IFUM-1092-FT

Many accelerating distorted black holes
Marco<EMAIL_ADDRESS>and Adriano
<EMAIL_ADDRESS>

aIstituto Nazionale di Fisica Nucleare (INFN), Sezione di Milano
Via Celoria 16, I-20133 Milano, Italy
bUniversità degli Studi di Milano

Via Celoria 16, I-20133 Milano, Italy

Abstract

An analytical solution of four-dimensional General Relativity, representing an
array of collinear and accelerating black holes, is constructed with the
inverse scattering method. The metric can be completely regularised from any
conical singularity, thanks to the presence of an external gravitational
field. Therefore the multi-black hole configuration can be maintained at
equilibrium without the need of string or struts. Some notable subcases such
as the accelerating distorted Schwarzschild black hole and the distorted
double C-metric are explicitly presented. The Smarr law and the thermodynamics
of these systems is studied. The Bonnor–Swaminarayan and the
Bičák–Hoenselaers–Schmidt particle metrics are recovered, through appropriate
limits, from the multi-black holes solutions.

###### Contents

  1. 1 Introduction
  2. 2 Inverse scattering method
    1. 2.1 The inverse scattering construction, a brief review
    2. 2.2 Accelerating multipolar gravitational background
  3. 3 Array of accelerating black holes in an external gravitational field
    1. 3.1 Behaviour at infinity and acceleration horizon
    2. 3.2 Regularisation
    3. 3.3 Smarr law
  4. 4 Accelerating Schwarzschild black hole in an external gravitational field
  5. 5 Double C-metric in an external gravitational field
    1. 5.1 Thermodynamics
  6. 6 Accelerating particles in an external gravitational field
    1. 6.1 The limit to the Bonnor–Swaminarayan metric
    2. 6.2 The limit to the Bičák–Hoenselaers–Schmidt metric
  7. 7 Summary and Conclusions

##  1 Introduction

Binary and multi-black hole systems are progressively acquiring more and more
relevance in the understanding of the large scale structure and interactions
of our universe. That is because gravitational waves detectors can most easily
capture gravitational radiation from very massive sources interplay. These
systems generally are not isolated but they belong to larger gravitational
structures such as galaxies, or clusters of galaxies which contribute to
deform the gravitational field around the black holes and also the behaviour
at large distances. For these reasons it is worth exploring the possibility to
relax the standard asymptotic boundary conditions: indeed global asymptotic
flatness is well known to constitute a big theoretical barrier against the
introduction of novel features in the black hole picture, basically because
the presence of no-hair theorems in four-dimensional General Relativity.
Global asymptotic flatness also represents a practical obstacle in building an
equilibrium, even just meta-stable, multi-black hole configuration, because of
the very attractive property of gravity which tends to aggregate and merge
matter, when it is not contrasted.

Recently it has been shown in [1, 2] that the introduction of an external
multipolar gravitational background can be useful to analytically build, in
pure Einstein gravity, black hole metrics of the Israel–Khan type [3], which
can remain at equilibrium without the need of singular matter, such as strings
or struts. In fact these objects plague the spacetime bringing conical defects
that violate all reasonable energy conditions or producing $\delta$-like
divergences of infinite length. We would like to avoid these objects because
they are theoretically problematic and because there is no clue of their
phenomenological existence, so far.

The multipolar gravitational field is not a novel background for (single)
black holes. First introduced by Doroshkevich, Zel’dovich and Novikov [4] in
the ’50s, these _distorted black holes_ have been later studied by several
authors including Chandrasekhar [5] and Geroch–Hartle [6] in the ’80s, and
several others in more recent times [7, 8, 9]. The novelty of our approach is
the use of this background to remove conical singularities which usually stem
superimposing more black sources. From a physical point of view it means that
the collapse is avoided thanks to the gravitational attraction of the matter
surrounding the binary system. Still, as in the single black hole case, these
metrics have to be considered fundamentally as local models. They are intended
to describe the physics of realistic black holes, which are more often
surrounded by gravitating matter other than isolated and idealised systems. In
fact, nowadays, because of their peculiar astrophysical nature, observed black
holes are mainly detected indirectly, thanks to their interaction with
bordering matter of the accretion disk. While these are pure vacuum Einstein
manifolds, usually one expects that the full solution may be complemented by
some matter distribution which models the presence of a galaxy around the
black holes and which provides the desired multipolar expansion.

In this article, inspired by an Ernst’s insight in [10], where he was able to
remove the conical singularity of an accelerating black hole thanks to the
first non-trivial term (the dipole) of the multipolar expansion of the
external gravitational field, we want to push forward his idea and, at the
same time, incorporate multi-black hole sources. More specifically, we want
$(i)$ generalise the regularised C-metric constructed by Ernst [10] including
the full multipolar expansion of the external gravitational field and $(ii)$
embed an arbitrary number of collinear accelerating black holes into a
gravitational background. The introduction of the acceleration in the deformed
black hole scenario, is not just for sake of generality. But the acceleration
parameter often brings with it an extra Killing horizon and a conformal
infinity factor, which can improve the local interpretation of these black
hole systems embedded in an external gravitational field. The idea we pursue
in this work was also pioneered by Gibbons, but by regularising with negative
mass particles [11].

We will focus only on axisymmetric and stationary spacetimes because the
solution generating technique we use to build these solution, the inverse
scattering method, briefly resumed in section 2, is available only under these
assumptions.

The resulting metrics will be analysed in section 3, some simple examples will
be explicitly worked out in sections 4 and 5, and finally some notable limits
to well known metrics are unveiled in section 6.

##  2 Inverse scattering method

The inverse scattering method [12, 13, 14] allows to superimpose black holes
on top of a background spacetime. For this reason, we start our discussion by
presenting the main features of such a solution generation technique and then
by setting the so-called “seed”, that is necessary to construct the
accelerating black hole spacetimes.

###  2.1 The inverse scattering construction, a brief review

The inverse scattering method takes advantage of integrability of the Einstein
equations for the class of stationary and axisymmetric metrics, which can be
described in Weyl coordinates [15] by

${ds}^{2}=f(\rho,z)({d\rho}^{2}+{dz}^{2})+g_{ab}(\rho,z){dx}^{a}{dx}^{b}\,,$
(2.1)

where $a,b=0,1$ and $x^{0}=t$, $x^{1}=\phi$. The spacetime (2.1) has two
commuting Killing vectors proportional to $\partial_{t}$ and
$\partial_{\phi}$. We postulate that the coordinate $\rho$ is chosen such that
$\det g=-\rho^{2}$.

The vacuum Einstein equations $R_{\mu\nu}=0$ can be equivalently written as

$\displaystyle U_{,\rho}+V_{,z}$ $\displaystyle=0\,,$ (2.2a)
$\displaystyle(\log f)_{,\rho}$
$\displaystyle=-\frac{1}{\rho}+\frac{1}{4\rho}\text{Tr}(U^{2}-V^{2})\,,$
(2.2b) $\displaystyle(\log f)_{,z}$
$\displaystyle=\frac{1}{2\rho}\text{Tr}(UV)\,,$ (2.2c)

where $U=\rho g_{,\rho}g^{-1}$ and $V=\rho g_{,z}g^{-1}$ are two $2\times 2$
matrices. We note that, by solving equation (2.2a) for $g$, one is able to
find the function $f$ in quadratures from equations (2.2b) and (2.2c). Thus,
the problem of integrating the vacuum Einstein equations is reduced to the
problem of constructing the matrix $g$.

One can prove that the integrability condition for the Einstein equations
(2.2) is equivalent to the linear eigenvalue equations

$D_{1}\Psi=\frac{\rho V-\lambda U}{\lambda^{2}+\rho^{2}}\Psi\,,\qquad
D_{2}\Psi=\frac{\rho U+\lambda V}{\lambda^{2}+\rho^{2}}\Psi\,,$ (2.3)

for the generating matrix $\Psi(\rho,z,\lambda)$, where the commuting
differential operators $D_{1}$ and $D_{2}$ are given by

$D_{1}\coloneqq\partial_{z}-\frac{2\lambda^{2}}{\lambda^{2}+\rho^{2}}\partial_{\lambda}\,,\quad
D_{2}\coloneqq\partial_{\rho}+\frac{2\lambda\rho}{\lambda^{2}+\rho^{2}}\partial_{\lambda}\,,$
(2.4)

and $\lambda$ is a complex spectral parameter.

In the inverse scattering technique we pick up a seed solution
$(g_{0},f_{0})$, and then construct a generating matrix $\Psi_{0}$ which
satisfies the linear equations (2.3). Given such a $\Psi_{0}$, we introduce
two functions

$\displaystyle\mu_{k}(\rho,z)$
$\displaystyle=\sqrt{\rho^{2}+(z-w_{k})^{2}}-(z-w_{k})\,,$ (2.5a)
$\displaystyle\bar{\mu}_{k}(\rho,z)$
$\displaystyle=-\sqrt{\rho^{2}+(z-w_{k})^{2}}-(z-w_{k})\,,$ (2.5b)

where $w_{k}$ are arbitrary (complex) constants, which are called poles.
$\mu_{k}$ and $\bar{\mu}_{k}$ are called solitons and anti-solitons,
respectively, and they satisfy the relation $\mu_{k}\bar{\mu}_{k}=-\rho^{2}$.

