aainstitutetext: Physique-Mathématique, Universite de Mons-Hainaut, Mons,
Belgiumbbinstitutetext: Departamento de Matemática da Universidade de Aveiro
and Center for Research and Development in Mathematics and Applications
(CIDMA)
Campus de Santiago, 3810-183 Aveiro, Portugal

# Kaluza-Klein monopole with scalar hair

Y. Brihaye b C. Herdeiro b J. Novo b and E. Radu

###### Abstract

We construct a new family of rotating black holes with scalar hair and a
regular horizon of spherical topology, within five dimensional ($d=5$)
Einstein’s gravity minimally coupled to a complex, massive scalar field
doublet. These solutions represent generalizations of the Kaluza-Klein
monopole found by Gross, Perry and Sorkin, with a twisted $S^{1}$ bundle over
a four dimensional Minkowski spacetime being approached in the far field. The
black holes are described by their mass, angular momentum, tension and a
conserved Noether charge measuring the hairiness of the configurations. They
are supported by rotation and have no static limit, while for vanishing
horizon size, they reduce to boson stars. When performing a Kaluza-Klein
reduction, the $d=5$ solutions yield a family of $d=4$ spherically symmetric
dyonic black holes with gauged scalar hair. This provides a link between two
seemingly unrelated mechanisms to endow a black hole with scalar hair: the
$d=5$ synchronization condition between the scalar field frequency and the
event horizon angular velocity results in the $d=4$ resonance condition
between the scalar field frequency and the electrostatic chemical potential.

###### Keywords:

black holes, higher dimensions, scalar hair

##  1 Introduction

The five dimensional ($d=5$) extension of Einstein’s theory of General
Relativity (GR) was introduced around one century ago by Kaluza kaluza  and
Klein Klein:1926tv  in an early attempt to unify the (then) known
interactions, namely gravity and electromagnetism. In the Kaluza-Klein (KK)
framework, the Universe has three non-compact spatial dimensions; one extra
dimension is compact, topologically a circle and sufficiently small as to
remain unobservable.

This simple idea has proven to be one of the most fruitful in theoretical
physics, and the original KK model has been extended in various directions,
such as including more (than one) compact extra-dimensions and starting from
higher dimensional theories that are not vacuum GR, including, $e.g.$, gauge
and scalar fields - see ACF  for a review and a large set of original
literature.

In the context of this work, a feature of the (initial) KK model is of
particular interest: the existence of gravitational solitons, $i.e.$ non-
singular, horizonless solutions of the vacuum Einstein field equations. While
no such solutions exist in four spacetime dimensions Einstein:1943ixi ;
Lichnerowicz ; Lichnerowicz1 , gravitational solitons exist in KK theory, as
found in the ’80s by Gross and Perry Gross:1983hb  and Sorkin Sorkin:1983ns .
In the simplest case, this corresponds to a regular solution, which, from a
four-dimensional perspective, describes a (gravitating) Abelian magnetic
monopole, a feature which has attracted much interest.

The Gross-Perry-Sorkin (GPS) soliton is the product between the $d=4$ NUT-
instanton Hawking:1976jb  and a time coordinate, being asymptotically locally
flat only. This defines a special type of squashed KK asymptotics, with a
twisted $S^{1}$ bundle over a four dimensional Minkowski spacetime being
approached in the far field. As expected, the vacuum GPS soliton possesses
Black Hole (BHs) generalizations111 There are also BH generalizations of the
GPS solution with gauge fields Ishihara:2005dp ; Yazadjiev:2006iv ;
Brihaye:2006ws ; Nakagawa:2008rm ; Nedkova:2012yn .  Chodos:1980df ;
Dobiasch:1981vh ; Pollard:1982gj ; Gibbons:1985ac ; Wang:2006nw , with an
event horizon of $S^{3}$ topology, geometrically being a squashed (rather
thand round) sphere. Such solutions are essentially higher dimensional near
the event horizon, but look like four-dimensional - with a compactified extra-
dimension -, at large distances.

An interesting question which arises in this context concerns the validity of
the ’no-hair’ conjecture Bekenstein:1996pn . In particular, do the BHs with
squashed KK asymptotics allow for scalar hair? In the last decade it became
clear that, at least for asymptotically flat Herdeiro:2014goa  or anti-de
Sitter Dias:2011at  BHs, there is a generic mechanism allowing for complex
scalar hair around rotating horizons. This mechanism relies on a
synchronization condition Herdeiro:2014goa ; Herdeiro:2014ima  which
guarantees that there is no scalar energy flux crossing the horizon.
Mathematically, this results in the following relation between the scalar
field frequency $\omega$ and the event horizon angular velocity $\Omega_{H}$:

$\displaystyle\omega=m\Omega_{H}~{},$ (1.1)

where $m$ is the winding number which enters the scalar ansatz and
$m\in\mathbb{Z}^{+}$ for the $d=4$ BHs in Herdeiro:2014goa ; Herdeiro:2014ima
. Eq. (1.1) means that the scalar field phase angular velocity matches the
horizon’s angular velocity; hence the name ’synchronization’. This mechanism
appears to possess a certain degree of universality, applying to both
asymptotically flat ($d=4$) neutral Herdeiro:2014goa  and electrically charged
Delgado:2016jxq  rotating BHs, as well as to BHs in $d>4$ dimensions
Brihaye:2014nba ; Herdeiro:2015kha , toroidal horizon topology
Herdeiro:2017oyt , or AdS asymptotics Dias:2011at , and even to other spin
fields Herdeiro:2016tmi ; Santos:2020pmh .

One of the main results of this work is to provide evidence that the same
mechanism holds as well for BHs with the same squashed KK asymptotics as the
GPS soliton. To do so, we consider $d=5$ Einstein’s gravity minimally coupled
to a massive complex scalar field doublet, with a special ansatz, originally
introduced in Hartmann:2010pm , which factorizes the angular dependence and
reduces the problem to solving a set of ordinary differential equations
(ODEs). By numerically solving the Einstein-Klein-Gordon (EKG) equations, we
find a four parameter family of regular (on and outside the horizon) BHs with
scalar hair and squashed KK asymptotics. The four continuous parameters are
the mass $M$, the angular momentum $J$, the tension ${\cal T}$ and the Noether
charge $Q$, which measures the scalar field outside the horizon. For vanishing
horizon size, the solutions reduce to solitonic Boson Stars (BSs).
Interestingly, some basic properties of these configurations are akin to those
of the $d=4$ BSs and BHs with scalar hair Liebling:2012fv ; Herdeiro:2015gia ,
rather than those of the known $d=5$ asymptotically Minkowski EKG solutions
Hartmann:2010pm ; Brihaye:2014nba .

We also consider the equivalent $d=4$ picture, obtained after performing a
standard KK reduction for both the metric and the scalar field, as discussed
$e.g.$ in Gross:1983hb . While the BSs result in $d=4$ singular configurations
- similarly to the dimensional reduction of the (vacuum) GPS monopole -, the
$d=5$ BHs correspond to asymptotically flat solutions of a specific $d=4$
Einstein-dilaton-Maxwell-(gauged) scalar (EdMgs) field model. They are
spherically symmetric and describe gravitating dyonic BHs with scalar hair.
Remarkably, the synchronization condition (1.1) in $d=5$ translates into the
$d=4$ resonance condition:

$\displaystyle\omega=q_{s}{\cal V},$ (1.2)

which has been found in the study of charged (non-spinning) BHs with gauged
scalar hair Hong:2019mcj ; Herdeiro:2020xmb ; Hong:2020miv ; Brihaye:2022afz .
In (1.2) $q_{s}$ is the gauge coupling constant, while ${\cal V}$ is the
electrostatic chemical potential, which, in the $d=5$ picture, corresponds to
the event horizon angular velocity.

This paper is organized as follows. In Section 2 we present the EKG model
together with a general framework, the ansatz taken - complemented by the
equations in Appendix A -, and discuss the computation of global charges,
together with the solutions of the Klein-Gordon equation in the probe limit.
The EKG solutions with squashed KK asymptotics are discussed in Section 3,
where we consider both the case of BSs and BHs. Section 4 is motivated by the
observation that the GPS soliton can be taken as an intermediate state between
the five dimensional Minkowski spacetime and the ’standard’ KK vacuum, $i.e.$
the direct product of four dimensional Minkowski spacetime and a circle.
Therefore, in Section 4 we consider a comparison between the EKG solutions in
Section 3 and those found for the other two spacetime asymptotics mentioned
above. In particular, the basic properties of the KK vortices in EKG model are
also discussed there for the first time in the literature. Section 5
reconsiders the results from a $d=4$ perspective and shows how the
$synchronized$ $d=5$ spinning hairy BHs (HBHs) become $d=4$ $resonant$
spherically symmetric BHs with gauged scalar hair. We conclude in Section 6
with a discussion and some further remarks. A brief review of the vacuum
spinning BH solution with squashed KK asymptotics Dobiasch:1981vh ;
Wang:2006nw  is presented in Appendix B, as well as an exact solution of the
KG equation on an extremal (vacuum) BH background.

##  2 The framework

###  2.1 Action and field equations

We consider the $d=5$ Einstein’s gravity minimally coupled to a massive
complex scalar field doublet $\Psi$, with action

$\mathcal{S}=\frac{1}{4\pi
G_{5}}\int_{\mathcal{M}}d^{5}x\sqrt{-g}\left[\frac{1}{4}R-\frac{1}{2}g^{ab}\left(\Psi_{,\,a}^{\dagger}\Psi_{,\,b}+\Psi_{,\,b}^{\dagger}\Psi_{,\,a}\right)-\mu^{2}\Psi^{\dagger}\Psi\right]-\frac{1}{8\pi
G_{5}}\int_{\partial\mathcal{M}}d^{4}x\sqrt{-h}K,$ (2.1)

where $\dagger$ denotes the complex transpose, $G_{5}$ is the five dimensional
Newton’s constant, which will be set to unity in the numerics, $\mu$ is the
scalar field mass, $h_{ij}$ is the induced metric on the boundary
$\partial\mathcal{M}$ of the spacetime $\mathcal{M}$, and $K_{ij}$ is the
extrinsic curvature of this boundary, with $K=K_{ij}h^{ij}$.

Variation of this action with respect to the metric and scalar field gives the
EKG equations:

$R_{ab}-\frac{1}{2}g_{ab}R=2~{}T_{ab}\,,~{}~{}\left(\Box-\mu^{2}\right)\Psi=0~{},$
(2.2)

where

$T_{ab}=\Psi_{,a}^{\dagger}\Psi_{,b}+\Psi_{,b}^{\dagger}\Psi_{,a}-g_{ab}\left[\frac{1}{2}g^{cd}\left(\Psi_{,c}^{\dagger}\Psi_{,d}+\Psi_{,d}^{\dagger}\Psi_{,c}\right)+\mu^{2}\Psi^{\dagger}\Psi\right]~{},$
(2.3)

is the stress-energy tensor of the scalar field.

###  2.2 The vacuum Gross-Perry-Sorkin solution

We start by introducing the squashed Kaluza-Klein (KK) geometry found in
Gross:1983hb ; Sorkin:1983ns , which captures some of the basic features of
the solutions constructed in this work. This metric solves the vacuum Einstein
equations, $\Psi=0$, and is the product between the $d=4$ (self-dual)
Euclidean Taub-NUT instanton Hawking:1976jb  and a time coordinate,

$ds^{2}=-dt^{2}+ds_{4}^{2}\ .$ (2.4)

The instanton metric $ds_{4}^{2}$ possesses an intrinsic length scale
$N\geqslant 0$, which is an input parameter: the NUT charge. The geometry can
be written with several different choices of the radial coordinate $r$, which
make more transparent various limits of interest.

The first form for the instanton metric we shall consider is

$\displaystyle
ds_{4}^{2}=\frac{r+N}{r-N}dr^{2}+(r^{2}-N^{2})(d\theta^{2}+\sin^{2}\theta
d\varphi^{2})+\frac{r-N}{r+N}4N^{2}(d\psi+\cos\theta d\varphi)^{2}.$ (2.5)

The range of the radial coordinate is $N\leqslant r<\infty$, while
$\theta,\,\varphi$ and $\psi$ are the usual Euler angles with ranges
$0\leqslant\theta\leqslant\pi$, $0\leqslant\varphi<2\pi$,
$0\leqslant\psi<4\pi$. This metric is asymptotically locally flat, in the
sense that the curvature goes to zero as $r\to\infty$. The surfaces of
constant $r$ are topologically $S^{3}$, although their metric is a deformed
3-sphere, with an $S^{1}$ fiber over $S^{2}$. Also, $r=N$ corresponds to the
origin of the coordinates, on $\mathbb{R}^{4}$, with the size of $S^{3}$
shrinking to zero.

The coordinate transformation

$\displaystyle r\to r-N$ (2.6)

leads to an equivalent form of (2.5),

$\displaystyle
ds_{4}^{2}=\left(1+\frac{2N}{r}\right)\left[dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\varphi^{2})\right]+\frac{4N^{2}}{1+\frac{2N}{r}}(d\psi+\cos\theta
d\varphi)^{2},$ (2.7)

but now with the usual range of the new radial coordinate, $0\leqslant
r<\infty$.

The $N\to 0$ limit of the $d=4$ NUT instanton corresponds to the flat
$\mathbb{R}^{3}\times S^{1}$ space. To take this limit, one defines a new
coordinate

$\displaystyle\psi=\frac{z}{2N}\,.$ (2.8)

Then, as $N\to 0$, one finds the line element

$\displaystyle ds^{2}=-dt^{2}+dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\varphi^{2})+dz^{2}\ ,$ (2.9)

which is the $\mathbb{M}^{1,3}\times S^{1}$ metric, with an arbitrary
periodicity $L$ for the coordinate $z$.

An alternative form of the instanton metric, which shall later be employed in
the construction of BSs, is obtained by taking the following coordinate
transformation in (2.5)

$\displaystyle r\to N+\frac{r^{2}}{8N}\,,$ (2.10)

with the new radial coordinate ranging from zero to infinity. This result in
the line element

$ds_{4}^{2}=\left(1+\frac{r^{2}}{16N^{2}}\right)\left[dr^{2}+\frac{r^{2}}{4}(d\theta^{2}+\sin^{2}\theta
d\varphi^{2})\right]+\frac{1}{4}\left(\frac{r^{2}}{1+\frac{r^{2}}{16N^{2}}}\right)(d\psi+\cos\theta
d\varphi)^{2}.$ (2.11)

Then the limit $N\to\infty$ corresponds to

$\displaystyle
ds=-dt^{2}+dr^{2}+\frac{r^{2}}{4}\left[d\theta^{2}+\sin^{2}d\varphi^{2}+(d\psi+\cos\theta
d\varphi)^{2}\right]\ ,$ (2.12)

which is the $d=5$ Minkowski spacetime $\mathbb{M}^{1,4}$.

As such, when varying the parameter $N$, one can consider the $d=5$ metric
(2.4) as interpolating between the ’standard’ KK vacuum, $i.e.$ the direct
product of $d=4$ Minkowski spacetime and a circle – the limit $N\to 0$ –, and
the $d=5$ Minkowski spacetime – the limit $N\to\infty$.

