}\frac{\partial V_{\rho}}{\partial
z}-\frac{\partial
V_{z}}{\partial\rho}\biggr{)}\hat{\phi}+\frac{1}{\rho}\biggl{(}\frac{\partial(\rho
V_{\phi})}{\partial\rho}-\frac{\partial
V_{\rho}}{\partial\phi}\biggr{)}\hat{z}\,,$ (1.9c) $\displaystyle\nabla^{2}f$
$\displaystyle=\frac{1}{\rho}\frac{\partial}{\partial\rho}\biggl{(}\rho\frac{\partial
f}{{\partial\rho}}\biggr{)}+\frac{1}{\rho^{2}}\frac{\partial^{2}f}{{\partial\phi}^{2}}+\frac{\partial^{2}f}{{\partial
z}^{2}}\,.$ (1.9d)

##### 1.1.1 Einstein–Maxwell equations

Let us define the Maxwell equations

$M^{\nu}\coloneqq\nabla_{\mu}F^{\mu\nu}\,.$ (1.10)

We notice that the $\phi$-component can be written as

$\begin{split}M^{\phi}&=e^{-2\gamma}f\biggl{[}\rho^{-2}f\bigl{(}\nabla^{2}A_{\phi}+\omega\nabla^{2}A_{t}\bigr{)}-2\rho^{-3}f\biggl{(}\frac{\partial
A_{\phi}}{\partial\rho}+\omega\frac{\partial A_{t}}{\partial\rho}\biggr{)}\\\
&\quad+\rho^{-2}f\nabla\omega\cdot\nabla A_{t}+\rho^{-2}\nabla
f\cdot\bigl{(}\nabla A_{\phi}+\omega\nabla A_{t}\bigr{)}\biggr{]}\equiv
0\,;\end{split}$ (1.11)

one can show that Eq. (1.11) is equivalent to

$\nabla\cdot\bigl{[}\rho^{-2}f\bigl{(}\nabla A_{\phi}+\omega\nabla
A_{t}\bigr{)}\bigr{]}=0\,.$ (1.12)

Analogously, we can manipulate the $t$-component of the Maxwell equations

$\begin{split}M^{t}&=e^{-2\gamma}f\biggl{[}\rho^{-2}f\omega\bigl{(}\nabla^{2}A_{\phi}+\omega\nabla^{2}A_{t}\bigr{)}-f^{-1}\nabla^{2}A_{t}+\rho^{-2}\omega\nabla
f\cdot\bigl{(}\nabla A_{\phi}+\omega\nabla A_{t}\bigr{)}\\\
&\quad+f^{-2}\nabla f\cdot\nabla
A_{t}+\rho^{-2}f\nabla\omega\cdot\bigl{(}\nabla A_{\phi}+2\omega\nabla
A_{t}\bigr{)}-2\rho^{-3}f\omega\biggl{(}\frac{\partial
A_{\phi}}{\partial\rho}+\omega\frac{\partial
A_{t}}{\partial\rho}\biggr{)}\biggr{]}\equiv 0\,,\end{split}$ (1.13)

which can be written as

$\nabla\cdot\bigl{[}-f^{-1}\nabla A_{t}+\rho^{-2}f\omega\bigl{(}\nabla
A_{\phi}+\omega\nabla A_{t}\bigr{)}\bigr{]}=0\,.$ (1.14)

Eqs. (1.12) and (1.14) are equivalent to the Maxwell equations.

Now we turn to the gravitational equations: we make use of the Einstein
equations in the form (1.6), and define

$E_{\mu\nu}\coloneqq R_{\mu\nu}-8\pi T_{\mu\nu}\,.$ (1.15)

We consider the $tt$-component

$\begin{split}E_{tt}&=-\frac{e^{-2\gamma}}{2\rho^{2}}\biggl{\\{}2\rho^{-2}f^{3}\bigl{[}(\nabla
A_{\phi})^{2}+\omega^{2}(\nabla A_{t})^{2}+2\omega\nabla A_{\phi}\cdot\nabla
A_{t}\bigr{]}+2f(\nabla A_{t})^{2}\\\ &\quad+(\nabla
f)^{2}-\rho^{-2}f^{4}(\nabla\omega)^{2}-f\nabla^{2}f\biggr{\\}}\equiv
0\,,\end{split}$ (1.16)

that can also be written as

$f\nabla^{2}f=(\nabla f)^{2}-\rho^{-2}f^{4}(\nabla\omega)^{2}+2f(\nabla
A_{t})^{2}+2\rho^{-2}f^{3}\bigl{(}\nabla A_{\phi}+\omega\nabla
A_{t}\bigr{)}^{2}\,,$ (1.17)

and the $t\phi$-component

$\begin{split}E_{t\phi}&=-\frac{e^{-2\gamma}}{2\rho^{2}}\biggl{\\{}2f^{3}\omega\bigl{[}(\nabla
A_{\phi})^{2}+\omega^{2}(\nabla A_{t})^{2}+2\omega\nabla A_{\phi}\cdot\nabla
A_{t}\bigr{]}+\rho^{2}\omega(\nabla f)^{2}-f^{4}\omega(\nabla\omega)^{2}\\\
&\quad-\rho f\bigl{[}2\rho\bigl{(}2\nabla A_{\phi}\cdot\nabla A_{t}+\nabla
f\cdot\nabla\omega\bigr{)}+\omega(2\rho(\nabla
A_{t})^{2}+\rho\nabla^{2}f\bigr{)}\bigr{]}\\\ &\quad-\rho
f^{2}\biggl{(}\rho\nabla^{2}\omega-2\frac{\partial\omega}{\partial\rho}\biggl{)}\biggr{\\}}\equiv
0\,.\end{split}$ (1.18)

Now we combine

$\begin{split}\omega
E_{tt}+E_{t\phi}&=-\frac{fe^{-2\gamma}}{2\rho^{2}}\biggl{[}4\rho^{-2}\nabla
A_{\phi}\cdot\nabla A_{t}+4\rho^{-2}\omega(\nabla A_{t})^{2}+2\rho^{-2}\nabla
f\cdot\nabla\omega\\\
&\quad+\rho^{-2}f\nabla^{2}\omega-2\rho^{-3}f\frac{\partial\omega}{\partial\rho}\biggr{]}\equiv
0\,,\end{split}$ (1.19)

which is equivalent to

$\nabla\cdot\bigl{[}\rho^{-2}f^{2}\nabla\omega+4\rho^{-2}fA_{t}\bigl{(}\nabla
A_{\phi}+\omega\nabla A_{t}\bigr{)}\bigr{]}=0\,.$ (1.20)

The other non-trivial Einstein equations define $\gamma$ by quadratures: such
equations are obtained from $E_{\rho z}$ and $E_{\rho\rho}-E_{zz}$, and they
are

$\begin{split}&\rho\partial_{z}\gamma+\rho^{2}f^{-1}\partial_{z}A_{t}\partial_{\rho}A_{t}-2f(\omega\partial_{z}A_{t}-\partial_{z}A_{\varphi})(\omega\partial_{\rho}A_{t}-\partial_{\rho}A_{\varphi})\\\
&\quad-\frac{1}{2}\rho^{2}f^{-2}\partial_{z}f\partial_{\rho}f+\frac{1}{2}f^{2}\partial_{z}\omega\partial_{\rho}\omega=0\,,\end{split}$
(1.21)

and

$\begin{split}&\rho\partial_{\rho}\gamma+\rho^{2}f^{-1}(\partial_{\rho}A_{t})^{2}-(\partial_{z}A_{t})^{2}\\\
&\quad+f\Bigl{\\{}(\partial_{z}A_{\varphi})^{2}-(\partial_{\rho}A_{\varphi})^{2}+\omega^{2}\Bigl{[}(\partial_{z}A_{t})^{2}-(\partial_{\rho}A_{t})^{2}\Bigr{]}+2\omega(\partial_{\rho}A_{t}\partial_{\rho}A_{\varphi}-\partial_{z}A_{t}\partial_{z}A_{\varphi})\Bigr{\\}}\\\
&\quad+\rho^{2}f^{-2}\Bigl{[}(\partial_{z}f)^{2}-(\partial_{\rho}f)^{2}\Bigr{]}+f^{2}\Bigl{[}(\partial_{z}\omega)^{2}-(\partial_{\rho}\omega)^{2}\Bigr{]}=0\,.\end{split}$
(1.22)

Once the other metric functions are known, $\gamma$ can be found by
integrating the latter equations.

##### 1.1.2 Ernst potentials

Now we can elaborate on the equations of motion (1.12), (1.14), (1.17) and
(1.20). We begin by noticing that (1.12) contains a total divergence: this
means that we can introduce a potential associated to the vector whose
divergence is zero. Let us recall that, for any function $h(\rho,z)$
sufficiently well behaved, it holds

$\nabla\cdot\bigl{(}\rho^{-1}\hat{\phi}\times\nabla h\bigr{)}=0\,,$ (1.23)

as it can be verified by a direct computation. This means that we can
introduce a so-called twisted potential $\tilde{A}_{\phi}$, such that

$\hat{\phi}\times\nabla\tilde{A}_{\phi}\coloneqq\rho^{-1}f\bigl{(}\nabla
A_{\phi}+\omega\nabla A_{t}\bigr{)}\,,$ (1.24)

which, thanks to (1.23), implies Eq. (1.12). Now we multiply the last equation
by $\hat{\phi}\times$

$\rho^{-1}\hat{\phi}\times\nabla
A_{\phi}=-f^{-1}\nabla\tilde{A}_{\phi}-\rho^{-1}\omega\hat{\phi}\times\nabla
A_{t}\,,$ (1.25)

and then apply $\nabla\cdot$, to obtain

$\nabla\cdot\bigl{(}f^{-1}\nabla\tilde{A}_{\phi}+\rho^{-1}\omega\hat{\phi}\times\nabla
A_{t}\bigr{)}=\nabla\cdot\bigl{(}\rho^{-1}\hat{\phi}\times\nabla
A_{\phi}\bigr{)}\overset{\eqref{potential}}{=}0\,.$ (1.26)

This latter equation substitutes (1.12) as an equation of motion.

We also notice that, implementing the definition of the potential (1.24), Eq.
(1.14) is written as

$\nabla\cdot\bigl{(}-f^{-1}\nabla
A_{t}+\rho^{-1}\omega\hat{\phi}\times\nabla\tilde{A}_{\phi}\bigr{)}=0\,.$
(1.27)

We have then shown that the Maxwell equations are equivalent, via the
potential $\tilde{A}_{\phi}$, to the system

$\displaystyle\nabla\cdot\bigl{(}f^{-1}\nabla\tilde{A}_{\phi}+\rho^{-1}\omega\hat{\phi}\times\nabla
A_{t}\bigr{)}$ $\displaystyle=0\,,$ (1.28a)
$\displaystyle\nabla\cdot\bigl{(}-f^{-1}\nabla
A_{t}+\rho^{-1}\omega\hat{\phi}\times\nabla\tilde{A}_{\phi}\bigr{)}$
$\displaystyle=0\,.$ (1.28b)

The two potentials $A_{t}$, $\tilde{A}_{\phi}$ can be effectively packed in a
complex potential

$\Phi\coloneqq A_{t}+i\tilde{A}_{\phi}\,,$ (1.29)

so that Eqs. (1.28) are written as a single complex equation, namely

$\nabla\cdot\bigl{(}f^{-1}\nabla\Phi+i\rho^{-1}\omega\hat{\phi}\times\nabla\Phi\bigr{)}=0\,.$
(1.30)

Now we turn to the Einstein equations (1.17) and (1.20): firstly, we note that
Eq. (1.23), with $h=A_{t}\tilde{A}_{\phi}$, gives

$\nabla\cdot\bigl{(}\rho^{-1}\hat{\phi}\times
A_{t}\nabla\tilde{A}_{\phi}\bigr{)}=-\nabla\cdot\bigl{(}\rho^{-1}\hat{\phi}\times\tilde{A}_{\phi}\nabla
A_{t}\bigr{)}\,.$ (1.31)

With this result, we can write Eq. (1.20) as

$\nabla\cdot\bigl{[}\rho^{-2}f^{2}\nabla\omega+2\rho^{-1}\hat{\phi}\times\operatorname{Im}\bigl{(}\Phi^{*}\nabla\Phi\bigr{)}\bigr{]}=0\,.$
(1.32)

This can be explicitly verified by expanding the product
$\operatorname{Im}\bigl{(}\Phi^{*}\nabla\Phi\bigr{)}$.

As in the case of the Maxwell equations, we can introduce a twisted
gravitational potential $\chi$ from equation (1.32), such that

$\hat{\phi}\times\nabla\chi\coloneqq-\rho^{-1}f^{2}\nabla\omega-2\hat{\phi}\times\operatorname{Im}\bigl{(}\Phi^{*}\nabla\Phi\bigr{)}\,.$
(1.33)

Multiplying by $\hat{\phi}\times$ and applying $\nabla\cdot$ as before, we
obtain

$\nabla\cdot\bigl{[}f^{-2}\nabla\chi+2f^{-2}\operatorname{Im}\bigl{(}\Phi^{*}\nabla\Phi\bigr{)}\bigr{]}=0\,,$
(1.34)

that substitutes Eq. (1.20). Finally, we elaborate Eq. (1.17) to implement the
potential $\chi$, and we find

$f\nabla^{2}f=(\nabla
f)^{2}-\bigl{[}\nabla\chi+2\operatorname{Im}\bigl{(}\Phi^{*}\nabla\Phi\bigr{)}]^{2}+2f\nabla\Phi\cdot\nabla\Phi^{*}\,.$
(1.35)

Summarising, the Einstein equations (1.17), (1.20) are replaced by the
equations

$\displaystyle\nabla\cdot\bigl{[}f^{-2}\nabla\chi+2f^{-2}\operatorname{Im}\bigl{(}\Phi^{*}\nabla\Phi\bigr{)}\bigr{]}=0\,,$
(1.36a) $\displaystyle f\nabla^{2}f=(\nabla
f)^{2}-\bigl{[}\nabla\chi+2\operatorname{Im}\bigl{(}\Phi^{*}\nabla\Phi\bigr{)}]^{2}+2f\nabla\Phi\cdot\nabla\Phi^{*}\,.$
(1.36b)

We introduce a gravitational complex potential

$\mathcal{E}\coloneqq f-\big{|}\Phi\big{|}^{2}+i\chi\,,$ (1.37)

which, together with $\Phi$, allows us to write the systems (1.28) and (1.36)
as the two complex equations

$\displaystyle\bigl{(}\operatorname{Re}\mathcal{E}+\big{|}\Phi\big{|}^{2}\bigr{)}\nabla^{2}\mathcal{E}$
$\displaystyle=\nabla\mathcal{E}\cdot\bigl{(}\nabla\mathcal{E}+2\Phi^{*}\nabla\Phi\bigr{)}\,,$
(1.38a)
$\displaystyle\bigl{(}\operatorname{Re}\mathcal{E}+\big{|}\Phi\big{|}^{2}\bigr{)}\nabla^{2}\Phi$
$\displaystyle=\nabla\Phi\cdot\bigl{(}\nabla\mathcal{E}+2\Phi^{*}\nabla\Phi\bigr{)}\,.$
(1.38b)

Eqs. (1.38) are called _Ernst equations_ , and $\mathcal{E}$ and $\Phi$ are
known as the _Ernst complex potentials_.