We associate a 2-components vector (the BZ vector) to each (anti-)soliton

$m_{a}^{(k)}=m_{0\,b}^{(k)}\bigl{[}\psi_{0}^{-1}(\mu_{k},\rho,z)\bigr{]}_{ba}\,,$
(2.6)

where $m_{0\,a}^{(k)}$ are arbitrary constants. Defined the matrix

$\Gamma_{kl}=\frac{m_{a}^{(k)}(g_{0})_{ab}m_{b}^{(l)}}{\rho^{2}+\mu_{k}\mu_{l}}\,,$
(2.7)

a new metric is found by adding $2N$ solitons to $(g_{0},f_{0})$ as

$\displaystyle g_{ab}$
$\displaystyle=\pm\rho^{-2N}\Biggl{(}\prod_{k=1}^{2N}\mu_{k}\Biggr{)}\Biggl{[}(g_{0})_{ab}-\sum_{k,l=1}^{2N}\frac{(\Gamma^{-1})_{kl}L_{a}^{(k)}L_{b}^{(l)}}{\mu_{k}\mu_{l}}\Biggr{]}\,,$
(2.8a) $\displaystyle f$
$\displaystyle=16C_{f}f_{0}\rho^{-(2N)^{2}/2}\Biggl{(}\prod_{k=1}^{2N}\mu_{k}^{2N+1}\Biggr{)}\Biggl{[}\prod_{k>l=1}^{2N}(\mu_{k}-\mu_{l})^{-2}\Biggr{]}\det\Gamma\,,$
(2.8b)

where $L_{a}^{(k)}=m_{c}^{(k)}(g_{0})_{ca}$ and $C_{f}$ is an arbitrary
constant. The new metric (2.8) fulfils by construction the Einstein equations
(2.2), and also $\det g=-\rho^{2}$.

###  2.2 Accelerating multipolar gravitational background

A natural way to obtain the C-metric by means of the inverse scattering
technique is to immerse a black hole in an accelerating background. This means
that two solitons have to be added to the Rindler spacetime, which is nothing
but Minkowski spacetime adapted to an accelerated observer. The Rindler
spacetime can be expressed in Weyl coordinates in the following way:

$\displaystyle g_{\text{acc}}$
$\displaystyle=\text{diag}\biggl{(}-\mu_{A},\frac{\rho^{2}}{\mu_{A}}\biggr{)}\,,$
(2.9a) $\displaystyle f_{\text{acc}}$
$\displaystyle=\frac{\mu_{A}}{\rho^{2}+\mu_{A}^{2}}\,,$ (2.9b)

where $\mu_{A}=\sqrt{\rho^{2}+(z-w_{A})^{2}}-(z-w_{A})$ is the soliton which
contains the acceleration parameter of the Rindler metric. A nice
parametrisation for the constant $w_{A}$ is indeed

$w_{A}=\frac{1}{2A}\,,$ (2.10)

where $A$ is the acceleration. The addition of two solitons to (2.9) gives the
standard C-metric, while more (even) solitons allow to construct the
accelerating multi-black hole metric discovered by Dowker and Thambyahpillai
[16].

We want to immerse many accelerating black holes in an external gravitational
field, hence the natural background is the accelerated version of the external
field background presented in [2]. The external gravitational field is
described by the metric

$\displaystyle g_{\text{ext}}$
$\displaystyle=\text{diag}\Biggl{[}-\exp\biggr{(}2\sum_{n=1}^{\infty}b_{n}r^{n}P_{n}\biggr{)},\rho^{2}\exp\biggl{(}-2\sum_{n=1}^{\infty}b_{n}r^{n}P_{n}\biggr{)}\Biggr{]}\,,$
(2.11a) $\displaystyle f_{\text{ext}}$
$\displaystyle=\exp\Biggl{[}2\sum_{n,p=1}^{\infty}\frac{npb_{n}b_{p}r^{n+p}}{n+p}\bigl{(}P_{n}P_{p}-P_{n-1}P_{p-1}\bigr{)}-2\sum_{n=1}^{\infty}b_{n}r^{n}P_{n}\Biggr{]}\,,$
(2.11b)

where $r\coloneqq\sqrt{\rho^{2}+z^{2}}$ and $P_{n}=P_{n}(z/r)$ are the
Legendre polynomials. The parameters $b_{n}$ are related to the multipole
momenta of the external field. Since the $b_{n}$ can be chosen at will, the
metric (2.11) can describe a gravitational field generated by a generic
distribution of matter [17]. The non-accelerating multi-black hole solutions
immersed in the background (2.11) were constructed in [2].

The metric which includes both the acceleration (2.9) and the external field
background (2.11) is naturally given by

$\displaystyle g_{0}$
$\displaystyle=\text{diag}\Biggl{[}-\mu_{A}\exp\biggr{(}2\sum_{n=1}^{\infty}b_{n}r^{n}P_{n}\biggr{)},\frac{\rho^{2}}{\mu_{A}}\exp\biggl{(}-2\sum_{n=1}^{\infty}b_{n}r^{n}P_{n}\biggr{)}\Biggr{]}\,,$
(2.12a)
$\displaystyle\begin{split}f_{0}&=\frac{\mu_{A}}{\rho^{2}+\mu_{A}^{2}}\exp\Biggl{[}2\sum_{n,p=1}^{\infty}\frac{npb_{n}b_{p}r^{n+p}}{n+p}\bigl{(}P_{n}P_{p}-P_{n-1}P_{p-1}\bigr{)}-2\sum_{n=1}^{\infty}b_{n}r^{n}P_{n}\\\
&\quad+\frac{\rho^{2}+\mu_{A}^{2}}{\mu_{A}}\sum_{n=1}^{\infty}b_{n}\sum_{l=0}^{n-1}w_{A}^{n-1-l}\,r^{l}P_{l}\Biggr{]}\,,\end{split}$
(2.12b)

We see that by turning off the external field (i.e. $b_{n}=0$), one is left
with the Rindler spacetime only. On the converse, by removing the
acceleration, in the limit $w_{A}\to\infty$, one recovers the background
(2.11).

Thus, we will take (2.12) as a background to construct our black hole
spacetime. Since the addition of solitons in the inverse scattering technique
is equivalent to the addition of black holes to the seed spacetime (2.12), it
is quite natural to interpret the resulting metric as a collection of many
black holes which are accelerating in an external gravitational field.
Following the discussion in section 2, we need the generating matrix
$\Psi_{0}$ to build a black hole spacetime on top of the background (2.12).
The function which satisfies equations (2.3) generalises the one presented in
[18], and it is given by

$\Psi_{0}(\rho,z,\lambda)=\begin{pmatrix}-(\lambda-\mu_{A})e^{F(\rho,z,\lambda)}&0\\\
0&(\lambda+\rho^{2}/\mu_{A})e^{-F(\rho,z,\lambda)}\end{pmatrix}\,,$ (2.13)

where

$F(\rho,z,\lambda)=2\sum_{n=1}^{\infty}b_{n}\Biggl{[}\sum_{l=0}^{\infty}\binom{n}{l}\biggl{(}\frac{-\rho^{2}}{2\lambda}\biggr{)}^{l}\biggl{(}z+\frac{\lambda}{2}\biggr{)}^{n-l}-\sum_{l=1}^{n}\sum_{k=0}^{[(n-l)/2]}\frac{(-1)^{k+l}2^{-2k-l}n!\lambda^{-l}}{k!(k+l)!(n-2k-l)!}\rho^{2(k+l)}z^{n-2k-l}\Biggr{]}\,.$
(2.14)

Now we can construct the BZ vectors (2.6): 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

$m^{(k)}=\biggl{(}-\frac{C_{0}^{(k)}}{\mu_{k}-\mu_{0}}e^{-F(\rho,z,\mu_{k})},\frac{C_{1}^{(k)}\mu_{0}}{\rho^{2}+\mu_{0}\mu_{k}}e^{F(\rho,z,\mu_{k})}\biggr{)}\,.$
(2.15)

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

##  3 Array of accelerating black holes in an external gravitational field

We construct the generalisation of the Dowker–Thambyahpillai solution [16],
which represents an array of collinear accelerating black holes. The
Dowker–Thambyahpillai metric is characterised by the presence of conical
singularities, which can not be removed by a fine tuning of the physical
parameters without admitting naked singularities.

Given the accelerating background (2.12) and the generating matrix (2.13), we
construct a new solution by adding $2N$ solitons (which correspond to $N$
black holes) with constants

$C_{0}^{(k)}=\begin{cases}1&k\text{ even}\\\ 0&k\text{ odd}\end{cases},\qquad
C_{1}^{(k)}=\begin{cases}0&k\text{ even}\\\ 1&k\text{ odd}\end{cases}\,.$
(3.1)

This choice guarantees a non-rotating metric, which is the one we are
interested in. A different choice for these constants allows the inclusion of
the rotation parameter $a$: for the $k$-th pair of BZ constants one takes

$\displaystyle
C_{0}^{(2k-1)}C_{0}^{(2k)}+C_{1}^{(2k-1)}C_{1}^{(2k)}=-\sqrt{m_{k}^{2}-a_{k}^{2}}\,,\quad$
$\displaystyle
C_{0}^{(2k-1)}C_{0}^{(2k)}-C_{1}^{(2k-1)}C_{1}^{(2k)}=-m_{k}\frac{1-A^{2}a_{k}^{2}}{1+A^{2}a_{k}^{2}}\,,\quad$
(3.2a) $\displaystyle
C_{0}^{(2k-1)}C_{1}^{(2k)}+C_{1}^{(2k-1)}C_{0}^{(2k)}=-\frac{2Am_{k}a_{k}}{1+A^{2}a_{k}^{2}}\,,\quad$
$\displaystyle C_{0}^{(2k-1)}C_{1}^{(2k)}-C_{1}^{(2k-1)}C_{0}^{(2k)}=a_{k}\,.$
(3.2b)

One can verify that, in the single black hole case and with no external field,
i.e. $b_{i}=0$ for all $i$, the above parametrisation leads to the standard
form of the rotating C-metric [19]. Likewise the no external gravitational
field with the choice (3.2), in the multi black hole case, leads to a vacuum
multi-Plebanski-Demianski metric111We do expand explicitly here the full
expression of multi-rotating-C-metric, neither with nor without the external
gravitational field, because it is quite lengthy, but it can be
straightforwardly written down from equations (2.8) and (3.2).. Being
interested in the phenomenological setting, we have not included the NUT
parameter in the above definitions; however, it is possible to include the NUT
charge as well in the inverse scattering formalism [20].

