that it generates. We will see that such a transformation embeds any given
stationary and axisymmetric seed spacetime into a rotating background, which
we will dub “swirling universe”, for its peculiar characteristic. Indeed, the
background can be interpreted as a gravitational whirlpool, and its frame
dragging turns a static seed solution into a stationary metric. The resulting
spacetime really looks similar to the Taub–NUT metric, however we will show
that it does not share the problematic features of its “electric” counterpart:
in fact, our new spacetime does not possess any Misner string, nor is affected
by the presence of conical singularities. The manifold is completely regular,
apart from the presence of the usual black hole curvature singularity.

Firstly, we generate the new metric via the “magnetic” Ehlers map. Then, we
test the consequences of the transformation on the spherical symmetric black
hole, which will be our seed metric, as done with the symmetries previously
analysed in the literature. In this way we generate a novel and analytic exact
solution of the Einstein equations, which generalises and deforms the
Schwarzschild spacetime. Of course, because of the well-known no-hair theorems
for black hole in four-dimensional General Relativity, the new solution can be
a black hole only by renouncing to asymptotic flatness, similarly to black
holes embedded in the external electromagnetic field of Melvin universe [130,
117, 223]. We also present the generalisation to the Kerr black hole in the
rotating background. Finally, we conclude the Chapter by establishing a
connection (via a double-Wick rotation) between our background metric and the
flat Taub–NUT spacetime. This Chapter is based on [224].

#### 4.1 Generation of the solution

As explained above, we consider the magnetic LWP metric of Eq. (1.48),

${ds}^{2}=\bar{f}(d\varphi-\bar{\omega}dt)^{2}+\bar{f}^{-1}\bigl{[}e^{2\bar{\gamma}}\bigl{(}{d\rho}^{2}+{dz}^{2}\bigr{)}-\rho^{2}{dt}^{2}\bigr{]}\,,$
(4.1)

and the Ehlers transformation (1.46c)

$\mathcal{E}^{\prime}=\frac{\mathcal{E}}{1+i\jmath\mathcal{E}}\,,$ (4.2)

where we dubbed with $\jmath$ the map parameter.

First of all we have to choose the seed. We start with the Schwarzschild black
hole, whose metric, in spherical coordinates, reads

${ds}^{2}=-\biggl{(}1-\frac{2m}{r}\biggr{)}{dt}^{2}+\frac{{dr}^{2}}{1-\frac{2m}{r}}+r^{2}{d\theta}^{2}+r^{2}\sin^{2}\theta{d\varphi}^{2}\,.$
(4.3)

The most convenient coordinates for the generating methods, in this case, are
the spherical ones $(r,\theta)$, related to the Weyl cylindrical coordinates
by

$\rho=\sqrt{r^{2}-2mr}\sin\theta\,,\quad z=(r-m)\cos\theta\,.$ (4.4)

The line element of the LWP metric (1.48) in spherical coordinates reads

${ds}^{2}=\bar{f}(d\varphi-\bar{\omega}dt)^{2}+\bar{f}^{-1}\biggl{[}-\rho^{2}{dt}^{2}+e^{2\bar{\gamma}}(r^{2}-2mr+m^{2}\sin^{2}\theta)\biggl{(}\frac{{dr}^{2}}{r^{2}-2mr}+{d\theta}^{2}\biggr{)}\biggr{]}\,.$
(4.5)

Comparing the seed (4.3) to the above metric we can identify the seed
structure functions

$\bar{f}(r,\theta)=r^{2}\sin^{2}\theta\,,\quad\bar{\omega}(r,\theta)=0\,.$
(4.6)

The value of $\bar{\gamma}$ is not fundamental because it is invariant under
Ehlers transformations, however we explicit it for completeness

$\bar{\gamma}_{0}(r,\theta)=\frac{1}{2}\log\biggl{(}\frac{r^{4}\sin^{2}\theta}{r^{2}-2mr+m^{2}\sin^{2}\theta}\biggr{)}\,.$
(4.7)

From definition of $\chi$ in (1.33), it is clear that $\bar{\chi}$ is at most
constant, but that constant can be reabsorbed via a coordinate transformation.
Therefore without loss of generality we can choose $\bar{\chi}=0$. Finally,
the seed Ernst gravitational potential takes the form

$\mathcal{E}(r,\theta)=\bar{f}(r,\theta)\,.$ (4.8)

The new solution, expressed in terms of the complex potential, is generated
via the Ehlers transformation (4.2), which gives

$\mathcal{E}^{\prime}=\frac{\mathcal{E}}{1+i\jmath\mathcal{E}}=\frac{r^{2}\sin^{2}\theta}{1+i\jmath\,r^{2}\sin^{2}\theta}\,.$
(4.9)

Note that in case we had used the LWP metric defined in (1.7), we would had
obtained, via the Ehlers transformation acting on the Schwarzschild seed, the
Taub–NUT spacetime, as explained in Chapter 1.

The solution, in metric form, is extracted from the definition of the
transformed Ernst potential $\mathcal{E}$. Hence, according to
$\mathcal{E}=f+i\chi$, we find

$f(r,\theta)=\frac{r^{2}\sin^{2}\theta}{1+\jmath^{2}r^{4}\sin^{4}\theta}\,,\qquad\chi(r,\theta)=\frac{\jmath\,r^{4}\sin^{4}\theta}{1+\jmath^{2}r^{4}\sin^{4}\theta}\,.$
(4.10)

$\omega$ has to be found from the definition of $\chi$, as in (1.33). The
result is

$\omega(r,\theta)=4\jmath(r-2m)\cos\theta+\omega_{0}\,,$ (4.11)

where $\omega_{0}$ is an integration constant related to the choice of
reference frame. Thus, recalling that $\gamma$ is not affected by the Ehlers
map, the full new metric is

$\begin{split}{ds}^{2}&=F(r,\theta)\biggl{[}-\biggl{(}1-\frac{2m}{r}\biggr{)}{dt}^{2}+\frac{dr^{2}}{1-\frac{2m}{r}}+r^{2}d\theta^{2}\biggr{]}\\\
&\quad+\frac{r^{2}\sin^{2}\theta}{F(r,\theta)}\biggl{\\{}d\varphi+\big{[}4\jmath(r-2m)\cos\theta+\omega_{0}\big{]}\
dt\biggr{\\}}^{2}\,,\end{split}$ (4.12)

where we have defined the function

$F(r,\theta)\coloneqq 1+\jmath^{2}r^{4}\sin^{4}\theta\,,$ (4.13)

We can immediately observe that the new metric (4.12) represents a non-
asymptotically flat deformation of the Schwarzschild black hole. Its structure
is quite similar to the Schwarzschild–Melvin spacetime [117], and indeed the
magnetic Ehlers map that we have used works in a similar fashion as the
Harrison transformation. For this reason we do not expect that the new
parameter can be considered as hair nor as conserved charge of the black hole.
The physical description of (4.12) will be analysed in detail in the next
Section.

Starting with a more general seed we can obtain generalisations of the metric
built above. In Sec. 4.4 we embed the Kerr black hole in the swirling
background, while in Appendix D we generate the Zipoy–Voorhees extension of
the spacetime (4.12).

#### 4.2 Analysis of the background spacetime

In the interpretation of the new spacetime (4.12), a fundamental point comes
from the physical meaning of the new parameter $\jmath$, which defines the
behaviour of the gravitational background and, at best of our knowledge, is
unknown. Thus, we firstly analyse the background metric obtained by turning
off the mass parameter $m$ in Eq. (4.12) and then, in Sec. 4.3, we study the
full black hole solution in his surrounding universe.

When the mass parameter $m$ vanishes the black hole disappears and we are left
with the rotating gravitational background only

${ds}^{2}=F\bigl{(}-{dt}^{2}+{dr}^{2}+r^{2}{d\theta}^{2}\bigr{)}+F^{-1}r^{2}\sin^{2}\theta(d\varphi+4\jmath\,r\cos\theta\
dt)^{2}\,.$ (4.14)

In cylindrical coordinates

$\rho=r\sin\theta\,,\quad z=r\cos\theta\,,$ (4.15)

the background takes the simpler form

${ds}^{2}=\bigl{(}1+\jmath^{2}\rho^{4}\bigr{)}\bigl{(}-{dt}^{2}+{d\rho}^{2}+{dz}^{2}\bigr{)}+\frac{\rho^{2}}{1+\jmath^{2}\rho^{4}}(d\varphi+4\jmath
zdt)^{2}\,.$ (4.16)

Such a metric has the very same form as the one presented in Appendix C of
[223]: however, it was not studied in that reference.

The spacetime (4.16) is characterised by the Killing vectors

$\partial_{t}\,,\quad\partial_{\varphi}\,,\quad
z\partial_{t}+t\partial_{z}-2\jmath\bigl{(}t^{2}+z^{2}\bigr{)}\partial_{\varphi}\,,\quad\partial_{z}-4\jmath
t\partial_{\varphi}\,,$ (4.17)

and by the Killing–Yano form

$-4\jmath\rho z\,dt\wedge
d\rho+\jmath\rho^{2}\bigl{(}1+\jmath^{2}\rho^{4})\,dt\wedge
dz+\rho\,d\rho\wedge d\varphi\,.$ (4.18)

It belongs to the Petrov type D class [119], and its Newman–Penrose spin
coefficient is equal to zero: these features allow us to infer that the metric
(4.16) belongs to the Kundt class (cf. Table 38.9 of [119]). We can indeed
explicitly express the background metric (4.16) in the standard Kundt form, by
performing the rescaling $t\to\jmath t$ and the change of coordinates

$q=2\jmath z\,,\quad p=\rho^{2}\,.$ (4.19)

Metric (4.16) then boils down to (after a rescaling of the conformal factor)

${ds}^{2}=(\gamma^{2}+p^{2})\bigl{(}-{dt}^{2}+{dq}^{2}\bigr{)}+\frac{\gamma^{2}+p^{2}}{\gamma^{2}p}{dp}^{2}+\frac{\gamma^{2}p}{\gamma^{2}+p^{2}}(d\varphi+2\gamma
qdt)^{2}\,,$ (4.20)

where we have defined $\gamma=1/\jmath$. This metric is equivalent to (16.27)
of [135], once we put $m=e=g=\Lambda=\alpha=\epsilon_{2}=k=0$,
$\epsilon_{0}=1$ and $n=\gamma^{2}/2$. One can check that the consistency
constraints of [135] are indeed satisfied.

The metric (4.20), despite being known for a long time (it was discovered by
Carter in [225]), does not have a clear physical interpretation. In
particular, the physical significance of the parameter $\gamma$ (i.e.
$\jmath$) is unknown: it has been called “anti-NUT” parameter by Plebański in
[226] because of its resemblance with the NUT parameter in the
Plebański–Demiański spacetime [134]111Actually, there exists a connection
between the rotating parameter $\jmath$ and the NUT parameter $\ell$ that will
be exploited in Sec. 4.5.. A generalisation in the presence of the
cosmological constant and some possible interpretations of this background are
given in Sec. 4.5.

An interesting limit is given by $p\to\infty$: by putting $\gamma=0$ after the
rescaling $t\to\gamma^{-2/3}t$, $q\to\gamma^{-2/3}q$,
$\varphi\to\gamma^{-2/3}\varphi$ and $p\to\gamma^{2/3}p$, one finds

${ds}^{2}=p^{2}\bigl{(}-{dt}^{2}+{dq}^{2}\bigr{)}+p\,{dp}^{2}+\frac{1}{p}{d\varphi}^{2}\,.$
(4.21)

This is nothing but the Levi-Civita metric [227] (cf. (10.9) of [182]), in the
limiting case when $\sigma=1/4$, which is locally isometric to the asymptotic
form of the Melvin spacetime, as shown in [228, 189, 229]; another connection
to the magnetic universe will be explored in Sec. 4.5. Finally, the above
limit can also be expressed in the Kasner-like form [230].

In order to gain some physical perspective is useful to investigate, in some
detail, the properties of the background metric by inspecting its geodesics.

##### 4.2.1 Geodesics in the background spacetime

We define the following geodesic Lagrangian from the background metric (4.16)

$\mathscr{L}=\bigl{(}1+\jmath^{2}\rho^{4}\bigr{)}\bigl{(}-\dot{t}^{2}+\dot{\rho}^{2}+\dot{z}^{2}\bigr{)}+\frac{\rho^{2}}{1+\jmath^{2}\rho^{4}}\bigl{(}\dot{\varphi}+4\jmath
z\dot{t}\bigr{)}^{2}\,,$ (4.22)

where the dots stand for the derivatives with respect to an affine parameter
$s$. We can define, via the Killing vectors $\xi=\partial_{t}$ and
$\Phi=\partial_{\varphi}$, the standard conserved quantities

$-E\coloneqq g_{\mu\nu}u^{\mu}\xi^{\nu}\,,\quad L\coloneqq
g_{\mu\nu}u^{\mu}\Phi^{\nu}\,,$ (4.23)

where $u^{\mu}$ is the four-momentum of the test particle, $E$ is the energy
and $L$ is the angular momentum. The explicit definitions for the conserved
quantities and the resulting Lagrangian can be found in Appendix C.1

The equations of motion derived from the Lagrangian are quite involved.
Qualitative results can be obtained from the normalisation of the four-
momentum, i.e. from equation $u^{\mu}u_{\mu}=\chi$, with $\chi=-1$ for
timelike geodesics and $\chi=0$ for null geodesics. The resulting
normalisation equation is (C.3). For large values of $\rho$ and for fixed $z$,
it follows from such equation that

$\dot{\rho}^{2}\approx L^{2}\rho^{-2}\,,$ (4.24)

which has solution $\rho(s)\propto\sqrt{2Ls}$: this means that $\rho$ is not
limited as the affine parameter grows.

We are interested in analysing the behaviour of $z$ as $\rho$ grows. We find
$\dot{z}^{2}=0$ by letting $\rho\to\infty$ in equation (C.3), therefore the
coordinate $z$ reaches a constant value as $\rho$ approaches infinity.
Moreover, the equation defining $L$, for large values of $\rho$, gives
$\phi\approx c^{2}L^{2}s^{2}$. Combining this with the approximate equation
for $\rho$, allows one to get the polar equation
$r\approx\sqrt{\frac{2}{c}}\phi^{1/4}$. Such an equation is the polar form of
the generalised Archimedean spiral with exponential 1/4. Therefore we expect
that a geodesic test particle follows a spiral-like path in the $(x,y)$ plane
and that it moves toward a constant value of $z$. These results are in good
agreement with the numerical evaluations, as can be observed from the plot in
Fig. 4.1, which shows the trajectory of a test particle in the $(x,y,z)$
space, where $x=\rho\cos\theta$, $y=\rho\sin\theta$.

Figure 4.1: Geodesic motion of a test particle in the gravitational background
for $E=1$, $L=1$, $\jmath=0.1$.

The statement that $z$ reaches a constant value can be verified by using the
equation of motion for $z$

$4\jmath^{2}\rho^{3}\dot{\rho}\dot{z}+(\jmath^{2}\rho^{4}+1)\ddot{z}=\frac{4\rho^{2}\jmath\dot{t}(\dot{\varphi}+4\jmath
z\dot{t})}{\jmath^{2}\rho^{4}+1}\,.$ (4.25)

By inspecting the equations for $\dot{t}$ and $\dot{\varphi}$ in Appendix C.1,
one can notice that $\dot{t}\propto 1/\rho^{4}$ and
$\dot{\varphi}\propto\rho^{2}$ as $\rho\to\infty$, therefore the r.h.s. of the
latter equation can be neglected for large values of $\rho$:

$4\jmath^{2}\rho^{3}\dot{\rho}\dot{z}+(\jmath^{2}\rho^{4}+1)\ddot{z}=0\,.$
(4.26)

Moreover, $1+\jmath^{2}\rho^{4}\approx\jmath^{2}\rho^{4}$ as $\rho$ approaches
infinity. By using the approximation $\rho(s)\approx\sqrt{2Ls}$ found above,
the equation becomes

$2\dot{z}+s\ddot{z}=0\,,$ (4.27)

whose solution is

$z(s)=\frac{D}{s}+C\,,$ (4.28)

where $C$, $D$ are integration constants. This result clearly shows that $z$
becomes constant as $s$ approaches infinity.