As expected, the GPS soliton possesses BH generalizations, with a squashed
horizon of $S^{3}$ topology and nonzero size, whose basic properties are
reviewed in Appendix A.

###  2.3 A general ansatz

The geometries studied in this work are described by a generic line
element222There is a residual metric gauge freedom in (2.13), to be fixed
later. The GPS metric (2.4), with the various choices of radial coordinate in
the spatial part, is of the form (2.13).

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle-{\cal
F}_{0}(r)dt^{2}+{\cal F}_{1}(r)dr^{2}+{\cal
F}_{2}(r)\left(\sigma_{1}^{2}+\sigma_{2}^{2}\right)+{\cal
F}_{3}(r)(\sigma_{3}-2W(r)dt)^{2}\ ,$ (2.13)

with the left-invariant 1-forms $\sigma_{i}$ on $S^{3}$,

$\displaystyle\sigma_{1}=\cos\psi d\theta+\sin\psi\sin\theta d\varphi,$
$\displaystyle\sigma_{2}=-\sin\psi d\theta+\cos\psi\sin\theta d\varphi,$
(2.14) $\displaystyle\sigma_{3}=d\psi+\cos\theta d\varphi,$

and $(\theta,\varphi,\psi)$ the usual Euler angles defined above, while $r$
and $t$ denote the radial and time coordinates, respectively333 An equivalent
form of this line element (used sometimes in the literature) is found by
defining the new coordinates $\Theta=\theta/2$,
$\varphi_{1}=(\psi-\varphi)/2$, $\varphi_{2}=(\psi+\varphi)/2$ (with
$0\leqslant\Theta\leqslant\pi/2$, $0\leqslant(\varphi_{1},\varphi_{2})<2\pi$),
yielding  $\displaystyle ds^{2}=-{\cal F}_{0}(r)dt^{2}+{\cal
F}_{1}(r)dr^{2}+4\bigg{[}{\cal F}_{2}(r)d\Theta^{2}+{\cal
F}_{3}(r)(\sin^{2}\Theta(d\varphi_{1}-Wdt)^{2}+\cos^{2}\Theta(d\varphi_{2}-Wdt)^{2})\bigg{]}\
\ \ \ $ (2.15) $\displaystyle+[{\cal F}_{2}(r)-{\cal
F}_{3}(r)]\sin^{2}(2\Theta)(d\varphi_{1}-d\varphi_{2})^{2}\ .\ \ \ $ . Apart
from the Killing vector $K_{0}=\partial_{\psi}$, the line element (2.13)
possesses three additional Killing vectors

$\displaystyle K_{1}=\sin\varphi\partial_{\theta}\,,\quad
K_{2}=-\cos\varphi\partial_{\theta}+\sin\varphi\cot\theta\partial_{\varphi}-\frac{\sin\varphi}{\sin\theta}\partial_{\psi}\,,\quad
K_{3}=\partial_{\varphi}\,,$

which obey an $SU(2)$ algebra.

Concerning the scalar sector, we shall consider a general ansatz, with

$\Psi=\phi\left(r\right)e^{-i\omega t}\hat{\Psi}_{s}\,,\quad{\rm
with}~{}~{}~{}s=0,1\,,$ (2.16)

where

$\displaystyle\hat{\Psi}_{0}=\left(\begin{array}[]{c}1\\\
0\end{array}\right)\,,\quad\hat{\Psi}_{1}=\left(\begin{array}[]{c}~{}\sin\frac{\theta}{2}~{}e^{-i\frac{\varphi}{2}}\\\
\cos\frac{\theta}{2}~{}e^{i\frac{\varphi}{2}}\end{array}\right)e^{i\frac{\psi}{2}}\,.$
(2.21)

The case $s=0$ is only compatible with a static line-element, in which case
similar results are found when considering a singlet scalar field, $i.e.$
$\Psi=\phi(r)e^{-i\omega t}$. In the rotating case, we shall use instead the
$s=1$ scalar ansatz, which was originally proposed in Hartmann:2010pm , albeit
for a parametrization of the 3-sphere in terms of
$\\{\Theta,\varphi_{1},\varphi_{2}\\}$ \- see Eq. (2.15).

Both solitons and BHs, can be studied by using the general metric form (2.13)
together with the scalar ansatz (2.16), (2.21). The corresponding expressions
of the Einstein and energy-momentum tensors are given in Appendix A; the
resulting EKG equations depend only on the radial variable $r$. The BHs have a
regular horizon located at some $r_{H}>0$, with ${\cal F}_{0}(r_{H})=0$. For
solitons, the horizon is replaced with a regular origin $r=0$, where both
${\cal F}_{2}$ and ${\cal F}_{3}$ vanish, while ${\cal F}_{0}$ and ${\cal
F}_{1}$ are finite and nonzero. In both cases, the solutions share the far
field asymptotics with the GPS metric (2.4), (2.5), with ${\cal F}_{0}\to 1$,
${\cal F}_{1}\to 1$, ${\cal F}_{2}\to r^{2}$, ${\cal F}_{3}\to 4N^{2}$, $W\to
0$ (and also $\phi\to 0$) as $r\to\infty$.

Finally, let us mention that the action (2.1) is invariant under the global
$U(1)$ transformation $\Psi\rightarrow e^{i\alpha}\Psi$, where $\alpha$ is a
constant. This implies that the current
$j^{a}=-i(\Psi^{*}\partial^{a}\Psi-\Psi\partial^{a}\Psi^{*})$ is conserved,
$i.e.$ $j^{a}_{\ ;a}=0$. Therefore integrating the timelike component of this
current on a spacelike slice $\Sigma$ yields a conserved quantity – the
Noether charge:

$\displaystyle
Q=\int_{\Sigma}~{}j^{t}=32\pi^{2}\int_{r_{0}}^{\infty}dr~{}F_{2}\sqrt{\frac{F_{1}F_{3}}{F_{0}}}(\omega-W)\phi^{2},$
(2.22)

where $r_{0}=\\{0,r_{H}\\}$ for solitons and BHs respectively.

###  2.4 The computation of global changes

Apart from the Noether charge, the solutions possess three more conserved
quantities: mass $M$, angular momentum $J$ and tension444 The tension of a
spacetime was first introduced in Traschen:2001pb ; Townsend:2001rg . This
global charge is associated with the translation symmetry along the extra-
dimension, in a similar way to the mass being related with the existence of a
timelike Killing vector field. ${\cal T}$, whose values are encoded in the far
field form of the metric functions. Given the non-standard asymptotics of the
solutions in this work, one way to compute their charges is to use the
quasilocal tensor of Brown and York Brown:1992br , augmented by the
counterterm formalism Kraus:1999di ; Lau:1999dp ; Mann:1999pc ;
Astefanesei:2005ad . This technique, inspired by the holographic
renormalization method in spacetimes with anti-de Sitter (AdS) asymptotics
Henningson:1998gx ; Balasubramanian:1999re  consists in adding a suitable
boundary counterterm $\mathcal{S}_{\rm ct}$ to the action of the theory; thus
the bulk equations of motion are not altered. $\mathcal{S}_{\rm ct}$ is built
up with curvature invariants of the induced metric on the boundary, which is
sent to spatial infinity after the integration. Unlike the background
substraction method (see below), this procedure is intrinsic to the spacetime
of interest and it is unambiguous once the counterterm is specified. In our
case, however, differently from the AdS case, this method has the drawback
that there is no rigorous justification for the choice of the counterterm.

In this work we shall use the counterterm proposed in Mann:2005cx  to compute
the mass of the KK monopole, with

$\mathcal{S}_{\rm ct}=\frac{1}{8\pi
G_{5}}\int_{\partial\mathcal{M}}d^{4}x\sqrt{-h}\sqrt{2\mathsf{R}}\,,$ (2.23)

where $\mathsf{R}$ is the Ricci scalar of the induced metric on the boundary.
The variation of this action $w.r.t.$ $h_{ij}$ results in the boundary stress-
energy tensor

$T_{ij}=\frac{1}{8\pi
G_{5}}\left(K_{ij}-Kh_{ij}-\Phi(\mathsf{R}_{ij}-\mathsf{R}h_{ij})-h_{ij}h^{kl}\Phi_{;kl}+\Phi_{;ij}\right)\,,$
(2.24)

where we defined $\Phi=\sqrt{2/{\mathsf{R}}}$. If the boundary geometry has an
isometry generated by a Killing vector $\xi^{i}$, then $T_{ij}\xi^{j}$ is
divergence free, from which it follows that the quantity

$\mathcal{Q}=\int_{\Sigma}d\Sigma_{i}T^{i}{}_{j}\xi^{j},$ (2.25)

associated with a closed surface $\Sigma$, is conserved. Physically, this
means that the observers on the boundary with the induced metric $h_{ij}$
measure the same value of $\mathcal{Q}$. For the considered framework, the
mass $M$, tension ${\cal T}$ and angular momentum555When considering the
coordinates of eq. (2.15), this angular momentum is related to equal rotations
$w.r.t.$ the angular directions $\varphi_{1}$ and $\varphi_{2}$. $J$ are
computed as the integrals of the boundary stress-energy tensor components
$T_{t}^{t}$, $T_{\psi}^{\psi}$ and $T_{\psi}^{t}$, respectively.
Interestingly, as found in Mann:2005cx , this approach predicts a non-zero
value for the mass and tension of the GPS soliton (2.4), respectively

$\displaystyle M=M_{0}=\frac{4\pi N^{2}}{G_{5}},~{}~{}{\cal T}={\cal
T}_{0}=-\frac{N}{G_{5}}~{},$ (2.26)

while the angular momentum is zero, as expected. When considering solutions of
the EKG equations with the same asymptotics, $M$ and ${\cal T}$ will contain
the above background contributions.

Apart from the boundary counterterm method, we have also computed $M,{\cal T}$
and $J$ by using the Abbott-Deser approach Abbott:1981ff . This was initially
proposed to address the issue of conserved charges of asymptotically (anti-)de
Sitter spacetime, but has also proved useful for other asymptotic behaviors.
In this approach, conserved charges are associated with isometries of some
background geometry and can be summarized as follows. First, the following
decomposition of the metric is introduced

$g_{ab}=\bar{g}_{ab}+\bar{h}_{ab}\,,$ (2.27)

where $\bar{g}_{ab}$ corresponds to the background metric and $\bar{h}_{ab}$
is a perturbation. Assuming that $\bar{g}_{ab}$ is a vacuum solution, then the
field equations can be written as:

$R^{ab}_{L}-\frac{1}{2}\bar{g}^{ab}R_{L}=\tilde{T}^{ab}\,,$ (2.28)

where the subscript $L$ denotes terms that are linear in the perturbation and
$\tilde{T}^{ab}$ collects all higher order terms in $\bar{h}$ (the tilde
serves to distinguish it from the boundary stress-energy tensor defined above
for the counterterm method). It can be shown that the left-hand side of the
above equation satisfies the Bianchi identity $w.r.t.$ the background metric
\- see $e.g.$ chap. 6 of Ortin:2004ms ; then the field equations imply:

$\bar{\nabla}_{a}\tilde{T}^{ab}=0\,,$ (2.29)

where the bar indicates that the covariant derivative is taken $w.r.t.$ the
background metric. If $\chi$ is a Killing vector of the background geometry,
the following conservation law holds:

$\bar{\nabla}_{a}\left(\tilde{T}^{ab}\chi_{b}\right)=0\,,$ (2.30)

which allows to define the associated conserved charge:

$\tilde{\mathcal{Q}}=\int_{\Sigma}d\Sigma_{i}\tilde{T}^{ij}\chi_{j}\,,$ (2.31)

where $\Sigma$ is a closed surface.

The Abbott-Deser method has been extensively used to compute conserved charges
on some non-asymptotically flat spacetimes, see $e.g.$ Cai:2006td ;
Wang:2006nw ; Peng:2017bjo  for the case of KK asymptotics. One should,
however, mention that the choice of the background metric $\bar{g}$ requires
special care. It is sometimes common to consider the asymptotic metric as the
background; however this is not always a solution to Einstein’s equations, in
which case the effective energy-momentum tensor $\tilde{T}^{\mu\nu}$ has
contributions not only from the perturbation, but also from the background.
For asymptotically squashed KK spacetimes, this issue has been explored in
Peng:2017bjo . The results there show that, when choosing the asymptotic form
of the metric (2.4), (2.5), as the background666 The large$-r$ limit of the
GPS metric (2.4), (2.5), results in the line element  $\displaystyle
ds^{2}=-dt^{2}+dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\varphi^{2})+4N^{2}(d\psi+\cos\theta d\varphi)^{2},$ (2.32) which, however,
does not solve the vacuum Einstein equations., the mass and tension of the GPS
soliton take the same (nonzero) values (2.26) as found for the counterterm
approach.

The global charges of the solutions in this work were computed using both
methods described above. We have found that the values of $M$, ${\cal T}$ and
$J$ obtained within the counterterm approach and Abbott-Deser approach \- with
the asymptotic metric (2.32) taken as the reference background -, agree777 As
expected, however, when using the Abbott-Deser approach with the GPS metric as
background, the terms $M_{0}$ and ${\cal T}_{0}$, $cf.$ eq. (2.26), are absent
in the expression of mass and tension of the EKG solutions.. But in order to
simplify the picture, and in particular to make clear the limit of a vanishing
scalar field, we have subtracted the $M_{0}$-term in all figures where the
mass of solutions with squashed KK asymptotics is displayed, which is
nonetheless in the corresponding equations.

###  2.5 The probe limit: no scalar clouds

Before discussing the solutions of the full system (2.2), it is of interest to
consider the solutions of the KG equation in the probe-limit case, $i.e.$
ignoring the backreaction on the spacetime geometry.

Starting with the case of a GPS background as given by (2.4) and (2.5), the KG
equation reads (where a prime denotes the derivative $w.r.t.$ the radial
coordinate $r$):

$\phi^{\prime\prime}+\frac{2\phi^{\prime}(r)}{r-N}+\frac{r+N}{r-N}\left(\omega^{2}-\mu_{\rm
eff}^{2}\right)\phi-\frac{s(r+5N)}{8N(r-N)^{2}}\phi=0,$ (2.33)

with $s=0,1$ for the scalar ansatz (2.16), (2.21). Also, in the above
expression we define

$\mu_{\rm eff}=\sqrt{\mu^{2}+\frac{s}{16N^{2}}}\,.$ (2.34)

That is, for $s=1$, the scalar field acquires a contribution to the mass
coming from the dependence on the $\psi$-direction, giving rise to an
effective mass, a features shared also by the gravitating solutions. The bound
state condition then needs to consider this effective mass, $i.e.$
$\omega^{2}\leqslant\mu_{\rm eff}^{2}$.