The metric resulting from the diagonal choice (3.1) is

$\displaystyle g_{N}$
$\displaystyle=\text{diag}\Biggl{[}-\mu_{A}\frac{\prod_{k=1}^{N}\mu_{2k-1}}{\prod_{l=1}^{N}\mu_{2l}}\exp\Biggl{(}2{\sum_{n=1}^{\infty}b_{n}r^{n}P_{n}}\Biggr{)},\frac{\rho^{2}}{\mu_{A}}\frac{\prod_{l=1}^{N}\mu_{2l}}{\prod_{k=1}^{N}\mu_{2k-1}}\exp\Biggl{(}-2{\sum_{n=1}^{\infty}b_{n}r^{n}P_{n}}\Biggr{)}\Biggr{]}\,,$
(3.3a)
$\displaystyle\begin{split}f_{N}&=16C_{f}\,f_{0}\,\frac{\mu_{A}^{2N+1}}{\rho^{2}+\mu_{A}^{2}}\Biggl{(}\prod_{k=1}^{2N}\mu_{k}^{2N+1}\Biggr{)}\Biggl{(}\prod_{k=1}^{N}\frac{1}{(\mu_{A}-\mu_{2k})^{2}}\Biggr{)}\Biggl{(}\prod_{k=1}^{N}\frac{1}{(\rho^{2}+\mu_{A}\mu_{2k-1})^{2}}\Biggr{)}\\\
&\quad\times\Biggl{(}\prod_{k=1}^{2N}\frac{1}{\rho^{2}+\mu_{k}^{2}}\Biggr{)}\Biggl{(}\prod_{k=1,l=1,3,\cdots}^{2N-1}\frac{1}{(\mu_{k}-\mu_{k+l})^{2}}\Biggr{)}\Biggl{(}\prod_{k=1,l=2,4,\cdots}^{2N-2}\frac{1}{(\rho^{2}+\mu_{k}\mu_{k+l})^{2}}\Biggr{)}\\\
&\quad\times\exp\Biggl{[}2\sum_{k=1}^{2N}(-1)^{k+1}F(\rho,z,\mu_{k})\Biggr{]}\,.\end{split}$
(3.3b)

Metric (3.3) is, by construction, a solution of the vacuum Einstein equations
(2.2), and it represents a collection of $N$ accelerating black holes, aligned
along the $z$-axis, and immersed in the external gravitational field (2.12).
Actually, as it usually happens for the C-metrics, the result (3.3) can be
interpreted as $N$ _pairs_ of black holes which accelerate in two opposite
directions [21]. However, since the black holes in each pair are causally
disconnected, being on two opposite sides of the acceleration horizon and
unable to communicate with each other, we restrict our attention to one of the
two sides only, and we focus mostly on the genuine $N$-black hole solution.

We consider real poles $w_{k}$, since it represents the physically relevant
situation. These constants are chosen with ordering
$w_{1}<w_{2}<\cdots<w_{2N-1}<w_{2N}<w_{A}$ and with
parametrisation222Parametrisation (3.4) slightly differs from the standard one
presented in [22], however it is coherent with [21]. The main advantage of
(3.4) is its close resemblance with the one used for the non-accelerating case
[2] and it is more suitable for non accelerating limits.

$w_{1}=z_{1}-m_{1}\,,\quad w_{2}=z_{1}+m_{1}\,,\quad\dotsc\quad
w_{2N-1}=z_{N}-m_{N}\,,\quad w_{N}=z_{N}+m_{N}\,,\quad w_{A}=\frac{1}{2A}\,.$
(3.4)

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

The black hole horizons correspond to the regions $w_{2k-1}<z<w_{2k}$
($k=1,\dotsc,N$), while the complementary regions are, in principle, affected
by the presence of conical singularities, as happens for the
Dowker–Thambyahpillai metric (cf. Fig. 1). The metric (3.3) constitutes an
extension of the multi-black hole solution presented in [2], because here we
incorporated an additional acceleration horizon which corresponds to the
region $z>w_{A}$ of the spacetime.

$w_{1}$$w_{2}$$w_{3}$$w_{2N-1}$$w_{2N}$$w_{A}$$z$$t$$\phi$ Figure 1: Rod
diagram for the multi-black hole spacetime (3.3). 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.

###  3.1 Behaviour at infinity and acceleration horizon

The multi-black hole solution (3.3) comes with the black holes curvature
singularities which are covered, as usual, by the event horizons. These
singularities do not represent a problem, being the usual ones which are
encountered in black hole spacetimes.

However, the external gravitational field metric (2.11) (and then (3.3)) may
be characterised by an unbounded growth of curvature invariants at spatial
infinity, which corresponds to $\sqrt{\rho^{2}+z^{2}}\to\infty$ in Weyl
coordinates. See [8] for a detailed study of possible curvature singularity in
the domain of outer communication, for the distorted Kerr black hole. This
feature is due to the presence of the sources that generate the external
gravitational field, and the latter are thought to be located at large
distances from the horizon. In this respect, as already remarked in [1] and
[2], this kind of metrics have to be considered _local_ , in the sense that
the global solution would correspond to the matching between the black hole
spacetime and an energy-momentum tensor which generates the external field
[17]. Hence the behaviour at infinity does not invalidate the physics in
proximity of the black holes: in fact the Smarr law, as well as the first law
of thermodynamics, can be achieved for such systems. Moreover, since these
systems are regular in the neighbourhood the horizons, one can even study the
second law of thermodynamics in a non-trivial setting [1].

The novelty of the spacetime (3.3) presented here, is that the curvature
unboundedness at infinity is covered by the acceleration horizon given by
$z>w_{A}$. Thus, the local interpretation of the spacetime is improved in our
setting, since the metric is mostly meaningful, between the event and the
acceleration horizon. Being the curvature singularity not directly accessible,
the spacetime is completely regular in the physical regions, where observers
enjoy the usual metric signature for the manifold and the local model for the
distorted multi-black hole system is supposed to hold.

###  3.2 Regularisation

The spacetime exhibits conical singularities when the ratio between the length
and the radius of small circles around the $z$-axis is different from $2\pi$.
A small circle around the $z$-axis has radius $R=\sqrt{g_{zz}}\rho$ and length
$L=2\pi\sqrt{g_{\phi\phi}}$ in Weyl coordinates [1]. Thus, the regularity
condition is nothing but $L/(2\pi R)\to 1$ as $\rho\to 0$. It is easy to prove
that, for the static and axisymmetric metric (2.1), such a condition is
equivalent to $\mathcal{P}\equiv fg_{tt}\to 1$ as $\rho\to 0$. In the case of
our multi-black hole metric (3.3), we can remove the angular defects by
choosing the gauge constant $C_{f}$, and by tuning the external field
parameters.

The constant $C_{f}$ is chosen as

$\begin{split}C_{f}&=2^{4N}\Biggl{[}\prod_{i=1}^{N}(w_{2i}-w_{2i-1})^{2}\Biggr{]}\Biggl{[}\prod_{k=1}^{N-1}\prod_{j=1}^{N-k}(w_{2k-1}-w_{2k+2j})^{2}(w_{2k}-w_{2k+2j-1})^{2}\Biggr{]}\\\
&\quad\times\Biggl{[}\prod_{l=1}^{N}(w_{A}-w_{2l-1})^{2}\Biggr{]}\exp\Biggl{(}-2\sum_{n=1}^{\infty}b_{n}w_{A}^{n}\Biggl{)}\,.\end{split}$
(3.5)

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

$\mathcal{P}_{k}=\Biggl{[}\prod_{i=k}^{N-1}\frac{(w_{A}-w_{2i+1})^{2}}{(w_{A}-w_{2i+2})^{2}}\Biggr{]}\Biggl{[}\prod_{i=1}^{2k}\prod_{j=2k+1}^{2N}(w_{j}-w_{i})^{2\,(-1)^{i+j+1}}\Biggr{]}\exp\Biggl{[}4\sum_{n=1}^{\infty}b_{n}\sum_{j=2k+1}^{2N}(-1)^{j+1}w_{j}^{n}\Biggl{]}\,,$
(3.6)

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

$\mathcal{P}_{0}=\Biggl{[}\prod_{i=1}^{N}\frac{(w_{A}-w_{2i-1})^{2}}{(w_{A}-w_{2i})^{2}}\Biggr{]}\exp\Biggl{[}4\sum_{n=1}^{\infty}b_{n}\sum_{j=1}^{2N}(-1)^{j+1}w_{j}^{n}\Biggl{]}\,,$
(3.7)

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

$\mathcal{P}_{N}=1\,,$ (3.8)

thanks to our choice of $C_{f}$.

The expressions (3.6), (3.7) provide a system of equations
$\mathcal{P}_{k}=1$, which can be solved for $b_{1},\dotsc,b_{N}$ to
completely regularise the spacetime.