#### 4.3 Schwarzschild black hole in a swirling universe

##### 4.3.1 Physical properties

The full black hole metric (4.12), that we report here for convenience

$\begin{split}{ds}^{2}&=F(r,\theta)\biggl{[}-\biggl{(}1-\frac{2m}{r}\biggr{)}{dt}^{2}+\frac{dr^{2}}{1-\frac{2m}{r}}+r^{2}d\theta^{2}\biggr{]}\\\
&\quad+\frac{r^{2}\sin^{2}\theta}{F(r,\theta)}\biggl{\\{}d\varphi+\big{[}4\jmath(r-2m)\cos\theta+\omega_{0}\big{]}\
dt\biggr{\\}}^{2}\,,\end{split}$ (4.29)

with $F(r,\theta)=1+\jmath^{2}r^{4}\sin^{4}\theta$, is a two parameters
metric, with $m$ and $\jmath$ related to the mass of the black hole and the
angular velocity of the background, respectively.

In view of the previous Section, the spacetime (4.29) can be interpreted as a
Schwarzschild black hole embedded into a swirling background. The main casual
structure is similar to the Schwarzschild case, as can be readily understood
by looking at some $\theta=$ constant slices of the conformal diagram. For
instance, the cases $\theta=\\{0,\pm\pi\\}$ precisely retrace the static
spherically symmetric black hole.

Indeed the metric (4.29) is characterised by a coordinate singularity located
at $r=2m$, which identifies the event horizon of the black hole. This latter
is a Killing horizon that has the same significance of the standard
Schwarzschild horizon. The presence of the rotating background deforms the
horizon geometry, making it more oblate, while maintaining exactly the same of
the Schwarzschild black hole, for the same values of the mass parameter $m$.
In Fig. 4.2 the deformation is pictured for different intensities of the
rotating gravitational background, governed by the new parameter $\jmath$
introduced by the Ehlers transformation.

(a)  $m=1$, $\jmath=0.05$

(b)  $m=1$, $\jmath=0.1$

(c)  $m=1$, $\jmath=0.3$

Figure 4.2: Embedding in Euclidean three-dimensional space $\mathbb{E}^{3}$ of
the event horizon of the black hole distorted by the rotating background, for
three different values of the background rotational parameter $\jmath$.

The solution (4.29) is free from axial conical singularities: to verify this,
it is sufficient to consider the ratio between the perimeter of a small circle
around the $z$-axis, both for $\theta=0$ and $\theta=\pi$, and its radius.
Such a ratio must be equal to $2\pi$, in case one wants to avoid angular
defects. It turns out that, for the metric (4.29), the ratios in the two
limits are equal to $2\pi$

$\lim_{\theta\to
0}\frac{1}{\theta}\int_{0}^{2\pi}\sqrt{\frac{g_{\varphi\varphi}}{g_{\theta\theta}}}d\varphi=2\pi=\lim_{\theta\to\pi}\
\frac{1}{\pi-\theta}\int_{0}^{2\pi}\sqrt{\frac{g_{\varphi\varphi}}{g_{\theta\theta}}}d\varphi\,.$
(4.30)

The metric function $\omega(r,\theta)$ is regular both asymptotically and on
the symmetry axis, thus implying the absence of Misner strings or NUT charges.
It is not only continuous, as we can appreciate by the following limits

$\lim_{\theta\to 0}\ \frac{g_{t\varphi}}{g_{tt}}=\lim_{\theta\to\pi}\
\frac{g_{t\varphi}}{g_{tt}}=0\,,$ (4.31)

but also its first and second derivatives are continuous.

A peculiar characteristic of this metric is that the angular velocity $\Omega$
on the $z$-axis is not constant, and it increases in opposite directions in
the two hemispheres

$\displaystyle\Omega\big{|}_{\theta=0}$ $\displaystyle=\lim_{\theta\rightarrow
0}\biggl{(}-\frac{g_{t\varphi}}{g_{\varphi\varphi}}\biggr{)}=-4\jmath(r-2m)+\omega_{0}\,,$
(4.32a) $\displaystyle\Omega\big{|}_{\theta=\pi}$
$\displaystyle=\lim_{\theta\rightarrow\pi}\biggl{(}-\frac{g_{t\varphi}}{g_{\varphi\varphi}}\biggr{)}=4\jmath(r-2m)+\omega_{0}\,.$
(4.32b)

This is a feature shared with magnetised Reissner–Nordström and magnetised
Kerr black holes solutions [223].

The frame dragging of the whole spacetime is given by [231]

$\frac{d\phi}{dt}=-\frac{g_{t\varphi}}{g_{\varphi\varphi}}=-4\jmath(r-2m)\cos\theta+\omega_{0}\,.$
(4.33)

Hence, outside the event horizon, for $r>2m$, the angular velocity coincides
with the asymptotic one $\omega_{0}$ for $\theta=\frac{\pi}{2}$, while for
$\theta\in(\frac{\pi}{2},\pi)$ it is bigger than $\omega_{0}$ and for
$\theta\in(0,\frac{\pi}{2})$ it is smaller than $\omega_{0}$. It is easy to
verify that for $r\to\infty$ the angular velocity grows unbounded and that it
is equal to $\omega_{0}$ on the event horizon: this would lead to the
conclusion that superluminal observers exist, since the value of the
gravitational dragging can easily exceed 1 (i.e. the speed of light, in our
units) and, then, it would violate causality. In this perspective, let us
study the possible occurrence of closed timelike curves (CTCs): considering
(4.29), curves in which $t$, $r$ and $\theta$ are constants are characterised
by

${ds}_{t,r,\theta=\text{const}}^{2}=F^{-1}(r,\theta)\,r^{2}\sin^{2}\theta{d\phi}^{2}\,.$
(4.34)

Such intervals are always space-like since the expression is always positive.
Therefore there are no CTCs and there are no related causality issues: thus
the “paradox” of the superluminal observers can be justified with the bad
choice of the coordinates222As it happens, for example, for the Alcubierre
spacetime [232]. A set of coordinates which is adapted to timelike observers
does not experience an unbounded growth of the angular velocity, as we will
see explicitly when studying the geodesics of the spacetime.

The Kretschmann scalar $R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$ suggests
that $r=2m$ is a coordinate singularity, while it is divergent for $r=0$, as
in the case of the static spherically symmetric black hole in pure General
Relativity. It is indeed possible to find an Eddington–Finkelstein coordinate
system that removes the $r=2m$ horizon. In particular, as $r\to 0$ we find

$R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\approx\frac{48m^{2}}{r^{6}}\,,$
(4.35)

which is exactly the Kretschmann scalar for the Schwarzschild spacetime. On
the other hand, the scalar invariant falls faster than the Schwarzschild
metric for large radial distances, indeed one finds, as $r\to\infty$,

$R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\approx\frac{192}{\jmath^{4}\sin^{12}\theta\,r^{12}}\,,$
(4.36)

therefore the solution (4.29) is locally asymptotically flat. We finally
notice that for $\theta=0,\pi$ the spacetime has an asymptotically constant
curvature: we find

$R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\big{|}_{\theta=0,\pi}\approx-192\jmath^{2}\quad\text{
as }r\to\infty\,,$ (4.37)

thus we see that on the $z$-axis, the spacetime is asimptotically of negative
constant curvature.

##### 4.3.2 Ergoregions

It is clear, just by inspection, that the $g_{tt}$ component of the metric
(4.29) becomes null on the event horizon, and that outside the horizon is not
everywhere negative. Therefore the spacetime presents some ergoregions,
analogously to Kerr or magnetised Reissner–Nordström black holes [223]. To
analyse these regions it is convenient to use the cylindrical coordinates as
defined in (4.15) to expand, for large $z$, the $g_{tt}$ part of the metric as
follows

$g_{tt}(\rho,z)\approx\frac{16\jmath^{2}\rho^{2}z(z-4m)}{1+\jmath^{2}\rho^{4}}\,.$
(4.38)

Hence, the ergoregions are located not only in the proximity of the event
horizon, as in Kerr spacetime, but also close to the $z$-axis, for large
values of $z$. A numerical analysis of the function $g_{tt}$ is represented in
Fig. 4.3: it shows how the ergoregions extend to infinity around the polar
axis, independently of the values of the integrating constants of the solution
$(m,\jmath)$. This behaviour is similar to what happens for magnetised
rotating black holes [223].

Figure 4.3: Ergoregions for the black hole embedded in a rotating universe,
with parameters $m=1$, $\jmath=0.3$ and $\omega_{0}=0$. The ergoregions extend
to infinity in the positive and negative $z$ directions, independently of the
choice of the parametrisation for the integrating constants.

##### 4.3.3 Petrov type

A standard procedure to determine the Petrov type of a spacetime consists in
computing the Weyl tensor in a null tetrad basis. We define a frame by

$\displaystyle e^{0}$
$\displaystyle=F^{1/2}\biggl{(}1-\frac{2m}{r}\biggr{)}^{1/2}\,dt\,,$ (4.39a)
$\displaystyle e^{1}$
$\displaystyle=F^{1/2}\biggl{(}1-\frac{2m}{r}\biggr{)}^{-1/2}\,dr\,,$ (4.39b)
$\displaystyle e^{2}$ $\displaystyle=rF^{1/2}\,d\theta\,,$ (4.39c)
$\displaystyle e^{3}$ $\displaystyle=r\sin\theta
F^{-1/2}\bigl{\\{}d\varphi+\big{[}4\jmath(r-2m)\cos\theta+\omega_{0}\big{]}\
dt\bigr{\\}}\,.$ (4.39d)

Given such a frame, the null tetrad is found as

$k_{\mu}=\frac{1}{\sqrt{2}}\bigl{(}e^{0}_{\mu}+e^{3}_{\mu}\bigr{)}\,,\quad
l_{\mu}=\frac{1}{\sqrt{2}}\bigl{(}e^{0}_{\mu}-e^{3}_{\mu}\bigr{)}\,,\quad
m_{\mu}=\frac{1}{\sqrt{2}}\bigl{(}e^{1}_{\mu}-ie^{2}_{\mu}\bigr{)}\,,\quad\bar{m}_{\mu}=\frac{1}{\sqrt{2}}\bigl{(}e^{1}_{\mu}+ie^{2}_{\mu}\bigr{)}\,.$
(4.40)

It is now possible to compute the components of the Weyl tensor in the null
basis, as

$\displaystyle\Psi_{0}$
$\displaystyle=C_{\mu\nu\rho\sigma}k^{\mu}m^{\nu}k^{\rho}m^{\sigma}\,,$
(4.41a) $\displaystyle\ \Psi_{1}$
$\displaystyle=C_{\mu\nu\rho\sigma}k^{\mu}l^{\nu}k^{\rho}m^{\sigma}\,,$
(4.41b) $\displaystyle\Psi_{2}$
$\displaystyle=C_{\mu\nu\rho\sigma}k^{\mu}m^{\nu}\bar{m}^{\rho}l^{\sigma}\,,$
(4.41c) $\displaystyle\Psi_{3}$
$\displaystyle=C_{\mu\nu\rho\sigma}l^{\mu}k^{\nu}l^{\rho}\bar{m}^{\sigma}\,,$
(4.41d) $\displaystyle\Psi_{4}$
$\displaystyle=C_{\mu\nu\rho\sigma}l^{\mu}\bar{m}^{\nu}l^{\rho}\bar{m}^{\sigma}\,,$
(4.41e)

where $C_{\mu\nu\rho\sigma}$ is the Weyl tensor.

One can easily show that $\Psi_{1}=\Psi_{3}=0$, while the other components are
more involved. The inspection of the scalar invariants

$I=\Psi_{0}\Psi_{4}-4\Psi_{1}\Psi_{3}+3\Psi_{2}^{2}\,,\quad
J=\det\begin{pmatrix}\Psi_{0}&\Psi_{1}&\Psi_{2}\\\
\Psi_{1}&\Psi_{2}&\Psi_{3}\\\ \Psi_{2}&\Psi_{3}&\Psi_{4}\end{pmatrix}\,,$
(4.42)

reveals that $I^{2}\neq 27J^{2}$: this implies that the spacetime is
algebraically general [119, 135]. Thus, the spacetime belongs to the general
Petrov type I, contrary to its background (4.14) or its generating seed, which
are both type D. Further, this result shows that the new black hole (4.29)
does not belong to the Plebański–Demiański class of spacetimes [134].

##### 4.3.4 Geodesics

We follow the same strategy as in the background case and define, from the
metric (4.29), the following Lagrangian (dropping the inessential $\omega_{0}$
term)

$\mathscr{L}=F\biggl{[}-\biggl{(}1-\frac{2m}{r}\biggr{)}\dot{t}^{2}+\frac{\dot{r}^{2}}{1-\frac{2m}{r}}+r^{2}\dot{\theta}^{2}\biggr{]}\\\
+F^{-1}r^{2}\sin^{2}\theta\Bigl{[}\dot{\varphi}+4\jmath(r-2m)\cos\theta\,\dot{t}\Bigr{]}^{2}\,,$
(4.43)

Proceeding in the same way as the background metric, we obtain the conserved
charges equations and the four-momentum normalisation equations, reported in
Appendix C.2.

We can extract some qualitative information, especially regarding the quantity
$(r-2m)\cos\theta$ that appears in the gravitational dragging. For stable
orbits $r$ is limited, hence the quantity $(r-2m)\cos\theta$ is limited as
well. For unstable orbits we analyse the geodesic motion as $r$ reaches
infinity, thus considering large values of $s$. We notice that $\dot{t}\approx
0$ and $\dot{\varphi}\approx\jmath^{2}Lr^{2}\sin^{2}\theta$, as $r\to\infty$,
and moreover $1-\frac{2m}{r}\approx 1$ and
$F\approx\jmath^{2}r^{4}\sin^{4}\theta$. These approximations simplify the
Lagrangian (4.43), that takes the form

$\mathscr{L}\approx\jmath^{2}r^{4}\sin^{4}\theta\bigl{(}\dot{r}^{2}+r^{2}\dot{\theta}^{2}\bigr{)}+\jmath^{2}L^{2}\,.$
(4.44)

The constant term is inessential and can be neglected. By changing to polar
coordinates $x=r\sin\theta$ $y=r\cos\theta$, the Lagrangian boils down to

$\mathscr{L}\approx\jmath^{2}x^{4}\bigl{(}\dot{x}^{2}+\dot{y}^{2}\bigr{)}\,.$
(4.45)

Being the Lagrangian independent of $y$, we find the conserved quantity

$A=\jmath^{2}x^{4}\dot{y}\,.$ (4.46)

This result can be plugged into the Lagrangian, and by noticing that it does
not depend explicitly on $s$, $d\mathscr{L}/ds=0$, the following equation is
derived:

$\jmath^{2}x^{4}\dot{x}^{2}+\frac{A^{2}}{\jmath^{2}x^{4}}=B\,,$ (4.47)

where $B$ is a real constant. From the last equation we find

$\dot{x}=\frac{\sqrt{B\jmath^{2}x^{4}-A^{2}}}{\jmath^{2}x^{4}}\approx\frac{\sqrt{B}}{\jmath
x^{2}}\,,$ (4.48)

where the numerator $\sqrt{B\jmath^{2}x^{4}-A^{2}}$ depends on $x$ which, by
our change of coordinate, is proportional to $r$, so when $r$ approaches
infinity so does $x$. Therefore the constant $A^{2}$ underneath the square
root can be neglected, thus justifying the approximation. Finally we get, by
integration,

$x(s)=\biggl{(}\frac{3\sqrt{B}}{\jmath}s+C\biggr{)}^{\frac{1}{3}}\,,$ (4.49)

with $C$ real constant. So $x\to\infty$ as $s\to\infty$, which means that
$\dot{y}\approx 0$ and $y\approx$ constant. These results can now be plugged
into the formula for the gravitational dragging, which gives

$-\frac{g_{t\varphi}}{g_{\varphi\varphi}}=-4\jmath\biggl{(}y-2m\frac{y}{\sqrt{x^{2}+y^{2}}}\biggr{)}\underset{s\to\infty}{\approx}-4\jmath
y\,.$ (4.50)

This result shows that, as $s\to\infty$, the angular velocity approaches a
constant value.