We can also express the equations for $\gamma$ in term of the Ernst
potentials, and they are given by

$\displaystyle\begin{split}\partial_{\rho}\gamma&=\frac{\rho}{4(\operatorname{Re}\mathcal{E}+\Phi\Phi^{*})^{2}}\Bigl{[}\bigl{(}\partial_{\rho}\mathcal{E}+2\Phi^{*}\partial_{\rho}\Phi\bigr{)}\bigl{(}\partial_{\rho}\mathcal{E}^{*}+2\Phi\partial_{\rho}\Phi^{*}\bigr{)}\\\
&\quad-\bigl{(}\partial_{z}\mathcal{E}+2\Phi^{*}\partial_{z}\Phi\bigr{)}\big{(}\partial_{z}\mathcal{E}^{*}+2\Phi\partial_{z}\Phi^{*}\bigr{)}\Bigr{]}\\\
&\quad-\frac{\rho}{\operatorname{Re}\mathcal{E}+\Phi\Phi^{*}}\big{(}\partial_{\rho}\Phi\partial_{\rho}\Phi^{*}-\partial_{z}\Phi\partial_{z}\Phi^{*}\big{)}\,,\end{split}$
(1.39)
$\displaystyle\begin{split}\partial_{z}\gamma&=\frac{\rho}{4(\operatorname{Re}\mathcal{E}+\Phi\Phi^{*})^{2}}\Big{[}\big{(}\partial_{\rho}\mathcal{E}+2\Phi^{*}\partial_{\rho}\Phi\big{)}\big{(}\partial_{z}\mathcal{E}^{*}+2\Phi\partial_{z}\Phi^{*}\big{)}\\\
&\quad+\big{(}\partial_{z}\mathcal{E}+2\Phi^{*}\partial_{z}\Phi\big{)}\big{(}\partial_{\rho}\mathcal{E}^{*}+2\Phi\partial_{\rho}\Phi^{*}\big{)}\Big{]}\\\
&\quad-\frac{\rho}{\operatorname{Re}\mathcal{E}+\Phi\Phi^{*}}\big{(}\partial_{\rho}\Phi\partial_{z}\Phi^{*}-\partial_{z}\Phi\partial_{\rho}\Phi^{*}\big{)}\,.\end{split}$
(1.40)

As before, we notice that such equations can be solved by quadratures.

We have formally reduced the problem of solving the Einstein–Maxwell equations
for a stationary and axisymmetric spacetime, to the problem of solving the
Ernst equations (1.38). We notice that Eqs. (1.38) represent an effective
three-dimensional problem, and as such we can forget about the four-
dimensional origin of the problem. Furthermore, the writing (1.38) allows us
to study in an efficient way the symmetries of the Einstein–Maxwell equations,
and to make use of such symmetries to construct new solutions starting from a
seed. We will develop this topic in the following.

##### 1.1.3 Symmetries of the Ernst equations

One can notice that the Ernst equations (1.38) can be derived from an
effective three-dimensional action, given by

$S=\int
d^{3}x\,\frac{(\nabla\mathcal{E}+2\Phi^{*}\nabla\Phi)\cdot(\nabla\mathcal{E}^{*}+2\Phi\nabla\Phi^{*})-\nabla\Phi\cdot\nabla\Phi^{*}}{\operatorname{Re}\mathcal{E}+\big{|}\Phi\big{|}^{2}}\,.$
(1.41)

This is a quite useful result, because now we can find the symmetries of the
Ernst equations (1.38) just by studying the symmetries of the Ernst action
(1.41)555Actually, not all the symmetries of the equations of motion
correspond to the symmetries of the action. The equations of motion might
enjoy more symmetries than the action does. However, the symmetries of the
action are surely symmetries of the equations of motion, and in this case they
coincide.

A smart way to study the symmetries of the action (1.41) is to consider the
quadratic form associated to (1.41), i.e. the associated metric [119]. Let us
consider $\mathcal{E}$ and $\Phi$ as complex coordinates, and introduce the
real coordinates $(x,y,u,v)$ as

$\mathcal{E}=x+iy\,,\quad\Phi=u+iv\,.$ (1.42)

This way, the metric associated to the action (1.41) is

${ds}^{2}=\frac{1}{4(u^{2}+v^{2}+x)^{2}}\bigl{[}{dx}^{2}+{dy}^{2}-4x\bigl{(}{du}^{2}+{dv}^{2}\bigr{)}-4vdydu+4udydv+4vdxdv+4udxdu\bigr{]}\,.$
(1.43)

Such a representation is particularly useful, since the Killing vectors of the
metric (1.43) are equivalent to the infinitesimal generators of the symmetries
of the action (1.41). The Killing vectors are defined by the Killing equation
$\nabla_{(i}\xi_{j)}=0$, which can be solved for the four-dimensional metric
(1.43) to find

$\displaystyle\xi_{1}$
$\displaystyle=4xy\partial_{x}+2(y^{2}-x^{2})\partial_{y}+2(xv+yu)\partial_{z}+2(yv-
xu)\partial_{v}\,,$ (1.44a) $\displaystyle\xi_{2}$
$\displaystyle=2(xv+yu)\partial_{x}+2(yv-
xu)\partial_{y}+(4uv-y)\partial_{z}+(2v^{2}-2u^{2}+x)\partial_{v}\,,$ (1.44b)
$\displaystyle\xi_{3}$ $\displaystyle=2(xu-
yv)\partial_{x}+2(xv+yu)\partial_{y}+(z^{2}-v^{2}+x)\partial_{z}+(4uv+y)\partial_{v}\,,$
(1.44c) $\displaystyle\xi_{4}$
$\displaystyle=4x\partial_{x}+4y\partial_{y}+2z\partial_{z}+2v\partial_{v}\,,$
(1.44d) $\displaystyle\xi_{5}$ $\displaystyle=-v\partial_{z}+z\partial_{v}\,,$
(1.44e) $\displaystyle\xi_{6}$
$\displaystyle=-2v\partial_{x}+2u\partial_{y}+\partial_{v}\,,$ (1.44f)
$\displaystyle\xi_{7}$
$\displaystyle=2u\partial_{x}+2v\partial_{y}-\partial_{z}\,,$ (1.44g)
$\displaystyle\xi_{8}$ $\displaystyle=4\partial_{y}\,.$ (1.44h)

Thus, we find eight Killing vectors for the quadratic form (1.43), that
correspond to eight infinitesimal symmetries for the action (1.41).

We are actually interested in the finite transformations generated by the
Killing vectors (1.44): we need to integrate the flow generated by such
vectors in order to find them. Practically, the equations that define the flow
and that we have to integrate are

$\frac{\partial x^{i}}{\partial\epsilon}=\xi^{i}\bigl{(}x^{i}\bigr{)}\,,$
(1.45)

where $x^{i}$ identifies the coordinates, i.e. $x^{i}=(x,y,u,v)$, and
$\epsilon$ is the flow parameter of the finite transformation. More details
about the connection between the infinitesimal and the finite transformations
can be found in [120].

We can integrate the infinitesimal transformations (1.44), and find the finite
transformations [119]

$\displaystyle\mathcal{E}^{\prime}$
$\displaystyle=|\lambda|^{2}\mathcal{E}\,,\qquad\qquad\qquad\quad\;\;\Phi^{\prime}=\lambda\Phi\,,$
(1.46a) $\displaystyle\mathcal{E}^{\prime}$
$\displaystyle=\mathcal{E}+ib\,,\qquad\qquad\qquad\quad\,\Phi^{\prime}=\Phi\,,$
(1.46b) $\displaystyle\mathcal{E}^{\prime}$
$\displaystyle=\frac{\mathcal{E}}{1+ic\mathcal{E}}\,,\qquad\qquad\qquad\;\,\Phi^{\prime}=\frac{\Phi}{1+ic\mathcal{E}}\,,$
(1.46c) $\displaystyle\mathcal{E}^{\prime}$
$\displaystyle=\mathcal{E}-2\beta^{*}\Phi-|\beta|^{2}\,,\qquad\quad\\!\Phi^{\prime}=\Phi+\beta\,,$
(1.46d) $\displaystyle\mathcal{E}^{\prime}$
$\displaystyle=\frac{\mathcal{E}}{1-2\alpha^{*}\Phi-|\alpha|^{2}\mathcal{E}}\,,\qquad\Phi^{\prime}=\frac{\alpha\mathcal{E}+\Phi}{1-2\alpha^{*}\Phi-|\alpha|^{2}\mathcal{E}}\,,$
(1.46e)

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

One can prove [119, 121] that transformations (1.46a), (1.46b) and (1.46d) are
actually gauge transformations, i.e. diffeomorphisms which do not change the
nature of the solutions. They can be reabsorbed by a redefinition of the
coordinates, and as such they are not interesting, since they do not produce
any new solutions. The interesting transformations are given by (1.46c) and
(1.46e), that are called _Ehlers transformation_ [114] and _Harrison
transformation_ [115], respectively.

The transformations (1.46) form a group, that one can show to be SU(2,1) [122,
123]. Indeed, it is known that SU(2,1) represents the symmetry group of the
Einstein–Maxwell equations.

###### Discrete transformation

We have analysed the symmetry transformations for the ansatz given by the
metric (1.7) and the Maxwell potential (1.8), which together represent the
class of stationary and axisymmetric spacetimes.

Actually, they do not represent the unique choice of ansatz for such a class
of solutions. There is another (non-equivalent) ansatz that one can consider,
which is related to the first one by a discrete double-Wick rotation

$t\to i\psi\,,\quad\phi\to i\tau\,,$ (1.47)

that produces the following metric

${ds}^{2}=f(d\psi-\omega
d\tau)^{2}+f^{-1}\bigl{[}e^{2\gamma}\bigl{(}{d\rho}^{2}+{dz}^{2}\bigr{)}-\rho^{2}{d\tau}^{2}\bigr{]}\,,$
(1.48)

and Maxwell field

$A=iA_{\tau}d\tau+iA_{\psi}d\psi\,.$ (1.49)

The imaginary unit in (1.49) does not represent an issue, since it can be
easily reabsorbed into a redefinition of the electric (or magnetic) charge.

One can reproduce the Ernst computations that we showed previously, for the
metric (1.48) and the potential (1.49). Almost everything works as above,
except for the definition of the twisted potential, that now reads

$\hat{\phi}\times\nabla\tilde{A}_{\tau}\coloneqq\rho^{-1}f\bigl{(}\nabla
A_{\tau}+\omega\nabla A_{\psi}\bigr{)}\,,$ (1.50)

and the definition of the Ernst complex potential

$\Phi\coloneqq A_{\psi}+i\tilde{A}_{\tau}\,.$ (1.51)

The roles of the time and azimuthal coordinates are exchanged, as one might
expect.

One recovers again the Ernst transformations (1.46), where the Ehlers map
(1.46c) and the Harrison map (1.46e) are the only relevant transformations. In
this case, however, their action will produce different solutions with respect
to the previous case, being the seed solution non-equivalent to the ansatz
(1.7) and (1.8). This means that we have four non-trivial maps, two Ehlers and
two Harrison transformations.

Thus, we will be considering two different ansätze for the application of the
Ehlers and the Harrison transformations: the _electric_ ansatz

$\displaystyle{ds_{e}}^{2}$ $\displaystyle=-f(dt-\omega
d\phi)^{2}+f^{-1}\bigl{[}e^{2\gamma}\bigl{(}{d\rho}^{2}+{dz}^{2}\bigr{)}+\rho^{2}{d\phi}^{2}\bigr{]}\,,$
(1.52a) $\displaystyle A_{e}$ $\displaystyle=A_{t}dt+A_{\phi}d\phi\,,$ (1.52b)

and the _magnetic_ ansatz

$\displaystyle{ds_{m}}^{2}$ $\displaystyle=f(d\phi-\omega
dt)^{2}+f^{-1}\bigl{[}e^{2\gamma}\bigl{(}{d\rho}^{2}+{dz}^{2}\bigr{)}-\rho^{2}{dt}^{2}\bigr{]}\,,$
(1.53a) $\displaystyle A_{m}$ $\displaystyle=A_{t}dt+A_{\phi}d\phi\,.$ (1.53b)

The terminology “electric” and “magnetic” is not common in the literature: we
adopt it because of the way the Harrison transformation works with the
different seeds (1.52) and (1.53), when the transformation parameter is real.
In the first case, the Harrison map adds an electric charge, while in the
second one it adds an external magnetic field; then it is quite natural to use
the words “electric” and “magnetic”.

In the remaining part of the Section, we will analyse the effects of the non-
trivial transformations given by (1.46c) and (1.46e) for the two different
kinds of metric, the electric (1.52) and the magnetic (1.53) one.

##### 1.1.4 Ehlers transformation

We now analyse the action of the Harrison transformation (1.46e), that we
report here for convenience:

$\mathcal{E}^{\prime}=\frac{\mathcal{E}}{1+ic\mathcal{E}}\,,\qquad\Phi^{\prime}=\frac{\Phi}{1+ic\mathcal{E}}\,.$
(1.54)

As noted above, such a map will produce a different result if applied to the
electric ansatz (1.52) or the magnetic one (1.53). We recall that the
parameter $c$ is real.

The effect of the Ehlers map on the electric ansatz is to add the NUT
parameter to the spacetime under consideration, as shown for the first time in
[116]. On the other hand, the action of the Ehlers transformation on the
magnetic ansatz has never been investigated before, and it will be the subject
of Chapter 4.

###### Electric ansatz

We start by considering the electric metric (1.52), and deal with the explicit
example of the Schwarzschild spacetime

${ds}^{2}=-\biggl{(}1-\frac{2M}{r}\biggr{)}{dt}^{2}+\frac{{dr}^{2}}{1-2M/r}+r^{2}\bigl{(}{d\theta}^{2}+\sin^{2}\theta{d\phi}^{2}\bigr{)}\,.$
(1.55)

In this case, the Maxwell field is zero, so $\Phi=0$. Being $f=1-2M/r$, the
gravitational Ernst potential is given by

$\mathcal{E}=1-\frac{2M}{r}\,.$ (1.56)

We now apply the Ehlers transformation, and obtain the new Ernst potentials

$\mathcal{E}^{\prime}=\frac{r^{2}-2Mr}{r^{2}+c^{2}(r-2M)^{2}}-i\frac{c(r-2M)^{2}}{r^{2}+c^{2}(r-2M)^{2}}\,,\qquad\Phi^{\prime}=0\,.$
(1.57)

We see that the Ehlers map does not add an electromagnetic field to a seed
solution that does not possess it from the beginning. Therefore, it maps a
vacuum solution into a vacuum solution.

From the potential $\mathcal{E}$ we are able to read the new functions

$f^{\prime}=\frac{r^{2}-2Mr}{r^{2}+c^{2}(r-2M)^{2}}\,,\quad\chi^{\prime}=-\frac{c(r-2M)^{2}}{r^{2}+c^{2}(r-2M)^{2}}\,,$
(1.58)

and to integrate $\omega^{\prime}$ from the definition of the twist potential
$\chi^{\prime}$ (1.33):

$\omega^{\prime}=4cM\cos\theta\,.$ (1.59)

Regarding the function $\gamma$, it can be easily verified that the Ehlers
transformation does not modify it, i.e. $\gamma^{\prime}=\gamma$.

Collecting all the information, we can write down the resulting metric

$\begin{split}ds^{\prime
2}&=-\frac{r^{2}-2Mr}{r^{2}+c^{2}(r-2M)^{2}}\bigl{(}dt-4Mc\cos\theta
d\phi\bigr{)}^{2}+\frac{r^{2}+c^{2}(r-2M)^{2}}{r^{2}-2Mr}{dr}^{2}\\\
&\quad+\bigl{(}r^{2}+c^{2}(r-2M)^{2}\bigr{)}\bigl{(}{d\theta}^{2}+\sin^{2}\theta{d\phi}^{2}\bigr{)}\,.\end{split}$
(1.60)

Obviously, for $c=0$ we recover the Schwarzschild metric, i.e. the seed
solution.