###  3.3 Smarr law

Let us derive the thermodynamic parameters which appear in the Smarr law.
Firstly, we compute the mass of the spacetime by means of the Komar–Tomimatsu
integral [23, 24]. The result for the $k$-th black hole (i.e. the black hole
in the interval $w_{2k-1}<z<w_{2k}$) is

$M_{k}=\alpha\int_{w_{2k-1}}^{w_{2k}}\rho
g_{tt}^{-1}\partial_{\rho}g_{tt}=\frac{\alpha}{2}(w_{2k}-w_{2k-1})=\alpha
m_{k}\,,$ (3.9)

where $\alpha$ is a constant which takes into account the proper normalisation
of the timelike Killing vector, generator of the horizon,
$\xi=\alpha\partial_{t}$. In general $\alpha$ is not unitary for not
asymptotically flat spacetimes.

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

$S_{k}=\frac{1}{4}\lim_{\rho\to
0}\int_{0}^{2\pi}d\phi\int_{w_{2k-1}}^{w_{2k}}dz\sqrt{fg_{\phi\phi}}=\pi
m_{k}W\exp\Biggl{[}2\sum_{n=1}^{\infty}b_{n}\sum_{j=2k}^{2N}(-1)^{j+1}w_{j}^{n}\Biggl{]}\,,$
(3.10)

where

$\log W=\lim_{\rho\to 0}\log\sqrt{fg_{\phi\phi}}=\log
2+\sum_{i=1}^{2k-1}\sum_{j=2k}^{2N}(-1)^{i+j+1}\log|w_{j}-w_{i}|+\sum_{i=2k}^{2N}(-1)^{i+1}\log|w_{A}-w_{i}|\,.$
(3.11)

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

$T_{k}=\frac{\alpha}{2\pi}\lim_{\rho\to
0}\rho^{-1}\sqrt{\frac{g_{tt}}{f}}=\frac{\alpha}{2\pi}\lim_{\rho\to
0}\frac{1}{\sqrt{fg_{\phi\phi}}}=\frac{\alpha m_{k}}{2S_{k}}\,.$ (3.12)

It is easy to show, by using (3.9), (3.10) and (3.12), that the Smarr law is
satisfied:

$\sum_{k=1}^{N}M_{k}=2\sum_{k=1}^{N}T_{k}S_{k}\,.$ (3.13)

The thermodynamics quantities just computed, can be compared with the standard
ones in the absence of the external gravitational field [25].

##  4 Accelerating Schwarzschild black hole in an external gravitational field

The first specialization of the multi-source metric proposed above it is worth
discussing is the single black hole case, i.e. $N=1$. The very first prototype
of these kind of metrics were built by Ernst [10] with the aim of regularising
the conical singularity of the C-metric, through the presence of an external
gravitational field possessing only the first term of the external multipolar
expansion, the dipole333The zeroth-order of the external multipolar expansion
is just a constant that can be reabsorbed. While the dipole term in the
standard internal multipolar expansion is often, under certain assumptions,
washed away thanks to a coordinate shift to the center of mass, in this
external multipolar expansion this change of reference cannot erase the dipole
contribution.. The $N=1$ characterization of the metric (3.3) represents the
full multipolar expansion of the external gravitational field with respect to
the Ernst solution. Moreover our metric carries an extra parameter $z_{1}$
which describes the position of the black hole with respect to the
multipoles444This differs with respect to the Ernst metric, which considered a
fixed position for the black hole in the center of the coordinate system, that
is for $-m/A<z<m/A$.. The metric is quite simple and can be written as (2.1)
with

$\displaystyle g_{\text{1}}$
$\displaystyle=\text{diag}\Biggl{[}-\frac{\mu_{1}\mu_{A}}{\mu_{2}}\exp\biggr{(}2\sum_{n=1}^{2}b_{n}r^{n}P_{n}\biggr{)},\rho^{2}\frac{\mu_{2}}{\mu_{1}\mu_{A}}\exp\biggl{(}-2\sum_{n=1}^{2}b_{n}r^{n}P_{n}\biggr{)}\Biggr{]}\,,$
(4.1a) $\displaystyle f_{\text{1}}$
$\displaystyle=-\frac{16C_{f}f_{0}w_{1}^{2}\mu_{1}^{3}\mu_{2}^{3}\exp\left[2F(\rho,z,\mu_{1})-2F(\rho,z,\mu_{2})\right]}{(\rho^{2}+\mu_{1}^{2})(\rho^{2}+\mu_{2}^{2})(\mu_{A}-\mu_{1})^{2}(\mu_{A}-\mu_{2})^{2}(\mu_{1}-\mu_{2})^{2}}\,,$
(4.1b)

where the sum in $F$ (2.14) is limited to the second order. The rod diagram
remains the same of the standard C-metric, since the poles are not affected by
the presence of the external gravitational field, as can be appreciated in
figure 2.

$w_{1}$$w_{2}$$w_{A}$$z$$t$$\phi$ Figure 2: Rod diagram for the C-metric
immersed in the external field (4.1). We notice that the presence of the
external field does not affect the rod diagram, i.e. the structure of the
poles. Being the rods defined by the poles $w_{n}$ (see [26]), we obtain the
usual C-metric diagram.

The limit to the standard C-metric in spherical coordinates
$(t,r,\theta,\varphi)$

$\begin{split}{ds}^{2}&=\frac{1}{(1+Ar\cos\theta)^{2}}\biggl{[}-\biggl{(}1-\frac{2m}{r}\biggr{)}\bigl{(}1-A^{2}r^{2}\bigr{)}{dt}^{2}+\frac{dr^{2}}{\bigl{(}1-\frac{2m}{r}\bigr{)}\bigl{(}1-A^{2}r^{2}\bigr{)}}\\\
&\quad+\frac{r^{2}{d\theta^{2}}}{(1+2Am\cos\theta)}+r^{2}(1+2Am\cos\theta)\sin^{2}\theta{d\phi}^{2}\biggr{]}\,,\end{split}$
(4.2)

is obtained thanks to the rescaling of time $t\to\sqrt{A}\,t$, azimuthal angle
$\phi\to\phi/\sqrt{A}$ and the following change of coordinates

$\rho=\frac{\sin\theta\sqrt{r(r-2m)(1-A^{2}r^{2})(1+2Am\cos\theta)}}{(1+Ar\cos\theta)^{2}}\,,\quad
z=z_{1}+\frac{(Ar+\cos\theta)(r-m+Amr\cos\theta)}{(1+Ar\cos\theta)^{2}}\,,$
(4.3)

vanishing $z_{1}$ and the whole multipolar expansion: $b_{n}=0,\,\,\forall n$.
To obtain exactly Eq. (4.2), one has to fix the solitonic constants in terms
of the physical quantities

$w_{1}=z_{1}-m\,,\quad w_{2}=z_{1}+m\,,\quad w_{A}=\frac{1}{2A}\,,\quad
C_{f}=\frac{1}{8A^{3}}\,.$ (4.4)

These coordinates and parametrization are also useful for describing the
accelerating and distorted black hole metric.

However, in order to remove the two conical singularities, which are generally
present in the $z$-axis of the single accelerating black hole above, some
constants have to be properly constrained, as explained in section 3.2. For
simplicity, in this section we will focus on the first two terms of the
multipolar expansion: $b_{n}=0,\,\forall n>2$. In this case we can explicitly
write down the values of the physical parameters which regularise the metric
for $z\in(z_{1}+m,1/2A)$ and $z\in(-\infty,z_{1}-m)$ respectively:

$\displaystyle\begin{split}C_{f}&=-\frac{(w_{1}-w_{2})^{2}(w_{1}-w_{A})^{2}}{8w_{1}^{2}}\exp\bigl{[}-2w_{A}(b_{1}+b_{2}w_{A})\bigr{]}\\\
&=-\frac{m+2Am(m-z_{1})^{2}}{8A^{2}(m-z_{1})^{2}}\exp\biggl{(}-\frac{2Ab_{1}+b_{2}}{2A^{2}}\biggr{)}\,,\end{split}$
(4.5) $\displaystyle b_{1}$
$\displaystyle=\frac{2b_{2}(w_{2}^{2}-w_{1}^{2})+\log(\frac{w_{2}-w_{A}}{w_{1}-w_{A}})}{2(w_{1}-w_{2})}=-2b_{2}z_{1}-\frac{1}{4m}\log\biggl{[}\frac{1-2A(m+z_{1})}{1+2A(m-z_{1})}\biggr{]}\,.$
(4.6)

Note that the $w_{i}$ remain defined as in (4.4), while in the presence of the
external gravitational field $C_{f}$ can be upgraded to eliminate the
conicity, as done in (4.5). We remark that this external multipolar distortion
is usually designed to model black holes locally. Nevertheless, thanks to the
above regularisation, the spacetime remains completely regular in the physical
regions. In fact, between the event horizon and the accelerating horizon,
where the signature is $(-+++)$, the metric is free from both conical and
curvature singularities, for any finite value of the accelerating parameter
$A$ satisfying the rod ordering relation $z_{1}-m<\frac{1}{2A}$.