We also plot the geodesic motion in spacetime (4.29): this amounts to
numerically integrate the geodesic equations reported in Appendix C.2, and the
results are shown in Fig. 4.4, 4.5 and C.1. More precisely, Fig. 4.4 compares
geodesics in Schwarzschild spacetime (i.e. $\jmath=0$) and geodesics in our
swirling spacetime (4.29). Fig. 4.5 shows the geodesics around the black hole
for different initial conditions. Finally, in Appendix C.2, Fig. C.1 pictures
unstable geodesic motion for two different values of the test particle angular
momentum.

(a) $m=1$, $\jmath=0$ (b) $m=1$, $\jmath=10^{-8}$

Figure 4.4: Geodesic motion around the black hole. The left panel shows the
Schwarzschild spacetime, while the right panel shows the new black hole
solution (4.29). The plots share the same initial conditions with $E=1$,
$L=12$.

(a) $m=1$, $\jmath=0.01$ (b) $m=1$, $\jmath=0.01$

Figure 4.5: Embedding diagram and geodesics for the new metric (4.29) with
different initial conditions, $\theta_{0}=\frac{\pi}{2}$,
$\dot{\theta}_{0}=-\frac{1}{2}$, $L=12$ for the l.h.s. diagram and
$\theta_{0}=\frac{4}{7}\pi$,
$\dot{\theta}_{0}=-\frac{1}{2}\sin^{-1}(\frac{4}{7}\pi)$, $L=-12$ for the
r.h.s. diagram. Both representations share the following data: $E=1$,
$r_{0}=3$, $\dot{r}_{0}=1$ and $\phi_{0}=\pi$.

##### 4.3.5 Charges and thermodynamics

The total mass of the spacetime can be computed by means of the surface
charges provided by the phase space formalism [233, 234]. We perturb the
metric with respect to the parameters of the solution, and we name that
variation $h_{\mu\nu}\coloneqq\delta g_{\mu\nu}$333In the particular case
under consideration the parameter space is spanned by the mass parameter of
the black hole $m$ and by the magnitude of the rotational whirlpool dragging,
$\jmath$. Thus the variation takes the form $h_{\mu\nu}=\delta
g_{\mu\nu}(m,\jmath)=\partial_{m}g(m,\jmath)\delta
m+\partial_{\jmath}g(m,\jmath)\delta\jmath$.. Then we find the local variation
of the charge $K_{\xi}$ computed along a given Killing direction $\xi^{\mu}$.

The local variation of the charge must be integrated between the parametric
reference background $\bar{\Psi}$ and the actual parametric configuration
labelled by $\Psi$, on a $D-2$ dimensional surface $\mathcal{S}$ containing
the event horizon
$\bigl{(}d^{D-2}x\bigr{)}_{\mu\nu}=\frac{1}{2(D-2)!}\epsilon_{\mu\nu\alpha_{1}...\alpha_{D-2}}dx^{\alpha_{1}}\wedge\cdots\wedge
dx^{\alpha_{D-2}}$. When the variation of the charge is integrable, all the
parametric paths between the reference background and the solution are
equivalent444In case the variation of the charge is not integrable, we still
have some gauge degree of freedom in defining the frame of reference, or the
normalisation of the time coordinate, to recover integrability..

The result gives the total surface charge $Q_{\xi}$, defined, as in [234,
235], by

$Q_{\xi}=\frac{1}{8\pi}\int_{\bar{\Psi}}^{\Psi}\int_{\mathcal{S}}K_{\xi}=\frac{1}{8\pi}\int_{\bar{\Psi}}^{\Psi}\int_{\mathcal{S}}K_{\xi}^{\mu\nu}\bigl{(}d^{D-2}x\bigr{)}_{\mu\nu}\,,$
(4.51)

where

$K_{\xi}^{\mu\nu}=\xi^{\mu}\nabla_{\sigma}h^{\nu\sigma}-\xi^{\mu}\nabla^{\nu}h-\xi_{\sigma}\nabla^{\mu}h^{\nu\sigma}-\frac{1}{2}h\nabla^{\mu}\xi^{\nu}+\frac{1}{2}h^{\sigma\mu}(\nabla_{\sigma}\xi^{\nu}-\nabla^{\nu}\xi_{\sigma})\,,$
(4.52)

and where $h\coloneqq h^{\mu}_{\;\;\mu}$. If we want to compute the mass of
the black hole, we have to consider the timelike Killing vector
$\xi=\partial_{t}$, then we find

$M=Q_{\partial_{t}}=m\,,$ (4.53)

as for the Schwarzschild black hole. In this case the presence of the
background does not modify the seed black hole mass, similarly to what happens
within the context of black holes embedded in an external electromagnetic
field [165]. Following this analogy we expect to observe some stronger
coupling with the background in case of more general black hole seeds, as it
happens in the next Section.

The angular momentum can be found analogously, just considering the Killing
vector which generates the rotational symmetry $\partial_{\varphi}$. In this
case one gets null angular momentum

$J=Q_{\partial_{\varphi}}=0\,,$ (4.54)

even though the solution is clearly rotating. In fact the angular momentum
refers just to the dipole term in the rotational multipolar expansion of the
metric at large distances. The fact that the metric is rotating, as its non-
diagonal form suggests, can be appreciated by the subsequent terms of the
multipolar expansion: the quadrupole, the octupole, etc555However, note that
in contexts where the asymptotia is not of globally constant curvature, the
notion of the gravitational multipolar expansion needs some further analysis
to be clearly defined..

We compute the entropy and the temperature of the event horizon, in order to
study the Smarr law and the thermodynamics of the black hole. The area of the
even horizon is found by integrating the $(\theta,\phi)$ part of the metric,
hence

$\mathcal{A}=\int_{0}^{2\pi}d\varphi\int_{0}^{\theta}d\theta\sqrt{g_{\theta\theta}g_{\phi\phi}}\big{|}_{r=2m}=16\pi
m^{2}\,.$ (4.55)

The entropy is given by the Bekenstein–Hawking formula $S=\mathcal{A}/4$. The
validity of the area law also for this unconventional background is confirmed
by the conformal field theory dual to the near-horizon geometry of the black
hole. The temperature can be easily obtained via the surface gravity,
$\kappa=\sqrt{-(\nabla_{\mu}\xi_{\nu})^{2}/2}\big{|}_{r=2m}$, where
$\xi=\partial_{t}$. We find

$T=\frac{\kappa}{2\pi}=\frac{1}{8\pi m}\,.$ (4.56)

Note that the entropy and the temperature of the black hole embedded into the
swirling background are unaffected by the spacetime rotation: they remain the
same of the
Schwarzschild seed. Again this is peculiar of the Lie point symmetry we used
to generate the solution, a general feature shared with the Harrison
transformation666Note that this is true only when the seed does not couple
with the background brought in by the transformation..

We can easily verify the validity of the Smarr law

$M=2TS\,.$ (4.57)

Further, the conserved charges satisfy the first law of thermodynamics

$\delta M=T\delta S\,.$ (4.58)

#### 4.4 Kerr black hole in a swirling universe

The generating techniques discussed in Sec. 4.1 can be also exploited to embed
a rotating black hole in a background endowed with its own rotation. By using
the Kerr metric in Boyer–Lindquist coordinates as a seed, we obtain

${ds}^{2}=F(d\varphi-\omega
dt)^{2}+F^{-1}\biggl{[}-\rho^{2}{dt}^{2}+\Sigma\sin^{2}\theta\biggl{(}\frac{{dr}^{2}}{\Delta}+{d\theta}^{2}\biggr{)}\biggr{]}\,,$
(4.59)

where the functions $F^{-1}$ and $\omega$ can be expanded in finite power
series of $\jmath$

$\displaystyle F^{-1}$
$\displaystyle=\chi_{(0)}+\jmath\chi_{(1)}+\jmath^{2}\chi_{(2)}\,,\qquad\omega=\omega_{(0)}+\jmath\omega_{(1)}+\jmath^{2}\omega_{(2)}\,,$
(4.60)

with

$\displaystyle\chi_{(0)}$
$\displaystyle=\frac{R^{2}}{\Sigma\sin^{2}\theta}\,,$ (4.61a)
$\displaystyle\chi_{(1)}$
$\displaystyle=\frac{4am\,\Xi\cos\theta}{\Sigma\sin^{2}\theta}\,,$ (4.61b)
$\displaystyle\chi_{(2)}$
$\displaystyle=\frac{4a^{2}m^{2}\Xi^{2}\cos^{2}\theta+\Sigma^{2}\sin^{4}\theta}{R^{2}\Sigma\sin^{2}\theta}\,,$
(4.61c)

and

$\displaystyle\omega_{(0)}$ $\displaystyle=-\frac{2amr}{\Sigma}+\omega_{0}\,,$
(4.62a) $\displaystyle\omega_{(1)}$
$\displaystyle=\frac{4\cos\theta[-a\Omega(r-m)+ma^{4}-r^{4}(r-2m)-\Delta
a^{2}r]}{-\Sigma}\,,$ (4.62b)
$\displaystyle\begin{split}\omega_{(2)}&=-\frac{2m}{\Sigma}\Bigl{\\{}3ar^{5}-a^{5}(r+2m)+2a^{3}r^{2}(r+3m)-r^{3}(\cos^{2}\theta-6)\Omega\\\
&\quad+a^{2}[\cos^{2}\theta(3r-2m)-6(r-m)]\Omega\Bigr{\\}}\,,\end{split}$
(4.62c)

where

$\displaystyle\Delta$
$\displaystyle=r^{2}-2mr+a^{2}\,,\qquad\qquad\qquad\qquad\qquad\\!\\!\\!\\!\\!\\!\\!\\!\rho^{2}=\Delta\sin^{2}\theta\,,$
(4.63a) $\displaystyle\Sigma$ $\displaystyle=(r^{2}+a^{2})^{2}-\Delta
a^{2}\sin^{2}\theta\,,\qquad\qquad\qquad\\!\\!\\!\\!\\!\\!\Omega=\Delta
a\cos^{2}\theta\,,$ (4.63b) $\displaystyle\Xi$
$\displaystyle=r^{2}(\cos^{2}\theta-3)-a^{2}(1+\cos^{2}\theta)\,,\qquad
R^{2}=r^{2}+a^{2}\cos^{2}\theta\,.$ (4.63c)

When $\jmath=0$ we recover the seed metric, i.e. the Kerr black hole. For
$\jmath\neq 0$ we have the direct generalisation of the metric (4.12).

However we notice that in this case, because of the spin-spin interaction
between the black hole and the background frame dragging, an extra force acts
on the axis of symmetry. But since it is not symmetric on the two hemispheres,
the metric is affected by non-removable conical singularities, indeed

$\lim_{\theta\to
0}\frac{1}{\theta}\int_{0}^{2\pi}\sqrt{\frac{g_{\varphi\varphi}}{g_{\theta\theta}}}d\varphi=\frac{2\pi}{(1-4am\jmath)^{2}}\neq\lim_{\theta\to\pi}\
\frac{1}{\pi-\theta}\int_{0}^{2\pi}\sqrt{\frac{g_{\varphi\varphi}}{g_{\theta\theta}}}d\varphi=\frac{2\pi}{(1+4am\jmath)^{2}}\,.$
(4.64)

In fact, even though the background spinning parameter $\jmath$ couples to the
Kerr angular momentum (for unit of mass) $a$, it is not possible to find a
relation among the physical parameters to remove simultaneously both angular
defects, unless of course for known subcases such as Kerr, for $\jmath=0$, the
spacetime discussed in Sec. 4.3 for $a=0$, or the rotating background for
$m=0$. The presence of a non-removable conical singularity implies that a
cosmic string or a strut (with their $\delta$-like stress-energy-momentum
tensor on a portion of the $z$-axis) have to be postulated in order to
compensate the “force” effect induced by the spin-spin interaction of the
black hole with the background, which would tent to add acceleration to the
black hole777The metric considered in this Section does not posses the
acceleration parameter: one should work with the rotating C-metric to
consistently include the acceleration. That is why, in this Section, the role
of the string uniquely results in the effect of compensating the spin
coupling..

In the case one wants to immerse the Kerr–Newman black hole into this spinning
universe, one has to use the charged generalised version of the Ehlers
transformation, as described in Chapter 1.

#### 4.5 Double-Wick rotation of the background: flat Taub–NUT spacetime

Given the analogies between the rotating background (4.16) and the Melvin
spacetime, and given that the analytical continuation of the Melvin universe
corresponds to the Reissner–Nordström metric with a flat base manifold, it is
natural to inquire about an analog analytical continuation for the rotating
background. At this scope, we implement a double Wick rotation between time
and the azimuthal angle $t\to i\phi$, $\varphi\to i\tau$ of the metric (4.16).
Redefining the integration constant of the rotating background as
$\jmath=m/2\ell^{3}$, changing the coordinate $\rho=\ell\sqrt{2r/m}$ and after
the rescaling of the other three coordinates we obtain

${ds}^{2}=-\frac{2mr}{r^{2}+\ell^{2}}(dt-2\ell\theta
d\phi)^{2}+\frac{r^{2}+\ell^{2}}{2mr}{dr}^{2}+(r^{2}+\ell^{2})({d\theta}^{2}+{d\phi}^{2})\,.$
(4.65)

It is not hard to recognise the Taub–NUT spacetime with a flat, or possibly
cylindrical if we keep the azimuthal angle identification, base manifold. In
fact, the flat Taub–NUT metric can be generated via the Ehlers
transformation888In this Section we are referring to transformations applied
to the _magnetic_ LWP metric, as explained in Sec. 4.1. from the Schwarzschild
metric, previously composed with a double-Wick rotation999While this is true
for a generic sign of the constant curvature of the seed base manifold, only
the metric with positive curvature can be interpreted as a black hole in
Einstein gravity.. Note that the Ehlers transformation can be used to build,
from the Minkowski seed, the rotating background, just considering $m=0$ in
the procedure of Sec. 4.1. This is analogous to what happens to the Melvin
universe, which can be obtained from Minkowski spacetime via the Harrison
transformation and whose double-Wick dual corresponds to the flat
Reissner–Nordström metric. This fact strengthens the link between the Melvin
universe and our rotating background. Actually this correspondence can be
summarized by the following proportion:

Melvin Universe : Harrison transformation = Rotating Universe : Ehlers
transformation (4.66)

This formal analogy can be exploited to build new solutions, even outside the
range of the generating technique based on the Lie point symmetries of the
Ernst equations: that is important because the symmetry transformations such
as the Ehlers and the Harrison maps break in the presence of the cosmological
constant. However, as noted in [132], the Melvin universe can still be
generalised when the cosmological constant is not zero, and it still preserves
its relation with the flat Reissner–Nordström metric with a constant curvature
base manifold101010Metrics without a topological spherical base manifold are
interpreted as black holes only in the presence of the cosmological constant..
Therefore, thanks to the analogy with the Melvin case, we have in our hand a
procedure to generalise the rotating background (4.16) in the presence of the
cosmological constant.