For the scalar Ansatz (2.16), (2.21) with $s=0$, the general solution of the
equation (2.33) reads $\phi(r)=c_{1}\phi_{1}(r)+c_{2}\phi_{2}(r)$ where
$c_{1},c_{2}$ are arbitrary constants, and

$\displaystyle\phi_{1}(r)=\frac{N}{r-N}\mbox{e}^{-\left(r-N\right)}U\left(N\sqrt{\mu^{2}-\omega^{2}},0,2(r-N)\sqrt{\mu^{2}-\omega^{2}}\right)$
(2.35)
$\displaystyle\phi_{2}(r)=\frac{N}{r-N}\mbox{e}^{-\left(r-N\right)}L_{-N\sqrt{\mu^{2}-\omega^{2}}}^{-1}\left(2(r-N)\sqrt{\mu^{2}-\omega^{2}}\right)$

Here $U$ is the confluent hypergeometric function and $L$ is the generalized
Laguerre polynomial. For solutions satisfying the bound state condition, the
function $\phi_{1}(r)$ diverges at $r=N$ while $\phi_{2}(r)$ diverges at
infinity888 In the limiting case with $\omega^{2}=\mu^{2}$, the solution
(2.35) takes the simpler form $\phi(r)=\frac{c_{1}}{r-N}+c_{2}\,$. .

For the scalar ansatz (2.16), (2.21) with an explicit dependence of angular
coordinates ($i.e.$ $s=1$), the general solution is
$\phi(r)=c_{1}\phi_{1}(r)+c_{2}\phi_{2}(r)$ , where

$\displaystyle\phi_{1}(r)$
$\displaystyle=\sqrt{\frac{r}{N}-1}e^{-(r-N)\sqrt{\mu_{\rm
eff}^{2}-\omega^{2}}}U\left(\frac{c}{2}3,2(r-N)\sqrt{\mu_{\rm
eff}^{2}-\omega^{2}}\right)\,,$ (2.36) $\displaystyle\phi_{2}(r)$
$\displaystyle=\sqrt{\frac{r}{N}-1}e^{-(r-N)\sqrt{\mu_{\rm
eff}^{2}-\omega^{2}}}L_{-\frac{c}{2}}^{2}\left(2(r-H)\sqrt{\mu_{\rm
eff}^{2}-\omega^{2}}\right)$

with $c=3+1/{(8N\sqrt{\mu_{\rm eff}^{2}-\omega^{2}})}+2N\sqrt{\mu_{\rm
eff}^{2}-\omega^{2}}$, which is again divergent at $r=N$ or at infinity999 In
the special case with $\omega^{2}=\mu_{\rm eff}^{2}$, the general solution
(2.36) simplifies to
$\phi(r)=\frac{4\sqrt{2N}}{\sqrt{r-N}}\left[c_{1}I_{2}\left(\sqrt{\frac{r-N}{2N}}\right)+c_{2}K_{2}\left(\sqrt{\frac{r-N}{1N}}\right)\right],$
(2.37) where $I$ and $K$ are Bessel functions, which is again divergent..
Therefore we conclude that there are no scalar clouds on a GPS soliton
background, a situation also found for the $\mathbb{M}^{1,3}\times S^{1}$ or
$\mathbb{M}^{1,4}$ cases.

In $d=4$, (real frequency) bound states are found for a particular set of Kerr
BHs. These configurations are at the threshold of the superradiant
instability, the scalar field satisfying the synchronization condition (1.1)
Hod:2012px ; Hod:2013zza . Inspired by this result, we have looked for scalar
clouds on a spinning vacuum BH with squashed KK asymptotics, whose metric is
presented in Appendix B.2, the scalar field ansatz being the case $s=1$ in
(2.21). Unfortunately, in the generic, non-extremal case, the resulting
equation for the scalar amplitude $\phi(r)$ could not be solved analytically.
Therefore we have considered a numerical approach, employing similar methods
as the ones described $e.g.$ in Benone:2014ssa . After imposing the
synchronization condition, Eq. (1.1), we looked for scalar configurations
which are regular on and outside the horizon and vanish at infinity. We found
no indication that such solution exist, despite not being able to provide a
non-existence proof. This result is also supported by the exact solutions
(B.13) found for the extremal (spinning) BH background, which is displayed in
Appendix B.2. Unlike the case of an extremal Kerr background in $d=4$
Hod:2012px , this solution is singular at the horizon or at infinity.

This non-existence result implies the absence of an existence line for the
HBHs with squashed KK asymptotics, which would be given by the set of vacuum
BH configurations allowing for scalar clouds. We mention that a similar result
Cardoso:2005vk ; Kunduri:2006qa  is found for for a test massive scalar field
on the background of an asymptotically $\mathbb{M}^{1,4}$ Myers-Perry BHs
Myers:1986un .

##  3 The solutions

###  3.1 The numerical approach

The numerical methods employed here are similar to those used in
Brihaye:2014nba  to study $d=5$ EKG solutions with equal-magnitude angular
momenta and $\mathbb{M}^{1,4}$ asymptotics. The BH problem possesses four
input parameters (we recall that $G_{5}=1$): two of them belong to a specific
model – the scalar field mass $\mu$ and the NUT parameter $N$; and two specify
a solution – the field frequency $\omega$ and the horizon radius $r_{H}$ (with
$r_{H}=0$ for solitons). In practice, dimensionless variables and global
quantities are introduced by using natural units set by the scalar field mass
$\mu$, $e.g.$ $r\to r/\mu$, $\omega\to\omega/\mu$ and $N\to N\mu$, which
reduce to taking $\mu=1$ in the input of the numerical code101010In principle,
the effective mass relation for the scalar field, Eq. (2.34), would allow for
the existence of spinning solutions with $\mu=0$. However, so far these could
not be obtained. . As such, we are left with three (two) input parameters for
BHs (solitons).

The system of five non-linear coupled ODEs for the metric functions and the
scalar amplitude, subject to the boundary conditions described below, was
solved using two different solvers. The BH solutions and a part of the BS sets
were found by using a professional package based on the Newton-Raphson method
schoen . In this case we have introduced a compactified coordinate $x$, where
$0\leqslant x\leqslant 1$; the relation between the usual radial coordinate
$r$ and $x$ being $r=(r_{H}+cx)/(1-x)$, with $c$ a suitable chosen constant,
usually of order one. Typical grids used have around $800$ points, distributed
equidistantly in $x$.

Most of the BSs solutions were found by using another package, which employs a
collocation method for boundary-value ordinary differential equations and a
damped Newton method of quasi-linearization COLSYS . The meshes here are non-
equidistant and use around 300 points in the interval $0\leqslant r<r_{\rm
max}$, with $r_{\rm max}$ around $10^{4}$.

We have compared a number of BS solutions constructed with these two different
methods and found a very good agreement between them. In both cases, the
constraint Einstein equation and the Smarr relation (3.14) have been used to
test the accuracy of the results. Based on that, we estimate a typical
relative error $<10^{-5}$ for the solutions reported herein. The numerical
accuracy, however, decreases close to the maximal value of the frequency and
also for solutions close to the center of the $(\omega,M)$-spiral - see below.

Finally, let us mention that in this work we report results for a nodeless
scalar field amplitude only, although solutions with nodes exist as well.

###  3.2 Boson Stars

Before discussing the BH solutions, it is useful to first describe the
properties of their solitonic limit, $i.e.$ of the BSs. In the numerical
construction of these horizonless solutions, we have found useful to use the
following metric ansatz111111 Since $g^{tt}=-e^{-2F_{0}(r)}<0$, the
$t-$coordinate provides a global time function and the spacetime is free of
causal pathologies, which is also the case for BHs.

Figure 1:  The profile of a typical static (left panel) and spinning (right
panel) BS with the same input parameters.

\- which can be viewed as a deformation of (2.11) -, in terms of four unknown
functions $(F_{0},F_{1},F_{2},W)$:

$\displaystyle ds^{2}=-e^{2F_{0}(r)}dt^{2}$
$\displaystyle+e^{2F_{1}(r)}\left(1+\frac{r^{2}}{16N^{2}}\right)\bigg{[}dr^{2}+\frac{1}{4}r^{2}\left(\sigma_{1}^{2}+\sigma_{2}^{2}\right)\bigg{]}$
$\displaystyle+\frac{e^{2F_{2}(r)}}{4}\left(\frac{r^{2}}{1+\frac{r^{2}}{16N^{2}}}\right)\left(\sigma_{3}-2W(r)dt\right)^{2}\,.$
(3.1)

Near the origin ($r=0$), the solutions possess a formal power series
expansion, whose first terms are given by

$\displaystyle
F_{0}(r)=f_{00}^{(0)}+f_{02}^{(0)}r^{2}+\dots,~{}~{}F_{1}(r)=f_{10}^{(0)}+f_{12}^{(0)}r^{2}+\dots,~{}~{}$
(3.2) $\displaystyle
F_{2}(r)=f_{10}^{(0)}+f_{22}^{(0)}r^{2}+\dots,~{}~{}W(r)=w_{0}+w_{2}^{(0)}r^{2}+\dots,~{}~{}\phi(r)=\phi_{1}r+\phi_{3}r^{3}+\dots,~{}~{}$

with all coefficients fixed by $f_{00}^{(0)}$, $f_{10}^{(0)}$, $f_{22}^{(0)}$,
$w_{0}$ and $\phi_{1}$, $e.g.$ $f_{12}^{(0)}=-f_{22}^{(0)}-2\phi_{1}^{2}/3$.

For the far field expansion of the solutions, one finds

$\displaystyle
F_{0}(r)=\frac{f_{02}}{r^{2}}+\dots,~{}~{}F_{1}(r)=\frac{f_{12}}{r^{2}}+\dots,~{}~{}F_{2}(r)=-\frac{(f_{02}+f_{12})}{r^{2}}+\dots,~{}~{}$
$\displaystyle
W(r)=\frac{w_{2}}{r^{2}}+\dots,~{}~{}\phi(r)=c_{1}\frac{e^{-r^{2}\sqrt{\mu^{2}_{\rm
eff}-\omega^{2}}}}{r^{2}}+\dots,~{}~{}$ (3.3)

with the parameters $f_{02}$, $f_{12}$, $w_{2}$ and $c_{1}$ being fixed by
numerics.

The mass, tension and angular momentum are defined in terms of the asymptotic
coefficients by (note the presence of the background terms (2.26) in $M,{\cal
T}$)

$\displaystyle
M=\frac{4\pi}{G_{5}}\left[N^{2}+\frac{1}{8}(f_{12}-f_{02})\right],~{}{\cal
T}=-\frac{1}{G}_{5}\left[N+\frac{1}{4N}(f_{12}+\frac{1}{2}f_{02})\right],~{}~{}J=\frac{2\pi
N^{2}w_{2}}{G_{5}}\,.~{}~{}{~{}~{}~{}}$ (3.4)

One can easily show that, as with BSs with other asymptotics, the Noether
charge and the angular momenta of the spinning BSs are not independent
quantities, with

$\displaystyle Q=2J\,.$ (3.5)

Figure 2:  The frequency-mass diagram is shown for static (left panel) and
spinning (right panel) families of BSs with several values of the parameter
$N$. In all plots for solutions with squashed KK asymptotics shown in this
work, the background contribution $M_{0}$ \- $cf.$ Eq. (2.26) - is subtracted
from the mass $M$, such that $M=0$ for horizonless, vacuum solutions.

Figure 3:  The mass $M$ and Noether charge $Q$ (angular momentum $J$) are
shown as a function of the central value of the scalar field $\phi(0)$ (left
panel, static solutions) and as a function of the first derivative of the
scalar field at the origin $\phi^{\prime}(0)$ (right panel, spinning
solutions). The insets shows how $\phi(0)$ (or $\phi^{\prime}(0)$) is related
to the frequency parameter $\omega$.

The static solutions are constructed for a version of (3.1) with $W=0$ and
$s=0$ in the scalar field ansatz (2.21). While their far field behavior is
still given by (3.3), with $\mu_{\rm eff}=\mu$, the scalar field does not
vanish at $r=0$, with $\phi(r)=\phi_{0}+{\cal O}(r).$

The profile of typical static and spinning BS solutions, with the same values
of the input parameters $N,\mu$ and $\omega$, are shown in Figure 1. One
remarks that the metric functions $F_{0},F_{1},F_{2}$ (and $W$ in the rotating
case) interpolate monotonically between some constant value at the origin and
zero at infinity, without presenting a local extremum.

Taking $\omega$ as a control parameter, the numerical results show that both
static and spinning BSs exist for a limited range of frequencies, $\omega_{\rm
min}<\omega<\mu_{\rm eff}$, with $\omega_{\rm min}$ depending on the value of
the NUT parameter $N$. In the limit $\omega\rightarrow\mu_{\rm eff}$, the
mass, tension and the Noether charge go to zero121212Recall that the
background contributions $M_{0},\,\mathcal{T}_{0}$ are subtracted such that
$M=\mathcal{T}=0$ in the absence of a scalar field, $cf.$ the discussion in
Section 2.4.. One remarks that this is the behavior also found in the $d=4$
asymptotically flat case Liebling:2012fv .

As can be seen in Figure 2, the BS mass first increases as $\omega$ is
decreased from $\mu_{\rm eff}$ approaching a maximal value, $M_{\rm max}$, for
some $\omega_{0}$ (with both $M_{\rm max}$ and $\omega_{0}$ increasing with
$N$). Then, the mass decreases, and, after some $\omega_{\rm min}$, a
backbending is observed in the $M(\omega)$-diagram. Further following the
curve, there is an inspiralling behaviour, towards a limiting configuration at
the center of the spiral, which occurs for a frequency $\omega_{\rm cr}$
(which is also a function of $N$). This central inspiralling behaviour appears
to be generic for BS solutions in EKG model, being also found in $d=4$
dimensions, or for $d=5$ solutions with $\mathbb{M}^{1,4}$ or even AdS
asymptotics. A similar diagram is recovered for the curve $Q(\omega)$; for
static BSs, one finds $M<\mu Q$ for a range of frequencies between $\mu$ and a
critical value marked with a black dot in Figure 2 (left) (see also the
inset). Rather unexpectedly, we have found that $M>\mu Q$ for all considered
spinning configurations, which suggests that these solutions are unstable.

Further insights on the properties of the BSs can be taken from Figure 3,
where we plot the mass $M$ and Noether charge $Q$ as a function of the central
value of the scalar field $\phi(0)$ (static case) and $\phi^{\prime}(0)$
(spinning BSs). Again, one notices a rather similar picture to that found for
static BSs in $d=4$ Kaup:1968zz ; Ruffini:1969qy .

###  3.3 Synchronized hairy Black Holes

The BH solutions are constructed for a slightly more complicated metric
ansatz, which fixes the behaviour at the horizon and at infinity, and also can
be taken as a deformation of the static vacuum BH (B.2), namely131313 The
spinning BS solutions can also be studied within the metric ansatz (3.6) with
$r_{H}=0$. However, the numerics is more difficult as compared to the metric
choice (3.1), the small-$r$ expansion of the scalar field starting with a
$\sqrt{r}$-term.