To have a better intuition on the physics introduced by the external external
field, it is instructive to analyse the weak field limit of the accelerating
metric (4.1), that is when the black hole mass parameter $m$ is small. In this
case we can appreciate the contribution of the external multipoles on the
acceleration imparted to inertial observer, which we consider located in the
origin of the (spherical) coordinates for simplicity. The tetra-dimensional
timelike worldline of an observer with proper time $\lambda$ and constant
radial $\bar{r}$ and polar coordinates $\bar{x}=\cos\bar{\theta}$ is

$y^{\mu}(\lambda)=\Biggl{(}\frac{1+A\bar{r}\bar{x}}{\sqrt{1-A^{2}\bar{r}^{2}}}e^{-\frac{\bar{r}}{2(1+A\bar{r}\bar{x})}\bigl{[}2b_{2}\bar{r}(A\bar{r}+\bar{x})^{2}+2b_{1}(A\bar{r}+\bar{x})(1+A\bar{r}\bar{x})^{2}-b_{2}\bar{r}(1-A^{2}\bar{r}^{2})(1-\bar{x}^{2})\bigr{]}}\lambda,\bar{r},0,0,\Biggr{)}\,.$
(4.7)

This choice fulfils the normalisation property of the four-velocity,
$u_{\mu}u^{\mu}=-1$, where $u^{\mu}\coloneqq dy^{\mu}(\lambda)/d\lambda$. The
absolute value of the four-acceleration,
$a^{\mu}\coloneqq(\nabla_{\nu}u^{\mu})u^{\nu}$, for this observer is given by

$|a|=\sqrt{a_{\mu}a^{\mu}}\
\Big{|}_{\bar{r}=0}=|A-b_{1}|\exp\biggl{(}-\frac{b_{2}+2Ab_{1}}{4A^{4}}\biggr{)}\,.$
(4.8)

Note that, because $a^{\mu}u_{\mu}=0$, $|a|$ corresponds also to the magnitude
of the three-acceleration in the rest frame of the observer, the external
gravitational field has a non-trivial role in the background acceleration
quantified by (4.8). In the vanishing multipoles limit, $b_{i}=0$, the
standard C-metric acceleration $|a|=A$ is retrieved from (4.8).

The non-relativistic limit, i.e. for small values of the accelerating
parameter $A\approx 0$, of the regularising condition (4.6) can provide some
further understanding of the multipolar deformations. In fact in this
approximation we expect to retrieve a Newtonian picture: the force felt by a
massive monopole in an uniform gravitational field is

$mA\approx\frac{1-e^{-4m(b_{1}+2b_{2}z_{1})}}{4}\,.$ (4.9)

One obtains a very simple expression when the external field is weak, i.e.
$b_{1}\approx b_{2}\approx 0$: in such a case the exponential in (4.9) can be
expanded and

$mA\approx m(b_{1}+2b_{2}z_{1}).$ (4.10)

The last equation is nothing but the Newton law $\vec{F}=m\vec{a}$, hence
$b_{1}+2b_{2}z_{1}$ is interpreted, in the Newtonian limit, as a constant
external gravitational field strength. We see that the regularisation
condition has a nice and physically transparent limit, which is consistent
with the analysis performed by Bonnor [27] on the Ernst solution.

The metric described in this section has interesting applications in the realm
of black hole pair creation. Usually it is speculated that in the presence of
a strong electromagnetic field the C-metric can describe a couple of casually
disconnected and charged black holes (eventually rotating [28]) popping-out
from vacuum and accelerating away [29, 30, 31]. In that picture the
regularising interaction between the electric charge of the black hole and the
background electromagnetic Bonnor–Melvin universe [32, 33] reveals to be
crucial. On the other hand, the metric presented in this section provides the
regularisation by means of the external gravitational field. Therefore, in
this setting the pair creation of black holes is fostered by the energy of the
external gravitational field. Thus this picture seems to be more
phenomenological, because it does not require the black holes to be constitute
by charged matter, an occurrence that appears outside empirical observations,
at the moment. In fact the metric (4.1) allows the pair creation of neutral
black holes in a gravitational background. The rate of the pair creation is
proportional to the intensity of the external gravitational field, but its
computation is outside the scope of this paper and will be done elsewhere
[34].

The thermodynamics quantities and the Smarr law follow directly from the
general multiple case presented in the section 3.3. Otherwise, the first law
of black hole thermodynamics can be retrieved as a trivial specialisation of
the double configuration studied in the next section, when one of the two
masses of the double configuration vanishes.

##  5 Double C-metric in an external gravitational field

The accelerating external gravitational background provides us a second
interesting opportunity to generalise the solution describing a binary black
hole system at equilibrium, as presented in [1, 2]. This extension not only
represents an enrichment of the physical model features, but it provides also
a mechanism to protect the most physical region of the black hole against the
unbounded growth of the scalar curvature invariants at spatial infinity. In
fact, for finite values of the radial coordinate $r$, an observer will
encounter the accelerating Killing horizon and the conformal infinity before
reaching spatial infinity, further enforcing the local nature of the model.

As explained in section 3, the metric can be analytically generated for the
whole multipolar series, which includes infinite independent terms, each one
with its independent integration constant. However in this section we work out
explicitly a simple example on a truncated multipolar expansion, namely
keeping only the dipole and quadrupole deformations. Indeed these two
quantities are sufficient to regularise the metric without constraining the
proper physical parameters of the black holes configuration555$F$ in $f_{2}$
is defined according to (2.14), but only up to the second order.. That’s
because the number of constraints on the physical parameters of the metric
coincides with the number of casually connected black holes, and we are now
considering a double accelerating c-metric. In this case the two blocks of the
spacetime metric can be written as follows666A Mathematica worksheet with this
metric can be found among the files of this arXiv paper and on the web-page
https://sites.google.com/site/marcoastorino/.

$\displaystyle g_{2}$
$\displaystyle=\text{diag}\Biggl{[}-\frac{\mu_{1}\mu_{3}\mu_{A}}{\mu_{2}\mu_{4}}\exp\biggr{(}2\sum_{n=1}^{2}b_{n}r^{n}P_{n}\biggr{)},\frac{\rho^{2}\mu_{2}\mu_{4}}{\mu_{1}\mu_{3}\mu_{A}}\exp\biggl{(}-2\sum_{n=1}^{2}b_{n}r^{n}P_{n}\biggr{)}\Biggr{]}\,,$
(5.1a) $\displaystyle f_{2}$
$\displaystyle=\frac{f_{1}w_{3}^{2}\mu_{1}^{2}\mu_{2}^{2}\mu_{3}^{5}\mu_{4}^{5}\exp[2F(\rho,z,\mu_{3})-2F(\rho,z,\mu_{4})]}{(\rho^{2}+\mu_{1}\mu_{3})(\rho^{2}+\mu_{3}^{2})(\rho^{2}+\mu_{2}\mu_{4})(\rho^{2}+\mu_{4}^{2})(\mu_{2}-\mu_{3})^{2}(\mu_{A}-\mu_{3})^{2}(\mu_{1}-\mu_{4})^{2}(\mu_{3}-\mu_{4})^{2}(\mu_{A}-\mu_{4})^{2}}\,,$
(5.1b)

where $f_{1}$ is the single-black hole value for the non-Killing elements of
the metric encountered in previous section. We have to properly tune three
parameters777Two of these parameters, the $b_{i}$, are relate to physical
quantities, while $C_{f}$ is a gauge constant of the metric. of the solution
to obtain a metric devoid of angular defects, one for each sector, on the
$z$-axis, between the timelike rods of Fig. 1. A possible choice in terms of
the poles $w_{i}$ is888A more physical parametrisation is achieved with the
constants of (3.4).

$\displaystyle C_{f}$
$\displaystyle=\frac{8e^{-2w_{A}(b_{1}+b2w_{A})}}{w_{1}^{2}w_{3}^{2}}(w_{1}-w_{2})^{2}(w_{2}-w_{3})^{2}(w_{1}-w_{4})^{2}(w_{3}-w_{4})^{2}(w_{1}-w_{A})^{2}(w_{3}-w_{A})^{2}\,,$
(5.2) $\displaystyle b_{1}$
$\displaystyle=\frac{4b_{2}(w_{2}^{2}-w_{1}^{2}+w_{4}^{2}-w_{3})+\log\Bigl{[}\frac{(w_{A}-w_{2})^{2}(w_{A}-w_{4})^{2}}{(w_{A}-w_{1})^{2}(w_{A}-w_{3})^{2}}\Bigr{]}}{4(w_{1}-w_{2}+w_{3}-w_{4})}\,,$
(5.3) $\displaystyle b_{2}$
$\displaystyle=\frac{2(w_{1}-w_{2}+w_{3}-w_{4})\log\Bigl{[}\frac{(w_{1}-w_{3})(w_{2}-w_{4})(w_{4}-w_{A})}{(w_{2}-w_{3})(w_{1}-w_{4})(w_{3}-w_{A})}\Bigr{]}+(w_{4}-w_{3})\log\Bigl{[}\frac{(w_{1}-w_{3})(w_{2}-w_{4})(w_{4}-w_{A})}{(w_{2}-w_{3})(w_{1}-w_{4})(w_{3}-w_{A})}\Bigr{]}}{4(w_{1}-w_{2})(w_{3}-w_{4})(-w_{1}-w_{2}+w_{3}+w_{4})}\,.$
(5.4)

Physically this choice can be interpreted as the specific deformation of the
external gravitational field multipoles required to support a generic binary
configuration. Other possible choices may have different physical
interpretation: for instance, fixing the position of the holes through $z_{i}$
instead of $b_{1}$ to eliminate the conical singularities, corresponds in
adjusting the position of the black holes in a given multipolar configuration.
Henceforth we will considered the value of the three parameters $b_{i}$ and
$C_{f}$ constrained as in (5.2)-(5.3) to ensure the spacetime to be free from
angular defects anywhere.