We can perform a change of coordinates and a parameter redefinition, in order
to identify the new metric. Let us consider the change of coordinates 666The
transformation of the radial coordinate is suggested by the $S^{2}$ element in
the metric.

$R=r\sqrt{1+c^{2}}-\frac{2Mc^{2}}{\sqrt{1+c^{2}}}\,,\quad
T=\frac{t}{\sqrt{1+c^{2}}}\,,$ (1.61)

and the parameter redefinitions

$M=-\frac{n}{2c}\sqrt{1+c^{2}}\,,\quad c=\frac{m-\sqrt{m^{2}+n^{2}}}{n}\,,$
(1.62)

that give the following metric:

$\begin{split}{ds^{\prime}}^{2}&=-\frac{R^{2}-2mR-n^{2}}{R^{2}+n^{2}}\bigl{(}dT-2n\cos\theta
d\phi\bigr{)}^{2}+\frac{R^{2}+n^{2}}{R^{2}-2mR-n^{2}}{dR}^{2}\\\
&\quad+\bigl{(}R^{2}+n^{2}\bigr{)}\bigl{(}{d\theta}^{2}+\sin^{2}\theta{d\phi}^{2}\bigr{)}\,.\end{split}$
(1.63)

This solution is the well known Taub–NUT metric [124, 125], where $m$ is the
mass and $n$ is the so-called NUT charge. Thus, we conclude that the Ehlers
transformation adds the NUT parameter to a given seed solution. One can
explicitly check, e.g., that the application of the Ehlers map to the
Reissner–Nordström black hole produces the charged Taub–NUT metric.

The Taub–NUT metric has been extensively studied in the literature, starting
with the famous paper by Misner [126], and recently it has been subjected to a
renewed interest [127], which led to the study of its thermodynamical
properties by many authors.

###### Magnetic ansatz

The Ehlers transformation applied to the magnetic ansatz (1.53) produces a new
solution that will be investigated in Chapter 4.

##### 1.1.5 Harrison transformation

The Harrison transformation is given by

$\mathcal{E}^{\prime}=\frac{\mathcal{E}}{1-2\alpha^{*}\Phi-|\alpha|^{2}\mathcal{E}}\,,\qquad\Phi^{\prime}=\frac{\alpha\mathcal{E}+\Phi}{1-2\alpha^{*}\Phi-|\alpha|^{2}\mathcal{E}}\,,$
(1.64)

where $\alpha$ is a complex parameter.

The effect of the Harrison map on the electric ansatz is to add a dyon, i.e.
an electric and a magnetic charge. On the other hand, the action of the
Harrison transformation on the magnetic ansatz embeds the given spacetime in
the Melvin universe, which is a spacetime filled with “uniform”
electromagnetic field.

###### Electric ansatz

We consider again the electric ansatz (1.52) with the Schwarzschild metric

${ds}^{2}=-\biggl{(}1-\frac{2M}{r}\biggr{)}{dt}^{2}+\frac{{dr}^{2}}{1-2M/r}+r^{2}\bigl{(}{d\theta}^{2}+\sin^{2}\theta{d\phi}^{2}\bigr{)}\,.$
(1.65)

We observe, just by looking at the structure of the Harrison map, that it
always produces a non-trivial electromagnetic field, even if starting with a
vacuum spacetime.

Here, it is again

$\mathcal{E}=1-\frac{2M}{r}\,,$ (1.66)

but now

$\Phi^{\prime}=\alpha\frac{r-2M}{r-|\alpha|^{2}(r-2M)}\,,$ (1.67)

and

$\mathcal{E}^{\prime}=\frac{r-2M}{r-|\alpha|^{2}(r-2M)}\,,$ (1.68)

from which we read (notice that $\mathcal{E}$ is real)

$f^{\prime}=\frac{r(r-2M)}{\bigl{[}2M|\alpha|^{2}+r(1-|\alpha|^{2})\bigr{]}^{2}}\,,\quad\chi^{\prime}=0\,.$
(1.69)

The remaining function $\gamma$ is again left unchanged,
$\gamma^{\prime}=\gamma$.

From the Ernst potential $\Phi^{\prime}$ and by integrating the definition of
the electromagnetic twist potential (1.24), we find the components of the new
Maxwell potential

$A^{\prime}=\alpha_{R}\frac{r-2M}{r-|\alpha|^{2}(r-2M)}dt+2\alpha_{I}M\cos\theta
d\phi\,,$ (1.70)

where $\alpha\coloneqq\alpha_{R}+i\alpha_{I}$. On the other hand, the metric
is

$\begin{split}{ds^{\prime}}^{2}&=-\frac{r(r-2M)}{\bigl{[}2M|\alpha|^{2}+r(1-|\alpha|^{2})\bigr{]}^{2}}{dt}^{2}+\frac{\bigl{[}2M|\alpha|^{2}+r(1-|\alpha|^{2})\bigr{]}^{2}}{r(r-2M)}{dr}^{2}\\\
&\quad+\bigl{[}2M|\alpha|^{2}+r(1-|\alpha|^{2})\bigr{]}^{2}\bigl{(}{d\theta}^{2}+\sin^{2}\theta{d\phi}^{2}\bigr{)}\,.\end{split}$
(1.71)

Being $\chi^{\prime}=0$, we found that $\omega^{\prime}=0$.

We perform the change of coordinates

$R=r(1-|\alpha|^{2})+2M|\alpha|^{2}\,,\quad T=\frac{t}{1-|\alpha|^{2}}\,,$
(1.72)

the reparametrisation

$p=2\alpha_{I}M\,,\quad q=-\frac{2\alpha_{R}M}{1-|\alpha|^{2}}\,,\quad
m=M(1+|\alpha|^{2})\,,$ (1.73)

and define the dyon $e^{2}\coloneqq q^{2}+p^{2}$, to end up with

$\displaystyle{ds^{\prime}}^{2}$
$\displaystyle=-\biggl{(}1-\frac{2m}{R}+\frac{e^{2}}{R^{2}}\biggr{)}{dt}^{2}+\biggl{(}1-\frac{2m}{R}+\frac{e^{2}}{R^{2}}\biggr{)}^{-1}{dR}^{2}+R^{2}\bigl{(}{d\theta}^{2}+\sin^{2}\theta{d\phi}^{2}\bigr{)}\,,$
(1.74) $\displaystyle A^{\prime}$ $\displaystyle=\frac{q}{R}dt+p\cos\theta
d\phi\,.$ (1.75)

We clearly recognise the dyonic Reissner–Nordström black hole [128, 129],
where $m$ is the mass, $q$ is the electric charge and $p$ is the magnetic
charge. Thus, the Harrison transformation adds a dyonic charge to a seed
solution of the form (1.7).

###### Magnetic ansatz

We now turn to the consideration of the magnetic ansatz (1.53), again for the
example of the Schwarzschild metric. The difference from the previous case
stands in the function $f=r^{2}\sin^{2}\theta$, that gives

$\mathcal{E}=r^{2}\sin^{2}\theta\,.$ (1.76)

A non-null electromagnetic field is generated via the Harrison map

$\Phi^{\prime}=\frac{\alpha
r^{2}\sin^{2}\theta}{1-|\alpha|^{2}r^{2}\sin^{2}\theta}\,,$ (1.77)

and

$\mathcal{E}^{\prime}=\frac{r^{2}\sin^{2}\theta}{1-|\alpha|^{2}r^{2}\sin^{2}\theta}\,.$
(1.78)

We can read the Maxwell field from
$\Phi^{\prime}=A^{\prime}_{\phi}+i\tilde{A}^{\prime}_{t}$, and integrating the
Maxwell twist potential (1.24)

$A^{\prime}=2\alpha_{I}(r-2M)\cos\theta
dt+\alpha_{R}\frac{r^{2}\sin^{2}\theta}{1-|\alpha|^{2}r^{2}\sin^{2}\theta}d\phi\,,$
(1.79)

where again $\alpha\coloneqq\alpha_{R}+i\alpha_{I}$.

We choose $\alpha_{R}=-B/2$ and $\alpha_{I}=E/2$, so that the metric and the
Maxwell field become

$\displaystyle{ds^{\prime}}^{2}$
$\displaystyle=\Lambda^{2}\biggl{[}-\biggl{(}1-\frac{2M}{r}\biggr{)}{dt}^{2}+\frac{{dr}^{2}}{1-2M/r}+r^{2}{d\theta}^{2}\biggr{]}+\Lambda^{-2}r^{2}\sin^{2}\theta{d\phi}^{2}\,,$
(1.80) $\displaystyle A^{\prime}$ $\displaystyle=E(r-2M)\cos\theta
dt-\frac{B}{2}\Lambda^{-1}r^{2}\sin^{2}\theta d\phi\,,$ (1.81)

where

$\Lambda\coloneqq 1-\frac{E^{2}+B^{2}}{4}r^{2}\sin^{2}\theta\,.$ (1.82)

This solution is known as the Schwarzschild–Melvin spacetime [117], and it
represents a static black hole embedded in an electromagnetic universe, i.e. a
universe filled of a “uniform” electric field $E$ and magnetic field $B$. The
most known version of such a solution is the purely magnetic one, with $E=0$.

The background is recovered by putting $M=0$

$\displaystyle{ds^{\prime}}^{2}$
$\displaystyle=\Lambda^{2}\bigl{(}-{dt}^{2}+{dr}^{2}+r^{2}{d\theta}^{2}\bigr{)}+\Lambda^{-2}r^{2}\sin^{2}\theta{d\phi}^{2}\,,$
(1.83) $\displaystyle A^{\prime}$ $\displaystyle=Er\cos\theta
dt-\frac{B}{2}\Lambda^{-1}r^{2}\sin^{2}\theta d\phi\,,$ (1.84)

which is the electromagnetic generalisation of the so-called Melvin universe
[130]. Hence, the Harrison transformation, applied to a magnetic seed (1.53),
adds an external electromagnetic field to the spacetime.

We point out that the Reissner–Nordström and the Melvin spacetimes are related
by an analytic continuation, similarly to what happens to the electric and
magnetic seeds: that was proven by Gibbons and Wiltshire in [131], however not
in the context of the Harrison transformation. Such an analytic continuation
was later implemented in [132] to add the cosmological constant to the Melvin
spacetime.

#### 1.2 Inverse scattering method in vacuum

The inverse scattering construction relies on the identification of a linear
eigenvalue equation, whose integrability condition corresponds to the non-
linear equations one aims to solve. We will describe the procedure developed
by Belinski and Zakharov for the Einstein equations in vacuum, so we will work
with pure General Relativity with action

$S=\int d^{4}x\sqrt{-g}\,R\,,$ (1.85)

and Einstein equations

$R_{\mu\nu}=0\,.$ (1.86)

This section is based on the original papers by Belinski and Zakharov [52, 53]
and on the book [133].

##### 1.2.1 Integrable ansatz

We work with the following metric ansatz in Weyl coordinates
$(t,\phi,\rho,z)$:

${ds}^{2}=f(\rho,z)\bigl{(}{d\rho}^{2}+{dz}^{2}\bigr{)}+g_{ab}(\rho,z)dx^{a}dx^{b}\,.$
(1.87)

Such a metric is the most general stationary and axisymmetric spacetime: it is
written in a form suitable for our purposes, and it contains the
Lewis–Weyl–Papapetrou metric (1.7) as a subcase [118]. Here, $f$ is a
function777$f$ should not be confused with the function appearing in (1.7)!
and $g$ is a $2\times 2$ matrix. The Latin indices take the values $0,1$,
which correspond to $t$ and $\phi$.

The freedom in the choice of the coordinates $\rho$ and $z$ can be used,
without loss of generality, to impose

$\det g=-\rho^{2}\,.$ (1.88)

This is a fundamental property, that must be preserved by the generation
technique, as we will see in the following.

It is convenient to rewrite the vacuum Einstein equations in matrix form, in
order to apply the inverse scattering formalism. The vacuum equations
naturally split into two groups, one for the matrix $g$ and the other for the
function $f$. We start by exploiting the first group.

One can easily show that the Einstein equations

$R^{t}_{\;\;t}=0\,,\quad R^{\phi}_{\;\;t}=0\,,\quad
R^{t}_{\;\;\phi}=0\,,\quad$ (1.89)

correspond, respectively, to the components $(0,0)$, $(0,1)$ and $(1,1)$ of
the second order matrix equation

$\partial_{\rho}\bigl{(}\rho\partial_{\rho}g\,g^{-1}\bigr{)}+\partial_{z}\bigl{(}\rho\partial_{z}g\,g^{-1}\bigr{)}=0\,.$
(1.90)

On the other hand, the Einstein equations

$R_{\rho\rho}-R_{zz}=0\,,\quad R_{\rho z}=0\,,$ (1.91)

determine the first order equations for $f$

$\displaystyle\partial_{\rho}\log f$
$\displaystyle=-\frac{1}{\rho}+\frac{1}{4\rho}\mathrm{Tr}\bigl{(}U^{2}-V^{2}\bigr{)}\,,$
(1.92a) $\displaystyle\partial_{z}\log f$
$\displaystyle=\frac{1}{2\rho}\mathrm{Tr}\bigl{(}UV\bigr{)}\,,$ (1.92b)

where we have defined the $2\times 2$ matrices

$U\coloneqq\rho\bigl{(}\partial_{\rho}g\bigr{)}g^{-1}\,,\quad
V\coloneqq\rho\bigl{(}\partial_{z}g\bigr{)}g^{-1}\,.$ (1.93)

With this first step, we managed to rewrite the Einstein equations in matrix
form. Such a form is particularly useful to recognise an integrability
condition for the equations. We explicitly notice that, as it happened in Sec.
1.1, the equations for the $(\rho,z)$ part of the metric, i.e. for the
function $f$ (1.92), completely decouple from the other Einstein equations.
This implies that we can forget about $f$ and work only on $g$: once we have
found $g$, we can simply integrate Eqs. (1.92) by quadratures.

##### 1.2.2 Integration scheme

We want to translate (1.90) into an equivalent system consisting of the
relations (1.93) and two first order matrix equations for the matrices $U$ and
$V$. The first equation of such a system is obviously obtained by rewriting
(1.93) in terms of $U$ and $V$:

$\partial_{\rho}U+\partial_{z}V=0\,.$ (1.94)

The second equation is obtained as the integrability condition of (1.93) with
respect to $g$: with this, we mean a differential equation for $U$ and $V$
which is identically satisfied when it is written in terms of $g$. Thus we
find

$\partial_{z}U-\partial_{\rho}V+\frac{1}{\rho}[U,V]+\frac{1}{\rho}V=0\,,$
(1.95)

where the square brackets denote the commutator. One can indeed verify that
(1.95) is satisfied for any $g$, given (1.93).

The key step of the inverse scattering procedure consists in representing the
first order Eqs. (1.94) and (1.95) as the compatibility conditions of an over-
determined system of matrix equations related to an eigenvalue problem for
some linear differential operator. Such a system will depend on a complex
spectral parameter $\lambda$, and the solutions of the original problem for
$g$, $U$ and $V$ will be determined by the analytic structure of the
eigenfunction in the complex $\lambda$-plane.