$\displaystyle
ds^{2}=-e^{2F_{0}(r)}\frac{\left(1-\frac{r_{H}}{r}\right)^{4}}{\left(1+\frac{r_{H}}{r}\right)^{2}}dt^{2}$
$\displaystyle+e^{2F_{1}(r)}H(r)\left(1+\frac{r_{H}}{r}\right)^{4}\bigg{[}dr^{2}+r^{2}\left(\sigma_{1}^{2}+\sigma_{2}^{2}\right)\bigg{]}$
$\displaystyle+e^{2F_{2}(r)}\frac{4N^{2}}{H(r)}[\sigma_{3}-2W(r)dt]^{2}\,,$
(3.6)

with $F_{0},F_{1},F_{2}$ and $W$ resulting from numerics, and the background
function

$\displaystyle
H(r)=1+\frac{2\left(\sqrt{N^{2}+r_{H}^{2}}-r_{H}\right)r}{(r+r_{H})^{2}}.$
(3.7)

As with the BSs, one can write an approximate form of the solutions at the
limits of the $r$-interval. The essential coefficients in these expansions
determine most of the quantities of interest, either horizon quantities
($r=r_{H}$) or global quantities ($r\to\infty$). At the horizon, the Killing
vector141414 $\xi\Psi=0$ is the only symmetry of the full solution (geometry
and scalars) and is generated by $\xi$. Also, the BSs are invariant under the
action of $\hat{\xi}=\partial_{t}+2\omega\partial_{\psi}$.
$\xi=\partial_{t}+\Omega_{H}\partial_{\psi}$ becomes null, with
$\Omega_{H}=2W(r_{H})$ the event horizon angular velocity.

Figure 4:  (Left panel) The metric functions $(F_{i},W)$ and the scalar field
amplitude $\phi$ of a HBH solution are shown as a function of the radial
coordinate. (Right panel) The metric functions $g_{tt}(r)$ and
$g_{\psi\psi}(r)$ are shown for the same solution. One notices the existence
of two ergo-regions ($g_{tt}>0$); also the metric function $g_{\psi\psi}(r)$
interpolates smoothly between the horizon value $g_{\psi\psi}(r_{H})=0.0076$
and the asymptotic value $g_{\psi\psi}(\infty)=4N^{2}$.

The following (formal) power series holds there (where $i=0,1,2$):

$\displaystyle F_{i}(r)=f_{i0}^{(H)}+f_{i2}^{(H)}(r-r_{H})^{2}+\dots,$
$\displaystyle
W(r)=\frac{1}{2}\Omega_{H}+w_{2}^{(H)}(r-r_{H})^{2}+\dots,~{}\phi(r)=\phi_{0}^{(H)}+\phi_{2}^{(H)}(r-r_{H})^{2}+\dots,~{}~{}$
(3.8)

with the following relation between frequency and event horizon angular
velocity

$\displaystyle\omega=W\big{|}_{r_{H}}=\frac{1}{2}\Omega_{H}~{},$ (3.9)

which is just the condition (1.1) with $m=1/2$, as implied by the employed
scalar ansatz.

The shape of the event horizon can be read off from the induced horizon metric

$\displaystyle
d\Sigma_{H}^{2}=8e^{2f_{10}^{(H)}}r_{H}^{2}\Bigg{(}1+\sqrt{1+\frac{N^{2}}{r_{H}^{2}}}\,\Bigg{)}\left(\sigma_{1}^{2}+\sigma_{2}^{2}\right)+\frac{8N^{2}e^{2f_{20}^{(H)}}}{1+\sqrt{1+\frac{N^{2}}{r_{H}^{2}}}}\sigma_{3}^{2}\,,$
(3.10)

which describes a squashed $S^{3}$ geometry. The expansions at the horizon,
(3.8), can be used to compute the event horizon’s area $A_{H}$ and Hawking
temperature $T_{H}$, which are given by

$\displaystyle
A_{H}=256\sqrt{2}\pi^{2}Nr_{H}^{2}e^{2f_{10}^{(H)}+f_{20}^{(H)}}\sqrt{1+\sqrt{1+\frac{N^{2}}{r_{H}^{2}}}}\,,~{}~{}T_{H}=\frac{1}{16\pi
r_{H}}\frac{\sqrt{2}~{}e^{f_{00}^{(H)}-f_{10}^{(H)}}}{\sqrt{1+\sqrt{1+\frac{N^{2}}{r_{H}^{2}}}}}\,.\
\ \ $ (3.11)

An approximate solution can also be constructed for large $r$, with

$\displaystyle
F_{0}(r)=\frac{f_{01}}{r}+\dots,~{}~{}F_{1}(r)=\frac{f_{11}}{r}+\dots,~{}~{}F_{2}(r)=-\frac{f_{11}+f_{01}}{r}+\dots,~{}~{}$
$\displaystyle
W(r)=\frac{w_{1}}{r}+\dots,~{}~{}~{}~{}\phi(r)=c_{1}\frac{e^{-r\sqrt{\mu^{2}_{\rm
eff}-\omega^{2}}}}{r}+\dots.$ (3.12)

Figure 5:  The mass $M$, angular momentum $J$, Hawking temperature $T_{H}$ and
hairiness parameter $q=Q/(2J)$ are shown as a function of event horizon area
$A_{H}$ for two sets of solutions with field frequency $\omega$ (or,
equivalently, constant angular velocity $\Omega_{H}$ ). In the two cases the
HBHs interpolate between a BS and $(i)$ another BS or, $(ii)$ an extremal HBH.

With these expressions, the computation of the mass, tension and angular
momentum is straightforward, with

$\displaystyle
M=\frac{4\pi}{G_{5}}N\left(\sqrt{N^{2}+r_{H}^{2}}+3r_{H}+f_{11}-f_{01}\right),~{}~{}~{}J=\frac{16\pi
N^{3}w_{1}}{G_{5}},$ (3.13) $\displaystyle{\cal
T}=\frac{1}{G_{5}}\left(\sqrt{N^{2}+r_{H}^{2}}+f_{11}+\frac{1}{2}f_{01}\right).$

As usual in BH mechanics (without a cosmological term), the temperature,
horizon area and the global charges are related through a Smarr mass formula
Smarr:1972kt ; Bardeen:1973gs , whose general form for the squashed KK
asymptotics reads

$\displaystyle M=\frac{1}{2}{\cal
T}L+\frac{3}{2}T_{H}\frac{A_{H}}{4G_{5}}+\frac{3}{2}\Omega_{H}\left(J-\frac{1}{2}Q\right)+M^{(\Psi)},$
(3.14)

where $L=8\pi N$ is the length of the twisted $S^{1}$ fiber at infinity, and

$\displaystyle
M^{(\Psi)}=-\frac{3}{2}\int_{\Sigma}\sqrt{-g}d^{4}x\left(T_{t}^{t}-\frac{1}{3}T_{a}^{a}\right),$
(3.15)

is the mass stored in the matter field(s) outside the horizon.

As with other BHs with synchronized hair, to measure the ’hairiness’ of the
solutions, we introduce a $normalized$ Noether charge $q$, with $q=0$ for a
vanishing scalar field and $q=1$ for BSs:

$\displaystyle q=\frac{Q}{2J}~{}.$ (3.16)

The complete classification of the solutions in the space of input parameters
$\\{N;\omega,r_{H}\\}$ is a considerable task which is beyond the scope of
this paper.

Figure 6:  The domain of existence (shaded area) of HBHs is shown in a mass
$vs.$ frequency (left panel) and horizon area $vs.$ frequency (right panel)
plot. The vacuum, spinning BHs exist below the blue dotted line (in which case
one takes $\Omega_{H}=2\omega$).

In what follows we show results for $N=1$, while a very similar phase diagram
has been found for $N=0.5$, these being the only values of $N$ for which we
have attempted for a systematic investigation of the solutions. However, we
have also constructed HBHs with $N=0.1,~{}2$ and $5$, and the displayed
picture for $N=1$ appears to be generic.

The profile of a BH solution which smoothly interpolates between the
asymptotic expansions (3.8), (3.12) is shown in Figure 4. An interesting
feature there is the existence of two distinct ergo-regions ($i.e.$ with
$g_{tt}>0$), a feature which is found also for $d=4$ BHs with synchronized
scalar hair Herdeiro:2014jaa . This is not, however, the generic behaviour,
since a single ergo-region is found for a large part of the parameter space.

Given $N\neq 0$, the domain of existence of solutions is obtained by
considering sequences of solutions at constant $\omega=\Omega_{H}/2$ and
varying the event horizon radius $r_{H}$. As expected, a (small) BH can be
added at the center of any spinning BS with a given $\omega$; this is the
starting point for any aforementioned sequence. However, the end point depends
on the value the frequency parameter (see Figure 5). For $\omega_{\rm
min}<\omega<\omega_{i}$, the sequence ends in another BS with the same
frequency (case $(i)$ in Figure 5), where we introduced $\omega_{i}$ to denote
the minimum frequency possible for extremal hairy BHs, $i.e.$ the zero
temperature configurations, see the black dotted curve in Figure 6. A
different picture is found for $\omega_{i}<\omega<\mu_{\rm eff}$ (case
$(ii)$), the sequences ending on extremal BHs with a nonvanishing horizon area
and hairiness parameter $q$.

In Figure 6 (left panel), we exhibit the domain of existence of the HBHs
(shaded region), in a $M(\omega)$ diagram, based on around two thousands of
solution points, a similar diagram being found for $J(\omega)$. The picture
possesses some similarities with the one found for spinning BHs with scalar
hair in $d=4$ Herdeiro:2014goa ; Herdeiro:2015gia . As with that case, the BS
curve (red solid line in Figure 6) forms a boundary of the domain; in
particular the BSs set the maximal value of the BHs’ mass.

Figure 7:  The mass-angular momentum diagram is shown for vacuum BHs (left
panel) and for HBHs (right panel).

There is also a curve of extremal HBs (black line), which appears to inspiral
towards a central value, where, we conjecture, it meets the endpoint of the BS
spiral. In $d=4$, the extremal BHs curve ends in a particular vacuum Kerr
solution, where it joins the existence line – a particular set of Kerr BHs
allowing for scalar clouds Herdeiro:2014goa ; Herdeiro:2015gia . However, in
the absence of an existence line for $d=5$ \- no scalar clouds on a vacuum BH
backgound -, the extremal HBH curve continues all the way to the maximal
frequency, $\omega=\mu_{\rm eff}$.

Now consider the horizon area $vs.$ frequency diagram, see Figure 6 (right
panel). Differently from the $d=4$ case Herdeiro:2015gia , one notices the
existence there of a vertical line segment with $\omega=\mu_{\rm eff}$ and
nonzero horizon area. That is, these solutions exist for a given range of
$r_{H}\geqslant 0$; there the scalar field spreads and tends to zero as
$\omega$ increases towards $\mu_{\rm eff}$, and also the values of the Noether
charge $Q$ and of the mass stored in the field $M^{(\Psi)}$ both tend to zero.
At the same time, the geometry does not trivialize, becoming that of a vacuum
(spinning) KK BH, with the input parameters151515The range of event horizon
radius here is $0\leqslant r_{H}\leqslant r_{H}^{\mathrm{ext}}$, the limits
corresponding to the GPS soliton and the extremal vacuum BH, respectively.
$\Omega_{H}=2\mu_{\rm eff}$ and $r_{H}$ (and nonzero global charges $M,{\cal
T}$ and $J$).

Further insight can be found in Figure 7 (right panel), where we plot the
domain of existence of HBHs in the ($J,M$)-plane Again, despite the absence of
an existence line, the overall picture is rather similar to that found in
Herdeiro:2015gia  for the $d=4$ HBHs counterparts. Observe, however, the
existence in Figure 7 (also in the inset) of a green line, which corresponds
to the limiting solutions with $\omega=\mu_{\rm eff}$; this line starts at
vacuum and ends in an extremal vacuum BH with (for $N\mu=1$) $M=15.0334$ and
$J=4.9348$.

##  4  Boson stars and Black Holes with squashed Kaluza-Klein asymptotics
$vs.$ solutions with $\mathbb{M}^{1,4}$ and $\mathbb{M}^{1,3}\times S^{1}$
asymptotics

One of the main goals of this work is to identify the effects of considering
squashed KK asymptotics on the properties of BSs and hairy BHs solutions of
the EKG system. Here we recall that, as discussed in Section 2.2, the GPS
soliton - whose asymptotics provide the background of our solutions - can be
seen as a vacuum state interpolating between the $\mathbb{M}^{1,4}$ and the
(standard) $\mathbb{M}^{1,3}\times S^{1}$ vacua, which are approached in the
limit of an infinite $N$ and vanishing $N$, respectively. As such, it is of
interest to contrast the picture found above for EKG solutions with $N\neq 0$,
with that for BSs and BHs solutions of the same model (2.1), which, however,
approach at infinity a background given by (2.12) or (2.9). This comparison
will be the main subject of this Section.

###  4.1 Solutions with $\mathbb{M}^{1,4}$ asymptotics

The case of EKG solutions with $\mathbb{M}^{1,4}$ asymptotics is better
understood, being the subject of several studies. Starting with static,
spherically symmetric solutions, we consider the scalar ansatz (2.16) and the
following metric form with two functions $U(r)$ and $\delta(r)$

$\displaystyle
ds=-e^{-2\delta(r)}U(r)dt^{2}+\frac{dr^{2}}{U(r)}+\frac{r^{2}}{4}\left[d\theta^{2}+\sin^{2}\theta
d\varphi^{2}+(d\psi+\cos\theta d\varphi)^{2}\right],$ (4.1)

the line-element (2.12) being approached asymptotically. The horizonless
solitonic solutions have been discussed in Hartmann:2010pm , describing
spherically symmetric BSs, and share all basic properties of the spinning
stars discussed below.