The equilibrium is achieved for finite proper distance of the two black hole
sources, as it can be checked from

$\ell=\int_{w_{2}}^{w_{3}}dz\sqrt{g_{zz}(\rho,z)}\Big{|}_{\rho=0}<\infty\,.$
(5.5)

In figure 3 the regularised black holes event and accelerating horizons can be
appreciated. It is also apparent how the Rindler horizon is deformed, on the
axis of symmetry, by the tidal forces of the binary system.

Figure 3: Embedding diagram in $\mathbb{E}^{3}$ of the binary accelerating
black hole horizons for the parametric values $w_{1}=1$, $w_{2}=2.5$,
$w_{3}=3$, $w_{4}=4$, $w_{A}=6.5$ expressed in Solar mass units $M_{\odot}$.
The blue surfaces are the event horizons of the binary black system, while the
yellow surface represents a small cylindrical section of the the accelerating
horizon. All the horizon are smooth surfaces thanks to constraints (5.2),
(5.3), (5.4). The accelerating horizon for $z<0$ is not visible because in
that sector it is shifted towards conformal infinity, as usually occurs for
C-metrics.

###  5.1 Thermodynamics

The mass of each member of the double configuration can be evaluated
integrating on their respective rod, as done for the general case (3.9):
$M_{i}=\alpha m_{i}$, thus the total mass is given by $M=M_{1}+M_{2}$. The
total entropy of the system is taken as the quarter of the two black hole
surfaces, as described in section 3.3: $S=S_{1}+S_{2}$, where

$\displaystyle\begin{split}S_{1}&=\frac{\pi(w_{1}-w_{2})^{2}(w_{1}-w_{4})(w_{3}-w_{A})}{2(w_{1}-w_{3})(w_{2}-w_{A})(w_{5}-w_{A})}e^{-2b_{1}(w_{2}-w_{3}+w_{4})-2b_{2}(w_{2}^{2}-w_{3}^{2}+w_{4}^{2})}\\\
&=\frac{4Ae^{-2b_{1}(m_{1}+2m_{2}+z_{1})-2b_{2}(m_{1}^{2}+2m_{1}z_{1}+z_{1}^{2}+4m_{2}z_{2})}m_{1}^{2}\pi[1+2A(m_{2}-z_{2})](m_{1}+m_{2}-z_{1}+z_{2})}{[-1+2A(m_{1}+z_{1})](m_{1}-m_{2}-z_{1}+z_{2})(-1+2A(m_{2}+z_{2}))}\,,\end{split}$
(5.6a)
$\displaystyle\begin{split}S_{2}&=\frac{\pi(w_{3}-w_{4})^{2}(w_{4}-w_{1})}{2(w_{2}-w_{4})(w_{4}-w_{A})}e^{-2w_{4}(b_{1}+b_{2}w_{4})}\\\
&=-\frac{4Ae^{-2(m_{2}+z_{2})(b_{1}+b_{2}(m_{2}+z_{2}))}m_{2}^{2}\pi(m_{1}+m_{2}-z_{1}+z_{2})}{(-m_{1}+m_{2}-z_{1}+z_{2})[-1+2A(m_{2}+z_{2})]}\,.\end{split}$
(5.6b)

The two horizon temperatures, computed as in (3.12), simply result
$T_{i}=M_{i}/(2S_{i})$, fulfilling straightforwardly the Smarr law both for
each single black source and for the binary configuration.

The first law of black hole thermodynamics can also be verified, once the
normalisation of the timelike Killing vector is chosen to make the mass of the
system integrable and continuously connected with the known cases, i.e. the
non-accelerating configuration and the single black hole configuration. The
Christodoulou–Ruffini mass formula [35] suggests the use of an integrating
factor $\alpha$ such that

$\sum_{i=1}^{2}M_{i}=\sum_{i=1}^{2}\sqrt{\frac{S_{i}}{4\pi}}\,.$ (5.7)

It can be verified that this occurs whenever

$w_{A}=\frac{e^{2b_{1}(w_{2}-w_{3}+w_{4})+2b_{2}(w_{2}^{2}-w_{3}^{2}+w_{4}^{2})}w_{2}(w_{3}-w_{2})+e^{2w_{4}(b_{1}+b_{2}w_{4})}w_{3}(w_{2}-w_{4})}{e^{2b_{1}(w_{2}-w_{3}+w_{4})+2b_{2}(w_{2}^{2}-w_{3}^{2}+w_{4}^{2})}(w_{3}-w_{2})+e^{2w_{4}(b_{1}+b_{2}w_{4})}(w_{2}-w_{4})}\,.$
(5.8)

The resulting value of the integrating factor is

$\alpha=e^{-w_{4}(b_{1}+b_{2}w_{4})}\sqrt{\frac{w_{4}-w_{2}}{2(w_{4}-w_{2})(w_{5}-w_{4})}}\,.$
(5.9)

Then the first law of black hole thermodynamics holds for each member of the
black hole configuration as follows

$\delta M_{i}=T_{i}\,\delta S_{i}\,.$ (5.10)

In the presence of thermodynamic equilibrium between the two horizons,
$T_{1}=T_{2}$, which can be achieved constraining another integrating constant
of the solution (e.g. $w_{4}$), a first law for the whole black hole
configuration can be written

$\delta M=T\,\delta S\,.$ (5.11)

##  6 Accelerating particles in an external gravitational field

There exist many particle-like solutions in General Relativity that have been
extensively studied over the years. These solutions are related to the
Curzon–Chazy family of metrics [36, 37], and they represent the gravitational
field generated by point-like particles. The particles themselves are nothing
but naked singularities999The structure of these curvature singularities is
quite complicated and depends on the direction one approaches it. See [21] and
references therein.. It is quite easy to construct a metric that contains a
collection of many Curzon–Chazy particles: these multi-particle solutions are
affected by the presence of conical singularities [38], as expected on
physical grounds.

The accelerated version of two Curzon–Chazy particles was found by Bonnor and
Swaminarayan [39]: in such a solution the particles are accelerated by two
cosmic strings101010Actually, the conical singularities disappear when one of
the two particles has negative mass.. The metric, in the case of a single
accelerating particle, was later regularised, following the lines of [10], by
Bičák, Hoenselaers and Schmidt [40] with the introduction of an external
field, by which they were able to remove the conical singularities. Moreover,
they showed that their external field, that actually corresponds to our
multipolar expansion (2.11) when $b_{n}=0$ for $n\geq 2$, can be obtained by
sending to infinity the second particle of the Bonnor–Swaminarayan solution.
This corroborates the idea that the field background is generated by sources
located at infinity.

It is worth mentioning an alternative approach, pursued by Gibbons [11], who
managed to regularise an accelerating black hole by means of a negative-mass
Curzon–Chazy particle.

We will show that the generalisation of the Bičák–Hoenselaers–Schmidt solution
to $N$ particles and with the generic gravitational field (2.9) can be
achieved by means of an appropriate limit of our multi-black hole solution
(3.3).

###  6.1 The limit to the Bonnor–Swaminarayan metric

Let us begin by considering the limit to the Bonnor–Swaminarayan solution,
i.e. two accelerating particles with no external gravitational field. We
consider then the metric (3.3) with $b_{n}=0$ for all $n$. To clarify the
limit, we specialise to the case $N=2$, nonetheless the generalisation any $N$
is straightforward.

It is useful to rewrite the metric (3.3) in the canonical Weyl form

${ds}^{2}=-e^{2\psi(\rho,z)}{dt}^{2}+e^{-2\psi(\rho,z)}\bigl{[}e^{2\gamma(\rho,z)}\bigl{(}{d\rho}^{2}+{dz}^{2}\bigr{)}+\rho^{2}{d\phi}^{2}\bigr{]}\,,$
(6.1)

and to work with the potential

$\psi=\frac{1}{2}\log\biggl{(}\frac{\mu_{1}}{\mu_{2}}\biggr{)}+\frac{1}{2}\log\biggl{(}\frac{\mu_{3}}{\mu_{4}}\biggr{)}+\frac{1}{2}\log\mu_{A}\,.$
(6.2)

Noticing that

$\frac{\mu_{k}}{\mu_{k+1}}=\frac{\mu_{k}-\bar{\mu}_{k+1}}{\mu_{k+1}-\bar{\mu}_{k}}=\frac{R_{k}+R_{k+1}-2m_{k}}{R_{k}+R_{k+1}+2m_{k}}\,,$
(6.3)

where

$R_{k}=\sqrt{\rho^{2}+(z+m_{k}-z_{k})^{2}}\,,\quad
R_{k+1}=\sqrt{\rho^{2}+(z-m_{k}-z_{k})^{2}}\,,$ (6.4)

we can write

$\psi=\frac{1}{2}\log\biggl{(}\frac{R_{1}+R_{2}-2m_{1}}{R_{1}+R_{2}+2m_{1}}\biggr{)}+\frac{1}{2}\log\biggl{(}\frac{R_{3}+R_{4}-2m_{2}}{R_{3}+R_{4}+2m_{2}}\biggr{)}+\frac{1}{2}\log\mu_{A}\,.$
(6.5)

One recognises two Schwarzschild potentials (the first two terms) and the
Rindler potential (the last term). Indeed, at the level of the Weyl potential
a superposition principle holds; the non-linearity is encoded in the function
$\gamma$, that we do not explicitly write here.

$w_{1}$$w_{2}$$w_{3}$$w_{4}$$w_{A}$$z$$t$$\phi$$(a)$$w_{2}\to
w_{1}\,,\,\,w_{4}\to w_{3}$$w_{1}=w_{2}$$w_{3}=w_{4}$$w_{A}$$z$$t$$\phi$$(b)$
Figure 4: Rod diagrams for $(a)$ the double C-metric in an external field
(5.1) and for $(b)$ the Bonnor–Swaminarayan metric (6.9). We see that the
limit considered in the main text $w_{2k}\to w_{2k-1}$ corresponds to
shrinking the timelike rods representing the event horizons. The horizons
disappear through that limit, and the resulting objects are naked
singularities which are represented by points in the rod diagram.