It is sufficient to operate a double-Wick rotation of the cosmological version
of the flat Taub–NUT metric (4.65)

$\begin{split}{ds}^{2}&=-\frac{\frac{\Lambda}{3}r^{4}+2\ell^{2}\Lambda
r^{2}+2mr-\ell^{4}\Lambda}{r^{2}+\ell^{2}}(dt-2\ell\theta d\phi)^{2}\\\
&\quad+\frac{(r^{2}+\ell^{2})}{\frac{\Lambda}{3}r^{4}+2\ell^{2}\Lambda
r^{2}+2mr-\ell^{4}\Lambda}{dr}^{2}+(r^{2}+\ell^{2})({d\theta}^{2}+{d\phi}^{2})\,.\end{split}$
(4.67)

Thus, using the same change of coordinates and parametrisation of the case
above, we get

$\begin{split}{ds}^{2}&=(1+\jmath^{2}\rho^{4})\biggl{(}-{d\tau}^{2}+\frac{\rho^{2}}{\frac{\Lambda}{4\jmath^{2}}+\rho^{2}-\frac{\Lambda}{2}\rho^{4}-\frac{\jmath^{2}\Lambda}{12}\rho^{8}}{d\rho}^{2}+{dz}^{2}\biggr{)}\\\
&\quad+\biggl{(}\frac{\Lambda}{4\jmath^{2}}+\rho^{2}-\frac{\Lambda}{2}\rho^{4}-\frac{\jmath^{2}\Lambda}{12}\rho^{8}\biggr{)}\frac{(d\psi+4\jmath
zd\tau)^{2}}{1+\jmath^{2}\rho^{4}}\,.\end{split}$ (4.68)

It is not difficult to realise that this metric still corresponds, up to a
change of coordinates, to the non-expanding and non-accelerating Kuntdt class
of the Plebański–Demiański family presented in Eq. (16.26) of [135]. The
explicit change of coordinates works as in Sec. 4.2, namely $q=2\jmath z$ and
$p^{2}=\rho$, together with the rescaling $t\to\jmath t$ and the redefinitions
$\gamma=1/\jmath$, $\tilde{\Lambda}=4\Lambda$. Then, metric (4.68) becomes

${ds}^{2}=(\gamma^{2}+p^{2})\biggl{(}-{d\tau}^{2}+\frac{dp^{2}}{\mathcal{P}}+{dq}^{2}\biggr{)}+\frac{\mathcal{P}}{\gamma^{2}+p^{2}}(d\psi+2\gamma
qd\tau)^{2}\,,$ (4.69)

where

$\mathcal{P}=\gamma^{4}\tilde{\Lambda}+\gamma^{2}p-2\gamma^{2}\tilde{\Lambda}p^{2}-\frac{\tilde{\Lambda}}{3}p^{4}\,.$
(4.70)

The latter corresponds to Eq. (16.26) of [135], where
$m=e=g=\alpha=\epsilon_{2}=0$, $\epsilon_{0}=1$,
$k=\gamma^{4}\tilde{\Lambda}$, $\epsilon=2\gamma^{2}\tilde{\Lambda}$ and
$n=\gamma^{2}/2$.

The analogy between the rotating background and the Taub–NUT spacetime can be
pushed further: it is known [217] that the Melvin spacetime corresponds to a
couple of magnetically charged Reissner–Nordström black holes moved towards
infinity. In this sense, the magnetic field which permeates the Melvin
spacetime is nothing but the field generated by two black hole sources at
infinity. Thus, it is natural to ask ourselves if a similar construction also
holds for the rotating background (4.14), i.e. if it can be obtained as a
limit of a double black hole metric.

By relying on the above considerations and, more specifically, on the
proportion (4.66), the natural candidate for an “ancestor” metric is the
double-Taub–NUT spacetime with opposite NUT parameters [26]: the rotation of
the two counter-rotating Taub–NUT black holes, once they are pushed at
infinity, should produce the rotation of the background that is experienced in
the background spacetime. This interpretation is also consistent with the
behaviour of the angular velocity (4.32): we noticed that the angular velocity
increases in opposite directions in the two hemisphere, coherently with the
fact that the two black holes rotate in different directions. Moreover this
picture is enforced by the geometry of the ergoregions, since the latter
thrive for large values of $z$ on the axis of symmetry.

#### 4.6 Outlook

The transformation we studied in this Chapter, which consists in a proper
composition of the Ehlers transformation with a discrete symmetry, allows us
to take advantage of the Ernst solution generating technique to non-linearly
superpose the Schwarzschild black hole and a swirling universe. The background
geometry can be interpreted as a gravitational whirlpool generated by a couple
of counter-rotating sources at infinity. Its frame dragging transforms the
static Schwarzschild metric into a stationary one, removing the asymptotic
flatness, but without drastically altering the black hole causal structure nor
introducing pathological features. The analogies between the swirling
background and the Melvin universe are numerous, like the metric structure,
the ergoregions and the deformations engraved on the event horizon: in fact
the former universe can be considered as the rotating counterpart of the
latter.

For this reason we expect that this spinning background can be used as a
regularising instrument for metric with conical singularities, exactly as the
electromagnetic background brought by the Harrison transformation does. In the
former case to have non-trivial physical effects one needs to exploit the
interplay between the coupling of the electromagnetic field of the seed with
the one of the background, as suggested by the analysis of the transformed
Kerr metric, in Sec. 4.4. Indeed the interaction between the Kerr parameter
$a$ and the background parameter $\jmath$ generates an additional “force”
which impels the system to accelerate. Unfortunately, the geometry of the
spacetime is not general enough to accommodate this physical feature into that
metric, yielding a conical singularity which compensates the mutual rotational
coupling. On the other hand we count that the spin-spin interaction between
the seed and the background environment can play a relevant role into the
regularisation of gravitational models which otherwise would be mathematically
defective and physically incomplete. For instance, in the same way as Ernst
showed that the electromagnetic background can remove the conical defect of
the accelerating and charged black hole, we foresee that the procedure
presented in this paper can remove the axial singularities of the rotating
C-metric, providing at the same time a reasonable physical explication for its
acceleration [236]. Also this model furnishes alternative scenarios for black
hole nucleation and pair creation, without relying on the electromagnetic
field, as discussed in the literature so far [187, 188, 185].

Clearly this procedure may be relevant for other systems, not necessarily
accelerating, such as balancing multi-black hole sources to reach an
equilibrium configuration. Also in that case the frame dragging of the
background can play a role in removing cosmic strings or strut from the
singular spacetime. On the other hand, preliminary studies suggest that the
spin-spin interaction between the swirling universe and a Taub–NUT spacetime
are not sufficient to mend also the singular behaviour of that metric, i.e. to
remove the Misner string111111Obviously we are referring to the non-compact
time representation of the Taub–NUT metric, because when one considers proper
periodic identification of the temporal coordinate the spacetime can be
regularised. Unfortunately, the latter interpretation violates causality
because of the appearance of closed timelike curves, which makes this picture
nonphysical..

From a phenomenological point of view our rotating background might be of some
interest in the description of black holes surrounded by interacting matter,
which produces intense frame dragging, such as the one caused by the collision
of counter-rotating galaxies.

Since this construction is based on a symmetry transformation of the Ernst
equations, it can be directly generalised to the Einstein–Maxwell case, to the
minimally and conformally coupled scalar field case and, more generally, to
scalar-tensor theories such as Brans–Dicke, just by using the adequate Ehlers
transformation as described in Sec. 1.1 and in [237] respectively. The
embedding method presented here may reveal useful in estabilishing and
improving traversability of wormhole spacetimes.

### Conclusions

Black holes represent one of the most fascinating areas of research in
contemporary physics: they are bizarre objects that challenge our common
sense, and whose existence represented a mystery for many years. They are also
the natural candidate for studying quantum gravity, as the region close to a
curvature singularity is characterised by strong gravity on very small scales.
As explained in the Introduction, binary and, more generally, multi-black hole
systems are relevant for the experiments and for the structure of the
gravitational theories. Thus, it is important to find regular multi-source
solutions that can describe actual astrophysical black holes and, from the
theoretical point of view, to properly define the thermodynamics of such
systems: indeed, the thermodynamics can give important hints about the
microscopic origin of the entropy and the temperature.

In this Thesis, we have studied some configurations that allow one to
regularise multi-black hole spacetimes, with the aid of solution generating
techniques.

We began by introducing such techniques and explaining the details of their
functioning in Chapter 1, which provided us with the machinery necessary to
the construction of our solutions. We also reported an historical perspective
on the development of the various solution generating methods over the years,
commenting on the peculiar features and on the pros and cons of the different
approaches.

Then, in Chapter 2, we introduced an external gravitational field as a
regularising background, and discussed its properties and multipolar
character. We provided many examples of singularity-free multi-black hole
spacetimes: static, rotating, charged an accelerating systems of black holes
were studied and analysed in detail. They share the common feature of being
regularised by an appropriate tuning of the external field parameters, i.e. by
choosing the multipole coefficients in order to balance the attraction among
the black holes. We inspected the action of the external field on the geometry
of these “deformed” black holes, and we verified that the thermodynamics of
these objects was well defined: this allowed us to conclude that these multi-
black hole systems are physically well posed. Moreover, the external fields
have a phenomenological interest since they can model matter surrounding black
holes.

Chapter 3 introduced the bubbles of nothing to the business of the multi-black
hole regularisation: the bubbles, which are usually studied in relation to the
vacuum stability, were considered as time-dependent expanding spacetimes that
can win against the mutual collapse of many black holes. We showed that
bubbles of nothing have a deep connection with black holes, as it can be
appreciated by taking some appropriate limits which bring a bubble into a
black hole, and viceversa (an analog construction works for bubbles in five
dimensions and black rings). Further, we embedded multi-black holes (in four
dimensions) and black rings (in five dimensions) in bubble spacetimes, and
explicitly showed that they reach an equilibrium configuration. We also
considered other examples that involve accelerating black holes (in four
dimensions) and black Saturn and di-ring (in five dimensions), and how the
manipulation of their rod diagrams naturally allows one to move from one
configuration to another. Thus, bubbles revealed to be very powerful
solutions, which permit to obtain solutions free of conical singularities, and
moreover we proved that their connection to elementary black hole solutions in
General Relativity is deeper than one may expect.

Finally, in Chapter 4 we introduced a transformation that add a “swirling”
background to a given spacetime. Such a rotational character acts as a sort of
whirlpool, which drags the observers that are at rest in the spacetime, as
shown by the embedding diagrams of the geodesics. A static black hole embedded
in the swirling geometry is (perhaps surprisingly) not affected by conical
singularities: it is possible to properly define conserved charges and
thermodynamics, and to study ergoregions and geometrical properties. On the
other hand, the swirling rotation couples to the angular momentum of a black
hole, as shown by the Kerr solution: in that case, conical singularities
appear and the solution is no more regular. This drawback, however, paves the
way to a possible mechanism to regularise a multi-black hole spacetime: the
embedding of a double-Kerr solution into the swirling background and the
resulting coupling between the swirl parameter and the spin-spin interaction
of the holes, might provide a mechanism to remove the conical singularities.
This is a very promising idea, that can be pursued in a future research.

## Appendices

###  Appendix A Conical singularities and energy conditions

Conical singularities, beyond making the spacetime manifold ill-defined from a
mathematical point of view, give also rise to energy issues. In general, such
singularities can be interpreted as strings or struts whose energy-momentum
tensor has a $\delta$-like nature. We show what are the physical issues that
the conical singularities bring in when they are present in the spacetime.

Let us consider Minkowski spacetime with an wedge of angle $2\pi\alpha\prime$
artificially removed. By defining $C=1-\alpha\prime$, we can write the metric
as

${ds}^{2}=-{dt}^{2}+{dr}^{2}+C^{2}r^{2}{d\varphi}^{2}+{dz}^{2}\,,$ (A.1)

where $0\leq\varphi<2\pi$. One can regard this spacetime as a field sourced by
a cosmic string or a strut [238], whose non-vanishing energy-momentum tensor
components are

$T^{t}_{t}=T^{\varphi}_{\varphi}=2\pi\mu\,\delta(x,y)\,,$ (A.2)

where

$\mu=\frac{1-C}{4C}\,,$ (A.3)

and $\delta(x,y)$ is the two-dimensional delta function depending on the
coordinates orthogonal to the $z$-axis $(x,y)$. $\mu$ is interpreted as the
tension of the filament source.

This result can be generalised to a generic four-dimensional spacetime [239],
for which one finds that the Einstein equations $G_{\mu\nu}=8\pi T_{\mu\nu}$
give rise to an energy-momentum tensor

$T^{0}_{0}=T^{3}_{3}=2\pi\mu\,\delta(x,y)\,,$ (A.4)

where now

$\mu=\frac{2\pi-C}{4C}\,.$ (A.5)

$C$ represents again the angular excess/deficit for the azimuthal angle.

Let us consider, e.g., a two-black hole spacetime from (2.18) ($N=2$): the
result (A.4) clearly shows that for $\mu>0$ (positive tension), the source
acts as a string that pull a black hole. This is the behaviour of the conical
singularities that one finds at $z<w_{1}$ and $z>w_{4}$ There are no negative-
energy issues in this case, but the string extends to infinity and the
$\delta$ function gives rise to a divergent energy-momentum tensor.

In the case of $\mu<0$ (negative tension), the conical singularity in
$w_{2}<z<w_{3}$ acts as a strut which pushes apart the two black holes. The
energy density associated to the energy-momentum tensor is negative (i.e. the
strut is composed of anti-gravitational matter) and there is again a
divergence due to the $\delta$ function.