By adapting a general theorem put forward in Pena:1997cy , one can show that,
as with the $d=4$ case and $\mathbb{M}^{1,3}$ asymptotics, there are no
static, spherically symmetric BHs with scalar hair (here we assume the
existence of an horizon with $U(r_{H})=0$ and $\delta(r_{H})$ finite). The
starting point is the conservation of the energy-momentum tensor of the scalar
field

$\displaystyle\nabla_{a}T^{a}_{b}=0~{},$ (4.2)

which, for $b\equiv r$ and the considered ansatz (note that we take $s=0$ in
(2.21)), results in

$\displaystyle
e^{\delta}(e^{-\delta}T_{r}^{r})^{\prime}=-\frac{3}{r}T_{r}^{r}+\frac{1}{2}\frac{dg_{ab}}{dr}T^{ab},$
(4.3)

with

$\displaystyle T_{r}^{r}=U\phi^{\prime
2}+\left(\frac{e^{2\delta}\omega^{2}}{U}-\mu^{2}\right)\phi^{2},$ (4.4)

and

$\displaystyle\frac{1}{2}\frac{dg_{ab}}{dr}T^{ab}$ $\displaystyle=$ (4.5)
$\displaystyle-\frac{3U\phi^{\prime 2}}{r}$ $\displaystyle+$
$\displaystyle\left(\frac{e^{2\delta}\omega^{2}}{U}-\mu^{2}\right)\frac{3\phi^{2}}{r}+\left[U\phi^{\prime
2}+\left(\mu^{2}+\frac{e^{2\delta}\omega^{2}}{U}\right)\phi^{2}\right]\delta^{\prime}-\left(\phi^{\prime
2}+\frac{e^{2\delta}\omega^{2}\phi^{2}}{U^{2}}\right)U^{\prime}.$

However, $U^{\prime}$ and $\delta^{\prime}$ can be eliminated from the above
relation by using a suitable combination of the Einstein equations161616 The
Einstein equations implies
$\displaystyle\delta^{\prime}+\frac{4}{3}r\left(\phi^{\prime
2}+\frac{e^{2\delta}\omega^{2}\phi^{2}}{U^{2}}\right)=0,~{}~{}~{}~{}~{}[r^{2}(1-U)]^{\prime}=\frac{4}{3}r^{2}\left[U\phi^{\prime
2}+\left(\mu^{2}+\frac{e^{2\delta}\omega^{2}}{U}\right)\phi^{2}\right].$ (4.6)
, which results in the following form of Eq. (4.3)

$\displaystyle
e^{\delta}(e^{-\delta}T_{r}^{r})^{\prime}=-\frac{2e^{2\delta}w^{2}(1-U)\phi^{2}}{rU^{2}}-\frac{2(1+2U)\phi^{\prime
2}}{r}.$ (4.7)

One notices that, since the $U$-equation in (4.6) implies $U<1$, the $r.h.s.$
of the above relation is a strictly negative quantity. Therefore
$e^{-\delta}T_{r}^{r}$ is a strictly decreasing function. Moreover, the
equation (4.7) implies the following relation (where we use the fact that the
scalar field decay exponentially at infinity and thus
$T^{r}_{r}\left(\infty\right)=0$):

$T^{r}_{r}(r_{H})=e^{\delta}(r_{H})\intop_{r_{H}}^{\infty}\mathrm{d}\bar{r}\;e^{-\delta}\left(\frac{2e^{2\delta}w^{2}(1-U)\phi^{2}}{\bar{r}U^{2}}+\frac{2(1+2U)\phi^{\prime
2}}{\bar{r}}\right)\geqslant 0~{}.$ (4.8)

To analyze the near horizon limit of Eq. (4.7), one introduces a proper radial
distance $x$, which is regular at the horizon, with $dx=\frac{dr}{\sqrt{U}}$.
In terms of this coordinate, the Eq. (4.7) becomes:

$\displaystyle
e^{\delta}\frac{d(e^{-\delta}T_{r}^{r})}{dx}=-\frac{2e^{2\delta}w^{2}(1-U)\phi^{2}}{rU^{3/2}}-\frac{2\sqrt{U}}{r}(1+2U)\phi^{\prime
2}~{}.$ (4.9)

For regular solutions, the $r.h.s.$ of this equation should remain finite as
$U\rightarrow 0$. It follows that the quantity
$\frac{2e^{2\delta}w^{2}\phi^{2}}{rU^{3/2}}$ must remain finite as $r\to
r_{H}$. Thus the term $\frac{e^{2\delta}\omega^{2}}{U}\phi^{2}$ will vanish in
the same limit, which, from (4.4), implies $T_{r}^{r}(r_{H})\leqslant 0$.
However, this would contradicts the Eq. (4.8) unless $T_{r}^{r}(r_{H})=0$.
Moreover, since the integrand of the $r.h.s.$ in (4.8) has a negative sign, it
follows that $\phi=0$, $i.e.$ the absence of scalar hair.

Turning to the case of the scalar field with dependence on the coordinates on
the three-sphere, $i.e.$ the ansatz (2.21) with $s=1$, the corresponding
spinning BSs have been discussed in Hartmann:2010pm . Their most striking
property is that they do not trivialize as the maximal frequency is reached,
$\omega\to\mu$ (note that this also holds for static BSs). While in this limit
the scalar field spreads and tends to zero point-wise and the geometry becomes
arbitrarily close to that of $\mathbb{M}^{1,4}$, the BS mass (and Noether
charge/angular momentum) remains finite and nonzero in that limit, see the red
curve in Fig. 8 (left panel). This implies the existence of a gap between the
$\omega\to\mu$ limiting configurations and the $\phi=0$ (vacuum)
$\mathbb{M}^{1,4}$ ground state. This is very different from the case of a
$\mathbb{M}^{1,3}$ background, where all BS charges vanish as $\omega\to\mu$,
both in the static and in the spinning cases171717Moreover, a similar
behaviour is found for the $d=4$ static non-spherically symmetric BSs reported
in Herdeiro:2020kvf , which can be axially symmetric chains or even
configurations with no spatial isometries.. An analytical argument which helps
to understand this different behavior was presented in Hartmann:2010pm , which
we shall briefly review here. This relies on the special scaling properties of
the EKG system, which are dimension dependent. Basically, as $\omega\to\mu$,
the radial coordinate and the scalar field scale as $r=\tilde{r}/\xi$,
$\phi=\xi^{2}\tilde{\phi}$ where $\mu^{2}=\omega^{2}+\xi^{2}\hat{w}_{c}^{2}$,
with $\xi$ a small parameter and $\hat{w}_{c}$ a constant. Then, in $d=4$ the
integral $\int_{0}^{\infty}dr~{}r^{2}\phi^{2}$ (which determines the Noether
charge) vanishes as $\xi\to 0$, while the $d=5$ corresponding expression
$\int_{0}^{\infty}dr~{}r^{3}\phi^{2}$ remains finite and nonzero. The same
reasoning explains the different behavior of the scalar field mass-integral
for $d=4,~{}5.$

Figure 8:  Left: The (frequency-mass) domain of existence of hairy BHs with
$\mathbb{M}^{1,4}$ (adapted from Brihaye:2014nba ). Right: The mass, Noether
charge and tension are shown as a function of frequency for EKG static
vortices with $\mathbb{M}^{1,3}\times S^{1}$ asymptotics.

This argument also helps to partially understand the behavior we have found
above for the BSs with squashed KK asymptotics. Since the size of $S^{3}$ in
the generic ansatz (2.13) becomes proportional in the far field with $r^{2}$
only, the solutions are effectively four dimensional and thus the Noether
charge integral (or the integral for the mass stored in the field) vanishes as
$\omega\to\mu_{\rm eff}$.

Differently from the spherically symmetric case, the scalar field ansatz
(2.21) with $s=1$ allows for hairy spinning BH solutions Brihaye:2014nba ,
which are found for the same ansatz employed in this work, and also obey the
synchronization condition (3.9). As with the BH solutions in Section 3, the
hair of the asymptotically $\mathbb{M}^{1,4}$ is intrinsically non-linear,
without the existence of scalar clouds on a vacuum Myers-Perry BH background
Myers:1986un  ($i.e.$ of an existence line). Additionally, and naturally, the
asymptotically $\mathbb{M}^{1,4}$ BH solutions inherit from the solitonic
limit a gap for the mass, angular momentum and Noether charge - Fig. 8 (left
panel).

###  4.2 The $\mathbb{M}^{1,3}\times S^{1}$ case: EKG vortices and no hairy
Black Strings

To our best knowledge, the case of EKG solutions with $\mathbb{M}^{1,3}\times
S^{1}$ asymptotics (see Eq. (2.9)), has not yet been considered in the
literature. Such solutions, if they exist, describe EKG vortices and Black
Strings.

To study them, we consider the scalar ansatz (2.16) together with the
following line element

$\displaystyle
ds^{2}=e^{-a\psi(r)}\left(-\mbox{e}^{-2\delta(r)}U(r)dt^{2}+\frac{dr^{2}}{U(r)}+r^{2}d\Omega_{2}^{2}\right)+e^{2a\psi(r)}dz^{2}~{},$
(4.10)

which makes contact with the $d=4$ picture discussed in the next Section, with
$a=2/\sqrt{3}$ and an arbitrary periodicity $L$ for the $z$-coordinate. The
metric functions $\psi(r)$, $\delta(r)$, $U(r)$, and the scalar amplitude
$\phi(r)$ are solutions of the equations

$\displaystyle(e^{-\delta}r^{2}U\psi^{\prime})^{\prime}+2ae^{-a\psi-\delta}\mu^{2}r^{2}\phi^{2}=0\;,~{}~{}\delta^{\prime}+r\psi^{\prime
2}+2r\left(\phi^{\prime
2}+\frac{e^{2\delta}\omega^{2}\phi^{2}}{U^{2}}\right)=0,$
$\displaystyle[r(1-U)]^{\prime}-2r^{2}\left(U\phi^{\prime
2}+(\mu^{2}e^{-a\psi})+\frac{e^{2\delta}\omega^{2}}{U}\right)\phi^{2}-r^{2}U\psi^{\prime
2}=0,$ (4.11)
$\displaystyle(e^{-\delta}r^{2}U\phi^{\prime})^{\prime}+r^{2}\left(\frac{e^{-\delta}\omega^{2}}{U}-e^{-a\psi-\delta}\mu^{2}\right)\psi=0~{}.$

The vortices have no horizon, the size of the $S^{2}$ sector in (4.10)
shrinking to zero as $r\to 0$, while the size of the $z$-circle remains
finite, with $\psi(r)=\psi_{0}+{\cal O}(r^{2})$, $U(r)=1+{\cal O}(r^{2})$,
$\delta(r)=\delta_{0}+{\cal O}(r^{2})$ and $\phi(r)=\phi_{0}+{\cal O}(r^{2})$
close to $r=0$ (where $\psi_{0}$, $\delta_{0}$, $\phi_{0}$ are parameters
fixed by numerics). The behaviour for large-$r$ is

$\displaystyle\psi=\frac{\psi_{1}}{r}+\dots,~{}U=1+\frac{h_{1}}{r}+\dots,~{}\delta=\frac{\psi_{1}^{2}}{2r^{2}}+\dots,~{}\phi=c_{1}\frac{e^{-r\sqrt{\mu^{2}-\omega^{2}}}}{r}+\dots,~{}$
(4.12)

with the free parameters $\psi_{1}$, $h_{1}$ and $c_{1}$.

The vortices posses a nonvanishing mass, tension181818 The mass and tension
are computed following Harmark:2003dg ; Harmark:2004ch . However, a similar
result is found within the counterterm approach, with the same boundary term
(2.23). and Noether charge, with

$\displaystyle M=-\frac{h_{1}L}{2G_{5}},~{}~{}{\cal
T}=-\frac{h_{1}+3a\psi_{1}}{4G_{5}},~{}~{}Q=8\pi\omega
L\int_{0}^{\infty}dr\frac{r^{2}e^{\delta}\phi^{2}}{U}~{}.$ (4.13)

The frequency-mass diagram of these solutions is shown in Figure 8 (right
panel). The picture there strongly resembles that for $d=4$ spherically
symmetric BSs in the (pure) EKG model Liebling:2012fv . This can be understood
by noticing that, when performing a KK reduction $w.r.t.$ the $z$-direction,
these EKG vortices become $d=4$ BSs in a EKG-dilaton model - see Section 5.

As expected, no Black Strings with complex scalar hair exist in this case.
This can be shown following the same Pẽna-Sudarsky-type argument Pena:1997cy
as in the previous subsection. It is straightforward to show that the
conservation of the stress-energy tensor together with the Einstein equations
implies the following relation

$\displaystyle
e^{\delta+a\psi}(e^{-\delta-a\psi}T_{r}^{r})^{\prime}=-\frac{e^{2\delta+a\psi}\omega^{2}(1-U)\phi^{2}}{rU^{2}}-\frac{e^{a\psi}(1+3U)\phi^{\prime
2}}{r}-\mu^{2}r\phi^{2}\psi^{\prime
2}+a\mu^{2}\phi^{2}\psi^{\prime}.~{}~{}~{}~{}{~{}~{}~{}~{}}$ (4.14)

Since the $\psi$-equation in (4.11) implies that $\psi^{\prime}<0$, we
conclude that $T_{r}^{r}$ is a strictly decreasing function, with
$T_{r}^{r}(r_{H})\geqslant 0$. However, by using similar arguments as employed
above to rule out the existence of (spherically symmetric) BHs with
$\mathbb{M}^{1,4}$ asymptotics, that is, by rewriting the eq. (4.14 in terms
of the proper radial distance $dx=\frac{dr}{\sqrt{U}}$), one finds that
$T_{r}^{r}(r_{H})\leqslant 0$. This leads to a contradiction, and we conclude
that the scalar field necessarily vanishes.

The case of solutions with the $s=1$ scalar field ansatz (2.21) and
$\mathbb{M}^{1,3}\times S^{1}$ asymptotics is unclear and we leave it for
future studies.

##  5 The Kaluza-Klein reduction and the four dimensional picture

The solutions with squashed KK asymptotics in Section 3 and also the above
discussed vortices can be considered from a $d=4$ perspective, upon KK
reduction on a circle. Following Gross:1983hb , let us consider a generic KK
metric ansatz

$\displaystyle ds_{5}^{2}$
$\displaystyle=e^{-a\psi(x)}ds_{4}^{2}+e^{2a\psi(x)}(dz+2A_{i}(x)dx^{i})^{2}\,,$
(5.1) $\displaystyle\quad{\rm
with}~{}~{}ds_{4}^{2}=g_{ij}^{(4)}(x)dx^{i}dx^{j}~{}~{}{\rm
and}~{}~{}a=\frac{2}{\sqrt{3}},~{}{~{}~{}}$ (5.2)

where in this section $x^{i}$ denote $d=4$ coordinates, with time $t$ being
one of them, and $z$ is the (compact) fifth-dimension, which has some
periodicity $L$.

For the scalar field (which can be a multiplet), one only assumes that it has
a specific $z$-dependence, which disappears at the level of energy-momentum
tensor and equations of motion, a single mode being excited,

$\Psi=\Phi(x)e^{ikz},$ (5.3)

where the function $\Phi$ can be complex, and $k=2\pi m/L$ ($m=0,\pm 1,\pm
2,\dots$).

As such, the $d=5$ EKG system admits an equivalent four dimensional
description, the function $g_{zz}$ determining the dilaton $\psi$, while the
metric components $g_{iz}$ resulting in a U(1) field, with the field strength
tensor $F_{ij}=\partial_{i}A_{j}-\partial_{j}A_{i}$. That is, after
integrating over the $z-$coordinate and dropping a boundary term, the
resulting four dimensional action reads

$\displaystyle\mathcal{S}_{4}=\frac{1}{4\pi
G_{4}}\int_{\mathcal{M}}d^{4}x\sqrt{-g^{(4)}}\bigg{[}\frac{1}{4}R^{(4)}-\frac{1}{4}e^{3a\psi}F_{ij}F^{ij}-\frac{1}{2}\partial_{i}\psi\partial^{i}\psi$
(5.4)
$\displaystyle{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}-\frac{1}{2}g^{ij(4)}\left(D_{i}\Phi^{\dagger}D_{j}\Phi+D_{j}\Phi^{\dagger}D_{i}\Phi\right)-U(|\Phi|,\psi\big{)}\bigg{]}~{},$

with $G_{4}=G_{5}/L$. Observe that the $d=4$ scalar field $\Phi$ is $gauged$
$w.r.t.$ the U(1) field $A_{i}$, with the gauged derivative

$\displaystyle D_{j}\Phi=(\partial_{j}-iq_{s}A_{j})\Phi,~{}~{}$ (5.5)

the gauge coupling constant being $q_{s}=2k$ and a potential191919 For a
dilaton $\psi$ which vanishes aymptotically, the $d=4$ scalar $\Phi$ possesses
an effective mass $\mu_{\rm eff}^{2}=\mu^{2}+k^{2}.$

$\displaystyle
U(|\Phi|,\psi)=\mu^{2}|\Phi|^{2}e^{-a\psi}+k^{2}e^{-3a\psi}|\Phi|^{2},$ (5.6)

which depends on both $\Phi$ and $\psi$. This EdMgs model reduces to the
standard KK Einstein-dilaton-Maxwell (EdM) model for $\Phi=0$. For vanishing
gauge potential, $A_{i}=0$, it reduces to an Einstein-dilaton-Klein-Gordon
(EdKG) model.