Now we consider the limit in which the finite timelike rods of Fig. 1 are
pinched to a point, i.e. when $w_{2k}\to w_{2k-1}$: this is equivalent to
consider $m_{k}\to 0$. We expand for small $m_{k}$ to order
$O\bigl{(}m_{k}^{2}\bigr{)}$, to find

$\psi\approx-\frac{m_{1}}{\sqrt{\rho^{2}+(z-z_{1})^{2}}}-\frac{m_{2}}{\sqrt{\rho^{2}+(z-z_{2})^{2}}}+\frac{1}{2}\log\mu_{A}\,.$
(6.6)

Again, we recognise the various terms in the last expression: the first two
are Curzon–Chazy potentials and represent point-like particles, while the last
term is still the Rindler one. Then it is natural to interpret the potential
as the one corresponding to two accelerating particles.

We cast the potential in the usual Bonnor–Swaminarayan by performing the
change of coordinate $\bar{z}=1/(2A^{2})-z$, by which

$\mu_{A}=\sqrt{\rho^{2}+\bar{z}^{2}}+\bar{z}\,,$ (6.7)

and then defining the new constants

$2\alpha_{1}^{2}=\frac{2A^{2}}{2A^{2}z_{1}-1}\,,\quad
2\alpha_{2}^{2}=\frac{2A^{2}}{2A^{2}z_{2}-1}\,.$ (6.8)

One finally finds

$\psi=-\frac{m_{1}}{\sqrt{\rho^{2}+(\bar{z}-\frac{1}{2\alpha_{1}^{2}})^{2}}}-\frac{m_{2}}{\sqrt{\rho^{2}+(\bar{z}-\frac{1}{2\alpha_{2}^{2}})^{2}}}+\frac{1}{2}\log\bigl{(}\sqrt{\rho^{2}+\bar{z}^{2}}+\bar{z}\bigr{)}\,,$
(6.9)

which is the Bonnor–Swaminarayan potential (cf. [39] and [41]). The
generalisation to $N$ accelerating particles is easily found as

$\psi=-\sum_{k=1}^{N}\frac{m_{k}}{\sqrt{\rho^{2}+(\bar{z}-\frac{1}{2\alpha_{k}^{2}})^{2}}}+\frac{1}{2}\log\bigl{(}\sqrt{\rho^{2}+\bar{z}^{2}}+\bar{z}\bigr{)}\,.$
(6.10)

The $\gamma$ function, which completes the Weyl metric (6.1), is found by
quadratures.

One can check that the Bonnor–Swaminarayan metric (6.9), for generic values of
the parameters, is affected by conical singularities. Such singularities can
be removed only when $m_{2}<0$: in this case the axis is everywhere regular,
except at the locations of the point particles. More explicitly, the
regularisation is achieved for [41]

$m_{1}=-m_{2}=\frac{\bigl{(}\alpha_{1}^{2}-\alpha_{2}^{2})^{2}}{4\alpha_{1}^{3}\alpha_{2}^{3}}\,.$
(6.11)

$Z$$T$ Figure 5: A spacetime diagram of the Bonnor–Swaminarayan in the boost-
rotation coordinates of [42], for the section $\rho=0$. The hyperbolae
represent the worldlines of two pairs of (causally disconnected) accelerating
particles.

###  6.2 The limit to the Bičák–Hoenselaers–Schmidt metric

The inclusion of the external gravitational field is now a simple matter: this
can be done following the lines of section 2, i.e. by means if the inverse
scattering method, or via the same limiting procedure of the multi-black hole
metric (3.3) of the previous subsection, now with the field parameters $b_{n}$
turned on. In both cases, the resulting Weyl potential is

$\psi=-\sum_{k=1}^{N}\frac{m_{k}}{\sqrt{\rho^{2}+(\bar{z}-\frac{1}{2\alpha_{k}^{2}})^{2}}}+\frac{1}{2}\log\bigl{(}\sqrt{\rho^{2}+\bar{z}^{2}}+\bar{z}\bigr{)}+\sum_{n=1}^{\infty}b_{n}r^{n}P_{n}\,.$
(6.12)

The meaning of the terms in the potential is clear, and the function $\gamma$
can be found again by quadratures. This potential specialises to the
Bičák–Hoenselaers–Schmidt one [40] for $N=1$ and $b_{1}\neq 0$, $b_{n}=0$ for
$n\geq 2$, that reads

$\psi=-\frac{m_{1}}{\sqrt{\rho^{2}+(\bar{z}-\frac{1}{2\alpha_{1}^{2}})^{2}}}+\frac{1}{2}\log\bigl{(}\sqrt{\rho^{2}+\bar{z}^{2}}+\bar{z}\bigr{)}+b_{1}z\,.$
(6.13)

The limiting procedure that leads to (6.12) does not affect the external
field, hence it can be fine tuned again to support the particles attraction
against the gravitational collapse and to remove the conical singularities.

It is worth mentioning that Bičák–Hoenselaers–Schmidt also found two
accelerating particles described by internal multipole momenta [43], in
analogy with the Erez–Rosen metric [44]. The multipole metric, initially
obtained through a coalescing limit of the Bonnor–Swaminarayan solution, can
be regularised everywhere on the $z$-axis (except at the two particles), thus
obtaining a regular accelerating metric without the need of any external
field. Relying on the discussion presented in [2], one can easily write down
the most general potential for $N$ accelerating particles with arbitrary
multipole momenta and immersed in an external gravitational field:

$\psi=-\sum_{k=1}^{N}\frac{m_{k}}{\sqrt{\rho^{2}+(\bar{z}-\frac{1}{2\alpha_{k}^{2}})^{2}}}+\frac{1}{2}\log\bigl{(}\sqrt{\rho^{2}+\bar{z}^{2}}+\bar{z}\bigr{)}+\sum_{n=1}^{\infty}\biggl{(}\frac{a_{n}}{r^{n+1}}+b_{n}r^{n}\biggr{)}P_{n}\,.$
(6.14)

$a_{n}$ are the internal momenta, that describe the deformations of the point-
like sources. We do not delve into the details of this solution, because it is
beyond the scope of this paper. However, it would be interesting to explicitly
write down the function $\gamma$ and to check that the conical singularities
can be removed by tuning the parameters $a_{n}$ and $b_{n}$.

##  7 Summary and Conclusions

In this article we constructed, thanks to the inverse scattering method, a
large family of new solutions which generalise the accelerating Israel–Khan
metric [16], by the introduction of an external gravitational field. We are
able to treat analytically the whole multipolar expansion of the gravitational
background introducing a countable number of integration constants,
characterising the gravitational multipoles of the external field, which are
useful to remove all the conical singularities typical of the accelerating
collinear multi-black hole configurations. As a concrete example, we
explicitly analyse some simple configurations such as the accelerating
Schwarzschild or the accelerating Bach–Weyl double black hole embedded in an
external gravitational field endowed with dipole and quadrupole moments. We
studied the thermodynamic of these solutions and verified the Smarr and the
first law.

We find these spacetimes relevant not only because they enrich our scarce
theoretical knowledge of multi-black hole solutions or because they are the
first accelerating black hole solutions which can be regularised without the
need of extra fields as the electromagnetic field111111Actually, in the
presence of Maxwell electrodynamics, Ernst showed also how to regularise the
charged C-metric, thanks to an external electromagnetic field such as the
Melvin universe. Even though axial magnetic fields in the center of the
galaxies can be of some prominence, charged black holes are not considered
plausible objects because matter in the universe is most often neutral., but
also because these metrics allow us to discuss some intriguing physical
processes. In fact regularised C-metrics can describe the pair creation of a
couple (or eventually four in case of the double C-metric) of black holes [34]
that accelerate in opposite direction remaining causally disconnected. This
process is propelled at expense of the external field, in our case the
multipolar gravitational background. The significance and the novelty of our
picture is given by the fact that the accelerating black hole couple can be
uncharged, a feature more in line with phenomenological observation.

As a by-product of our construction we show how to extend the vacuum
Plebański–Demiański class of metrics to the rotating and accelerating multi-
Kerr black holes, with or without the presence of the external gravitational
field.

We are also able to detect some notable known metrics as limits of our general
solution describing accelerating particles with or without the external
gravitational background such as the Bonnor–-Swaminarayan and the
Bičák–Hoenselaers–Schmidt solutions.

In general we have shown that, as the inverse scattering method predicts, in
practice basically any diagonal seed can be used as a background for the
solution generating technique. In particular this technique reveals to be
useful to embed and overlap a generic number of black hole sources, possibly
providing a mechanism to regularise the conical singularities that usually
afflicts these metrics. Of course it would be interesting to explore also
different backgrounds.

All these results can be extended to gravitational theories where the solution
generating techniques hold, from minimally to conformally coupled scalar
fields or other scalar tensor theories such as some classes of Brans–Dicke or
$f(R)$ gravity121212For details to adapt general relativity solution
generating techniques to these other theories see [45, 46]..

## Aknowledgments

This work was supported in part by Conicyt–Beca Chile n${}^{\textrm{o}}$
74200076, by MIUR-PRIN contract 2017CC72MK003 and also by INFN.