###  Appendix B Harrison and Kramer–Neugebauer charging trasformations

Both the Harrison and the Kramer–Neugebauer [173] transformations are two
symmetries of the Ernst equations for the Einstein–Maxwell theory presented in
Chapter 1. As it was explained there, the complex Ernst equations enjoy an
$SU(2,1)$ Lie-point symmetry group spanned by the finite transformations
(1.46)

The Harrison transformation is given by Eq. (1.46e), while the
Kramer–Neugebauer one, as defined in [240] to charge the Kerr metric embedded
in an external gravitational field, is

$\mathcal{E}^{\prime}=\frac{\mathcal{E}-\zeta^{2}}{1-\zeta^{2}\mathcal{E}}\,,\qquad\Phi^{\prime}=\frac{\zeta(\mathcal{E}-1)}{1-\zeta^{2}\mathcal{E}}\,.$
(B.1)

The latter transformation reduces to the one in (2.63) for static and
uncharged seeds. Since both the Kramer–Neugebauer transformation (B.1) and the
Harrison transformation (1.46e) have the same physical effects (they add an
electric monopole to an uncharged seed), we have the suspect that they are
basically the same transformation, up to gauge transformations. In fact it can
be shown that the subsequent composition of transformations (1.46a), (1.46d)
and (1.46e) to an Ernst seed $(\mathcal{E},\Phi)$ gives

$\displaystyle\mathcal{E}^{\prime}$
$\displaystyle=\frac{\lambda\lambda^{*}\mathcal{E}-\beta^{*}(\beta+2\lambda\Phi)}{1+\alpha^{2}(-2\beta+\alpha^{*}\beta\beta^{*}-\alpha\lambda\lambda^{*}\mathcal{E}+2\lambda(\alpha\beta^{*}-1)\Phi}\,,$
(B.2) $\displaystyle\Phi^{\prime}$
$\displaystyle=\frac{\beta-\alpha\beta\beta^{*}+\lambda\lambda^{*}\alpha\mathcal{E}+\Phi-2\lambda\alpha\beta^{*}\Phi}{1+\alpha^{2}(-2\beta+\alpha^{*}\beta\beta^{*}-\alpha\lambda\lambda^{*}\mathcal{E}+2\lambda(\alpha\beta^{*}-1)\Phi}\,.$
(B.3)

Then considering a null electromagnetic Ernst potential, $\Phi=0$, the
imaginary part of the parameters $\alpha$, $\beta$, $\lambda$ equal to zero
and choosing

$\lambda=\frac{1}{1-\zeta^{2}}\,,\qquad\alpha=\zeta\,,\qquad\beta=-\frac{\zeta}{1-\zeta^{2}}\,,$
(B.4)

we exactly recover the transformation (B.1). In case of static metrics the
latter further simplifies to Eqs. (2.63). Therefore the Kramer–Neugebauer and
the Harrison transfomration are basically equivalent, up to gauge
transformations, so they might be called collectively
Harrison–Kramer–Neugebauer transformation.

As an explicit example we show the efficacy of the charging transformation
(2.63) on an asymptotically flat, static and discharged metric: acting on the
Schwarzschild metric, we are able to produce the Reissner–Nordström black
hole.

For simplicity we take the seed in spherical symmetric coordinates

${ds}^{2}=-\biggl{(}1-\frac{2m}{r}\biggr{)}{dt}^{2}+\frac{{dr}^{2}}{1-\frac{2m}{r}}+r^{2}{d\theta}^{2}+r^{2}\sin^{2}\theta{d\varphi}^{2}\,,$
(B.5)

from which we can easily read

$e^{2\psi}=1-\frac{2m}{r}\,,\qquad A_{t}=0\,.$ (B.6)

After the charging transformation (2.63) we get the new solution

$e^{2\hat{\psi}}=\frac{r(r-2m)(1-\zeta^{2})^{2}}{[r+(2m-r)\zeta^{2}]^{2}}\,,\quad\hat{A}_{t}=-\frac{2m\zeta}{r+(2m-r)\zeta^{2}}\,.$
(B.7)

A shift of the radial coordinate

$r\to\hat{r}-M+\sqrt{M^{2}-q^{2}}\,,$ (B.8)

and a rescaling of the parameters

$\zeta\to\frac{M-\sqrt{M^{2}-q^{2}}}{q}\,,\qquad m\to\sqrt{M^{2}-q^{2}}\,,$
(B.9)

suffice to recognise the standard Reissner–Norström spacetime

$\displaystyle{d\hat{s}}^{2}$
$\displaystyle=-\biggl{(}1-\frac{2M}{\hat{r}}+\frac{q^{2}}{\hat{r}^{2}}\biggr{)}{dt}^{2}+\frac{{d\hat{r}}^{2}}{1-\frac{2M}{\hat{r}}+\frac{q^{2}}{\hat{r}^{2}}}+\hat{r}^{2}{d\theta}^{2}+\hat{r}^{2}\sin^{2}\theta{d\varphi}^{2}\,,$
(B.10) $\displaystyle\hat{A}$ $\displaystyle=-\frac{q}{\hat{r}}\,dt\,.$ (B.11)

###  Appendix C Geodesics of the swirling spacetime

We report here the explicit expression for the geodesic equation from Chapter
4, both for the background and the full black hole metric.

#### C.1 Background geodesics

The explicit expressions for the definitions of the conserved quantities
(4.23) are

$\dot{t}=\frac{E+4\jmath
Lz}{1+\jmath^{2}\rho^{4}}\,,\qquad\dot{\phi}=\jmath^{2}L\rho^{2}+\frac{L}{\rho^{2}}-4\jmath
z\frac{E+4\jmath Lz}{1+\jmath^{2}\rho^{4}}\,.$ (C.1)

By substituting these relations into the Lagrangian (4.22), we get

$\mathscr{L}=(1+\jmath^{2}\rho^{2})\biggl{[}\frac{L^{2}}{\rho^{2}}-\biggl{(}\frac{E+4\jmath
Lz}{1+\jmath^{2}\rho^{4}}\biggr{)}^{2}+\dot{\rho}^{2}+\dot{z}^{2}\biggr{]}\,.$
(C.2)

The equation coming from the normalisation of the four-momentum
$u_{\mu}u^{\mu}=\chi$ is

$\begin{split}\dot{\rho}^{2}+\dot{z}^{2}&=\frac{1}{(1+\jmath^{2}\rho^{4})^{2}}\Biggl{[}E^{2}-8\jmath
Lz(E+6\jmath
Lz)+\frac{L^{2}}{\rho^{2}}+2\jmath^{2}L^{2}\rho^{2}+\jmath^{4}L^{2}\rho^{6}\\\
&\quad+\biggl{(}4\sqrt{2}\jmath\rho z\frac{E+4\jmath
Lz}{1+\jmath^{2}\rho^{4}}\biggr{)}^{2}\Biggr{]}+\frac{\chi}{1+\jmath^{2}\rho^{4}}\,.\end{split}$
(C.3)

(a) $m=1$, $\jmath=0.01$, $L=1$ (b) $m=1$, $\jmath=0.01$, $L=36$

Figure C.1: Black hole (4.29) with two different unstable orbits. The first
one represents the case where $L$ is small and shows that the orbit approaches
an asymptotic value. The second one shows an orbit with large $L$: the test
particle on the plane defined by $\theta=\pi/2$ is attracted towards the black
hole.

#### C.2 Black hole geodesics

The conserved charges equations are

$\displaystyle\dot{t}$ $\displaystyle=\frac{r[E+4\jmath
L\cos\theta(r-2m)]}{(r-2m)(1+\jmath^{2}r^{4}\sin^{4}\theta)}\,,$ (C.1a)
$\displaystyle\dot{\phi}$ $\displaystyle=\frac{L-\jmath
r^{3}\sin^{2}\theta[4\cos\theta(E-8\jmath
Lm\cos\theta)-\jmath^{3}Lr^{5}\sin^{6}\theta+2\jmath
Lr(9\cos^{2}\theta-1)]}{r^{2}\sin^{2}\theta(1+\jmath^{2}r^{4}\sin^{4}\theta)}\,.$
(C.1b)

$L$ is the angular momentum and $E$ is the energy. From the conservation of
the four-momentum it follows

$\begin{split}&\bigl{(}1+\jmath^{2}r^{4}\sin^{4}\theta\bigr{)}\biggl{[}\biggl{(}\frac{r(E+4\jmath
L\cos\theta(r-2m))}{(r-2m)(1+\jmath^{2}r^{4}\sin^{4}\theta)}\biggr{)}^{2}+\frac{\dot{r}^{2}}{1-\frac{2m}{r}}+r^{2}\dot{\theta}^{2}\biggr{]}\\\
&\quad+\frac{L^{2}(1+\jmath^{2}r^{4}\sin^{4}\theta)^{2}}{r^{4}\sin^{4}\theta}=\chi\,.\end{split}$
(C.2)

###  Appendix D Zipoy–Voorhees spacetime embedded in the swirling universe

We apply the procedure described in Sec. 4.1 to a slightly more general metric
than the Schwarzschild black hole, the Zipoy–Voorhees metric [138, 241, 242].
This class of spacetime is relevant in General Relativity because, thanks to
its rich multipolar expansion, can be used to model the exterior gravitational
field of planets or stars. It is worth to mention, in connection with Chapter
2, that Kerns and Wild constructed a version of the Zipoy–Voorhees spacetime
embedded in the external gravitational field [243].

The Zipoy–Voorhees metric can be of some interest, when supported by a
conformally coupled scalar field, to build hairy black holes or wormholes such
as the Bekenstein black hole [74] or the Barcelo–Visser wormhole
[76]111Actually the associated complex Ernst field equations remain the same
of the pure general relativistic case, so as the main structure functions in
the metric. Only the decoupled function $\gamma$ has to be slightly modified
according to [75].. In particular, the presence of the swirling background
might be useful in the wormhole configuration to improve both the stability
and the traversability properties of the solution.

We start by casting the Zipoy–Voorhees seed in terms of the magnetic LWP
metric (1.48) in prolate spherical coordinates

${d\bar{s}}^{2}=\bar{f}\bigl{(}d\phi-\bar{\omega}d\tau\bigr{)}^{2}+\frac{1}{\bar{f}}\biggl{[}-\rho^{2}d\tau^{2}+\kappa^{2}(x^{2}-y^{2})e^{2\bar{\gamma}}\biggl{(}\frac{{dx}^{2}}{x^{2}-1}+\frac{{dy}^{2}}{1-y^{2}}\biggr{)}\biggr{]}\,,$
(D.1)

where

$\displaystyle\bar{f}(x,y)$
$\displaystyle=\kappa^{2}\biggl{(}\frac{x-1}{x+1}\biggr{)}^{-\delta}(x^{2}-1)(1-y^{2})\,,$
(D.2a) $\displaystyle\bar{\gamma}(x,y)$
$\displaystyle=\frac{1}{2}\log\left[\kappa^{2}\biggl{(}\frac{x-1}{x+1}\biggr{)}^{-2\delta}(x^{2}-1)(1-y^{2})\biggl{(}\frac{x^{2}-1}{x^{2}-y^{2}}\biggr{)}^{\delta^{2}}\right]\,,$
(D.2b) $\displaystyle\rho(x,y)$
$\displaystyle=\kappa\sqrt{(x^{2}-1)(y^{2}-1)}\,.$ (D.2c)

Clearly, this metric reduces to the static Schwarzschild black hole of Sec.
4.1 when $\delta=1$. For generic values of $\delta\neq 1$, the metric looses
the spherical symmetry and presents naked singularities outside the event
horizon, hence it is not suitable for describing legit black holes in pure
General Relativity. However, for $\delta=1/2$, when properly coupled with a
scalar field, it represents the first hairy black hole ever discovered [74].

Thanks to the Ehlers transformation (4.2), and following the same procedure
illustrated in Sec. 4.1, we are able to embed the Zipoy–Vorhees metric into
the swirling background. The $\bar{\gamma}$ function remains the same, while

$\displaystyle f(x,y)$
$\displaystyle=\frac{\kappa^{2}(1-y^{2})(x^{2}-1)^{1+\delta}}{(x-1)^{2\delta}+\jmath^{2}(1+x)^{2\delta}(x^{2}-1)^{2}(1-y^{2})^{2}}\,,$
(D.3a) $\displaystyle\omega(x,y)$
$\displaystyle=4\jmath\kappa^{2}y(x-\delta)+\omega_{0}\,.$ (D.3b)

The metric defined by (D.1) and (D.3) represents the $\delta$ extension of the
spacetime (4.12), therefore the Zipoy–Voorhees spacetime immersed in the
rotating background described in Sec. 4.2. Further generalisations with
angular momentum can be built straightforwardly, starting with seeds of the
family of the Tomimatsu–Sato solutions [244, 245].