###  5.1  Static, dyonic Black Holes with gauged scalar hair

Starting with the vacuum case $(\Psi=0)$, and the squashed KK asymptotics, let
us remark that, since $g_{zz}$ shrinks to zero as $r\to 0$, the dilaton
diverges there and the five-dimensional GPS soliton is singular from a four
dimensional perspective Gross:1983hb ; Sorkin:1983ns . However, the situation
is different with BHs; for example, the spinning solution in Section B results
in spherically symmetric dyonic BHs in a specific Einstein-Maxwell-dilaton
model discussed $e.g.$ in Gibbons:1985ac .

Turning now to the EKG solution in Section 3, the situation with the BSs is
similar to that of (vacuum) KK monopoles, being singular at $r=0$ in the $d=4$
picture. The spinning HBHs, on the other hand, result in a family of $d=4$
solutions of the model (5.4) which describe static spherically symmetric BHs
with resonant gauged scalar hair and a dyonic U(1) field. All properties of
these solutions follow from those of the $d=5$ EKG BHs. To make explicit this
correspondence, we have found useful to consider an (equivalent) version of
the $d=5$ line element with the following form of the last term in Eq. (3.6)

$\displaystyle e^{2F_{2}(r)}\frac{1}{H(r)}(dz+2N\cos\theta
d\varphi-4NW(r)dt)^{2},~{}~{}{\rm with}~{}~{}~{}z=2N\psi~{},$ (5.7)

such that the dilaton $\psi(r)={F_{3}(r)}/{a}$ vanishes
asymptotically202020Alternatively, one can work with the initial form in Eq.
(3.6), which implies a non-vanishing dilaton in the far field, and then
consider a rescaling of the $d=4$ line element.. The $d=4$ gauge field
describes a dyon, with $A=V(r)dt+Q_{m}\cos\theta d\varphi$, the $N$-parameter
becoming in the $d=4$ perspective the magnetic charge, $Q_{m}=N$, while the
$W(r)$-function associated with rotation determines the electric potential,
$V(r)=-2NW(r)$. The $d=4$ BH metric reads

$\displaystyle
ds_{4}^{2}=-S_{0}(r)dt^{2}+S_{1}(r)\big{[}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\varphi^{2})\big{]},~{}~{}{~{}~{}}$ (5.8) $\displaystyle{\rm
where}~{}~{}S_{0}(r)=e^{2F_{0}(r)+aF_{3}(r)}\frac{\left(1-\frac{r_{H}}{r}\right)^{4}}{\left(1+\frac{r_{H}}{r}\right)^{2}},~{}~{}S_{1}(r)=e^{2F_{1}(r)+aF_{3}(r)}H(r)\left(1+\frac{r_{H}}{r}\right)^{4},$

and $H(r)$ is given by Eq. (3.7). The expression of the $d=4$ scalar field
reads212121 Properties of this scalar ansatz (including regularity) are
discussed in a more general context in Gervalle:2022npx ; Gervalle:2022vxs .

$\Phi=\phi(r)\left(\begin{array}[]{c}~{}\sin\frac{\theta}{2}~{}e^{-i\frac{\varphi}{2}}\\\
\cos\frac{\theta}{2}~{}e^{i\frac{\varphi}{2}}\end{array}\right)e^{-i\omega
t}.$

The gauged coupling constant is fixed by the $N$-parameters, with
$q_{s}=1/(2N)$ (and $k=1/(4N)$. Then one can easily see that $d=5$
synchronization condition (1.1) translates into the $d=4$ resonance condition
(1.2), that is222222This result holds as well when considering a form of the
metric (3.6) with the coordinate $\psi$ replaced by $z$, $cf.$ Eq. (5.7).
Although in this case $\Omega_{H}=4NW({r_{H}})$, the synchronization condition
$\omega=W({r_{H}})$ still holds, since the $z$-dependence in (the new form of)
scalar field ansatz implies $m=1/(4N)$.

$\displaystyle\omega=m\Omega_{H}=W\big{|}_{r_{H}}=-\frac{1}{2N}V\big{|}_{r_{H}}=q_{s}{\cal
V},$ (5.9)

with ${\cal V}=V(\infty)-V(r_{H})$ the electrostatic chemical potential232323
Since $W(\infty)=0$, the $d=4$ BH solutions are found by fixing a (residual)
gauge freedom via $V(\infty)=0$, while Herdeiro:2020xmb  uses $V(r_{H})=0$.
The physical results are, of course, independent of the gauge choice..

There is a simple map between the all quantities of interest of the $d=5$ BHs
and those of the $d=4$ solutions, with $e.g.$

$\displaystyle
LG_{4}M^{(d=4)}=G_{5}M^{(d=5)},~{}LG_{4}Q_{e}^{(d=4)}=2G_{5}J^{(d=5)},~{}A_{H}^{(d=4)}=\frac{1}{L}A_{H}^{(d=5)},~{}T_{H}^{(d=4)}=T_{H}^{(d=5)},~{}~{}{~{}~{}~{}}$

where $Q_{e}$ is the electric charge.

Differently from the other BHs with resonant hair discussed in the literature
Hong:2019mcj ; Herdeiro:2020xmb ; Hong:2020miv ; Brihaye:2022afz , these $d=4$
solutions exist without self-interaction terms in $\Psi$, in the scalar
potential (5.6). Finally, we remark that the absence of an existence line for
the $d=5$ vacuum spinning BHs corresponds to the absence of charged scalar
bound states on the background of a dyonic BH in a KK Einstein-dilaton-Maxwell
(EdM) model, although this result should not perhaps be a surprise, given the
similar findings in Hod:2012wmy ; Hod:2013nn  for the Reissner-Nordström BH
case.

###  5.2  Other cases

The case of EKG vortices with $\mathbb{M}^{1,3}\times S^{1}$ asymptotics in
Section 4.2 is simpler, since the direct KK reduction leads to an (ungauged)
EdKG model, $i.e.$ with $A_{i}=0$ in (5.4), while the $d=4$ metric form and
dilaton are read directly from (4.10). We remark that, for the employed $s=0$
ansatz (2.16), (2.21), it is more natural to interpret the results as for a
model with a single scalar field, which has the same expression (and also
field mass $\mu$) in both $d=4,~{}5$.

$d=4$ solutions with a gauged scalar field can, nonetheless, be generated by
using the $d=5$ EKG vortices as seeds. The basic procedure is well known in
the literature - however, without also considering a complex scalar field -
and works as follows. Starting with any EKG vortex, we perform a boost in the
$(t,z)$-plane, with

$\begin{cases}t=\cosh\alpha~{}T-\sinh\alpha~{}Z\\\
z=\cosh\alpha~{}Z-\sinh\alpha~{}T\\\ \end{cases}\,,~{}~{}{\rm
where}~{}~{}\alpha\in\mathbb{R}\,.$ (5.10)

Then a KK reduction $w.r.t.$ the direction $Z$ results in the following
solution of the $d=4$ model (5.4)

$\displaystyle
ds_{4}^{2}=-\frac{e^{-2\delta(r)}N(r)}{\sqrt{S(r)}}dT^{2}+\sqrt{S(r)}\left(\frac{dr^{2}}{N(r)}+r^{2}d\Omega_{2}^{2}\right),~{}~{}A=V(r)dt,$
(5.11)

where

$\displaystyle
S(r)=\cosh^{2}\alpha-~{}e^{-\delta(r)-3a\psi(r)}N(r)\sinh^{2}\alpha,~{}~{}V(r)=\left(e^{-2\delta(r)-3a\psi(r)}N(r)-1\right)\frac{\sinh\alpha\cosh\alpha}{2S(r)}~{},$

while the $d=4$ dilaton field is $\psi(r)=\psi_{i}(r)+\frac{\log S(r)}{2a},$
with $\psi_{i}(r)$ the function which enters the seed $d=5$ solution, denoted
by $\psi$ in the metric (4.10). The $d=4$ scalar field $\Phi$ is

$\displaystyle\Phi=\phi(r)e^{-i\tilde{\omega}T},~{}~{}{\rm
with}~{}~{}\tilde{\omega}=\omega\cosh\alpha,$ (5.12)

while the gauge coupling constant is $g_{s}=2\omega\sinh\alpha$.

These solutions describe static, electrically charged, spherically symmetric
gauged BSs, which are a generalisation of the usual (uncharged) EKG BSs
Jetzer:1989av ; Jetzer:1992tog ; Pugliese:2013gsa , with an extra-dilaton
field, $cf.$ eq. (5.4). Again, it is straightforward to derive the map between
the quantities of interest of the $d=5$ and $d=4$ configurations.

Finally, we mention that the same setup can be used to construct a
generalization of the known $d=4$ spherically symmetric (ungauged) BSs by
including the effects of a background $U(1)$ magnetic field. In this case, one
starts again with the $d=5$ EKG vortices in Section 4.2; however, the
resulting $d=4$ configurations have axial symmetry only, and describe
(unguged) BSs which approach asymptotically a dilatonic Melvin Universe
background. The way to introduce a $d=4$ magnetic field in a KK setup involves
twisting the $z$-direction Dowker:1994up ; Dowker:1995gb , that is by taking
$\varphi\to\varphi+B_{0}z$ in the metric (4.10), with $B_{0}$ an arbitrary
real constant, and reidentifying points appropriately. Upon reduction, the
resulting $d=4$ solutions have a line element

$\displaystyle
ds_{4}^{2}=\sqrt{\Lambda(r,\theta)}\left(\frac{dr^{2}}{N(r)}+r^{2}d\theta^{2}-N(r)e^{-2\delta(r)}dt^{2}\right)+\frac{r^{2}\sin^{2}\theta
d\varphi^{2}}{\sqrt{\Lambda(r,\theta)}},$ (5.13)

with $\Lambda(r,\theta)=1+e^{-3a\psi(r)}B_{0}^{2}r^{2}\sin^{2}\theta$. The
U(1)-potential and the $d=4$ dilaton are

$\displaystyle
A=\frac{e^{-3a\psi(r)}B_{0}r^{2}\sin^{2}\theta}{2\Lambda(r,\theta)}d\varphi,~{}~{}\psi(r,\theta)=\psi_{i}(r)+\frac{1}{2a}\log\Lambda(r,\theta),$
(5.14)

while the scalar $\Phi$ has the same expression as $\Psi$ in five dimensions,
and remains $ungauged$.

##  6 Conclusions and remarks

The study of solitons and BH solutions in more than $d=4$ dimensions is a
subject of long standing interest in General Relativity, the case of a KK
theory with only one (compact) extra-dimension providing the simplest model.
Although the original proposal in kaluza ; Klein:1926tv  does not result in a
realistic theory of Nature, it still continues to provide insight into more
sophisticated theories, such as supergravity and string/M-theory.

In the context of this work, we were mainly interested in KK solutions of the
$d=5$ Einstein equations with a squashed sphere at infinity, the simplest case
being the vacuum soliton found by Gross and Perry Gross:1983hb  and Sorkin
Sorkin:1983ns  (GPS). As discussed by various authors, an horizon can be added
also for these asymptotics, which results in BHs with a squashed horizon of
$S^{3}$ topology Chodos:1980df ; Dobiasch:1981vh ; Pollard:1982gj ;
Gibbons:1985ac ; Wang:2006nw .

The main purpose of this paper was to extend the solutions in Gross:1983hb ;
Sorkin:1983ns ; Chodos:1980df ; Dobiasch:1981vh ; Pollard:1982gj ;
Gibbons:1985ac ; Wang:2006nw  by including a scalar field doublet, with a
special ansatz, originally proposed in Hartmann:2010pm , in the action of the
model; both solitons (BSs) and BHs were considered.

Our main results can be summarized as follows:

  * •

We have provided evidence for the existence of BHs with scalar hair, with the
same far field squashed KK asymptotics as the GPS soliton. These solutions
provide further evidence for the universality of the hair synchronization
mechanism. They satisfy the same specific condition between the scalar field
frequency and event horizon velocity known to hold for a variety of other BHs
with (complex) scalar hair, see $e.g.$ Herdeiro:2014goa -Herdeiro:2017oyt .
Moreover, similar to other cases, the BHs do not trivialize in the limit of a
vanishing horizon area, and become BS solutions with squashed KK asymptotics.

  * •

The basic properties of the considered BSs and BHs are a combination between
those of the known $d=4$ and $d=5$ solutions with Minkowski spacetime
asymptotics. For example, as with the $d=4$ case Liebling:2012fv , the global
charges of the BSs vanish as the maximal frequency is approached. On the other
hand, for BHs, there is no existence line, $i.e.$ of scalar clouds on a
vacuum, spinning BH background, a feature shared with the $d=5$ solutions in
Ref. Brihaye:2014nba .

  * •

As a new feature induced by the squashed KK asymptotics, the scalar field
possesses (in the spinning case) an effective mass $\mu_{\rm
eff}^{2}=\mu^{2}+\frac{1}{16N^{2}}$, where the second term is a geometric
contribution - $N$ is related to the size of the twisted $S^{1}$ fiber at
infinity. The bound state condition for the scalar field frequency is
$\omega\leqslant\mu_{\rm eff}$.

  * •

These $d=5$ solutions of the EKG equations possesses an equivalent $d=4$
description. While, as with the vacuum GPS case Gross:1983hb , the solitons
corresponds to $d=4$ singular configurations, the KK reduction of the BHs
result in static spherically symmetric dyonic BHs with gauged scalar hair, in
a specific EdMgs model.

As a byproduct of this study, we have also investigated EKG solutions with
standard $\mathbb{M}^{1,3}\times S^{1}$ asymptotics and established first the
absence of static Black Strings with scalar hair. However, horizonless
solutions do exist, corresponding to EKG vortices. After boosting and
considering a KK reduction, these configurations result in spherically
symmetric, charged BSs, generalizing for an extra-dilaton field the known
gauged BSs Jetzer:1989av ; Jetzer:1992tog ; Pugliese:2013gsa . Figure 9
provides an overview of the solutions studied in this paper and their
relations.

Figure 9:  Overview of the solutions used or constructed in this paper and
their inter-relations. The vertical green dashed arrows represent KK
reduction. The sections where the solutions are discussed in this paper are
also given.