We begin by introducing the differential operators

$D_{1}\coloneqq\partial_{z}-\frac{2\lambda^{2}}{\lambda^{2}+\rho^{2}}\partial_{\lambda}\,,\quad
D_{2}\coloneqq\partial_{\rho}+\frac{2\lambda\rho}{\lambda^{2}+\rho^{2}}\partial_{\lambda}\,,$
(1.96)

where $\lambda$ is the complex spectral parameter, independent of $\rho$ and
$z$. One can easily verify that the commutator of the operators $D_{1}$ and
$D_{2}$ vanishes,

$[D_{1},D_{2}]=0\,.$ (1.97)

We now introduce a complex matrix $\psi(\lambda,\rho,z)$, called the
generating matrix, and consider the system of equations

$\displaystyle D_{1}\psi=\frac{\rho V-\lambda U}{\lambda^{2}+\rho^{2}}\psi\,,$
(1.98a) $\displaystyle D_{2}\psi=\frac{\rho U+\lambda
V}{\lambda^{2}+\rho^{2}}\psi\,,$ (1.98b)

where $U$ and $V$ are real and do not depend on $\lambda$.

The fundamental property of the system (1.98) is that its compatibility
condition coincides exactly with Eqs. (1.94) and (1.95). This can be easily
verified by applying $D_{2}$ to (1.98a) and $D_{1}$ to (1.98b), and then by
subtracting the results: because of the commutativity of $D_{1}$ and $D_{2}$,
we get zero on the left hand side, while the right hand side is a rational
function of $\lambda$. Requiring that all the coefficients of the various
powers of $\lambda$ vanish, we get exactly Eqs. (1.94) and (1.95).

We notice that the system (1.98) gives, when $\lambda=0$,

$U=\rho\bigl{(}\partial_{\rho}\psi\bigr{)}\psi^{-1}\,,\quad
V=\rho\bigl{(}\partial_{z}\psi\bigr{)}\psi^{-1}\,,$ (1.99)

which is nothing but Eq. (1.93) with $g=\psi$. From this, we get the
fundamental property

$g(\rho,z)=\psi(0,\rho,z)\,,$ (1.100)

which implies that a solution of the eigensystem (1.98) not only guarantees
that the equations satisfied by $U$ and $V$ are true, but also gives a
solution $g$ of (1.93).

The integration procedure assumes that we know a “seed” solution $g_{0}$,
$f_{0}$: we denote $U_{0}$, $V_{0}$ and $\psi_{0}$ the associated matrices,
solutions of (1.93), (1.94) and (1.98). We look for a solution of the form

$\psi=\chi\psi_{0}\,,$ (1.101)

where $\chi(\lambda,\rho,z)$ is called dressing matrix. By inserting this
ansatz into the eigensystem (1.98), we find the equations

$\displaystyle D_{1}\chi$ $\displaystyle=\frac{\rho V-\lambda
U}{\lambda^{2}+\rho^{2}}\chi-\chi\frac{\rho V_{0}-\lambda
U_{0}}{\lambda^{2}+\rho^{2}}\,,$ (1.102a) $\displaystyle D_{2}\chi$
$\displaystyle=\frac{\rho U+\lambda
V}{\lambda^{2}+\rho^{2}}\chi-\chi\frac{\rho U_{0}+\lambda
V_{0}}{\lambda^{2}+\rho^{2}}\,,$ (1.102b)

These equations are not enough to guarantee that $g$ is real and symmetric: we
have to impose some supplementary conditions, which read

$\chi^{*}(\lambda^{*})=\chi(\lambda)\,,\quad\psi^{*}(\lambda^{*})=\psi(\lambda)\,,$
(1.103)

for the reality, and

$g=\chi(\lambda)g_{0}{\chi}^{T}(-\rho^{2}/\lambda)\,,$ (1.104)

for the symmetry. The last condition is not trivial, and follows from an
invariance property of the system (1.102). Furthermore, we require

$\chi(\infty)=\mathbf{1}\,,$ (1.105)

where $\mathbf{1}$ is the unit matrix, that in turn implies

$g=\chi(0)g_{0}\,.$ (1.106)

Thus, the problem consists of solving the system (1.102) and determining the
dressing matrix $\chi$ that fulfills the supplementary conditions (1.103),
(1.105).

It is worth emphasising that the new solution $g$ must still satisfy $\det
g=-\rho^{2}$. It follows another condition on $\chi$:

$\det\chi(0)=1\,.$ (1.107)

However, the best strategy is to not take into account this problem during the
procedure for the construction of the solution, and simply renormalise the
final result in order to obtain the correct functions. We will call such
correct functions the _physical_ functions.

##### 1.2.3 General $n$-soliton solution

The solution for the matrix $g$ that corresponds to the presence of pole
singularities in the dressing matrix $\chi(\lambda,\rho,z)$, in the complex
plane of the spectral parameters $\lambda$, is called the _soliton solution_.
We consider the case in which the matrix $\chi$ has $n$ simple poles888The
case of higher-order poles can be faced by the so-called _pole fusion_., and
is thus represented in the form

$\chi=\mathbf{1}+\sum_{k=1}^{n}\frac{R_{k}}{\lambda-\mu_{k}}\,,$ (1.108)

where the matrices $R_{k}$ and the functions $\mu_{k}$ depend only on $\rho$
and $z$. $\mu_{k}$ are the pole trajectories, i.e. the positions of the poles
as functions of $(\rho$, $z)$.

Now we can substitute the expansion (1.108) into the eigensystem (1.102), and
impose the constraints (1.103) and (1.105). These equations completely
determine the matrices $R_{k}$ and the pole trajectories $\mu_{k}$. In
particular, the requirement that there are no poles of order two at the points
$\lambda=\mu_{k}$ in (1.102), gives the differential equations

$\partial_{z}\mu_{k}=-\frac{2\mu_{k}^{2}}{\mu_{k}^{2}+\rho^{2}}\,,\quad\partial_{\rho}\mu_{k}=\frac{2\rho\mu_{k}}{\mu_{k}^{2}+\rho^{2}}\,.$
(1.109)

The solutions of such differential equations are the roots of the quadratic
algebraic equation

$\mu_{k}^{2}+2(z-w_{k})\mu_{k}-\rho^{2}=0\,,$ (1.110)

which gives the explicit expressions

$\displaystyle\mu_{k}$
$\displaystyle=\sqrt{\rho^{2}+(z-w_{k})^{2}}-(z-w_{k})\,,$ (1.111a)
$\displaystyle\bar{\mu}_{k}$
$\displaystyle=-\sqrt{\rho^{2}+(z-w_{k})^{2}}-(z-w_{k})\,,$ (1.111b)

where $w_{k}$ are arbitrary complex constants that are called _poles_. The
function (1.111a) is called a _soliton_ , while the function (1.111b) is
called an _anti-soliton_. The final solution is independent on the use of
solitons or anti-solitons: for definiteness, we will always make use of the
solitons.

One can prove, via the Eqs. (1.102), that the matrices $R_{k}$ are degenerate,
and as such they can be written in the form

$(R_{k})_{ab}=n_{a}^{(k)}m_{b}^{(k)}\,,$ (1.112)

where $n_{a}^{(k)}$ and $m_{a}^{(k)}$ are two-components vectors. The vectors
$m_{a}^{(k)}$ are found from the reality conditions (1.103) by requiring that
the equations are satisfied at the poles $\lambda=\mu_{k}$: then one finds

$m_{a}^{(k)}=m_{0\,b}^{(k)}\bigl{[}\psi_{0}^{-1}(\mu_{k},\rho,z)\bigr{]}_{ba}\,,$
(1.113)

where $m_{0\,b}^{(k)}$ are arbitrary constants, and the summation over the
repeated Latin indices is understood. The vectors $m_{a}^{(k)}$ are usually
called _Belinski–Zakharov vectors_ (BZ vectors).

The vectors $n_{a}^{(k)}$ are determined from the condition at infinity
(1.105), which gives the $n$-th order algebraic system

$\sum_{l=1}^{n}\Gamma_{kl}n_{a}^{(k)}=\mu_{k}^{-1}m_{c}^{(k)}(g_{0})_{ca}\,,$
(1.114)

for $k,l=1,2,\dotsc,n$. The symmetric matrix $\Gamma_{kl}$ is given by

$\Gamma_{kl}=\frac{m_{c}^{(k)}(g_{0})_{cb}m_{b}^{(l)}}{\rho^{2}+\mu_{k}\mu_{l}}\,,$
(1.115)

thus the vectors $n_{a}^{(k)}$ are

$n_{a}^{(k)}=\sum_{l=1}^{n}(\Gamma^{-1})_{lk}\mu_{l}^{-1}L_{a}^{(k)}\,,$
(1.116)

where

$L_{a}^{(k)}=m_{c}^{(k)}(g_{0})_{ca}\,.$ (1.117)

We are now ready to compute the matrix $g$: recalling that

$g=\psi(0)=\chi(0)\psi_{0}(0)=\chi(0)g_{0}=\Biggl{(}\mathbf{1}-\sum_{k=1}^{n}R_{k}\mu_{k}^{-1}\Biggr{)}g_{0}\,,$
(1.118)

and using what we have found up to now, we obtain

$g_{ab}=(g_{0})_{ab}-\sum_{k,l=1}^{n}(\Gamma^{-1})_{kl}\frac{L_{a}^{(k)}L_{b}^{(l)}}{\mu_{k}\mu_{l}}\,.$
(1.119)

We must guarantee that the condition on the determinant is satisfied by the
physical metric $g^{(ph)}$. The computation of the determinant gives

$\det g=(-1)^{n}\rho^{2n}\Biggl{(}\prod_{k=1}^{n}\mu_{k}^{-2}\Biggr{)}\det
g_{0}\,,$ (1.120)

from which follows that the number of the solitons $n$ must be even, being
$\det g_{0}=-\rho^{2}$. Therefore, the stationary and axisymmetric solutions
appear as even-soliton states. Eq. (1.90) must be satisfied by the physical
matrix $g^{(ph)}$ as well, and one can verify that

$g^{(ph)}=\pm\rho\sqrt{-\det g}\,g\,,$ (1.121)

does satisfy (1.90) and $\det g^{(ph)}=-\rho^{2}$. Putting together (1.120)
and (1.121), we finally get the physical matrix

$g^{(ph)}=\pm\rho^{-n}\Biggl{(}\prod_{k=1}^{n}\mu_{k}^{-2}\Biggr{)}g\,,\quad\det
g^{(ph)}=-\rho^{2}\,.$ (1.122)

The computation of the function $f$ is quite involved: one has to insert the
expression for $g^{(ph)}$ (1.122) into Eqs. (1.92), and in the end one gets

$f^{(ph)}=16C_{f}f_{0}\rho^{-n^{2}/2}\Biggl{(}\prod_{k=1}^{n}\mu_{k}^{-2}\Biggr{)}^{n+1}\Biggl{[}\prod_{k>l=1}^{n}(\mu_{k}-\mu_{l})^{-2}\Biggr{]}\det\Gamma_{kl}\,,$
(1.123)

where $C_{f}$ is an arbitrary gauge parameter and the factor 16 is put for
convenience.

Now we have the complete recipe for constructing soliton solutions from a seed
metric: once we have chosen a seed $g_{0}$, $f_{0}$, we need to solve the
linear eigensystem (1.98) to find $\psi_{0}$, and then we just have to perform
algebraic computations (by adding $n$ solitons to the seed) in order to find
the new solution $g^{(ph)}$, $f^{(ph)}$ from (1.122), (1.123). This is a very
powerful feature of the inverse scattering method: once we have solved for
$\psi_{0}$, the new metric is found by performing _algebraic_ computations.
This, of course, enormously reduces the difficulty of solving the equations of
motions, since the Einstein equations are non-linear while the eigensystem
(1.98) is linear.

We will apply the formalism that we have presented to construct the Kerr–NUT
spacetime starting with Minkowski spacetime and, as a further example, the
rotating C-metric starting with Rindler spacetime.

##### 1.2.4 Kerr–NUT spacetime

Let us consider Minkowski spacetime, in cylindrical coordinates, as a seed:

${ds}^{2}=-{dt}^{2}+\rho^{2}{d\phi}^{2}+{d\rho}^{2}+{dz}^{2}\,.$ (1.124)

We clearly see that $g_{0}=\mathrm{diag}\bigl{(}-1,\rho^{2}\bigr{)}$ and
$f_{0}=1$, and thus $\det g_{0}=-\rho^{2}$. By Eq. (1.93) one finds that
$U_{0}=\mathrm{diag}(0,2)$, while $V_{0}$ is zero. From the eigensystem (1.98)
we get the solution

$\psi_{0}=\begin{pmatrix}-1&0\\\ 0&\rho^{2}-2\lambda z-\lambda^{2}\\\
\end{pmatrix}\,,$ (1.125)

which satisfies the requirement $\psi_{0}(\lambda=0)=g_{0}$. By using (1.113),
we find the BZ vectors

$m^{(k)}=\Biggl{(}C_{0}^{(k)},\frac{C_{1}^{(k)}}{\mu_{k}}\Biggr{)}\,,$ (1.126)

where $C_{0}^{(k)}$ and $C_{1}^{(k)}$ are arbitrary constants. From (1.115) we
find

$\Gamma_{kl}=\frac{-C_{0}^{(k)}C_{0}^{(l)}+C_{1}^{(k)}C_{1}^{(l)}\mu_{k}^{-1}\mu_{l}^{-1}\rho^{2}}{\rho^{2}+\mu_{k}\mu_{l}}\,.$
(1.127)

We can use these quantities and Eqs. (1.122), (1.123) to construct a generic
$n$-soliton solution on the flat space background.

We explicitly compute the two-soliton solution on the flat background, which
corresponds to the Kerr–NUT spacetime. This means that we have two poles
$\lambda=\mu_{1}$ and $\lambda=\mu_{2}$. Firstly, we represent the constants
$w_{1}$ and $w_{2}$, that appear in the solitons, as

$w_{1}=\tilde{z}_{1}-\sigma\,,\quad w_{2}=\tilde{z}_{1}+\sigma\,,$ (1.128)

where $\tilde{z}_{1}$ and $\sigma$ are new arbitrary constants.
$\tilde{z}_{1}$ is real, and is interpreted as the position of the black hole
on the $z$-axis. We now introduce the spherical-like coordinates

$\rho=\sqrt{(r-m)^{2}-\sigma^{2}}\sin\theta\,,\quad
z=\tilde{z}_{1}+(r-m)\cos\theta\,,$ (1.129)

where $m$ is another (real) arbitrary constant. In these coordinates, the
solitons take the form

$\mu_{1}=(r-m-\sigma)(1-\cos\theta)\,,\quad\mu_{2}=(r-m+\sigma)(1-\cos\theta)\,.$
(1.130)

We impose, without loss of generality, the following conditions on the
constants $C_{0}^{(k)}$, $C_{1}^{(k)}$:

$\displaystyle C_{1}^{(1)}C_{0}^{(2)}-C_{0}^{(1)}C_{1}^{(2)}$
$\displaystyle=\sigma\,,\quad
C_{1}^{(1)}C_{0}^{(2)}+C_{0}^{(1)}C_{1}^{(2)}=-m\,,$ (1.131a) $\displaystyle
C_{0}^{(1)}C_{0}^{(2)}-C_{1}^{(1)}C_{1}^{(2)}$ $\displaystyle=n\,,\quad
C_{0}^{(1)}C_{0}^{(2)}+C_{1}^{(1)}C_{1}^{(2)}=a\,.$ (1.131b)

The first condition of (1.131a) take advantage of the non-physical
arbitrariness of the normalisation $C_{a}^{(k)}\to\zeta^{(k)}C_{a}^{(k)}$.
After such a transformation the metric $g$ does not change, hence the metric
does not depend on $\zeta^{(k)}$. The first relation of (1.131a) partially
fixes this arbitrariness. The second relation of (1.131a), on the other hand,
is just the definition of $m$. The two conditions (1.131b) define the new
constants $a$ and $n$. Furthermore, it follows from (1.131) that

$\sigma^{2}=m^{2}-a^{2}+n^{2}\,,$ (1.132)

which explicitly proves that we are introducing only three new parameters into
the physical metric.