### References

  * [1] A. Einstein.  Zur Allgemeinen Relativitätstheorie.  Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys. ), 1915:778–786, 1915.  [Addendum: Sitzungsber.Preuss.Akad.Wiss.Berlin (Math.Phys.) 1915, 799–801 (1915)].
  * [2] D. Hilbert.  Die Grundlagen der Physik. 1.  Gott. Nachr., 27:395–407, 1915.
  * [3] F.W. Dyson, A.S. Eddington, and C. Davidson.  A Determination of the Deflection of Light by the Sun’s Gravitational Field, from Observations Made at the Total Eclipse of May 29, 1919\.  Phil. Trans. Roy. Soc. Lond. A, 220:291–333, 1920.  doi:10.1098/rsta.1920.0009.
  * [4] B. Bertotti, L. Iess, and P. Tortora.  A test of general relativity using radio links with the Cassini spacecraft.  Nature, 425:374–376, 2003.  doi:10.1038/nature01997.
  * [5] T.E. Collett, L.J. Oldham, R.J. Smith, M.W. Auger, K.B. Westfall, D. Bacon, R.C. Nichol, K.L. Masters, K. Koyama, and R. van den Bosch.  A precise extragalactic test of General Relativity.  Science, 360:1342, 2018.  arXiv:1806.08300, doi:10.1126/science.aao2469.
  * [6] B.P. Abbott et al.  Observation of Gravitational Waves from a Binary Black Hole Merger.  Phys. Rev. Lett., 116(6):061102, 2016.  arXiv:1602.03837, doi:10.1103/PhysRevLett.116.061102.
  * [7] C.M. Will.  The Confrontation between General Relativity and Experiment.  Living Rev. Rel., 17:4, 2014.  arXiv:1403.7377, doi:10.12942/lrr-2014-4.
  * [8] K. Schwarzschild.  On the gravitational field of a mass point according to Einstein’s theory.  Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys. ), 1916:189–196, 1916.  arXiv:physics/9905030.
  * [9] M. Parsa, A. Eckart, B. Shahzamanian, V. Karas, M. Zajaček, J.A. Zensus, and C. Straubmeier.  Investigating the relativistic motion of the stars near the supermassive black hole in the galactic center.  The Astrophysical Journal, 845(1):22, aug 2017.  doi:10.3847/1538-4357/aa7bf0.
  * [10] A. Boehle, A.M. Ghez, R. Schödel, L. Meyer, S. Yelda, S. Albers, G.D. Martinez, E.E. Becklin, T. Do, J.R. Lu, K. Matthews, M.R. Morris, B. Sitarski, and G. Witzel.  An improved distance and mass estimate for sgr a* from a multistar orbit analysis.  The Astrophysical Journal, 830(1):17, oct 2016.  doi:10.3847/0004-637x/830/1/17.
  * [11] The GRAVITY Collaboration.  A geometric distance measurement to the galactic center black hole with 0.3% uncertainty.  A&A, 625:L10, 2019.  doi:10.1051/0004-6361/201935656.
  * [12] K. Akiyama et al.  First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole.  Astrophys. J. Lett., 875:L1, 2019.  arXiv:1906.11238, doi:10.3847/2041-8213/ab0ec7.
  * [13] K. Akiyama et al.  First M87 Event Horizon Telescope Results. II. Array and Instrumentation.  Astrophys. J. Lett., 875(1):L2, 2019.  arXiv:1906.11239, doi:10.3847/2041-8213/ab0c96.
  * [14] K. Akiyama et al.  First M87 Event Horizon Telescope Results. III. Data Processing and Calibration.  Astrophys. J. Lett., 875(1):L3, 2019.  arXiv:1906.11240, doi:10.3847/2041-8213/ab0c57.
  * [15] K. Akiyama et al.  First M87 Event Horizon Telescope Results. IV. Imaging the Central Supermassive Black Hole.  Astrophys. J. Lett., 875(1):L4, 2019.  arXiv:1906.11241, doi:10.3847/2041-8213/ab0e85.
  * [16] K. Akiyama et al.  First M87 Event Horizon Telescope Results. V. Physical Origin of the Asymmetric Ring.  Astrophys. J. Lett., 875(1):L5, 2019.  arXiv:1906.11242, doi:10.3847/2041-8213/ab0f43.
  * [17] K. Akiyama et al.  First M87 Event Horizon Telescope Results. VI. The Shadow and Mass of the Central Black Hole.  Astrophys. J. Lett., 875(1):L6, 2019.  arXiv:1906.11243, doi:10.3847/2041-8213/ab1141.
  * [18] R.P. Kerr.  Gravitational field of a spinning mass as an example of algebraically special metrics.  Phys. Rev. Lett., 11:237–238, 1963.  doi:10.1103/PhysRevLett.11.237.
  * [19] S. Hod.  Late time evolution of realistic rotating collapse and the no hair theorem.  Phys. Rev. D, 58:104022, 1998.  arXiv:gr-qc/9811032, doi:10.1103/PhysRevD.58.104022.
  * [20] B.P. Abbott et al.  GW151226: Observation of Gravitational Waves from a 22-Solar-Mass Binary Black Hole Coalescence.  Phys. Rev. Lett., 116(24):241103, 2016.  arXiv:1606.04855, doi:10.1103/PhysRevLett.116.241103.
  * [21] B.P. Abbott et al.  GW170104: Observation of a 50-Solar-Mass Binary Black Hole Coalescence at Redshift 0.2.  Phys. Rev. Lett., 118(22):221101, 2017.  [Erratum: Phys.Rev.Lett. 121, 129901 (2018)].  arXiv:1706.01812, doi:10.1103/PhysRevLett.118.221101.
  * [22] B.P. Abbott et al.  GWTC-1: A Gravitational-Wave Transient Catalog of Compact Binary Mergers Observed by LIGO and Virgo during the First and Second Observing Runs.  Phys. Rev. X, 9(3):031040, 2019.  arXiv:1811.12907, doi:10.1103/PhysRevX.9.031040.
  * [23] B.P. Abbott et al.  GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral.  Phys. Rev. Lett., 119(16):161101, 2017.  arXiv:1710.05832, doi:10.1103/PhysRevLett.119.161101.
  * [24] B.P. Abbott et al.  Gravitational Waves and Gamma-rays from a Binary Neutron Star Merger: GW170817 and GRB 170817A.  Astrophys. J. Lett., 848(2):L13, 2017.  arXiv:1710.05834, doi:10.3847/2041-8213/aa920c.
  * [25] J. Hennig.  On the balance problem for two rotating and charged black holes.  Class. Quant. Grav., 36(23):235001, 2019.  arXiv:1906.04847, doi:10.1088/1361-6382/ab4f41.
  * [26] D. Kramer and G. Neugebauer.  The superposition of two kerr solutions.  Physics Letters A, 75(4):259 – 261, 1980.
  * [27] G. Neugebauer.  A general integral of the axially symmetric stationary einstein equations.  Journal of Physics A: Mathematical and General, 13(2):L19–L21, feb 1980.  doi:10.1088/0305-4470/13/2/003.
  * [28] P.T. Chrusciel, M. Eckstein, L. Nguyen, and S.J. Szybka.  Existence of singularities in two-Kerr black holes.  Class. Quant. Grav., 28:245017, 2011.  arXiv:1111.1448, doi:10.1088/0264-9381/28/24/245017.
  * [29] G. Neugebauer and J. Hennig.  Non-existence of stationary two-black-hole configurations.  Gen. Rel. Grav., 41:2113–2130, 2009.  arXiv:0905.4179, doi:10.1007/s10714-009-0840-8.
  * [30] J. Hennig and G. Neugebauer.  Non-existence of stationary two-black-hole configurations: The degenerate case.  Gen. Rel. Grav., 43:3139–3162, 2011.  arXiv:1103.5248, doi:10.1007/s10714-011-1228-0.
  * [31] G. Neugebauer and J. Hennig.  Stationary two-black-hole configurations: A non-existence proof.  J. Geom. Phys., 62:613–630, 2012.  arXiv:1105.5830, doi:10.1016/j.geomphys.2011.05.008.
  * [32] V.S. Manko and E. Ruiz.  Metric for two arbitrary Kerr sources.  Phys. Lett. B, 794:36–40, 2019.  arXiv:1806.10408, doi:10.1016/j.physletb.2019.05.027.
  * [33] J. Hennig, C. Cederbaum, and M. Ansorg.  A Universal inequality for axisymmetric and stationary black holes with surrounding matter in the Einstein-Maxwell theory.  Commun. Math. Phys., 293:449–467, 2010.  arXiv:0812.2811, doi:10.1007/s00220-009-0889-y.
  * [34] A. Papapetrou.  A static solution of the equations of the gravitational field for an arbitary charge-distribution.  Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences, 51:191–204, 1945.  URL: http://www.jstor.org/stable/20488481.
  * [35] S.D. Majumdar.  A class of exact solutions of Einstein’s field equations.  Phys. Rev., 72:390–398, 1947.  doi:10.1103/PhysRev.72.390.
  * [36] D. Kastor and J.H. Traschen.  Cosmological multi - black hole solutions.  Phys. Rev. D, 47:5370–5375, 1993.  arXiv:hep-th/9212035, doi:10.1103/PhysRevD.47.5370.
  * [37] S. Chimento and D. Klemm.  Multicentered black holes with a negative cosmological constant.  Phys. Rev. D, 89(2):024037, 2014.  arXiv:1311.6937, doi:10.1103/PhysRevD.89.024037.
  * [38] K.S. Thorne.  Disk accretion onto a black hole. 2. Evolution of the hole.  Astrophys. J., 191:507–520, 1974.  doi:10.1086/152991.
  * [39] P.T. Chrusciel and P. Tod.  The Classification of static electro-vacuum space-times containing an asymptotically flat spacelike hypersurface with compact interior.  Commun. Math. Phys., 271:577–589, 2007.  arXiv:gr-qc/0512043, doi:10.1007/s00220-007-0191-9.
  * [40] G.A. Alekseev and V.A. Belinski.  Superposition of fields of two rotating charged masses in general relativity and existence of equilibrium configurations.  Gen. Rel. Grav., 51(5):68, 2019.  arXiv:1905.05317, doi:10.1007/s10714-019-2543-0.
  * [41] I. Cabrera-Munguia.  Metric for two unequal extreme Kerr-Newman black holes.  Phys. Lett. B, 826:136895, 2022.  arXiv:2110.04879, doi:10.1016/j.physletb.2022.136895.
  * [42] G.A. Alekseev.  Thirty years of studies of integrable reductions of Einstein’s field equations.  In 12th Marcel Grossmann Meeting on General Relativity, pages 645–666, 11 2010.  arXiv:1011.3846, doi:10.1142/9789814374552_0033.
  * [43] C.S. Gardner, J.M. Greene, M.D. Kruskal, and R.M. Miura.  Method for solving the Korteweg-deVries equation.  Phys. Rev. Lett., 19:1095–1097, 1967.  doi:10.1103/PhysRevLett.19.1095.
  * [44] F.J. Ernst.  New formulation of the axially symmetric gravitational field problem.  Phys. Rev., 167:1175–1179, 1968.  doi:10.1103/PhysRev.167.1175.
  * [45] F.J. Ernst.  New formulation of the axially symmetric gravitational field problem. ii.  Phys. Rev., 168:1415–1417, Apr 1968.  doi:10.1103/PhysRev.168.1415.
  * [46] R.P. Geroch.  A Method for generating new solutions of Einstein’s equation. 2.  J. Math. Phys., 13:394–404, 1972.  doi:10.1063/1.1665990.
  * [47] W. Kinnersley.  Symmetries of the Stationary Einstein-Maxwell Field Equations. 1.  J. Math. Phys., 18:1529–1537, 1977.  doi:10.1063/1.523458.
  * [48] W. Kinnersley and D.M. Chitre.  Symmetries of the Stationary Einstein-Maxwell Field Equations. 2.  J. Math. Phys., 18:1538–1542, 1977.  doi:10.1063/1.523459.
  * [49] W. Kinnersley and D.M. Chitre.  Symmetries of the Stationary Einstein-Maxwell Field Equations. 3.  J. Math. Phys., 19:1926–1931, 1978.  doi:10.1063/1.523912.
  * [50] W. Kinnersley and D.M. Chitre.  Symmetries of the stationary Einstein–Maxwell equations. IV. Transformations which preserve asymptotic flatness.  J. Math. Phys., 19:2037–2042, 1978.  doi:10.1063/1.523580.
  * [51] D. Maison.  Are the stationary, axially symmetric Einstein equations completely integrable?  Phys. Rev. Lett., 41:521, 1978.  doi:10.1103/PhysRevLett.41.521.
  * [52] 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.
  * [53] V.A. Belinsky and V.E. Sakharov.  Stationary Gravitational Solitons with Axial Symmetry.  Sov. Phys. JETP, 50:1–9, 1979.
  * [54] G.A. Alekseev.  N-soliton solutions of einstein-maxwell equations.  JETP Lett. (Engl. Transl.); (United States), 32:4, 8 1980.  URL: https://www.osti.gov/biblio/6803745.
  * [55] B.K. Harrison.  Bäcklund transformation for the ernst equation of general relativity.  Phys. Rev. Lett., 41:1197–1200, Oct 1978.  URL: https://link.aps.org/doi/10.1103/PhysRevLett.41.1197, doi:10.1103/PhysRevLett.41.1197.
  * [56] F.B. Estabrook and H.D. Wahlquist.  Prolongation Structures of Nonlinear Evolution Equations. 2.  J. Math. Phys., 17:1293–1297, 1976.  doi:10.1063/1.523056.
  * [57] G. Neugebauer.  Backlund transformations of axially symmetric stationary gravitational fields.  J. Phys. A, 12:L67–L70, 1979.  doi:10.1088/0305-4470/12/4/001.
  * [58] G. Neugebauer.  Recursive calculation of axially symmetric stationary einstein fields.  Journal of Physics A: Mathematical and General, 13(5):1737–1740, may 1980.  doi:10.1088/0305-4470/13/5/031.
  * [59] B. Julia.  Kac-Moody symmetry of gravitation and supergravity theories.  In American Mathematical Society summer seminar on Appication of Group Theory in Physics and Mathematical Physics, 9 1982.
  * [60] B. Julia.  Application of sueprgravity to gravitation theory.  In International School of Cosmology and Gravitation: 8th Course: Unified Field Theories of More than Four Dimensions, Including Exact Solutions, 8 1982.
  * [61] P. Breitenlohner and D. Maison.  On the Geroch Group.  Ann. Inst. H. Poincare Phys. Theor., 46:215, 1987.
  * [62] I. Hauser and F.J. Ernst.  Integral equation method for effecting Kinnersley-Chitre transformations.  Phys. Rev. D, 20:362–369, 1979.  doi:10.1103/PhysRevD.20.362.
  * [63] I. Hauser and F.J. Ernst.  Integral equation method for effecting Kinnersley-Chitre transformations. II.  Phys. Rev. D, 20:1783–1790, 1979.  doi:10.1103/PhysRevD.20.1783.
  * [64] I. Hauser and F.J. Ernst.  A homogeneous Hilbert problem for the Kinnersley-–Chitre transformations.  Journal of Mathematical Physics, 21(5):1126–1140, 1980.  doi:10.1063/1.524536.
  * [65] I. Hauser and F.J. Ernst.  A homogeneous hilbert problem for the kinnersley–chitre transformations of electrovac space-times.  Journal of Mathematical Physics, 21(6):1418–1422, 1980.
  * [66] I. Hauser and F.J. Ernst.  Proof of a geroch conjecture.  Journal of Mathematical Physics, 22(5):1051–1063, 1981.  doi:10.1063/1.525012.
  * [67] N.R. Sibgatullin.  Construction of the general solution of the system of einstein–maxwell equations for the stationary axisymmetric case.  In Soviet Physics-Doklady, volume 29, pages 802–804, 1984.
  * [68] N.R. Sibgatullin.  Oscillations and waves in strong gravitational and electromagnetic fields.  Springer-Verlag, 1991.
  * [69] G.A. Alekseev.  The Method of the Inverse Problem of Scattering and the Singular Integral Equations for Interacting Massless Fields.  Sov. Phys. Dokl., 30:565–568, 1985.
  * [70] G.A. Alekseev.  Exact solutions in the general theory of relativity.  Trudy Matematicheskogo Instituta imeni VA Steklova, 176:211–258, 1987.
  * [71] G.A. Alekseev.  Isomonodromy deformations and integrability of electrovacuum einstein-maxwell field equations with isometries.  In Solitons and applications (Dubna, 1989), pages 174–179. World Sci. Publ., River Edge, NJ, 1990.
  * [72] G.A. Alekseev.  Monodromy transform approach to solution of the Ernst equations in general relativity.  In International European Conference on Gravitation: Journees Relativistes 99, 9 1999.  arXiv:gr-qc/9912109.