As possible avenues for future research, we mention first the more systematic
investigation of the $d=5$ squashed BHs solutions, such as the study of
geodesic motion, their lensing properties or their thermodynamics, together
with a detailed study of the resulting $d=4$ configurations. It would be
interesting to investigate solutions with the same squashed KK asymptotics,
but which rotate also in $\varphi$-direction, for the coordinates in the
metric ansatz (2.13). This would result in EKG generalization of the vacuum
BHs discussed in Rasheed:1995zv ; Matos:1996km ; Larsen:1999pp ; their study,
however, requires solving a set of partial differential equations. Moreover,
the KK reduction of these HBHs would result in a generalization of the $d=4$
Kerr-Newman BHs with scalar hair studied in Delgado:2016jxq , with an extra-
dilaton field in the action and also with a nonzero magnetic charge.

Finally, we mention the case of KK dipolar asymptotics instead of monopolar,
as in this work. In the vacuum case, such a configuration has been considered
in Gross:1983hb , being again of the form (2.4), with $ds_{4}^{2}$ there
corresponding to the Kerr instanton metric. We anticipate the existence of
similar configurations in an EKG model with a single complex scalar field.
Differently from the vacuum case, which necessarily possess a ’bolt’ - $i.e.$
a minimal nonzero size of the $S^{2}$ part in $ds_{4}^{2}$ metric -, the EKG
system would allow also also for ’nutty’ solitons - that is, with the $S^{3}$
part in the $ds_{4}^{2}$ metric shrinking to zero at $r=0$, as with the BSs in
this work. We hope to return elsewhere with a study of these aspects.

###### Acknowledgements.

C.H., J.N. and E.R. would like to express their gratitude to Juan Carlos
Degollado for being an exceptional host during a visit at at the Instituto de
Ciencias Físicas, UNAM - Campus de Morelos, Mexico, where a part of this
project has been developed. This work is supported by the Center for Research
and Development in Mathematics and Applications (CIDMA) through the Portuguese
Foundation for Science and Technology (FCT – Fundação para a Ciência e a
Tecnologia), references UIDB/04106/2020 and UIDP/04106/2020. The authors
acknowledge support from the projects PTDC/FIS-AST/3041/2020, as well as
CERN/FIS-PAR/0024/2021 and 2022.04560.PTDC. This work has further been
supported by the European Union’s Horizon 2020 research and innovation (RISE)
programme H2020-MSCA-RISE-2017 Grant No. FunFiCO-777740 and by the European
Horizon Europe staff exchange (SE) programme HORIZON-MSCA-2021-SE-01 Grant No.
NewFunFiCO-101086251. J. N. is supported by the FCT grant 2021.06539.BD.

##  Appendix A The Einstein and energy-momentum tensors

To get an idea about the equations solved in practice, we display here the
expression of the non-vanishing components of the Einstein tensor
$E_{a}^{b}=R_{a}^{b}-\frac{1}{2}R\delta_{a}^{b}$ and of energy-momentum tensor
$T_{a}^{b}$ for the generic metric ansatz (2.13) and the scalar ansatz (2.16),
(2.21). For the Einstein tensor, one finds

$\displaystyle E^{t}_{t}$ $\displaystyle=\frac{1}{{\cal
F}_{2}}\left(\frac{{\cal F}_{3}}{4{\cal F}_{2}}-1\right)+\frac{1}{{\cal
F}_{1}}\left(\frac{{\cal F}_{2}^{\prime}{}^{2}}{4{\cal F}_{2}^{2}}+\frac{{\cal
F}_{3}^{\prime}{}^{2}}{4{\cal F}_{3}^{2}}+\frac{{\cal F}_{1}^{\prime}{\cal
F}_{2}^{\prime}}{{\cal F}_{1}F_{2}}-\frac{{\cal F}_{3}^{\prime}{\cal
F}_{2}^{\prime}}{{\cal F}_{2}F_{3}}+\frac{{\cal F}_{1}^{\prime}{\cal
F}_{3}^{\prime}}{2{\cal F}_{1}{\cal F}_{3}}\right)$
$\displaystyle-\frac{1}{{\cal F}_{1}}\left(\frac{2{\cal
F}_{3}WW^{\prime\prime}}{{\cal F}_{0}}+\frac{{\cal
F}_{2}^{\prime\prime}}{{\cal F}_{2}}+\frac{{\cal F}_{3}^{\prime\prime}}{2{\cal
F}_{3}}+\frac{{\cal F}_{3}W^{\prime 2}}{{\cal F}_{0}}\right)-\frac{{\cal
F}_{3}WW^{\prime}}{{\cal F}_{0}{\cal F}_{1}}\left(\frac{{\cal
F}_{0}^{\prime}}{{\cal F}_{0}}+\frac{{\cal F}_{1}^{\prime}}{{\cal
F}_{1}}-\frac{2{\cal F}_{2}^{\prime}}{{\cal F}_{2}}-\frac{3{\cal
F}_{3}^{\prime}}{{\cal F}_{3}}\right)\,,$ $\displaystyle E_{r}^{r}$
$\displaystyle=\frac{{\cal F}_{3}W^{\prime 2}}{{\cal F}_{0}{\cal
F}_{1}}+\frac{1}{2{\cal F}_{1}}\left(\frac{{\cal F}_{2}^{\prime 2}}{2{\cal
F}_{2}^{2}}+\frac{{\cal F}_{0}^{\prime}{\cal F}_{2}^{\prime}}{{\cal
F}_{0}{\cal F}_{2}}+\frac{{\cal F}_{3}^{\prime}{\cal F}_{2}^{\prime}}{{\cal
F}_{2}{\cal F}_{3}}+\frac{{\cal F}_{0}^{\prime}{\cal F}_{3}^{\prime}}{2{\cal
F}_{0}{\cal F}_{3}}\right)+\frac{1}{{\cal F}_{2}}\left(\frac{F_{3}}{4{\cal
F}_{2}}-1\right)\,,$ (A.1) $\displaystyle E_{\theta}^{\theta}$
$\displaystyle=E_{\varphi}^{\varphi}=(\frac{{\cal F}_{0}^{\prime\prime}}{{\cal
F}_{0}}+\frac{{\cal F}_{2}^{\prime\prime}}{{\cal F}_{2}}+\frac{{\cal
F}_{3}^{\prime\prime}}{{\cal F}_{3}})\frac{1}{2{\cal F}_{1}}-(\frac{{\cal
F}_{0}^{\prime 2}}{{\cal F}_{0}^{2}}+\frac{{\cal F}_{2}^{\prime 2}}{{\cal
F}_{2}^{2}}+\frac{{\cal F}_{3}^{\prime 2}}{{\cal F}_{3}^{2}})\frac{1}{4{\cal
F}_{1}}-\frac{{\cal F}_{3}W^{\prime 2}}{{\cal F}_{0}{\cal F}_{1}}-\frac{{\cal
F}_{3}}{4{\cal F}_{2}^{2}}$ $\displaystyle\qquad\qquad+(-\frac{{\cal
F}_{0}^{\prime}{\cal F}_{1}^{\prime}}{{\cal F}_{0}{\cal F}_{1}}+\frac{{\cal
F}_{0}^{\prime}{\cal F}_{2}^{\prime}}{{\cal F}_{0}{\cal F}_{2}}+\frac{{\cal
F}_{0}^{\prime}{\cal F}_{3}^{\prime}}{{\cal F}_{0}{\cal F}_{3}}-\frac{{\cal
F}_{1}^{\prime}{\cal F}_{2}^{\prime}}{{\cal F}_{1}{\cal F}_{2}}-\frac{{\cal
F}_{1}^{\prime}{\cal F}_{3}^{\prime}}{{\cal F}_{1}{\cal F}_{3}}+-\frac{{\cal
F}_{2}^{\prime}{\cal F}_{3}^{\prime}}{{\cal F}_{2}{\cal
F}_{3}})\frac{1}{4{\cal F}_{1}},$ $\displaystyle E^{\psi}_{\psi}$
$\displaystyle=\frac{1}{{\cal F}_{1}}\left(-\frac{2{\cal
F}_{3}WW^{\prime\prime}}{{\cal F}_{0}}+\frac{{\cal
F}_{0}^{\prime\prime}}{2{\cal F}_{0}}+\frac{{\cal F}_{2}^{\prime\prime}}{{\cal
F}_{2}}\right)-\frac{1}{{\cal F}_{1}}\left(\frac{3{\cal F}_{3}W^{\prime
2}}{{\cal F}_{0}}+\frac{{\cal F}_{0}^{\prime}{}^{2}}{4{\cal
F}_{0}^{2}}+\frac{{\cal F}_{2}^{\prime}{}^{2}}{4{\cal F}_{2}^{2}}\right)$
$\displaystyle\qquad+\frac{{\cal F}_{3}WW^{\prime}}{{\cal F}_{0}{\cal
F}_{1}}\left(\frac{{\cal F}_{0}^{\prime}}{{\cal F}_{0}}+\frac{{\cal
F}_{1}^{\prime}}{{\cal F}_{1}}-\frac{2{\cal F}_{2}^{\prime}}{{\cal
F}_{2}}-\frac{3{\cal F}_{3}^{\prime}}{{\cal F}_{3}}\right)-\frac{1}{2{\cal
F}_{1}}\left(\frac{{\cal F}_{0}^{\prime}{\cal F}_{1}^{\prime}}{2{\cal
F}_{0}{\cal F}_{1}}+\frac{{\cal F}_{2}^{\prime}{\cal F}_{1}^{\prime}}{{\cal
F}_{1}{\cal F}_{2}}-\frac{{\cal F}_{0}^{\prime}{\cal F}_{2}^{\prime}}{{\cal
F}_{0}{\cal F}_{2}}\right)$ $\displaystyle\qquad+\frac{1}{{\cal
F}_{2}}\left(\frac{3{\cal F}_{3}}{4{\cal F}_{2}}-1\right),$ $\displaystyle
E^{t}_{\psi}$
$\displaystyle=\frac{1}{\cos\theta}E^{t}_{\varphi}=\frac{W^{\prime}}{{\cal
F}_{1}}\left(\frac{{\cal F}_{3}{\cal F}_{0}^{\prime}}{2{\cal
F}_{0}^{2}}+\frac{{\cal F}_{3}{\cal F}_{1}^{\prime}}{2{\cal F}_{0}{\cal
F}_{1}}-\frac{{\cal F}_{3}{\cal F}_{2}^{\prime}}{{\cal F}_{0}{\cal
F}_{2}}-\frac{3{\cal F}_{3}^{\prime}}{2{\cal F}_{0}}\right)-\frac{{\cal
F}_{3}W^{\prime\prime}}{{\cal F}_{0}{\cal F}_{1}},$ $\displaystyle
E_{\varphi}^{\psi}$ $\displaystyle=\cos\theta\left[\frac{1}{2{\cal
F}_{1}}\left(-\frac{4{\cal F}_{3}WW^{\prime\prime}}{{\cal F}_{0}}+\frac{{\cal
F}_{2}^{\prime\prime}}{F_{2}}-\frac{{\cal F}_{3}^{\prime\prime}}{{\cal
F}_{3}}\right)+\frac{1}{{\cal F}_{2}}\left(\frac{{\cal F}_{3}}{{\cal
F}_{2}}-1\right)\right.$ $\displaystyle\qquad\qquad\qquad+\frac{{\cal
F}_{3}W^{\prime}}{{\cal F}_{1}}\left(\frac{W}{{\cal F}_{0}}\left(\frac{{\cal
F}_{0}^{\prime}}{{\cal F}_{0}}+\frac{{\cal F}_{1}^{\prime}}{{\cal
F}_{1}}-\frac{2{\cal F}_{2}^{\prime}}{{\cal F}_{2}}-\frac{3{\cal
F}_{3}^{\prime}}{{\cal F}_{3}}\right)-\frac{2W^{\prime}}{{\cal F}_{0}}\right)$
$\displaystyle\left.\qquad\qquad\qquad+\frac{1}{4{\cal
F}_{1}}\left(\frac{{\cal F}_{3}^{\prime}{}^{2}}{{\cal F}_{3}^{2}}-\frac{{\cal
F}_{0}^{\prime}{\cal F}_{3}^{\prime}}{{\cal F}_{0}{\cal F}_{3}}+\frac{{\cal
F}_{1}^{\prime}{\cal F}_{3}^{\prime}}{{\cal F}_{1}{\cal F}_{3}}-\frac{{\cal
F}_{2}^{\prime}{\cal F}_{3}^{\prime}}{{\cal F}_{2}{\cal F}_{3}}+\frac{{\cal
F}_{0}^{\prime}{\cal F}_{2}^{\prime}}{{\cal F}_{0}{\cal F}_{2}}-\frac{{\cal
F}_{1}^{\prime}{\cal F}_{2}^{\prime}}{{\cal F}_{1}{\cal
F}_{2}}\right)\right],$

while the non-vanishing components of the energy momentum tensor are

$\displaystyle T^{t}_{t}=-\mu^{2}\phi^{2}-\frac{\omega^{2}-sW^{2}}{{\cal
F}_{0}}\phi^{2}-\frac{s}{4}\left(\frac{1}{{\cal F}_{3}}+\frac{2}{{\cal
F}_{2}}\right)\phi^{2}-\frac{\phi^{\prime 2}}{F_{1}},$ $\displaystyle
T_{r}^{r}=\frac{\phi^{\prime 2}}{{\cal F}_{1}}-s\frac{1}{2}(\frac{1}{{\cal
F}_{2}}+\frac{1}{2{\cal F}_{3}})\phi^{2}+\frac{(\omega-sW)^{2}}{{\cal
F}_{0}}\phi^{2}-\mu^{2}\phi^{2},$ $\displaystyle
T_{\theta}^{\theta}=T_{\varphi}^{\varphi}=-\frac{\phi^{\prime 2}}{{\cal
F}_{1}}-s\frac{1}{4{\cal F}_{3}}\phi^{2}+\frac{(\omega-sW)^{2}}{{\cal
F}_{0}}\phi^{2}-\mu^{2}\phi^{2},$ $\displaystyle
T_{\psi}^{\psi}=\frac{\left(\omega^{2}-pW^{2}\right)}{{\cal
F}_{0}}\phi^{2}-\mu^{2}\phi^{2}+\frac{s}{4}\phi^{2}\left(\frac{1}{{\cal
F}_{3}}-\frac{2}{{\cal F}_{2}}\right)-\frac{\phi^{\prime 2}}{F_{1}},$
$\displaystyle T^{t}_{\psi}=s\frac{(\omega-W)}{{\cal
F}_{0}}\phi^{2},~{}~{}T^{t}_{\varphi}=T^{t}_{\psi}\cos\theta,$ (A.2)
$\displaystyle T^{\psi}_{\varphi}=s\left(\frac{2W(\omega-W)}{{\cal
F}_{0}}-\frac{1}{2{\cal F}_{2}}+\frac{1}{2{\cal
F}_{3}}\right)\cos\theta\;\phi^{2},$

where $s=0,1$ – $cf.$ the scalar ansatz (2.16), (2.21).