By using all the expressions that we have collected up to now, we can compute
the new metric with the formulae (1.122) and (1.123). The line element
$d\rho^{2}+dz^{2}$ is transformed, accordingly to the new spherical
coordinates, as

${d\rho}^{2}+{dz}^{2}=\bigl{[}(r-m)^{2}-\sigma^{2}\cos^{2}\theta\bigr{]}\biggl{[}\frac{{dr}^{2}}{(r-m)^{2}-\sigma^{2}}+{d\theta}^{2}\biggr{]}\,.$
(1.133)

The new metric is finally given, after the rotation $t\to t+2a\phi$, by

$\begin{split}{ds}^{2}&=-\frac{\Delta-a^{2}\sin^{2}\theta}{\Sigma}{dt}^{2}-\frac{\Delta(a\sin^{2}\theta+2n\cos\theta)^{2}-\sin^{2}\theta(r^{2}+a^{2}+n^{2})^{2}}{\Sigma}{d\phi}^{2}\\\
&\quad+\frac{4\Delta n\cos\theta-4a\sin^{2}\theta(mr+n^{2})}{\Sigma}dtd\phi-
C_{f}\biggl{(}\frac{\Sigma}{\Delta}{dr}^{2}+\Sigma{d\theta}^{2}\biggr{)}\,,\end{split}$
(1.134)

where

$\Sigma=r^{2}+(n-a\cos\theta)^{2}\,,\quad\Delta=r^{2}-2mr+a^{2}-n^{2}\,.$
(1.135)

One recognises the Kerr–NUT metric [134] in Boyer–Lindquist coordinates, when
$C_{f}=-1$. We notice that the constant $\tilde{z}_{1}$ disappeared in the
final solution (1.134): this is due to the fact that the solution is invariant
under translations along the $z$-axis. The solution with horizons corresponds
to the case in which $\sigma$ is real (i.e. $m^{2}-a^{2}+n^{2}>0$), so that
$w_{1}$, $w_{2}$ are real and the pole trajectories $\mu_{1}$, $\mu_{2}$ are
real as well along $\sigma$. If $\sigma$ is pure imaginary (i.e.
$m^{2}-a^{2}+n^{2}<0$), the constants $w_{1}$, $w_{2}$ and the pole
trajectories $\mu_{1}$, $\mu_{2}$ are complex and conjugate to each other, and
we find a solution without horizons.

The Schwarzschild metric corresponds to the choice $a=n=0$, evidently. One can
directly obtain the Schwarzschild solution from the inverse scattering
procedure by choosing

$C_{0}^{(1)}=C_{1}^{(2)}=0\,,\quad C_{0}^{(2)}=C_{1}^{(1)}=1\,.$ (1.136)

Actually, such a choice for the constants always guarantees a diagonal
physical metric for any seed. The construction of the Kerr–NUT metric shows
that a rotating black hole corresponds to a gravitational two-soliton
solution.

##### 1.2.5 Rotating C-metric

We start with Rindler spacetime in cylindrical coordinates [135]

${ds}^{2}=-\mu_{A}{dt}^{2}+\frac{\rho^{2}}{\mu_{A}}{d\phi}^{2}+\frac{\mu_{A}}{\rho^{2}+\mu_{A}^{2}}\bigl{(}{d\rho}^{2}+{dz}^{2}\bigr{)}\,,$
(1.137)

where $\mu_{A}$ is the soliton associated to the acceleration parameter $A$.

In this case $g_{0}=\mathrm{diag}\bigl{(}-\mu_{A},\rho^{2}/\mu_{A}\bigr{)}$
and $f_{0}=\mu_{A}/(\rho^{2}+\mu_{A}^{2})$, and again $\det g_{0}=-\rho^{2}$.
The eigensystem (1.98) provides the solution

$\psi_{0}=\begin{pmatrix}\lambda-\mu_{A}&0\\\
0&\lambda+\frac{\rho^{2}}{\mu_{A}}\\\ \end{pmatrix}\,,$ (1.138)

which satisfies again the constraint $\psi_{0}(\lambda=0)=g_{0}$. The BZ
vectors are

$m^{(k)}=\Biggl{(}\frac{C_{0}^{(k)}}{\mu_{k}-\mu_{A}},C_{1}^{(k)}\frac{\mu_{A}}{\rho^{2}+\mu_{k}\mu_{A}}\Biggr{)}\,,$
(1.139)

where $C_{0}^{(k)}$ and $C_{1}^{(k)}$ are arbitrary constants.

We now add, as in the previous Section, two solitons on top of the Rindler
background by means of Eqs. (1.122), (1.123), and explicitly show that the
rotating C-metric in the form presented by Hong and Teo [136] is recovered.

The change of coordinates is provided by [137] and, in order to simplify the
computations, it is particularly useful to adopt the trick explained in [91]:
we firstly switch to the coordinates $(u,v)$ defined by

$\rho=\frac{2\kappa^{2}\sqrt{(1-u^{2})(v^{2}-1)(1+\nu u)(1+\nu
v)}}{(u-v)^{2}}\,,\quad z=\frac{\kappa^{2}(1-uv)(2+\nu u+\nu
v)}{(u-v)^{2}}\,,$ (1.140)

with poles parametrised as

$w_{1}=-\nu\kappa^{2}\,,\qquad w_{2}=\nu\kappa^{2}\,,\qquad
w_{A}=\kappa^{2}\,,$ (1.141)

and the new parameters $\nu$, $\kappa$ chosen in such a way that
$w_{1}<w_{2}<w_{A}$. Then, we perform the Möbius transformation

$u=\frac{x+d}{1+dx}\,,\qquad
v=\frac{y+d}{1+dy}\,,\qquad\nu=\frac{c-d}{1-cd}\,,$ (1.142)

to work with the standard C-metric coordinates $(x,y)$, and to introduce two
new parameters $c$ and $d$.

We still have to parametrise the constants appearing in the BZ vectors: we
choose

$\displaystyle C_{1}^{(1)}C_{0}^{(2)}-C_{0}^{(1)}C_{1}^{(2)}$
$\displaystyle=-\sqrt{m^{2}-a^{2}}\,,\quad
C_{1}^{(1)}C_{0}^{(2)}+C_{0}^{(1)}C_{1}^{(2)}=m\frac{1-a^{2}A^{2}}{1+a^{2}A^{2}}\,,$
(1.143a) $\displaystyle C_{0}^{(1)}C_{0}^{(2)}-C_{1}^{(1)}C_{1}^{(2)}$
$\displaystyle=\frac{2amA}{1+a^{2}A^{2}}\,,\qquad\;C_{0}^{(1)}C_{0}^{(2)}+C_{1}^{(1)}C_{1}^{(2)}=a\,,$
(1.143b)

where we have introduced the mass $m$, the angular momentum $a$ and the
acceleration $A$. Finally, we set

$\displaystyle c$ $\displaystyle=A\Bigl{(}m+\sqrt{m^{2}-a^{2}}\Bigr{)}\,,\quad
d=A\Bigl{(}m-\sqrt{m^{2}-a^{2}}\Bigr{)}\,,$ (1.144a) $\displaystyle\kappa^{2}$
$\displaystyle=\frac{1-a^{2}A^{2}}{2A^{2}}\,,\qquad\qquad\quad\\!C_{f}=-\frac{1+a^{2}A^{2}}{A^{6}}\,.$
(1.144b)

With these definitions, the metric boils down to

$\begin{split}{ds}^{2}&=\frac{1}{A^{2}(x-y)^{2}}\biggl{[}\frac{G(y)}{1+a^{2}A^{2}x^{2}y^{2}}(dt-
aAx^{2}d\phi)^{2}-\frac{1+a^{2}A^{2}x^{2}y^{2}}{G(y)}{dy}^{2}\\\
&\quad+\frac{1+a^{2}A^{2}x^{2}y^{2}}{G(x)}{dx}^{2}+\frac{G(x)}{1+a^{2}A^{2}x^{2}y^{2}}(d\phi+aAy^{2}dt)^{2}\biggr{]}\,,\end{split}$
(1.145)

with

$G(\xi)=(1-\xi^{2})(1+r_{+}A\xi)(1+r_{-}A\xi)\,,$ (1.146)

and where $r_{\pm}$ are the usual Kerr horizons, i.e.
$r_{\pm}=m\pm\sqrt{m^{2}-a^{2}}$.

We did not show it explicitly, but it is possible to construct the
accelerating Kerr–NUT solution by including the NUT parameter in the
parametrisation of the solution constructed above, similarly to what we did
for the Kerr–NUT metric, thus recovering the vacuum Plebański–Demiański
solution [134].

##### 1.2.6 Multi-black hole solutions

We construct the double-Schwarzschild solution, also known as the Bach–Weyl
metric [138] (that is a subcase of the Israel–Kahn metric [139]). It is worth
to remind to the reader that, as mentioned in the Introduction, such a
solution is not regular, being affected by the presence of conical
singularities.

By following the discussion in the end of the previous Sections, we just have
to add four solitons to the Minkowski background, so to add two black holes to
the flat background. We choose the BZ constants as

$C_{0}^{(1)}=C_{1}^{(2)}=C_{0}^{(3)}=C_{1}^{(4)}=0\,,\quad
C_{0}^{(2)}=C_{1}^{(1)}=C_{0}^{(4)}=C_{1}^{(3)}=1\,,$ (1.147)

to have a diagonal solution. Then, by applying the inverse scattering formulae
(1.122) and (1.123), we find

$\displaystyle g^{(ph)}$
$\displaystyle=\mathrm{diag}\biggl{(}-\frac{\mu_{1}\mu_{3}}{\mu_{2}\mu_{4}},\rho^{2}\frac{\mu_{2}\mu_{4}}{\mu_{1}\mu_{3}}\biggr{)}\,,$
(1.148a) $\displaystyle f^{(ph)}$
$\displaystyle=16C_{f}\frac{\mu_{1}^{3}\mu_{2}^{5}\mu_{3}^{3}\mu_{4}^{5}}{W_{11}W_{22}W_{33}W_{44}W_{13}^{2}W_{24}^{2}Y_{12}^{2}Y_{14}^{2}Y_{23}^{2}Y_{34}^{2}}\,,$
(1.148b)

where we have defined

$W_{ij}\coloneqq\rho^{2}+\mu_{i}\mu_{j}\,,\quad
Y_{ij}\coloneqq\mu_{i}-\mu_{j}\,.$ (1.149)

We see that the structure of the double-black hole solution (at least at the
level of the matrix $g^{(ph)}$) is very simple: it is just an alternating
series of solitons, where in the $tt$ component the odd solitons are at the
numerator and the even solitons are at the denominator, while in the
$\phi\phi$ component the situation is reversed.

One can easily guess the structure of the complete Israel–Kahn solution [139],
i.e. the multi-black hole solution with $N$ black holes (that corresponds to
$n=2N$ solitons):

$\displaystyle g^{(ph)}$
$\displaystyle=\mathrm{diag}\Biggl{(}-\frac{\prod_{k=1}^{N}\mu_{2k-1}}{\prod_{l=1}^{N}\mu_{2l}},\rho^{2}\frac{\prod_{l=1}^{N}\mu_{2l}}{\prod_{k=1}^{N}\mu_{2k-1}}\Biggr{)}\,,$
(1.150a)
$\displaystyle\begin{split}f^{(ph)}&=16C_{f}f_{0}\Biggl{(}\prod_{k=1}^{N}\mu_{2k}^{2N+1}\mu_{2k-1}^{2N-1}\Biggr{)}\Biggl{(}\prod_{k=1}^{2N}\frac{1}{\rho^{2}+\mu_{k}^{2}}\Biggr{)}\Biggl{(}\prod_{k=1,l=1,3,\cdots}^{2N-1}\frac{1}{(\mu_{k}-\mu_{k+l})^{2}}\Biggr{)}\\\
&\quad\times\Biggl{(}\prod_{k=1,l=2,4,\cdots}^{2N-2}\frac{1}{(\rho^{2}+\mu_{k}\mu_{k+l})^{2}}\Biggr{)}\,.\end{split}$
(1.150b)

The resulting metric represents $N$ black holes aligned on the $z$-axis. The
standard
parametrisation for the poles $w_{k}$ is given by

$w_{1}=z_{1}-m_{1}\,,\quad w_{2}=z_{1}+m_{1}\,,\quad\dotsc\quad
w_{2N-1}=z_{N}-m_{N}\,,\quad w_{2N}=z_{N}+m_{N}\,,$ (1.151)

with obvious ordering $w_{1}<w_{2}<\cdots<w_{2N-1}<w_{2N}$. The parameters
$z_{k}$ represent the positions of the black holes on the $z$-axis, while
$m_{k}$ are the mass parameters. The black hole horizons correspond to the
regions $w_{2k-1}<z<w_{2k}$ ($k=1,\dotsc,N$), while the complementary regions
are affected by the presence of conical singularities, that guarantee the
stationarity of the solution by preventing the collapse of the system.

It is conceptually easy to extend the previous discussion to the construction
of the double-Kerr–NUT solution [140]: one can adopt the parametrisation

$\displaystyle C_{1}^{(1)}C_{0}^{(2)}-C_{0}^{(1)}C_{1}^{(2)}$
$\displaystyle=\sigma_{1}\,,\quad
C_{1}^{(1)}C_{0}^{(2)}+C_{0}^{(1)}C_{1}^{(2)}=-m_{1}\,,$ (1.152a)
$\displaystyle C_{0}^{(1)}C_{0}^{(2)}-C_{1}^{(1)}C_{1}^{(2)}$
$\displaystyle=n_{1}\,,\quad
C_{0}^{(1)}C_{0}^{(2)}+C_{1}^{(1)}C_{1}^{(2)}=a_{1}\,,$ (1.152b)

and

$\displaystyle C_{1}^{(3)}C_{0}^{(4)}-C_{0}^{(3)}C_{1}^{(4)}$
$\displaystyle=\sigma_{2}\,,\quad
C_{1}^{(3)}C_{0}^{(4)}+C_{0}^{(3)}C_{1}^{(4)}=-m_{2}\,,$ (1.153a)
$\displaystyle C_{0}^{(3)}C_{0}^{(4)}-C_{1}^{(3)}C_{1}^{(4)}$
$\displaystyle=n_{2}\,,\quad
C_{0}^{(3)}C_{0}^{(4)}+C_{1}^{(3)}C_{1}^{(4)}=a_{2}\,.$ (1.153b)

We simply double the parametric choice we made for the Kerr–NUT solution.
However, it is important to notice that the various parameters we have
introduced do not correspond to the physical charges, in general. For
instance, the angular momentum parameters $a_{1}$ and $a_{2}$ do not
corresponds to the actual angular momenta of the black holes. Such charges
will be, in general, functions of all the parameters of the solutions. This
implies that turning off $a_{1}=a_{2}=0$ does not guarantee that the angular
momenta of the black holes will be zero, since they will depend on other
parameters of the theory.