## References

  * [1] M. Astorino and A. Viganò.  Binary black hole system at equilibrium.  2021\.  arXiv:2104.07686.
  * [2] M. Astorino and A. Viganò.  Charged and rotating multi-black holes in an external gravitational field.  2021\.  arXiv:2105.02894.
  * [3] W. Israel and K.A. Khan.  Collinear particles and bondi dipoles in general relativity.  Il Nuovo Cimento, 33(2):331–344, 1964.
  * [4] A.G. Doroshkevich, Y.B. Zel’dovich, and I.D. Novikov.  Gravitational Collapse of Non-Symmetric and Rotating Bodies.  Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, 49:170, December 1965.
  * [5] S. Chandrasekhar.  The mathematical theory of black holes.  Clarendon Press, Oxford, 1985.
  * [6] R.P. Geroch and J.B. Hartle.  Distorted black holes.  J. Math. Phys., 23:680, 1982.  doi:10.1063/1.525384.
  * [7] N. Breton, A.A. Garcia, V.S. Manko, and T.E. Denisova.  Arbitrarily deformed Kerr–Newman black hole in an external gravitational field.  Phys. Rev. D, 57:3382–3388, 1998.  doi:10.1103/PhysRevD.57.3382.
  * [8] S. Abdolrahimi, J. Kunz, P. Nedkova, and C. Tzounis.  Properties of the distorted Kerr black hole.  JCAP, 12:009, 2015.  arXiv:1509.01665, doi:10.1088/1475-7516/2015/12/009.
  * [9] S. Fairhurst and B. Krishnan.  Distorted black holes with charge.  Int. J. Mod. Phys. D, 10:691–710, 2001.  arXiv:gr-qc/0010088, doi:10.1142/S0218271801001086.
  * [10] F.J. Ernst.  Generalized c-metric.  Journal of Mathematical Physics, 19(9):1986–1987, 1978.  doi:https://doi.org/10.1063/1.523896.
  * [11] G.W. Gibbons.  The Motion of black holes.  Commun. Math. Phys., 35:13–23, 1974.  doi:10.1007/BF01646451.
  * [12] V.A. Belinsky and V.E. Zakharov.  Integration of the Einstein Equations by the Inverse Scattering Problem Technique and the Calculation of the Exact Soliton Solutions.  Sov. Phys. JETP, 48:985–994, 1978.
  * [13] V.A. Belinsky and V.E. Sakharov.  Stationary Gravitational Solitons with Axial Symmetry.  Sov. Phys. JETP, 50:1–9, 1979.
  * [14] V. Belinski and E. Verdaguer.  Gravitational solitons.  Cambridge Monographs on Mathematical Physics. Cambridge University Press, 2005.  doi:10.1017/CBO9780511535253.
  * [15] H. Weyl.  Zur gravitationstheorie.  Annalen der Physik, 359(18):117–145, 1917.  doi:https://doi.org/10.1002/andp.19173591804.
  * [16] H.F. Dowker and S.N. Thambyahpillai.  Many accelerating black holes.  Class. Quant. Grav., 20:127–136, 2003.  arXiv:gr-qc/0105044, doi:10.1088/0264-9381/20/1/310.
  * [17] G.M. de Castro and P.S. Letelier.  Black holes surrounded by thin rings and the stability of circular orbits.  Class. Quant. Grav., 28:225020, 2011.  doi:10.1088/0264-9381/28/22/225020.
  * [18] P.S. Letelier.  Multipole stationary soliton solutions to the einstein equations.  Revista Brasileira de Fisica, 14:371–376, September 1984.
  * [19] K. Hong and E. Teo.  A New form of the rotating C-metric.  Class. Quant. Grav., 22:109–118, 2005.  arXiv:gr-qc/0410002, doi:10.1088/0264-9381/22/1/007.
  * [20] P.S. Letelier.  Static and stationary multiple soliton solutions to the einstein equations.  Journal of Mathematical Physics, 26(3):467–476, 1985.  doi:10.1063/1.526633.
  * [21] J.B. Griffiths, P. Krtous, and J. Podolsky.  Interpreting the C-metric.  Class. Quant. Grav., 23:6745–6766, 2006.  arXiv:gr-qc/0609056, doi:10.1088/0264-9381/23/23/008.
  * [22] K. Hong and E. Teo.  A New form of the C metric.  Class. Quant. Grav., 20:3269–3277, 2003.  arXiv:gr-qc/0305089, doi:10.1088/0264-9381/20/14/321.
  * [23] A. Komar.  Covariant conservation laws in general relativity.  Phys. Rev., 113:934–936, Feb 1959.  doi:10.1103/PhysRev.113.934.
  * [24] A. Tomimatsu.  Equilibrium of Two Rotating Charged Black Holes and the Dirac String.  Prog. Theor. Phys., 72:73, 1984.  doi:10.1143/PTP.72.73.
  * [25] R. Gregory, Z.L. Lim, and A. Scoins.  Thermodynamics of Many Black Holes.  Front. in Phys., 9:187, 2021.  arXiv:2012.15561, doi:10.3389/fphy.2021.666041.
  * [26] T. Harmark.  Stationary and axisymmetric solutions of higher-dimensional general relativity.  Phys. Rev. D, 70:124002, 2004.  arXiv:hep-th/0408141, doi:10.1103/PhysRevD.70.124002.
  * [27] W.B. Bonnor.  An exact solution of einstein’s equations for two particles falling freely in an external gravitational field.  General relativity and gravitation, 20(6):607–622, 1988.  doi:https://doi.org/10.1007/BF00758917.
  * [28] M. Astorino.  Pair Creation of Rotating Black Holes.  Phys. Rev. D, 89(4):044022, 2014.  arXiv:1312.1723, doi:10.1103/PhysRevD.89.044022.
  * [29] G.W. Gibbons.  Quantized flux tubes in Einstein-Maxwell theory and noncompact internal spaces.  In 22nd Winter School of Theoretical Physics: Fields and Geometry, 5 1986.
  * [30] D. Garfinkle, S.B. Giddings, and A. Strominger.  Entropy in black hole pair production.  Phys. Rev. D, 49:958–965, 1994.  arXiv:gr-qc/9306023, doi:10.1103/PhysRevD.49.958.
  * [31] S.W. Hawking, G.T. Horowitz, and S.F. Ross.  Entropy, Area, and black hole pairs.  Phys. Rev. D, 51:4302–4314, 1995.  arXiv:gr-qc/9409013, doi:10.1103/PhysRevD.51.4302.
  * [32] W.B. Bonnor.  Static magnetic fields in general relativity.  Proceedings of the Physical Society. Section A, 67(3):225, 1954\.  doi:https://doi.org/10.1088/0370-1298/67/3/305.
  * [33] M.A. Melvin.  Pure magnetic and electric geons.  Phys. Lett., 8:65–70, 1964.  doi:10.1016/0031-9163(64)90801-7.
  * [34] M. Astorino and A. Viganò.  Pair creation of black holes in gravitational background.  In preparation.
  * [35] D. Christodoulou and R. Ruffini.  Reversible transformations of a charged black hole.  Phys. Rev. D, 4:3552–3555, 1971.  doi:10.1103/PhysRevD.4.3552.
  * [36] H.E.J. Curzon.  Cylindrical solutions of einstein’s gravitation equations.  Proceedings of the London Mathematical Society, 2(1):477–480, 1925\.  doi:https://doi.org/10.1112/plms/s2-23.1.477.
  * [37] J. Chazy.  Sur le champ de gravitation de deux masses fixes dans la théorie de la relativité.  Bulletin de la Societe mathematique de France, 52:17–38, 1924.  doi:https://doi.org/10.24033/bsmf.1044.
  * [38] A. Einstein and N. Rosen.  Two-Body Problem in General Relativity Theory.  Phys. Rev., 49:404–405, 1936.  doi:10.1103/PhysRev.49.404.2.
  * [39] W.B. Bonnor and N.S. Swaminarayan.  An exact solution for uniformly accelerated particles in general relativity.  Zeitschrift für Physik, 177(3):240–256, 1964.  doi:https://doi.org/10.1007/BF01375497.
  * [40] J. Bicak, C. Hoenselaers, and B.G. Schmidt.  The solutions of the einstein equations for uniformly accelerated particles without nodal singularities. i. freely falling particles in external fields.  Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 390(1799):397–409, 1983.  doi:10.1098/rspa.1983.0138.
  * [41] J.B. Griffiths and J. Podolsky.  Exact Space-Times in Einstein’s General Relativity.  Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, 2009.  doi:10.1017/CBO9780511635397.
  * [42] J. Bicak and V. Pravda.  Spinning C metric: Radiative space-time with accelerating, rotating black holes.  Phys. Rev. D, 60:044004, 1999.  arXiv:gr-qc/9902075, doi:10.1103/PhysRevD.60.044004.
  * [43] J. Bicak, C. Hoenselaers, and B.G. Schmidt.  The solutions of the einstein equations for uniformly accelerated particles without nodal singularities. ii. self-accelerating particles.  Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 390(1799):411–419, 1983.  doi:10.1098/rspa.1983.0139.
  * [44] G. Erez and N. Rosen.  The gravitational field of a particle possessing a multipole moment.  Bull. Research Council Israel, Sect. F.8, 9 1959.
  * [45] M. Astorino.  Embedding hairy black holes in a magnetic universe.  Phys. Rev. D, 87(8):084029, 2013.  arXiv:1301.6794, doi:10.1103/PhysRevD.87.084029.
  * [46] M. Astorino.  Stationary axisymmetric spacetimes with a conformally coupled scalar field.  Phys. Rev. D, 91:064066, 2015.  arXiv:1412.3539, doi:10.1103/PhysRevD.91.064066.