  * [73] V.A. Belinsky.  One soliton cosmological waves.  Sov. Phys. JETP, 50:623–631, 1979.
  * [74] J.D. Bekenstein.  Black Holes with Scalar Charge.  Annals Phys., 91:75–82, 1975.  doi:10.1016/0003-4916(75)90279-1.
  * [75] 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.
  * [76] C. Barcelo and M. Visser.  Traversable wormholes from massless conformally coupled scalar fields.  Phys. Lett. B, 466:127–134, 1999.  arXiv:gr-qc/9908029, doi:10.1016/S0370-2693(99)01117-X.
  * [77] D.V. Galtsov, A.A. Garcia, and O.V. Kechkin.  Symmetries of the stationary Einstein-Maxwell dilaton theory.  Class. Quant. Grav., 12:2887–2903, 1995.  arXiv:hep-th/9504155, doi:10.1088/0264-9381/12/12/007.
  * [78] D.V. Galtsov and A. A. Garcia.  Hidden symmetries in dilaton gravity.  Phys. Rev. D, 52:3432–3439, 1995.  doi:10.1103/PhysRevD.52.3432.
  * [79] D.V. Galtsov and O.V. Kechkin.  Ehlers-Harrison type transformations in dilaton - axion gravity.  Phys. Rev. D, 50:7394–7399, 1994.  arXiv:hep-th/9407155, doi:10.1103/PhysRevD.50.7394.
  * [80] A. Garcia, D. Galtsov, and O. Kechkin.  Class of stationary axisymmetric solutions of the Einstein-Maxwell dilaton - axion field equations.  Phys. Rev. Lett., 74:1276–1279, 1995.  doi:10.1103/PhysRevLett.74.1276.
  * [81] D.V. Gal’tsov and P.S. Letelier.  Ehlers-Harrison transformations and black holes in dilaton - axion gravity with multiple vector fields.  Phys. Rev. D, 55:3580–3592, 1997.  arXiv:gr-qc/9612007, doi:10.1103/PhysRevD.55.3580.
  * [82] D.V. Gal’tsov and S.A. Sharakin.  Matrix Ernst potentials for EMDA with multiple vector fields.  Phys. Lett. B, 399:250–257, 1997.  arXiv:hep-th/9702039, doi:10.1016/S0370-2693(97)00295-5.
  * [83] A.A. Pomeransky.  Complete integrability of higher-dimensional Einstein equations with additional symmetry, and rotating black holes.  Phys. Rev. D, 73:044004, 2006.  arXiv:hep-th/0507250, doi:10.1103/PhysRevD.73.044004.
  * [84] A.A. Pomeransky and R.A. Sen’kov.  Black ring with two angular momenta, 12 2006.  arXiv:hep-th/0612005.
  * [85] R.C. Myers and M.J. Perry.  Black Holes in Higher Dimensional Space-Times.  Annals Phys., 172:304, 1986.  doi:10.1016/0003-4916(86)90186-7.
  * [86] R. Emparan and H.S. Reall.  A Rotating black ring solution in five-dimensions.  Phys. Rev. Lett., 88:101101, 2002.  arXiv:hep-th/0110260, doi:10.1103/PhysRevLett.88.101101.
  * [87] H. Iguchi and T. Mishima.  Solitonic generation of five-dimensional black ring solution.  Phys. Rev. D, 73:121501, 2006.  arXiv:hep-th/0604050, doi:10.1103/PhysRevD.73.121501.
  * [88] S. Tomizawa, Y. Morisawa, and Y. Yasui.  Vacuum solutions of five dimensional Einstein equations generated by inverse scattering method.  Phys. Rev. D, 73:064009, 2006.  arXiv:hep-th/0512252, doi:10.1103/PhysRevD.73.064009.
  * [89] S. Tomizawa and M. Nozawa.  Vacuum solutions of five-dimensional Einstein equations generated by inverse scattering method. II. Production of black ring solution.  Phys. Rev. D, 73:124034, 2006.  arXiv:hep-th/0604067, doi:10.1103/PhysRevD.73.124034.
  * [90] T. Mishima and H. Iguchi.  New axisymmetric stationary solutions of five-dimensional vacuum Einstein equations with asymptotic flatness.  Phys. Rev. D, 73:044030, 2006.  arXiv:hep-th/0504018, doi:10.1103/PhysRevD.73.044030.
  * [91] Y. Chen and E. Teo.  Rotating black rings on Taub-NUT.  JHEP, 06:068, 2012.  arXiv:1204.3116, doi:10.1007/JHEP06(2012)068.
  * [92] H. Iguchi, K. Izumi, and T. Mishima.  Systematic solution-generation of five-dimensional black holes.  Prog. Theor. Phys. Suppl., 189:93–125, 2011.  arXiv:1106.0387, doi:10.1143/PTPS.189.93.
  * [93] H. Elvang and P. Figueras.  Black Saturn.  JHEP, 05:050, 2007.  arXiv:hep-th/0701035, doi:10.1088/1126-6708/2007/05/050.
  * [94] H. Iguchi and T. Mishima.  Black di-ring and infinite nonuniqueness.  Phys. Rev. D, 75:064018, 2007.  [Erratum: Phys.Rev.D 78, 069903 (2008)].  arXiv:hep-th/0701043, doi:10.1103/PhysRevD.78.069903.
  * [95] J. Evslin and C. Krishnan.  The Black Di-Ring: An Inverse Scattering Construction.  Class. Quant. Grav., 26:125018, 2009.  arXiv:0706.1231, doi:10.1088/0264-9381/26/12/125018.
  * [96] H. Elvang and M.J. Rodriguez.  Bicycling Black Rings.  JHEP, 04:045, 2008.  arXiv:0712.2425, doi:10.1088/1126-6708/2008/04/045.
  * [97] J.V. Rocha, M.J. Rodriguez, and A. Virmani.  Inverse Scattering Construction of a Dipole Black Ring.  JHEP, 11:008, 2011.  arXiv:1108.3527, doi:10.1007/JHEP11(2011)008.
  * [98] R. Emparan and H.S. Reall.  Black Rings.  Class. Quant. Grav., 23:R169, 2006.  arXiv:hep-th/0608012, doi:10.1088/0264-9381/23/20/R01.
  * [99] R. Emparan and H.S. Reall.  Black Holes in Higher Dimensions.  Living Rev. Rel., 11:6, 2008.  arXiv:0801.3471, doi:10.12942/lrr-2008-6.
  * [100] S.S. Yazadjiev.  Completely integrable sector in 5-D Einstein-Maxwell gravity and derivation of the dipole black ring solutions.  Phys. Rev. D, 73:104007, 2006.  arXiv:hep-th/0602116, doi:10.1103/PhysRevD.73.104007.
  * [101] I. Bakas.  Solitons of axion - dilaton gravity.  Phys. Rev. D, 54:6424–6434, 1996.  arXiv:hep-th/9605043, doi:10.1103/PhysRevD.54.6424.
  * [102] I. Bakas.  2-D gravisolitons in string theory.  In 2nd International Sakharov Conference on Physics, pages 350–354, 6 1996.  arXiv:hep-th/9606030.
  * [103] A. Bouchareb, G. Clement, C.-M. Chen, D.V. Gal’tsov, N.G. Scherbluk, and T. Wolf.  G(2) generating technique for minimal D=5 supergravity and black rings.  Phys. Rev. D, 76:104032, 2007.  [Erratum: Phys.Rev.D 78, 029901 (2008)].  arXiv:0708.2361, doi:10.1103/PhysRevD.76.104032.
  * [104] D.V. Gal’tsov and N.G. Scherbluk.  Hidden symmetries in 5D supergravities and black rings.  PoS, BHGRS:016, 2008.  arXiv:0912.2770, doi:10.22323/1.075.0016.
  * [105] D.V. Gal’tsov and N.G. Scherbluk.  Generating technique for U(1)**3 5D supergravity.  Phys. Rev. D, 78:064033, 2008.  arXiv:0805.3924, doi:10.1103/PhysRevD.78.064033.
  * [106] D.V. Gal’tsov and N.G. Scherbluk.  Improved generating technique for D=5 supergravities and squashed Kaluza-Klein Black Holes.  Phys. Rev. D, 79:064020, 2009.  arXiv:0812.2336, doi:10.1103/PhysRevD.79.064020.
  * [107] P. Figueras, E. Jamsin, J.V. Rocha, and A. Virmani.  Integrability of Five Dimensional Minimal Supergravity and Charged Rotating Black Holes.  Class. Quant. Grav., 27:135011, 2010.  arXiv:0912.3199, doi:10.1088/0264-9381/27/13/135011.
  * [108] D. Katsimpouri, A. Kleinschmidt, and A. Virmani.  Inverse Scattering and the Geroch Group.  JHEP, 02:011, 2013.  arXiv:1211.3044, doi:10.1007/JHEP02(2013)011.
  * [109] D. Katsimpouri, A. Kleinschmidt, and A. Virmani.  An inverse scattering formalism for STU supergravity.  JHEP, 03:101, 2014.  arXiv:1311.7018, doi:10.1007/JHEP03(2014)101.
  * [110] D. Katsimpouri, A. Kleinschmidt, and A. Virmani.  An Inverse Scattering Construction of the JMaRT Fuzzball.  JHEP, 12:070, 2014.  arXiv:1409.6471, doi:10.1007/JHEP12(2014)070.
  * [111] G.A. Alekseev.  Integrability of generalized (matrix) Ernst equations in string theory.  Theor. Math. Phys., 144:1065–1074, 2005.  arXiv:hep-th/0410246, doi:10.1007/s11232-005-0136-4.
  * [112] G.A. Alekseev.  Integrability of the symmetry reduced bosonic dynamics and soliton generating transformations in the low energy heterotic string effective theory.  Phys. Rev. D, 80:041901, 2009.  arXiv:0811.1358, doi:10.1103/PhysRevD.80.041901.
  * [113] M. Godazgar and H.S. Reall.  Algebraically special axisymmetric solutions of the higher-dimensional vacuum Einstein equation.  Class. Quant. Grav., 26:165009, 2009.  arXiv:0904.4368, doi:10.1088/0264-9381/26/16/165009.
  * [114] J. Ehlers.  Konstruktionen und Charakterisierung von Losungen der Einsteinschen Gravitationsfeldgleichungen.  Phd thesis, Hamburg University, 1957.
  * [115] B.K. Harrison.  New solutions of the einstein‐maxwell equations from old.  Journal of Mathematical Physics, 9(11):1744–1752, 1968.  doi:10.1063/1.1664508.
  * [116] C. Reina and A. Treves.  Nut-like generalization of axisymmetric gravitational fields.  Journal of Mathematical Physics, 16(4):834–835, 1975.  doi:10.1063/1.522614.
  * [117] F.J. Ernst.  Black holes in a magnetic universe.  J. Math. Phys., 17(1):54–56, 1976.  doi:10.1063/1.522781.
  * [118] M. Heusler.  Black Hole Uniqueness Theorems.  Cambridge Lecture Notes in Physics. Cambridge University Press, 1996.  doi:10.1017/CBO9780511661396.
  * [119] H. Stephani, D. Kramer, M.A.H. MacCallum, C. Hoenselaers, and E. Herlt.  Exact solutions of Einstein’s field equations.  Cambridge Monographs on Mathematical Physics. Cambridge Univ. Press, Cambridge, 2003.  doi:10.1017/CBO9780511535185.
  * [120] H. Stephani.  Differential Equations: Their Solution Using Symmetries.  Cambridge University Press, 1990.  doi:10.1017/CBO9780511599941.
  * [121] R. Martelli.  The Action of the Axisymmetric and Stationary Symmetry Group of General Relativity on a Static Black Hole.  Bachelor thesis, Milan U., 2022.
  * [122] G. Neugebauer and D. Kramer.  Eine methode zur konstruktion stationärer einstein-maxwell-felder.  Annalen der Physik, 479(1-2):62–71, 1969.  doi:https://doi.org/10.1002/andp.19694790108.
  * [123] W. Kinnersley.  Generation of stationary einstein‐maxwell fields.  Journal of Mathematical Physics, 14(5):651–653, 1973.  doi:10.1063/1.1666373.
  * [124] A.H. Taub.  Empty space-times admitting a three parameter group of motions.  Annals Math., 53:472–490, 1951.  doi:10.2307/1969567.
  * [125] E. Newman, L. Tamburino, and T. Unti.  Empty space generalization of the Schwarzschild metric.  J. Math. Phys., 4:915, 1963.  doi:10.1063/1.1704018.
  * [126] C.W. Misner.  Taub-Nut Space as a counterexample to almost anything.  In J. Ehlers, editor, Relativity Theory and Astrophysics. Vol.1: Relativity and Cosmology, volume 8, page 160. Rhode Island: American Mathematical Society, 1967.
  * [127] G. Clément, D. Gal’tsov, and M. Guenouche.  Rehabilitating space-times with NUTs.  Phys. Lett. B, 750:591–594, 2015.  arXiv:1508.07622, doi:10.1016/j.physletb.2015.09.074.
  * [128] H. Reissner.  Über die eigengravitation des elektrischen feldes nach der einsteinschen theorie.  Annalen der Physik, 355(9):106–120, 1916.  doi:https://doi.org/10.1002/andp.19163550905.
  * [129] G. Nordström.  On the Energy of the Gravitation field in Einstein’s Theory.  Koninklijke Nederlandse Akademie van Wetenschappen Proceedings Series B Physical Sciences, 20:1238–1245, January 1918.
  * [130] M.A. Melvin.  Pure magnetic and electric geons.  Phys. Lett., 8:65–70, 1964.  doi:10.1016/0031-9163(64)90801-7.
  * [131] G.W. Gibbons and D.L. Wiltshire.  Space-Time as a Membrane in Higher Dimensions.  Nucl. Phys. B, 287:717–742, 1987.  arXiv:hep-th/0109093, doi:10.1016/0550-3213(87)90125-8.
  * [132] M. Astorino.  Charging axisymmetric space-times with cosmological constant.  JHEP, 06:086, 2012.  arXiv:1205.6998, doi:10.1007/JHEP06(2012)086.
  * [133] V. Belinski and E. Verdaguer.  Gravitational solitons.  Cambridge Monographs on Mathematical Physics. Cambridge University Press, 2005.  doi:10.1017/CBO9780511535253.
  * [134] J.F. Plebanski and M. Demianski.  Rotating, charged, and uniformly accelerating mass in general relativity.  Annals Phys., 98:98–127, 1976.  doi:10.1016/0003-4916(76)90240-2.
  * [135] 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.
  * [136] 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.
  * [137] T. Harmark.  Stationary and axisymmetric solutions of higher-dimensional general relativity.  Phys. Rev. D, 70:124002, 2004.  URL: 10.1103/PhysRevD.70.124002, arXiv:hep-th/0408141, doi:10.1103/PhysRevD.70.124002.
  * [138] R. Bach and H. Weyl.  Neue lösungen der einsteinschen gravitationsgleichungen.  Mathematische Zeitschrift, 13(1):134–145, 1922.
  * [139] W. Israel and K.A. Khan.  Collinear particles and bondi dipoles in general relativity.  Il Nuovo Cimento, 33(2):331–344, 1964.  doi:https://doi.org/10.1007/BF02750196.
  * [140] P.S. Letelier and S.R. de Oliveira.  Double Kerr-NUT space-times: Spinning strings and spinning rods.  Phys. Lett. A, 238:101–106, 1998.  doi:10.1016/S0375-9601(97)00730-5.
  * [141] G. Erez and N. Rosen.  The gravitational field of a particle possessing a multipole moment.  Bull. Research Council Israel, Sect. F.8, 9 1959.
  * [142] 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.
  * [143] R. Beig and R.M. Schoen.  On static n-body configurations in relativity.  Classical and Quantum Gravity, 26(7):075014, mar 2009.  doi:10.1088/0264-9381/26/7/075014.
  * [144] 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.
  * [145] S. Chandrasekhar.  The mathematical theory of black holes.  Clarendon Press, Oxford, 1985.
  * [146] R.P. Geroch and J.B. Hartle.  Distorted black holes.  J. Math. Phys., 23:680, 1982.  doi:10.1063/1.525384.
  * [147] F.J. Ernst.  Generalized c‐metric.  Journal of Mathematical Physics, 19(9):1986–1987, 1978.  doi:https://doi.org/10.1063/1.523896.
  * [148] R.M. Kerns and W.J. Wild.  Black hole in a gravitational field.  General Relativity and Gravitation, 14(1):1–4, 1982.
  * [149] M. Astorino and A. Viganò.  Binary black hole system at equilibrium.  Phys. Lett. B, 820:136506, 2021.  arXiv:2104.07686, doi:10.1016/j.physletb.2021.136506.
  * [150] M. Astorino and A. Viganò.  Charged and rotating multi-black holes in an external gravitational field.  Eur. Phys. J. C, 82(9):829, 2022.  arXiv:2105.02894, doi:10.1140/epjc/s10052-022-10787-y.
  * [151] M. Astorino and A. Viganò.  Many accelerating distorted black holes.  Eur. Phys. J. C, 81(10):891, 2021.  arXiv:2106.02058, doi:10.1140/epjc/s10052-021-09693-6.
  * [152] 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.