###  Chapter 2 Black holes in an external gravitational field

As we argued in the Introduction, multi-black holes solutions are intriguing
both from the theoretical and the phenomenological point of view. On the
theoretical side, these solutions disclose the non-linear nature of General
Relativity and represent an important playground in which testing the laws of
black hole mechanics. On the experimental side, the recent remarkable
observations of gravitational waves heavily rely on the interactions between
two black holes in a binary system: thus an analytical description of such a
spacetime is of utmost relevance for the interpretation of the measurements.
Of course one of the main obstacle in modelling a stationary multi-
gravitational sources system is to provide a mechanism to balance the
gravitational attraction of the bodies. Otherwise the system naturally tends
to collapse.

The purpose of this Chapter is to provide an example of a background that can
be used to regularise a multi-black hole spacetime. Such a background is given
by an external gravitational field: we will show that there exists a solution
of vacuum Einstein equations which represents a multipolar expansion of a
generic external gravitational field. Such a solution was discovered by Erez
and Rosen a long time ago [141], and it is the natural background to be used
to regularise a multi-black hole spacetime. Indeed, the gravitational
background provides the physical explanation of the regularity of a double-
black hole system: the external field might be tuned in such a way to balance
the gravitational attraction between the sources and, thus, to prevent the
collapse.

External gravitational fields represent a natural setting for multi-black
holes systems, as recent gravitational waves detection proceeding from the
center of galaxies confirms. In fact the observed astrophysical black holes
are not isolated systems, as they are always embedded in external
gravitational fields. In particular, it has been shown in [142] that the
multipolar gravitational field, we will deem here, can be produced by a
distribution of matter such as thin disks or rings, typical shapes of
gravitational objects such as galaxies or nebulae. Anyway, the solutions
considered in this Chapter will be pure vacuum solution, without any energy-
momentum tensor. In principle a distribution of matter might be possibly
considered very far away from the black holes: in this sense our solutions can
be interpreted as local models for binary or multi-black hole configurations.
In a certain sense the metrics presented in this Chapter have to be considered
as the gravitational analogous of the stationary black holes in Melvin
magnetic universe. In fact, also in this latter case, the solution is a pure
electrovacuum solution with no definite sources for the electromagnetic
external field, therefore their feasibility remains in the proximity domain of
the black bodies. The multipolar expansion is the key ingredient that allows
to circumvent the no-go theorem about the non-existence of static [143]
configurations of many-body systems, with a suitable separation condition.
Being the multipolar expansion non-asimptotically flat, one of the main
hypothesis of the theorem does not hold and regular multi-black hole solutions
are allowed to exist.

Single black holes in external multipolar gravitational field have been
pioneered, in the literature, by Doroshkevich, Zel’dovich and Novikov [144],
later studied by Chandrasekhar [145], and Geroch and Hartle [146]; these
solutions are known in the literature as deformed black holes. The novelty of
our proposal, based on an Ernst’s insight in the context of the C-metric
[147], and already implemented in the case of a single black hole [148], is to
take advantage of the external field to sustain the black bodies and prevent
their collision. From a mathematical point of view this means that the
solution can be regularised from conical singularities that usually affect
multi-black hole metrics.

We start by investigating the properties of the external gravitational field
and its representation as a seed for the inverse scattering procedure. Then,
we will construct an array of static black holes, which generalises the
Israel–Khan solution, and explicitly show that such a solution can be
regularised, by removing all the conical singularities via a tuning of the
external field parameters. Subsequently, we will specialise to a very
interesting subcase, i.e. the binary black hole system immersed in an external
gravitational field, and also present the charged and rotating generalisations
of that case. Finally, we will add acceleration and construct a chain of
accelerating black hole. All of these systems can be regularised, in order to
obtain legit multi-black hole spacetimes. This Chapter is based on the papers
[149, 150, 151].

#### 2.1 Background gravitational field

We discuss the general solution to the Einstein equations in vacuum, which
contemplates both the internal deformations of the source, and the
contributions which come from matter far outside the source. This general
solution finds its roots in the pioneeristic work of Erez and Rosen [141], and
it was lately discussed and expanded in [144, 146, 145] to include the
deformations due to an external gravitational field.

The general solution for the Weyl metric

${ds}^{2}=-e^{2\psi(\rho,z)}{dt}^{2}+e^{-2\psi(\rho,z)}\bigl{[}e^{2\gamma(\rho,z)}({d\rho}^{2}+{dz}^{2})+\rho^{2}{d\phi}^{2}\bigr{]}\,,$
(2.1)

is given, following the conventions in [152], by

$\displaystyle\psi$
$\displaystyle=\sum_{n=1}^{\infty}\biggl{(}\frac{a_{n}}{r^{n+1}}+b_{n}r^{n}\biggr{)}P_{n}\,,$
(2.2a) $\displaystyle\gamma$
$\displaystyle=\sum_{n,p=1}^{\infty}\biggl{[}\frac{(n+1)(p+1)a_{n}a_{p}}{(n+p+2)r^{n+p+2}}(P_{n+1}P_{p+1}-P_{n}P_{p})+\frac{npb_{n}b_{p}r^{n+p}}{n+p}(P_{n}P_{p}-P_{n-1}P_{p-1})\biggr{]}\,,$
(2.2b)

where $r\coloneqq\sqrt{\rho^{2}+z^{2}}$ defines the asymptotic radial
coordinate and $P_{n}=P_{n}(z/r)$ is the $n$-th Legendre polynomial. The real
constants $a_{n}$ describe the deformations of the source, while the real
parameters $b_{n}$ describe the external static gravitational field.

We observe that the “internal” part $a_{n}$, which is related to the
deformations of the source, is asymptotically flat: this seems to contradict
Israel’s theorem [153], which states that the only regular and static
spacetime in vacuum is the Schwarzschild black hole. Actually, the internal
deformations lead to curvature singularities not covered by a horizon [141],
in agreement with the theorem. Because of this feature, in the following we
will discard the internal contributions and focus on the external ones only.

On the converse, the “external” part $b_{n}$ is not asymptotically flat. This
is in agreement with the physical interpretation: this part of the metric
represents an external gravitational field generated by a distribution of
matter located at infinity. This interpretation parallels the Melvin spacetime
one, and in fact there is no stress-energy tensor here to model the matter
responsible of the field. The asymptotia is not flat because at infinity there
is the source matter and the spacetime is not empty. Actually, it can be shown
that the curvature invariants, such as the Kretschmann scalar, can grow
indefinitely at large distances111For the black hole model we are considering
below, these large distances can be quantified in several orders of magnitude
larger than the scale of the black holes. However, before reaching such
distances, large values of the curvature invariants can be covered by a
Killing horizon such as the one provided by the acceleration. in some
directions. In this sense, a spacetime containing such an external
gravitational field should be considered local, in the sense that the
description is meaningful in the neighborhood of the black holes that one
embeds in this background. In this regard, these metrics are not different
with respect to the usual single distorted black holes studied in the
literature [152, 154]. To have a completely physical solution, one should
match the black holes immersed in the external field with an appropriate
distribution of matter (such as galaxies), which possibly is asymptotic to
Minkowski spacetime at infinity. A model for matter content which is
consistent with the multipolar background expansion treated here, and based on
a thin ring distribution, can be found in [142]222The analysis of the sectors
where the scalar curvature invariants are larger, at least for the firsts
order of the multipole expansion, suggests that a ring matter distribution is
the more appropriate one for these backgrounds..

Accelerating generalisations of the multipolar gravitational background
studied in this Section naturally are endowed with a Killing horizon of
Rindler type. These accelerating horizon are typically located in between the
black hole sources and the far region, and they will be studied later on.

The constants $a_{n}$ and $b_{n}$ are related to the multipole momenta of the
spacetime. The relativistic definition of the multipole momenta was given by
Geroch [155], and lately refined by Hansen [156]. That definition applies for
a stationary, axisymmetric and asymptotically flat spacetime: the modified
Ernst potential associated to the spacetime is expanded at infinity, and the
first coefficients of the expansion correspond to the multipole momenta [157].
Clearly, the internal deformations asymptote a flat spacetime, while the
external ones do not.

In the following, we compute the multipole momenta associated to the above
deformations, in order to clarify the intepretation of $a_{n}$ and $b_{n}$. In
particular, we calculate the internal multipole contributions by means of the
standard definition, and then we propose a new approach for determining the
multipoles associated to external deformations. We shall see that the proposed
definition is analogous to the usual one and gives a consistent interpretation
of the external field parameters.

##### 2.1.1 Background field multipoles

Let us start with the internal deformations. If we turn off $b_{n}$, then we
are left with

$\psi_{\text{int}}=\sum_{n=1}^{\infty}\frac{a_{n}}{r^{n+1}}P_{n}\,.$ (2.3)

We define $\xi$ as a function of the Ernst potential

$\mathcal{E}=\frac{1-\xi}{1+\xi}\,,$ (2.4)

that in our case reads

$\xi_{\text{int}}\coloneqq\frac{1-e^{2\psi_{\text{int}}}}{1+e^{2\psi_{\text{int}}}}\,.$
(2.5)

We want to expand the above expression around infinity: following [157], we
bring infinity to a finite point by defining
$\bar{\rho}=\rho/(\rho^{2}+z^{2})$, $\bar{z}=z/(\rho^{2}+z^{2})$, and by
conformally rescaling $\xi_{\text{int}}$:

$\bar{\xi}=\frac{1}{\bar{\rho}^{2}+\bar{z}^{2}}\,\xi_{\text{int}}\,.$ (2.6)

One can prove that $\bar{\xi}$ is uniquely determined by its value on the
$z$-axis: since infinity corresponds to $\bar{\rho}=\bar{z}=0$ in the new
coordinates, then we expand $\bar{\xi}$ around $\bar{z}$ for $\rho=0$

$\bar{\xi}(\bar{\rho}=0)=\sum_{j=0}^{\infty}M_{j}\bar{z}^{j}\,,$ (2.7)

where $M_{j}$ are the expansion coefficients. The multipole momenta are
completely determined by the coefficients $M_{j}$, and in particular one can
show (see [157] and references therein) that the first four coefficients
$M_{0},\dotsc,M_{3}$ are exactly equal to the first four multipole momenta
$\mathcal{Q}_{0}^{\text{int}},\dotsc,\mathcal{Q}_{3}^{\text{int}}$. Thus, for
the internal deformations, we find

$\mathcal{Q}_{0}^{\text{int}}=M_{0}=0\,,\quad\mathcal{Q}_{1}^{\text{int}}=M_{1}=-a_{1}\,,\quad\mathcal{Q}_{2}^{\text{int}}=M_{2}=-a_{2}\,,\quad\mathcal{Q}_{3}^{\text{int}}=M_{3}=-a_{3}\,.$
(2.8)

We see that there is a direct correspondence between the coefficients $a_{n}$
and the multipole momenta, at least at the first orders.
$\mathcal{Q}_{0}^{\text{int}}$ is the monopole term, which is zero since no
source (e.g. no black hole) is present. $\mathcal{Q}_{1}^{\text{int}}$ is the
dipole, $\mathcal{Q}_{2}^{\text{int}}$ is the quadrupole and
$\mathcal{Q}_{3}^{\text{int}}$ is the octupole moment. The subsequent
multipole momenta can be still computed from the expansion (2.7) by means of a
recursive algorithm, but they will be non-trivial combinations of the
coefficients $M_{j}$ [157]. For instance, the 16-pole is given by
$\mathcal{Q}_{4}^{\text{int}}=M_{4}-1/7M_{0}^{2}M_{2}$.

The above construction can not work for the external deformations: the
asymptotia is not flat, and infinity is the place where the sources of the
external field are thought to be, hence it does not make sense to expand
there. On the converse, it is meaningful to detect the effects of the
deformation in the origin of the Weyl coordinates: thus, by paralleling the
Geroch–Hansen treatise, we propose to expand the modified Ernst potential in
the _origin_ of the cylindrical coordinates.

Now we consider only the external deformations

$\psi_{\text{ext}}=\sum_{n=1}^{\infty}b_{n}r^{n}P_{n}\,,$ (2.9)

with modified Ernst potential

$\xi_{\text{ext}}=\frac{1-e^{2\psi_{\text{ext}}}}{1+e^{2\psi_{\text{ext}}}}\,.$
(2.10)

Accordingly to the above discussion, we assume that $\xi_{\text{ext}}$ is
completely determined on the $z$-axis, so we expand around the origin for
$\rho=0$

$\xi_{\text{ext}}(\rho=0)=\sum_{j=0}^{\infty}N_{j}z^{j}\,,$ (2.11)

where $N_{j}$ are the expansion coefficients. We _define_ the first four
multipole momenta
$\mathcal{Q}_{0}^{\text{ext}},\dotsc,\mathcal{Q}_{3}^{\text{ext}}$ as the
coefficients $N_{0},\dotsc,N_{3}$, which are equal to

$\mathcal{Q}_{0}^{\text{ext}}=N_{0}=0\,,\quad\mathcal{Q}_{1}^{\text{ext}}=N_{1}=-b_{1}\,,\quad\mathcal{Q}_{2}^{\text{ext}}=N_{2}=-b_{2}\,,\quad\mathcal{Q}_{3}^{\text{ext}}=N_{3}=\frac{b_{1}^{3}}{3}-b_{3}\,.$
(2.12)

Again, the monopole moment is zero because of the absence of a source. We
observe, contrary to (2.8), that the octupole $\mathcal{Q}_{3}^{\text{ext}}$
is given by a non-trivial mixing of the constants $b_{1}$ and $b_{3}$. This
definition occurs only for the first momenta: a definition which takes into
account higher momenta can be achieved by generalising the procedure in [157].

For the time being, we content ourselves by proposing the following
interpretation: the coefficients $b_{n}$ are related to the multipole momenta
generated by the external gravitational field, similarly to what happens for
the internal deformations. Then, the presence of the external gravitational
field affects the momenta of a black hole source immersed in it.

##### 2.1.2 Seed for the inverse scattering construction

We are interested in the external deformations only, hence we consider
$a_{n}=0$ hereafter, and focus only on the contributions from $b_{n}$. We
express the external gravitational field metric in a form which is suitable
for the inverse scattering procedure, i.e.

$\displaystyle g_{0}$
$\displaystyle=\mathrm{diag}\Biggl{[}-\exp\biggr{(}2\sum_{n=1}^{\infty}b_{n}r^{n}P_{n}\biggr{)},\rho^{2}\exp\biggl{(}-2\sum_{n=1}^{\infty}b_{n}r^{n}P_{n}\biggr{)}\Biggr{]}\,,$
(2.13a) $\displaystyle f_{0}$
$\displaystyle=\exp\Biggl{[}2\sum_{n,p=1}^{\infty}\frac{npb_{n}b_{p}r^{n+p}}{n+p}\bigl{(}P_{n}P_{p}-P_{n-1}P_{p-1}\bigr{)}-2\sum_{n=1}^{\infty}b_{n}r^{n}P_{n}\Biggr{]}\,,$
(2.13b)

The parameters $b_{n}$ are related the multipole momenta of the external
field, as explained above. Metric (2.13) represents a generic static and
axisymmetric gravitational field.

Since we want to add black holes on the background represented by the
gravitational field, we take the metric (2.13) as a seed for the inverse
scattering procedure. Following the discussion in section 1.2, we need the
generating matrix $\psi_{0}$, which serves as starting point to build the
multi-black hole solution. The function which satisfies equations (1.98) is
[158]

$\psi_{0}(\rho,z,\lambda)=\begin{pmatrix}-e^{F(\rho,z,\lambda)}&0\\\
0&(\rho^{2}-2\lambda\rho-\lambda^{2})e^{-F(\rho,z,\lambda)}\end{pmatrix}\,,$
(2.14)

where

$\begin{split}F(\rho,z,\lambda)&=2\sum_{n=1}^{\infty}b_{n}\Biggl{[}\sum_{l=0}^{\infty}\binom{n}{l}\biggl{(}\frac{-\rho^{2}}{2\lambda}\biggr{)}^{l}\biggl{(}z+\frac{\lambda}{2}\biggr{)}^{n-l}\\\
&\quad-\sum_{l=1}^{n}\sum_{k=0}^{[(n-l)/2]}\frac{(-1)^{k+l}2^{-2k-l}n!\lambda^{-l}}{k!(k+l)!(n-2k-l)!}\rho^{2(k+l)}z^{n-2k-l}\Biggr{]}\,.\end{split}$
(2.15)

Now we can construct the BZ vectors (1.113): we parametrise
$m_{0}^{(k)}=\bigl{(}C_{0}^{(k)},C_{1}^{(k)}\bigr{)}$, where $C_{0}^{(k)}$,
$C_{1}^{(k)}$ are constants that will be eventually related to the physical
parameters of the solution. The BZ vectors are thus

$m^{(k)}=\biggl{(}-C_{0}^{(k)}e^{-F(\rho,z,\mu_{k})},\frac{C_{1}^{(k)}}{\mu_{k}}e^{F(\rho,z,\mu_{k})}\biggr{)}\,.$
(2.16)

Depending on the value of $C_{0}^{(k)}$ and $C_{1}^{(k)}$, the spacetime will
be static or stationary, as we will see in the following.

#### 2.2 Array of static black holes

We now proceed to the generalisation of the Israel–Khan solution [139], which
represents an array of collinear Schwarzschild black holes. The Israel–Khan
metric is plagued by the presence of conical singularities which can not be
removed by a fine tuning of the physical parameters333Actually, in the limit
of an infinite number of collinear black holes, the conical singularities
disappear and the metric is regular. See [159].. On the converse, we will see
that the external gravitational field will furnish the force necessary to
achieve the complete equilibrium among the black holes.

Given the seed metric (2.13) and the generating matrix (2.14), we construct a
new solution by adding $2N$ solitons with constants

$C_{0}^{(k)}=\begin{cases}1&k\text{ even}\\\ 0&k\text{ odd}\end{cases},\qquad
C_{1}^{(k)}=\begin{cases}0&k\text{ even}\\\ 1&k\text{ odd}\end{cases}\,.$
(2.17)

This choice guarantees a diagonal, and hence static, metric. Each couple of
solitons adds a black hole, then the addition of $2N$ solitons gives rise to a
spacetime containing $N$ black holes, whose metric is

$\displaystyle g_{N}$
$\displaystyle=\mathrm{diag}\Biggl{[}-\frac{\prod_{k=1}^{N}\mu_{2k-1}}{\prod_{l=1}^{N}\mu_{2l}}\exp\Biggl{(}2{\sum_{n=1}^{\infty}b_{n}r^{n}P_{n}}\Biggr{)},\rho^{2}\frac{\prod_{l=1}^{N}\mu_{2l}}{\prod_{k=1}^{N}\mu_{2k-1}}\exp\Biggl{(}-2{\sum_{n=1}^{\infty}b_{n}r^{n}P_{n}}\Biggr{)}\Biggr{]}\,,$
(2.18a)
$\displaystyle\begin{split}f_{N}&=16C_{f}f_{0}\Biggl{(}\prod_{k=1}^{N}\mu_{2k}^{2N+1}\mu_{2k-1}^{2N-1}\Biggr{)}\Biggl{(}\prod_{k=1}^{2N}\frac{1}{\rho^{2}+\mu_{k}^{2}}\Biggr{)}\Biggl{(}\prod_{k=1,l=1,3,\cdots}^{2N-1}\frac{1}{(\mu_{k}-\mu_{k+l})^{2}}\Biggr{)}\\\
&\quad\times\Biggl{(}\prod_{k=1,l=2,4,\cdots}^{2N-2}\frac{1}{(\rho^{2}+\mu_{k}\mu_{k+l})^{2}}\Biggr{)}\exp\Biggl{[}2\sum_{k=1}^{2N}(-1)^{k+1}F(\rho,z,\mu_{k})\Biggr{]}\,.\end{split}$
(2.18b)

Metric (2.18) is, by construction, a solution of the vacuum Einstein
equations, and it represents a collection of $N$ Schwarzschild black holes,
aligned along the $z$-axis, and immersed in the external gravitational field
(2.13).

We limit ourselves to the case of real poles $w_{k}$, since it represents the
physically most relevant situation. These constants are chosen with ordering
$w_{1}<w_{2}<\cdots<w_{2N-1}<w_{2N}$ and with parametrisation

$w_{1}=z_{1}-m_{1}\,,\quad w_{2}=z_{1}+m_{1}\,,\quad\dotsc\quad
w_{2N-1}=z_{N}-m_{N}\,,\quad w_{2N}=z_{N}+m_{N}\,.$ (2.19)

The constants $m_{k}$ represent the black hole mass parameters, while $z_{k}$
are the black hole positions on the $z$-axis.

The black hole horizons correspond to the regions $w_{2k-1}<z<w_{2k}$
($k=1,\dotsc,N$), while the complementary regions are affected by the presence
of conical singularities, as happens for the Israel–Khan solution (cf. Fig.
2.1), which can be recovered by setting $b_{n}=0$. Differently from that case,
our solution can be regularised, as we will show in the next subsection.

Figure 2.1: Rod diagram for the multi-black hole spacetime (2.18). The
horizons correspond to the timelike rods (thick lines of the $t$ coordinate),
while the conical singularities correspond to “bolts” where conical
singularities can be avoided by imposing an appropriate periodicity on the
angular coordinate.

##### 2.2.1 Conical singularities and regularisation

The infinite multipole momenta $b_{n}$ allow us to regularise the metric, i.e.
to remove all the conical singularities, by tuning their values. More
precisely, given $N$ black holes, there are exactly $N+1$ conical
singularities: two cosmic strings, one rear the first black hole and one ahead
the last black hole, and $N-1$ struts located between the $N$ black holes.
Hence, one needs at least $N+1$ parameters to fix the singularities.

The manifold exhibits angular defects when the ratio between the length and
the radius of small circles around the $z$-axis is different from $2\pi$.
Working in Weyl coordinates, a small circle around the $z$-axis has radius
$R=\sqrt{g_{zz}}\rho$ and length $L=2\pi\sqrt{g_{\phi\phi}}$ [149]. The
regularity condition corresponds then to $L/(2\pi R)\to 1$ as $\rho\to 0$. It
is easy to prove that, for the static and axisymmetric metrics of the class we
are considering, the above condition is equivalent to $\mathcal{P}\equiv
fg_{tt}\to 1$ as $\rho\to 0$.

We choose for convenience the gauge parameter $C_{f}$ as

$C_{f}=2^{2(2N+1)}\Biggl{[}\prod_{i=1}^{N}(w_{2i}-w_{2i-1})^{2}\Biggr{]}\Biggl{[}\prod_{k=1}^{N-1}\prod_{j=1}^{N-k}(w_{2k-1}-w_{2k+2j})^{2}(w_{2k}-w_{2k+2j-1})^{2}\Biggr{]}\,.$
(2.20)

The quantity $\mathcal{P}=fg_{tt}$ is equal to

$\mathcal{P}_{k}=\Biggl{[}\prod_{i=1}^{2k}\prod_{j=2k+1}^{2N}(w_{j}-w_{i})^{2\,(-1)^{i+j+1}}\Biggr{]}\exp\Biggl{[}4\sum_{n=1}^{\infty}b_{n}\sum_{j=2k+1}^{2N}(-1)^{j+1}w_{j}^{n}\Biggl{]}\,,$
(2.21)

between the $k$-th and $(k+1)$-th black holes (i.e. $w_{2k}<z<w_{2k+1}$), for
$1\leq k<N$. In the region $z<w_{1}$ we find

$\mathcal{P}_{0}=\exp\Biggl{[}4\sum_{n=1}^{\infty}b_{n}\sum_{j=1}^{2N}(-1)^{j+1}w_{j}^{n}\Biggl{]}\,,$
(2.22)

while for $z>w_{2N}$ we simply have

$\mathcal{P}_{N}=1\,,$ (2.23)

thanks to our choice of $C_{f}$. These expressions are the natural
generalisations of the conical singularities for the Israel–Khan metric [160].

The above expressions provide a system of equations $\mathcal{P}_{k}=1$ for
$0\leq k<N$, which can be solved, e.g., for the parameters
$b_{1},\dotsc,b_{N}$, with the result of fixing all the conical singularities.
Hence the solution can be made completely regular outside the black hole
horizons.

##### 2.2.2 Smarr law

We investigate the Smarr law for spacetime (2.18): to this end, we compute the
total mass of the spacetime and the entropy and temperature of the black
holes.

The mass is easily found by means of the Komar–Tomimatsu integral [161, 162].
The result for the $k$-th black hole (i.e. the black hole in the interval
$w_{2k-1}<z<w_{2k}$) is

$M_{k}=\alpha\int_{w_{2k-1}}^{w_{2k}}\rho
g_{tt}^{-1}\partial_{\rho}g_{tt}=\frac{\alpha}{2}(w_{2k}-w_{2k-1})=\alpha
m_{k}\,,$ (2.24)

where $\alpha$ is a constant which takes into account the proper normalisation
of the timelike Killing vector, generator of the horizon,
$\xi=\alpha\partial_{t}$ associated to (2.18).

The entropy of a black hole is related to the area as
$S_{k}=\mathcal{A}_{k}/4$, hence

$S_{k}=\frac{1}{4}\lim_{\rho\to
0}\int_{0}^{2\pi}d\phi\int_{w_{2k-1}}^{w_{2k}}dz\sqrt{fg_{\phi\phi}}=\pi
m_{k}W\exp\Biggl{[}2\sum_{n=1}^{\infty}b_{n}\sum_{j=2k}^{2N}(-1)^{j+1}w_{j}^{n}\Biggl{]}\,,$
(2.25)

where

$\log W=\lim_{\rho\to 0}\log\sqrt{fg_{\phi\phi}}=\log
2+\sum_{i=1}^{2k-1}\sum_{j=2k}^{2N}(-1)^{i+j+1}\log|w_{j}-w_{i}|\,.$ (2.26)

The product $\sqrt{fg_{\phi\phi}}$ is independent of $z$ in the limit $\rho\to
0$, and that is crucial in the derivation of (2.25).

Finally, the temperature is found via the Wick-rotated metric, and the result
is

$T_{k}=\frac{\alpha}{2\pi}\lim_{\rho\to
0}\rho^{-1}\sqrt{\frac{g_{tt}}{f}}=\frac{\alpha}{2\pi}\lim_{\rho\to
0}\frac{1}{\sqrt{fg_{\phi\phi}}}=\frac{\alpha m_{k}}{2S_{k}}\,.$ (2.27)

It is easily shown, by using (2.24), (2.25) and (2.27), that the Smarr law is
satisfied:

$\sum_{k=1}^{N}M_{k}=2\sum_{k=1}^{N}T_{k}S_{k}\,.$ (2.28)

We notice that the explicit value of $\alpha$ is not needed for (2.28) to
work, while it is relevant in the study of the thermodynamics [149].

##### 2.2.3 Schwarzschild black hole

The simplest non-trivial example we can consider for the complete external
multipolar expansion from the general solution (2.18), is clearly the single
black hole configuration for $N=1$. In that case the functions that appear in
the Weyl static metric (2.18) take the form

$\displaystyle g_{1}$
$\displaystyle=\mathrm{diag}\biggl{[}-\frac{\mu_{1}}{\mu_{2}}\exp\biggr{(}2\sum_{n=1}^{\infty}b_{n}r^{n}P_{n}\biggr{)},\rho^{2}\frac{\mu_{2}}{\mu_{1}}\exp\biggr{(}-2\sum_{n=1}^{\infty}b_{n}r^{n}P_{n}\biggr{)}\biggr{]}\,,$
(2.29a) $\displaystyle f_{1}$
$\displaystyle=\frac{16C_{f}\,f_{0}\,\mu_{1}\mu_{2}^{3}\,e^{2[F(\mu_{1})-F(\mu_{2})]}}{(\mu_{1}-\mu_{2})^{2}(\rho^{2}+\mu_{1}^{2})(\rho^{2}+\mu_{2}^{2})}\,.$
(2.29b)

This spacetime represents a static black hole embedded in an external
gravitational field. The limit to the Schwarzschild metric is clear: it is
obtained by switching off all the multipoles $b_{n}=0$. In order to recover
the standard Schwarzschild metric in spherical coordinates, the following
transformation is needed

$\rho=\sqrt{r(r-2m)}\sin\theta\,,\quad z=z_{1}+(r-m)\cos\theta\,.$ (2.30)

A solution of this kind is not completely new, since it was already present in
Chandrasekhar’s book [145], see also [152]. However, the form we are writing
here is more general because, thanks to the extra parameter $z_{1}$, allows to
place the black hole in any point of the $z$-axis. In the absence of the
external field the location of the black hole is irrelevant because the
solution is symmetric under a finite shift of $z_{1}$. But, when the external
gravitational field is not zero, a translation along the $z$-axis is
significant, since the relative position of the black hole with respect to the
external multipolar sources has some non-trivial effects on the geometry and
on the physics of the black hole.

(a) $m=0.5$, $b_{2}=0.4$, $z_{1}=2$ (b) $m=1$, $b_{2}=-0.1$, $z_{1}=2$ (c)
$m=1$, $b_{2}=-0.5$, $z_{1}=2$

Figure 2.2: Embeddings in Euclidean three-dimensional space $\mathbb{E}^{3}$
of the event horizon of single black holes distorted by the external dipolar
and quadrupolar gravitational field, for three different sets of physical
parameters. $b_{1}$ and $C_{f}$ are fixed by the regularity constraint (2.31).

In particular the translation along the $z$-axis affects the event horizon
shape. In fact, as can be evaluated by computing the equatorial and polar
circles around the event horizon, it is possible to understand how the horizon
surface stretches or contracts depending on the position of the black hole and
the values of the external parameters. Some pictorial examples of the black
hole horizon deformation for different external gravitational backgrounds are
given in Fig. 2.2.

(a) Black hole horizon with conical singularity (b) Black hole horizon
regularised on both poles

Figure 2.3: Embedding in $\mathbb{E}^{3}$ of a single static black hole
horizon (orange surface) immersed into a dipolar and quadrupolar external
gravitational field, for $m=0.6$, $z_{1}=-1.3$, $b_{2}=0.3$. A section is
taken to appreciate the deformation w.r.t. the null external field case, drawn
in green as a reference: the standard spherical Schwarzschild horizon, which
is everywhere covered by the horizon swollen by the presence of the external
gravitational field. The two black holes in external field differ only for the
value of $b_{1}$, which in panel $(b)$ is chosen according to Eq. (2.31) to
remove also the second conical singularity, while in panel $(a)$ is
$b_{1}=0.5$.

Physically, the solution is affected by conical singularities because the
arbitrary external field deforms and attempts to accelerate the black hole,
which however does not contain an acceleration parameter, thus causing the
appearance of the defects. Only a fine tuned field, that is able to maintain
the non-accelerating character of the solution, can avoid this pathological
behaviour.