  * [153] W. Israel.  Event horizons in static vacuum space-times.  Phys. Rev., 164:1776–1779, 1967.  doi:10.1103/PhysRev.164.1776.
  * [154] 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.
  * [155] R.P. Geroch.  Multipole moments. II. Curved space.  J. Math. Phys., 11:2580–2588, 1970.  doi:10.1063/1.1665427.
  * [156] R. O. Hansen.  Multipole moments of stationary space-times.  J. Math. Phys., 15:46–52, 1974.  doi:10.1063/1.1666501.
  * [157] G. Fodor, C. Hoenselaers, and Z. Perjés.  Multipole moments of axisymmetric systems in relativity.  Journal of Mathematical Physics, 30(10):2252–2257, 1989.  doi:10.1063/1.528551.
  * [158] P.S. Letelier.  Multipole stationary soliton solutions to the einstein equations.  Revista Brasileira de Fisica, 14:371–376, September 1984.
  * [159] R.C. Myers.  Higher Dimensional Black Holes in Compactified Space-times.  Phys. Rev. D, 35:455, 1987.  doi:10.1103/PhysRevD.35.455.
  * [160] 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.
  * [161] A. Komar.  Covariant conservation laws in general relativity.  Phys. Rev., 113:934–936, Feb 1959.  doi:10.1103/PhysRev.113.934.
  * [162] 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.
  * [163] M. Henneaux and C. Teitelboim.  Asymptotically anti-De Sitter Spaces.  Commun. Math. Phys., 98:391–424, 1985.  doi:10.1007/BF01205790.
  * [164] A. Ashtekar, S. Fairhurst, and B. Krishnan.  Isolated horizons: Hamiltonian evolution and the first law.  Phys. Rev. D, 62:104025, 2000.  arXiv:gr-qc/0005083, doi:10.1103/PhysRevD.62.104025.
  * [165] M. Astorino, G. Compère, R. Oliveri, and N. Vandevoorde.  Mass of Kerr-Newman black holes in an external magnetic field.  Phys. Rev. D, 94(2):024019, 2016.  arXiv:1602.08110, doi:10.1103/PhysRevD.94.024019.
  * [166] J.B. Hartle and S.W. Hawking.  Path Integral Derivation of Black Hole Radiance.  Phys. Rev. D, 13:2188–2203, 1976.  doi:10.1103/PhysRevD.13.2188.
  * [167] L. Smarr.  Mass formula for Kerr black holes.  Phys. Rev. Lett., 30:71–73, 1973.  [Erratum: Phys.Rev.Lett. 30, 521–521 (1973)].  doi:10.1103/PhysRevLett.30.71.
  * [168] 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.
  * [169] M.M. Caldarelli, G. Cognola, and D. Klemm.  Thermodynamics of Kerr-Newman-AdS black holes and conformal field theories.  Class. Quant. Grav., 17:399–420, 2000.  arXiv:hep-th/9908022, doi:10.1088/0264-9381/17/2/310.
  * [170] M. Astorino.  Thermodynamics of Regular Accelerating Black Holes.  Phys. Rev. D, 95(6):064007, 2017.  arXiv:1612.04387, doi:10.1103/PhysRevD.95.064007.
  * [171] A. Tomimatsu.  Distorted rotating black holes.  Physics Letters A, 103(8):374–376, 1984.  URL: https://www.sciencedirect.com/science/article/pii/0375960184901348, doi:https://doi.org/10.1016/0375-9601(84)90134-8.
  * [172] M. Astorino.  Enhanced Ehlers Transformation and the Majumdar-Papapetrou-NUT Spacetime.  JHEP, 01:123, 2020.  arXiv:1906.08228, doi:10.1007/JHEP01(2020)123.
  * [173] D. Kramer and G. Neugebauer.  Eine exakte stationäre lösung der einstein-maxwell-gleichungen.  Annalen der Physik, 479(1‐2):59–61, 1969.  doi:https://doi.org/10.1002/andp.19694790107.
  * [174] G.A. Alekseev and V.A. Belinski.  Equilibrium configurations of two charged masses in General Relativity.  Phys. Rev. D, 76:021501, 2007.  arXiv:0706.1981, doi:10.1103/PhysRevD.76.021501.
  * [175] V.S. Manko.  The Double-Reissner-Nordstrom solution and the interaction force between two spherically symmetric charged particles.  Phys. Rev. D, 76:124032, 2007.  arXiv:0710.2158, doi:10.1103/PhysRevD.76.124032.
  * [176] R. Emparan and E. Teo.  Macroscopic and microscopic description of black diholes.  Nucl. Phys. B, 610:190–214, 2001.  arXiv:hep-th/0104206, doi:10.1016/S0550-3213(01)00319-4.
  * [177] V.S. Manko and N.R. Sibgatullin.  Construction of exact solutions of the Einstein-Maxwell equations corresponding to a given behaviour of the Ernst potentials on the symmetry axis.  Classical and Quantum Gravity, 10(7):1383–1404, July 1993.  doi:10.1088/0264-9381/10/7/014.
  * [178] C.A.R. Herdeiro and C. Rebelo.  On the interaction between two Kerr black holes.  JHEP, 10:017, 2008.  arXiv:0808.3941, doi:10.1088/1126-6708/2008/10/017.
  * [179] G.W. Gibbons.  The Motion of black holes.  Commun. Math. Phys., 35:13–23, 1974.  doi:10.1007/BF01646451.
  * [180] 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.
  * [181] 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.
  * [182] 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.
  * [183] 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.
  * [184] 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.
  * [185] M. Astorino.  Pair Creation of Rotating Black Holes.  Phys. Rev. D, 89(4):044022, 2014.  arXiv:1312.1723, doi:10.1103/PhysRevD.89.044022.
  * [186] 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.
  * [187] 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.
  * [188] 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.
  * [189] 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.
  * [190] 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.
  * [191] 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.
  * [192] 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.
  * [193] 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.
  * [194] 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.
  * [195] 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.
  * [196] 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.
  * [197] W. Kinnersley and M. Walker.  Uniformly accelerating charged mass in general relativity.  Phys. Rev. D, 2:1359–1370, 1970.  doi:10.1103/PhysRevD.2.1359.
  * [198] E. Witten.  Instability of the Kaluza-Klein Vacuum.  Nucl. Phys. B, 195:481–492, 1982.  doi:10.1016/0550-3213(82)90007-4.
  * [199] O. Aharony, M. Fabinger, G.T. Horowitz, and E. Silverstein.  Clean time dependent string backgrounds from bubble baths.  JHEP, 07:007, 2002.  arXiv:hep-th/0204158, doi:10.1088/1126-6708/2002/07/007.
  * [200] R. Emparan and H.S. Reall.  Generalized Weyl solutions.  Phys. Rev. D, 65:084025, 2002.  arXiv:hep-th/0110258, doi:10.1103/PhysRevD.65.084025.
  * [201] G.T. Horowitz and K. Maeda.  Colliding Kaluza-Klein bubbles.  Class. Quant. Grav., 19:5543–5556, 2002.  arXiv:hep-th/0207270, doi:10.1088/0264-9381/19/21/317.
  * [202] H. Elvang and G.T. Horowitz.  When black holes meet Kaluza-Klein bubbles.  Phys. Rev. D, 67:044015, 2003.  arXiv:hep-th/0210303, doi:10.1103/PhysRevD.67.044015.
  * [203] H. Elvang, T. Harmark, and N.A. Obers.  Sequences of bubbles and holes: New phases of Kaluza-Klein black holes.  JHEP, 01:003, 2005.  arXiv:hep-th/0407050, doi:10.1088/1126-6708/2005/01/003.
  * [204] S. Tomizawa, H. Iguchi, and T. Mishima.  Rotating Black Holes on Kaluza-Klein Bubbles.  Phys. Rev. D, 78:084001, 2008.  arXiv:hep-th/0702207, doi:10.1103/PhysRevD.78.084001.
  * [205] H. Iguchi, T. Mishima, and S. Tomizawa.  Boosted black holes on Kaluza-Klein bubbles.  Phys. Rev. D, 76:124019, 2007.  [Erratum: Phys.Rev.D 78, 109903 (2008)].  arXiv:0705.2520, doi:10.1103/PhysRevD.78.109903.
  * [206] D. Kastor, S. Ray, and J. Traschen.  The Thermodynamics of Kaluza-Klein Black Hole/Bubble Chains.  Class. Quant. Grav., 25:125004, 2008.  arXiv:0803.2019, doi:10.1088/0264-9381/25/12/125004.
  * [207] J. Kunz and S. Yazadjiev.  Charged black holes on a Kaluza-Klein bubble.  Phys. Rev. D, 79:024010, 2009.  arXiv:0811.0730, doi:10.1103/PhysRevD.79.024010.
  * [208] S.S. Yazadjiev and P.G. Nedkova.  Sequences of dipole black rings and Kaluza-Klein bubbles.  JHEP, 01:048, 2010.  arXiv:0910.0938, doi:10.1007/JHEP01(2010)048.
  * [209] P.G. Nedkova and S.S. Yazadjiev.  Rotating black ring on Kaluza-Klein bubbles.  Phys. Rev. D, 82:044010, 2010.  arXiv:1005.5051, doi:10.1103/PhysRevD.82.044010.
  * [210] J. Kunz, P.G. Nedkova, and C. Stelea.  Charged black holes on Kaluza-Klein bubbles.  Nucl. Phys. B, 874:773–791, 2013.  arXiv:1304.7020, doi:10.1016/j.nuclphysb.2013.06.013.
  * [211] M. Astorino, R. Emparan, and A. Viganò.  Bubbles of nothing in binary black holes and black rings, and viceversa.  JHEP, 07:007, 2022.  arXiv:2204.09690, doi:10.1007/JHEP07(2022)007.
  * [212] L. Susskind.  Computational Complexity and Black Hole Horizons.  Fortsch. Phys., 64:24–43, 2016.  [Addendum: Fortsch.Phys. 64, 44–48 (2016)].  arXiv:1403.5695, doi:10.1002/prop.201500092.
  * [213] B. Kleihaus, J. Kunz, and E. Radu.  d greater than or equal to five static black holes with S**2 x S**(d-4) event horizon topology.  Phys. Lett. B, 678:301–307, 2009.  arXiv:0904.2723, doi:10.1016/j.physletb.2009.06.039.
  * [214] B. Kleihaus, J. Kunz, E. Radu, and M.J. Rodriguez.  New generalized nonspherical black hole solutions.  JHEP, 02:058, 2011.  arXiv:1010.2898, doi:10.1007/JHEP02(2011)058.
  * [215] R. Emparan.  Black diholes.  Phys. Rev. D, 61:104009, 2000.  arXiv:hep-th/9906160, doi:10.1103/PhysRevD.61.104009.
  * [216] R. Emparan.  Rotating circular strings, and infinite nonuniqueness of black rings.  JHEP, 03:064, 2004.  arXiv:hep-th/0402149, doi:10.1088/1126-6708/2004/03/064.
  * [217] R. Emparan and M. Gutperle.  From p-branes to fluxbranes and back.  JHEP, 12:023, 2001.  arXiv:hep-th/0111177, doi:10.1088/1126-6708/2001/12/023.
  * [218] O.J.C. Dias, G.W. Gibbons, J.E. Santos, and B. Way.  TBD.  In preparation.
  * [219] H. Weyl.  The theory of gravitation.  Annalen Phys., 54:117–145, 1917.  doi:10.1007/s10714-011-1310-7.
  * [220] H.S. Tan and E. Teo.  Multi - black hole solutions in five-dimensions.  Phys. Rev. D, 68:044021, 2003.  arXiv:hep-th/0306044, doi:10.1103/PhysRevD.68.044021.
  * [221] M. Astorino and A. Viganò.  TBD.  In preparation.
  * [222] R. Emparan, T. Harmark, V. Niarchos, and N.A. Obers.  New Horizons for Black Holes and Branes.  JHEP, 04:046, 2010.  arXiv:0912.2352, doi:10.1007/JHEP04(2010)046.
  * [223] G.W. Gibbons, A.H. Mujtaba, and C.N. Pope.  Ergoregions in Magnetised Black Hole Spacetimes.  Class. Quant. Grav., 30(12):125008, 2013.  arXiv:1301.3927, doi:10.1088/0264-9381/30/12/125008.
  * [224] M. Astorino, R. Martelli, and A. Viganò.  Black holes in a swirling universe.  Phys. Rev. D, 106(6):064014, 2022.  arXiv:2205.13548, doi:10.1103/PhysRevD.106.064014.
  * [225] B. Carter.  Hamilton-Jacobi and Schrodinger separable solutions of Einstein’s equations.  Commun. Math. Phys., 10(4):280–310, 1968.  doi:10.1007/BF03399503.
  * [226] J.F. Plebañski.  A class of solutions of Einstein-Maxwell equations.  Annals Phys., 90(1):196–255, 1975.  doi:10.1016/0003-4916(75)90145-1.
  * [227] T. Levi-Civita.  ${ds}^{2}$ einsteiniani in campi newtoniani.  IX. L’analogo del potenziale logaritmico, Rend. Acc. Lincei, 28:101, 1919.
  * [228] W.B. Bonnor.  Certain exact solutions of the equations of general relativity with an electrostatic field.  Proceedings of the Physical Society. Section A, 66(2):145, 1953\.
  * [229] K.S. Thorne.  Absolute stability of melvin’s magnetic universe.  Physical Review, 139(1B):B244, 1965.
  * [230] E. Kasner.  Geometrical theorems on einstein’s cosmological equations.  American Journal of Mathematics, 43(4):217–221, 1921.
  * [231] E. Poisson.  A Relativist’s Toolkit: The Mathematics of Black-Hole Mechanics.  Cambridge University Press, 12 2009.  doi:10.1017/CBO9780511606601.
  * [232] M. Alcubierre.  The Warp drive: Hyperfast travel within general relativity.  Class. Quant. Grav., 11:L73–L77, 1994.  arXiv:gr-qc/0009013, doi:10.1088/0264-9381/11/5/001.
  * [233] J. Lee and R.M. Wald.  Local symmetries and constraints.  J. Math. Phys., 31:725–743, 1990.  doi:10.1063/1.528801.
  * [234] G. Barnich.  Boundary charges in gauge theories: Using Stokes theorem in the bulk.  Class. Quant. Grav., 20:3685–3698, 2003.  arXiv:hep-th/0301039, doi:10.1088/0264-9381/20/16/310.
  * [235] G. Compère, K. Murata, and T. Nishioka.  Central Charges in Extreme Black Hole/CFT Correspondence.  JHEP, 05:077, 2009.  arXiv:0902.1001, doi:10.1088/1126-6708/2009/05/077.
  * [236] M. Astorino.  Removal of conical singularities from rotating C-metrics and dual CFT entropy, 7 2022.  arXiv:2207.14305.
  * [237] 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.
  * [238] B. Linet.  The static metrics with cylindrical symmetry describing a model of cosmic strings.  General Relativity and Gravitation, 17(11):1109–1115, 1985.  doi:https://doi.org/10.1007/BF00774211.
  * [239] J.A.G. Vickers.  Generalized cosmic strings.  Class. Quant. Grav., 4:1–9, 1987.  doi:10.1088/0264-9381/4/1/004.
  * [240] V.S. Manko and I.D. Novikov.  Generalizations of the kerr and kerr-newman metrics possessing an arbitrary set of mass-multipole moments.  Classical and Quantum Gravity, 9(11):2477–2487, nov 1992.  doi:10.1088/0264-9381/9/11/013.
  * [241] D.M. Zipoy.  Topology of some spheroidal metrics.  Journal of Mathematical Physics, 7(6):1137–1143, 1966.  arXiv:https://doi.org/10.1063/1.1705005, doi:10.1063/1.1705005.
  * [242] B.H. Voorhees.  Static axially symmetric gravitational fields.  Phys. Rev. D, 2:2119–2122, Nov 1970.  URL: https://link.aps.org/doi/10.1103/PhysRevD.2.2119, doi:10.1103/PhysRevD.2.2119.
  * [243] R.M. Kerns and W.J. Wild.  Generalized Zipoy-Voorhees metric.  Phys. Rev. D, 26:3726–3727, 1982.  doi:10.1103/PhysRevD.26.3726.
  * [244] C.M. Cosgrove.  New family of exact stationary axisymmetric gravitational fields generalising the tomimatsu-sato solutions.  Journal of Physics A: Mathematical and General, 10(9):1481–1524, sep 1977.  doi:10.1088/0305-4470/10/9/010.
  * [245] C. Hoenselaers, W. Kinnersley, and B.C. Xanthopoulos.  Symmetries of the stationary einstein–maxwell equations. vi. transformations which generate asymptotically flat spacetimes with arbitrary multipole moments.  Journal of Mathematical Physics, 20(12):2530–2536, 1979.  arXiv:https://doi.org/10.1063/1.524058, doi:10.1063/1.524058.