The equilibrium constraint which removes the conical defects can be loosen
with respect to the one usually found in the literature [152]. The imposition
usually requested in the literature, i.e $\sum_{n=1}^{\infty}b_{2n+1}=0$, is
not fundamental when the black hole can be adjusted coherently with the
external gravitational field. In fact, when $z_{1}\neq 0$ the two regularising
constraints we have to impose to avoid conical singularities become, as seen
above,

$C_{f}=\frac{(w_{1}-w_{2})^{2}}{4}\,,\quad\sum_{n=1}^{\infty}b_{n}\bigl{(}w_{1}^{n}-w_{2}^{n}\bigr{)}=0\,.$
(2.31)

When $b_{n}=0$ for $n>2$, we obtain a special subcase of the solution [149],
in the limit in which one of the black hole vanishes or where the two horizon
rods merge, remaining only with a single black hole configuration. Fig. 2.3 in
this section refers, for simplicity, to this truncated expansion of the
external field. A qualitatively analogous behaviour of the black hole horizon
occurs in the full multipolar expansion.

#### 2.3 Binary system: a case study

We consider an important subcase of the multi-black hole spacetime described
in the previous section, namely the binary black hole configuration. The
importance of this subcase comes from its phenomenological relevance, since
black holes usually present themselves in pairs; moreover, they are the main
sources of gravitational waves events. It is known that gravitational waves
are mainly generated in binary black hole systems, thus it is worth to
construct and study regular exact solutions that might describe such
configurations.

From the full solution (2.18) we can extract, by choosing $N=2$ and $b_{n}=0$
for $n>2$, the binary black hole system immersed in an external dipole-
quadrupole field

$\displaystyle g$
$\displaystyle=\frac{\mu_{1}\mu_{3}}{\mu_{2}\mu_{4}}\exp\biggl{[}2b_{1}z+2b_{2}\biggl{(}z^{2}-\frac{\rho^{2}}{2}\biggr{)}\biggr{]}\,,$
(2.32a)
$\displaystyle\begin{split}f&=16C_{f}\frac{\mu_{1}^{3}\mu_{2}^{5}\mu_{3}^{3}\mu_{4}^{5}}{W_{11}W_{22}W_{33}W_{44}W_{13}^{2}W_{24}^{2}Y_{12}Y_{14}Y_{23}Y_{34}}\exp\biggl{[}-b_{1}^{2}\rho^{2}+\frac{b_{2}^{2}}{2}\bigl{(}\rho^{2}-8z^{2}\bigr{)}\rho^{2}-4b_{1}b_{2}z\rho^{2}\\\
&\quad+2b_{1}(-z+\mu_{1}-\mu_{2}+\mu_{3}-\mu_{4})\\\
&\quad+b_{2}\bigl{(}-2z^{2}+\rho^{2}+4z(\mu_{1}-\mu_{2})+\mu_{1}^{2}-\mu_{2}^{2}+(\mu_{3}-\mu_{4})(4z+\mu_{3}+\mu_{4})\bigr{)}\biggr{]}\,.\end{split}$
(2.32b)

where $W_{ij}=\rho^{2}+\mu_{i}\mu_{j}$, $Y_{ij}=(\mu_{i}-\mu_{j})^{2}$. The
parametrisation is inherited from the one in the previous Section, and the two
event horizons extend in the regions $w_{1}<z<w_{2}$ and $w_{3}<z<w_{4}$ when
$\rho=0$.

The conical singularities are removed as in the previous Section: in this
case, we can specialise the general formulas found above, and express the
regularisation conditions in terms of the physical parameters

$\displaystyle C_{f}$
$\displaystyle=256m_{1}^{2}m_{2}^{2}(m_{1}+m_{2}+z_{1}-z_{2})^{2}(m_{1}+m_{2}-z_{1}+z_{2})^{2}\,,$
(2.33) $\displaystyle b_{1}$
$\displaystyle=-\frac{(m_{1}z_{1}+m_{2}z_{2})}{4m_{1}m_{2}(z_{1}-z_{2})}\log\biggl{[}\frac{(m_{1}-m_{2}+z_{1}-z_{2})(m_{1}-m_{2}-z_{1}+z_{2})}{(m_{1}+m_{2}+z_{1}-z_{2})(m_{1}+m_{2}-z_{1}+z_{2})}\biggr{]}\,,$
(2.34) $\displaystyle b_{2}$
$\displaystyle=\frac{(m_{1}+m_{2})}{8m_{1}m_{2}(z_{1}-z_{2})}\log\biggl{[}\frac{(m_{1}-m_{2}+z_{1}-z_{2})(m_{1}-m_{2}-z_{1}+z_{2})}{(m_{1}+m_{2}+z_{1}-z_{2})(m_{1}+m_{2}-z_{1}+z_{2})}\biggr{]}\,.$
(2.35)

It is understood that $C_{f}$, $b_{1}$, $b_{2}$ will assume the above values,
from now on. Note that the physical parameters characterising the black holes
were left unconstrained: this means that the physical properties of the binary
black hole system are completely generic. This fact guarantees a wide
flexibility in a possible phenomenological scenario 444In fact the presented
solution is not the minimal one which can be regularised. It is sufficient to
consider only one multipole term for the external gravitational field to
remove all the singularities from the double black hole configuration, at the
price of fixing the position or the mass of a black hole.. Of course one could
alternatively keep the $b_{n}$ generic, to model an arbitrary external
gravitational background: in that case the intrinsic black holes parameters
would adjust to fit the given background.

##### 2.3.1 Near-horizon limit

We consider the near-horizon limit of the binary metric (2.32), in order to
show that it contains two distorted Schwarzschild black holes. We zoom in to
the first black hole horizon by performing the change of coordinates

$\rho=\sqrt{r(r-2m_{1})}\sin\theta\,,\quad z=z_{1}+(r-m_{1})\cos\theta\,,$
(2.36)

and by taking the limit $r\to 2m_{1}$, by which the metric (2.32) boils down
to

$\begin{split}{ds}^{2}&\simeq
h(\theta)\biggl{[}-\biggl{(}1-\frac{2m_{1}}{r}\biggr{)}e^{F_{1}(\theta)}{dt}^{2}+\frac{D^{2}e^{F_{2}(\theta)}}{1-2m_{1}/r}{dr}^{2}\biggr{]}\\\
&\quad+(2m_{1})^{2}\biggl{[}D^{2}h(\theta)e^{F_{2}(\theta)}{d\theta}^{2}+\frac{\sin^{2}\theta}{h(\theta)}e^{-F_{1}(\theta)}{d\phi}^{2}\biggr{]}\,,\end{split}$
(2.37)

where

$\displaystyle h(\theta)$
$\displaystyle=\frac{m_{1}\cos\theta+m_{2}+z_{1}-z_{2}}{m_{1}\cos\theta-
m_{2}+z_{1}-z_{2}}\,,$ (2.38) $\displaystyle F_{1}(\theta)$
$\displaystyle=2[b_{1}+b_{2}(z_{1}+m_{1}\cos\theta)](z_{1}+m_{1}\cos\theta)\,,$
(2.39)
$\displaystyle\begin{split}F_{2}(\theta)&=2b_{1}(m_{1}\cos\theta-2m_{1}-4m_{2}-z_{1})\\\
&\quad+2b_{2}(m_{1}^{2}\cos^{2}\theta+2m_{1}z_{1}\cos\theta-2m_{1}^{2}-z_{1}^{2}-4m_{1}z_{1}-8m_{2}z_{2})\,,\end{split}$
(2.40)

and

$D=\frac{m_{1}+m_{2}-z_{1}+z_{2}}{m_{1}-m_{2}-z_{1}+z_{2}}\,.$ (2.41)

One clearly recognises the structure of a distorted Schwarzschild black hole
[146]. Actually the first black hole horizon is deformed by the presence of
both the external field and the second black hole. Indeed, when the external
field and the second black hole vanish, one recovers the standard
Schwarzschild metric. Obviously, a similar description holds for the second
black hole as well. A pictorial representation of the deformation that the two
horizons undergo is given in the embedding diagram of Fig. 2.4.

Figure 2.4: Embedding diagram in $\mathbb{E}^{3}$ of the surfaces of the two
black hole event horizons for the parametric values $z_{1}=5$, $z_{2}=15$,
$m_{1}=4$, $m_{2}=4$. This picture shows the deformation of the horizons due
to both the external gravitational field and the mutual interaction between
the sources. The horizon surface is smooth because of the absence of any
conical singularity.

Another element about the geometry of the black hole horizon can be clarified
by computing the length of the equatorial and polar circles in the near-
horizon geometry (2.37). The equator ($\theta=\pi/2$) length is given by

$L_{\text{equator}}=2m_{1}\int_{0}^{2\pi}\frac{e^{-F_{1}(\pi/2)/2}}{\sqrt{h(\pi/2)}}d\phi=4\pi
m_{1}\sqrt{\frac{z_{1}-z_{2}-m_{2}}{z_{1}-z_{2}+m_{2}}}e^{-2(b_{1}+b_{2}z_{1})z_{1}}\,,$
(2.42)

while the polar length is

$L_{\text{polar}}=4m_{1}D\int_{0}^{\pi}\sqrt{h(\theta)}e^{F_{2}(\theta)/2}d\theta\,.$
(2.43)

It is not possible to analytically perform the latter integral, nevertheless
we can consider a numerical comparison between (2.42), (2.43) and
$L_{\text{Schwarzschild}}=4\pi m_{1}$. The result is

$L_{\text{equator}}>L_{\text{Schwarzschild}}\,,\qquad
L_{\text{polar}}>L_{\text{Schwarzschild}}\,.$ (2.44)

Thus, not only a deformation along the $z$-axis occurs, but there is also an
enlargement of both the equatorial and the polar circle with respect to the
Schwarzschild one. This is consistent with the behaviour of the black hole
temperature, as we will see below.

##### 2.3.2 Thermodynamics

Going back to the exact metric (2.32), we now compute some physical quantities
for that spacetime, in order to discuss the thermodynamics of the system.

The mass of the two black holes is found by means of the Komar–Tomimatsu
integral [161, 162]. The formula for the conserved mass, in the case of the
static spacetime (2.32), takes the form

$M=\alpha\int_{w_{2i-1}}^{w_{2i}}dz\,\rho\,g_{tt}^{-1}\,\partial_{\rho}g_{tt}\big{|}_{\rho=0}\,,$
(2.45)

where $\alpha$ is a constant that takes into account the normalisation of the
timelike Killing vector $\xi=\alpha\partial_{t}$, which generates the
stationary symmetry. It is well known that, in the absence of asymptotic
flatness, $\alpha$ is not necessarily equal to one, as happens for black holes
in AdS [163] or Melvin [164, 165] backgrounds. The integration is performed
over the intervals $w_{1}<z<w_{2}$ and $w_{3}<z<w_{4}$, which correspond to
the black holes, and the result is

$M_{1}=\alpha m_{1}\,,\qquad M_{2}=\alpha m_{2}\,.$ (2.46)

The horizons areas are found integrating over the horizon surfaces, i.e.

$\mathcal{A}=\int_{0}^{2\pi}d\phi\int_{w_{2i-1}}^{w_{2i}}dz\,\sqrt{g_{zz}g_{\phi\phi}}\Big{|}_{\rho=0}\,,$
(2.47)

thus

$\displaystyle\mathcal{A}_{1}$ $\displaystyle=16\pi
m_{1}^{2}\frac{m_{1}+m_{2}-z_{1}+z_{2}}{m_{1}-m_{2}-z_{1}+z_{2}}e^{-2b_{1}(2m_{2}+m_{1}+z_{1})-2b_{2}((m_{1}+z_{1})^{2}+4m_{2}z_{2})}\,,$
(2.48) $\displaystyle\mathcal{A}_{2}$ $\displaystyle=16\pi
m_{2}^{2}\frac{m_{1}+m_{2}-z_{1}+z_{2}}{m_{2}-m_{1}-z_{1}+z_{2}}e^{-2b_{1}(m_{2}+z_{2})-2b_{2}(m_{2}+z_{2})^{2}}\,.$
(2.49)

The temperature is obtained from the surface gravity as $T=\kappa/(2\pi)$.
Recalling that $\kappa^{2}=-\frac{1}{2}(\nabla_{\mu}\xi_{\nu})^{2}$, the
metric (2.32) gives rise to

$\kappa^{2}=-\frac{\alpha^{2}}{4}\frac{(\partial_{z}V)^{2}+(\partial_{\rho}V)^{2}}{fV}\bigg{|}_{\rho=0}\,.$
(2.50)

The temperatures are then

$\displaystyle T_{1}$ $\displaystyle=\frac{\alpha}{8\pi
m_{1}}\frac{m_{1}-m_{2}-z_{1}+z_{2}}{m_{1}+m_{2}-z_{1}+z_{2}}e^{2b_{1}(2m_{2}+m_{1}+z_{1})+2b_{2}((m_{1}+z_{1})^{2}+4m_{2}z_{2})}\,,$
(2.51) $\displaystyle T_{2}$ $\displaystyle=\frac{\alpha}{8\pi
m_{2}}\frac{m_{2}-m_{1}-z_{1}+z_{2}}{m_{1}+m_{2}-z_{1}+z_{2}}e^{2b_{1}(m_{2}+z_{2})+2b_{2}(m_{2}+z_{2})^{2}}\,.$
(2.52)

One can verify that the same results are found via the Euclidean method [166].
We notice that the presence of the external field lowers the black holes
temperature, with respect to the Schwarzschild one, and hence the surface
gravity. A lower surface gravity means a lower gravitational “pressure” on the
horizon, which then can swell up. This feature is in agreement with (2.44) and
with the related observations. Moreover, it explains how the external
gravitational field acts, providing an external pressure in the region of the
holes, to sustain the mutual gravitational collapse of the binary system.

Defining the entropy as $S=\mathcal{A}/4$, the above quantities satisfy the
Smarr law [167] both for the individual black holes $M_{i}=2T_{i}S_{i}$
($i=1,2$) and for the double configuration

$M_{1}+M_{2}=2T_{1}S_{1}+2T_{2}S_{2}\,.$ (2.53)

This result holds regardless of the value of the constant $\alpha$.
Nevertheless, a choice for $\alpha$ must be done in order to study the
thermodynamics of the system.

We are interested in the first law of thermodynamics from a local point of
view: the involved quantities are evaluated on the horizons, therefore the
sources at infinity (which generate the external field) are not accessible to
local observers near the black holes. Hence we will discard work terms, in the
first law, due to the variation of the parameters $b_{n}$ [146].

We consider the system at thermal equilibrium from now on, i.e.
$T_{1}=T_{2}\equiv\bar{T}$. This condition is satisfied by imposing
$m_{1}=m_{2}$. We furthermore choose

$\alpha=\sqrt{\frac{z_{1}-z_{2}-2m_{1}}{z_{1}-z_{2}}}e^{-(b_{1}+b_{2}(m_{1}+z_{2}))(m_{1}+z_{2})}\,,$
(2.54)

in order to fulfill a Christodoulou–Ruffini mass formula [168], as it happens
for regular metrics in which the asymptotic symmetry is different from the
flat one [165, 169, 170]. For black hole configurations endowed with $N$
disconnected horizons the best proposal is an additive generalisation such
that

$\sum_{i=1}^{N}M_{i}=\sum_{i=1}^{N}\sqrt{\frac{\mathcal{A}_{i}}{16\pi}}\,.$
(2.55)

It is worth noting that $m_{1}=m_{2}$ is not the only possibility for thermal
equilibrium, but it is clearly the simplest one. Moreover, this choice
guarantees the integrability of the masses (2.46).