In the numerics, we choose a metric gauge with ${\cal F}_{2}=\lambda{\cal
F}_{1}r^{2}$ with $\lambda=1/4,1$ for solitons and BHs, respectively. Then
$E_{\theta}^{\theta}$ is a linear combination of $E_{\psi}^{\psi}$ and
$E_{\varphi}^{\psi}$ (and also for the corresponding $T_{a}^{b}$) and we are
left with five Einstein equations for four metric functions. However, the
$(r,r)$-Einstein equation is treated as a constraint, being satisfied once the
remaining equations are zero. Therefore we conclude that the considered ansatz
is consistent.

For completeness, we include here also the general equation satisfied by the
scalar amplitude:

$\displaystyle\phi^{\prime\prime}$ $\displaystyle+\frac{1}{2}\left(\frac{{\cal
F}_{0}^{\prime}}{{\cal F}_{0}}-\frac{{\cal F}_{1}^{\prime}}{{\cal
F}_{1}}+\frac{2{\cal F}_{2}^{\prime}}{{\cal F}_{2}}+\frac{{\cal
F}_{3}^{\prime}}{{\cal F}_{3}}\right)\phi^{\prime}$ (A.3)
$\displaystyle+\left[\frac{(\omega-sW)^{2}}{{\cal
F}_{0}}-\mu^{2}-\frac{s}{2}\left(\frac{1}{{\cal F}_{2}}+\frac{1}{2{\cal
F}_{3}}\right)\right]{\cal F}_{1}\phi=0.$

##  Appendix B The vacuum Black Hole solution

###  B.1 The static Black Hole

The GPS solution allows for BH generalizations Chodos:1980df ; Dobiasch:1981vh
; Pollard:1982gj ; Gibbons:1985ac ; Wang:2006nw . The static case has a
particularly simple form, with

$ds=-\bigg{(}1-\frac{r_{h}}{r}\bigg{)}dt^{2}+\left(1+\frac{2\bar{N}}{r}\right)\left[\frac{dr^{2}}{1-\frac{r_{h}}{r}}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\varphi^{2})\right]+\frac{4N^{2}}{1+\frac{2\bar{N}}{r}}(d\psi+\cos\theta
d\varphi)^{2}$

with $\bar{N}=\sqrt{N^{2}+\frac{1}{16}r_{h}^{2}}-\frac{1}{4}r_{h}.$ The
coordinate transformation (with $r_{h}=4r_{H})$

$\displaystyle r\to r\left(1+\frac{r_{H}}{r}\right)^{2}\ ,$ (B.1)

leads to the following equivalent form of (B.1) in isotropic coordinates,
which was used as the background for the non-extremal HBH solutions reported
in Section 3.3:

$ds^{2}=-\frac{\left(1-\frac{r_{H}}{r}\right)^{4}}{\left(1+\frac{r_{H}}{r}\right)^{2}}dt^{2}+\left(1+\frac{r_{H}}{r}\right)^{4}\bigg{[}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\varphi^{2})\bigg{]}+\frac{4N^{2}}{H(r)}(d\psi+\cos\theta d\varphi)^{2}\
,~{}~{}{~{}~{}~{}}$ (B.2)

with the function $H(r)$ given by (3.7) and $r_{H}>0$ the event horizon
radius. Note that the Schwarzschild Black String is recovered as $N\to 0$,
while $r_{H}=0$ results in the GPS solution with the form (2.7) of
$ds_{4}^{2}$ part of the metric. The corresponding expressions of various
quantities of interest results straightforwardly from those of the spinning
BHs displayed below.

###  B.2 The rotating Black Hole

####  B.2.1 The general case

The rotating generalization of the static line element (B.2) has been derived
in Dobiasch:1981vh ; Wang:2006nw . It can be written in the generic form
(2.13), where we have found useful to use a form of the metric functions with

$\displaystyle{\cal
F}_{0}=\frac{(1+u^{2}(1-y))}{y}\frac{1-\frac{(1-y)P_{1}(r)}{P_{2}^{2}(r)}}{1+\frac{u^{2}(1-y)P_{1}^{2}(r)}{P_{2}^{2}(r)}}\,,\qquad{\cal
F}_{1}=\frac{P_{1}(r)}{P_{2}(r)}\frac{y}{1-(1-y)\frac{P_{1}(r)}{P_{2}^{2}(r)}},~{}~{}$
$\displaystyle{\cal F}_{2}=r^{2}P_{1}(r)P_{2}(r)\,,\quad{\cal
F}_{3}=\frac{r_{0}^{2}\left(1+u^{2}(1-y)\frac{P_{1}^{2}(r)}{P_{2}^{2}(r)}\right)P_{2}(r)}{(1+u^{2})^{2}yP_{1}(r)}\,,$
(B.3) $\displaystyle
W(r)=\frac{(1+u^{2})^{3/2}(y-1)u}{2r_{0}\sqrt{1+(1-y)u^{2}}}\frac{1-\frac{P_{2}^{2}(r)}{P_{1}^{2}(r)}}{(y-1)u^{2}-\frac{P_{2}^{2}(r)}{P_{1}^{2}(r)}},$

where

$\displaystyle
P_{1}(r)=1+\frac{r_{0}}{r},~{}~{}P_{2}(r)=1+\frac{r_{0}}{r}\frac{u^{2}}{1+u^{2}},~{}~{}$
(B.4)

$\\{r_{0},u,y\\}$ being three parameters. The static limit is recovered for
$u=0$, resulting in the metric form (B.1) (with $r_{h}=r_{0}(1-y)/y$ and
$N=r_{0}/(2\sqrt{y})$).

Returning to the spinning case, we notice first the absence of a rotating
generalization of the horizonless (vacuum) GPS soliton242424 Note, however,
the existence of such spinning solitons in a model with a $U(1)$ field
Tomizawa:2008rh .. To better understand the BH properties, we express the
$y$-parameter in terms of the horizon radius $r_{H}$, with

$\displaystyle
y=\frac{r_{0}(r_{H}-(r_{0}+r_{H})u^{4})}{r_{H}(r_{0}+r_{H})(1+u^{2})^{2}}~{},$
(B.5)

and define

$\displaystyle r_{0}=sr_{H}~{}.$ (B.6)

As such, the input parameters become $u,s$ and $r_{H}>0$, with $0\leqslant
u\leqslant 1$ and $1-(1+s)u^{2}\geqslant 0$.

One can easily verify that metric has the right asymptotics, with $W\to 0$ as
$r\to\infty$, while the following relation holds between the horizon radius
and the NUT-parameter:

$\displaystyle
r_{H}=2N\left(1+u^{2}\right)\sqrt{\frac{1-(1+s)u^{4}}{s(1+s+(3+2s)u^{2}+3(1+s)u^{4}+(1+s)^{2}u^{6})}}~{}.$
(B.7)

The computation of various quantities which enter the thermodynamic
description of this rotating BH solution is straightforward, with

$\displaystyle M=$ $\displaystyle\frac{4\pi
N^{2}}{G_{5}}\frac{\left(2+s+2(1+s)u^{2}\right)\left(1+2u^{2}+(1+s)u^{4}\right)}{\left(1+s+(3+2s)u^{2}+3(1+s)u^{4}+(1+s)^{2}u^{6}\right)^{3/2}\sqrt{s(1-(1+s)u^{4})}\
}$
$\displaystyle\times\left(1+s+(1+2s)u^{2}-(1+s)u^{4}-(1+s)^{2}u^{6}\right)\,,$
$\displaystyle J=$ $\displaystyle\frac{16\pi
N^{3}}{G_{5}}\sqrt{1+s}u(1+(1+s)u^{2})^{2}$
$\displaystyle\times\left(\frac{1+u^{2}}{1+s+(3+2s)u^{2}+3(1+s)u^{4}+(1+s)^{2}u^{6})}\right)^{3/2},$
$\displaystyle{\cal T}=$
$\displaystyle\frac{N}{2G_{5}}\frac{\left(1+s+(1+2s)u^{2}-(1+s)u^{4}-(1+s)^{2}u^{6}\right)}{\sqrt{s(1-(1+s)u^{4})}(1+s+(3+2s)u^{2}+3(1+s)u^{4}+(1+s)^{2}u^{4})^{3/2}}$
$\displaystyle\times\left(s^{2}u^{4}(u^{2}-1)+(1+u^{2})^{3}+2s(1+u^{2})(1+u^{4})\right)\,,$
$\displaystyle A_{H}=$ $\displaystyle
128\pi^{2}N^{3}(1+\frac{1}{s})(1+(1+s)u^{2})^{2}(1-(1+s)u^{4})$ (B.8)
$\displaystyle\times\left(\frac{1+u^{2}}{1+s+(3+2s)u^{2}+3(1+s)u^{4}+(1+s)^{2}u^{6})}\right)^{3/2}\,,$
$\displaystyle T_{H}=$ $\displaystyle\frac{1}{8\pi
N}\frac{s(1-(1+s)u^{2})(1+s+(3+2s)u^{2}+3(1+s)u^{4}+(1+s)^{2}u^{6})}{(1+s)\sqrt{s(1+u^{2})}(1+(1+s)u^{2})(1-(1+s)u^{4})^{3/2}}\,,$
$\displaystyle\Omega_{H}=$
$\displaystyle\frac{1}{2N}\frac{u(2+s+2(1+s)u^{2}}{1+(1+s)u^{2}}\frac{\sqrt{s}}{\sqrt{(1+s)(1+u^{2})(1-(1+s)u^{4})}}\,.$

This solution has a variety of interesting properties, some of them different
from the case of asymptotically $\mathbb{M}^{1,4}$ Myers-Perry BHs (with the
same symmetries). Here we mention only the existence, for a given value of the
$N$, of an upper bound of the spinning parameter $J$, with $J_{\rm
max}=\frac{8\pi N^{3}}{G_{5}}$. For $J=J_{\rm max}$, the mass can take an
arbitrary value $M>M_{c}=\frac{8\pi\sqrt{2}N^{2}}{G_{5}}$, see Figure 7 (left
panel). Note that the solution with $M=M_{c}$, $J=J_{\rm max}$ corresponds to
an extremal BH. Also, in the context of this work, it is interesting to
consider the issue of solutions with constant $\Omega_{H}$. These sequences
start at the horizonless GPS soliton limit; however, their end point can be
different. For $0<\Omega_{H}\leqslant 1/(\sqrt{2}N)$ they reach an extremal BH
solution ($T_{H}=0$) - see the inset in Figure 7 (left panel); the extremal BH
is marked there with a red dot. The situation is different for larger angular
velocities, and a sequence of BHs at constant $\Omega_{H}$ ends on the set of
critical solutions with $J=J_{\rm max}$ and $M>M_{c}$.

####  B.2.2 The extremal limit and an exact solution of the Klein-Gordon
equation

The extremal BH limit ($T_{H}=0$) is found for

$\displaystyle s=\frac{1}{u^{2}}-1~{},$ (B.9)

in which case the solution takes a much simpler form. One finds

$\displaystyle{\cal
F}_{1}=\frac{\left(1-\left(1-\frac{1}{u^{2}}\right)\frac{r_{H}}{r}\right)\left(1+\frac{1-u^{2}}{1+u^{2}}\frac{r_{H}}{r}\right)}{\left(1-\frac{r_{H}}{r}\right)^{2}},~{}~{}{\cal
F}_{2}=r^{2}\left(1-\left(1-\frac{1}{u^{2}}\right)\frac{r_{H}}{r}\right)\left(1+\frac{1-u^{2}}{1+u^{2}}\frac{r_{H}}{r}\right)),$
$\displaystyle{\cal
F}_{3}=\frac{r_{H}^{2}\left(1+\frac{2r_{H}}{r}+\frac{5r_{H}^{2}}{r^{2}}+2\left(1-\frac{r_{H}}{r}\right)\left(1+\frac{5r_{H}}{r}\right)u^{2}+5\left(1-\frac{r_{H}}{r}\right)^{2}u^{4}\right)}{u^{4}\left(1+u^{2}\right)^{2}\left(1+\frac{1-u^{2}}{1+u^{2}}\frac{r_{H}}{r}\right)\left(1-(1-\frac{1}{u^{2}}\right)\frac{r_{H}}{r})},$
(B.10) $\displaystyle
W=\frac{2u}{r}\sqrt{\frac{1+u^{2}}{1+2u^{2}+5u^{4}}}\frac{2u^{2}\left(1+u^{2}\right)+\left(1-u^{2}\right)\left(1+2u^{2}\right)\frac{r_{H}}{r}}{1+\frac{2r_{H}}{r}+\frac{5r_{H}^{2}}{r^{2}}+2\left(1-\frac{r_{H}}{r}\right)\left(1+\frac{5r_{H}}{r}\right)u^{2}+5\left(1-\frac{r_{H}}{r}\right)^{2}u^{4}}~{},$

for the metric functions, with the main quantities of interest are

$\displaystyle M=\frac{4\pi
N^{2}}{G_{5}}\frac{\left(1+3u^{2}\right)^{3}}{\left(1_{2}u^{2}+5u^{4}\right)^{3/2}}\,,~{}~{}~{}J=\frac{64\pi
N^{3}u^{3}}{G_{5}}\left(\frac{1+u^{2}}{1+2u^{2}+5u^{4}}\right)^{3/2}\,,$
$\displaystyle
A_{H}=512\pi^{2}N^{2}u^{3}\left(\frac{1+u^{2}}{1+2u^{2}+5u^{4}}\right)^{3/2},~{}~{}~{}\Omega_{H}=\frac{1}{4N}\frac{1+3u^{2}}{u\sqrt{1+u^{2}}},$
(B.11) $\displaystyle{\cal
T}=\frac{N}{G_{5}}\frac{\left(1+3u^{2}\right)\left(1+3u^{4}\right)}{\left(1+2u^{2}+5u^{4}\right)^{3/2}}.$

The Klein-Gordon equation (A.3) (with $s=1$) takes a relatively simple form
for the above (extremal) background

$\displaystyle\left((r-r_{H})^{2}\phi^{\prime}\right)^{\prime}-\left(s_{2}(r-r_{H})^{2}+s_{1}(r-r_{H})+s_{0}\right)\phi=0$
(B.12)

where we note

$\displaystyle
s_{0}=\frac{3}{8}+\frac{2\mu^{2}r_{H}^{2}}{u^{2}(1+u^{2})}\,,\quad
s_{1}=\left(\frac{\mu^{2}r_{H}}{u^{2}(1+u^{2})}-\frac{1}{16r_{H}}\right)\left(1+3u^{2}\right)\,,~{}~{}s_{2}=\mu^{2}-\frac{u^{2}\left(1+u^{2}\right)}{16r_{H}^{2}}\,.$

The general solution of the above equation reads

$\displaystyle\phi(r)=$
$\displaystyle\mbox{e}^{-\sqrt{s_{2}}(r-r_{H})}(r-r_{H})^{\frac{1}{2}\left(\sqrt{1+4s_{0}}-1\right)}$
$\displaystyle\times\bigg{[}c_{1}L_{-\kappa}^{\sqrt{4s_{0}+1}}\left(2\sqrt{s_{2}}(r-r_{H})\right)$
$\displaystyle\quad+c_{2}U\left(\kappa,1+\sqrt{1+4s_{0}},2\sqrt{s_{2}}(r-r_{H})\right)\bigg{]}$
(B.13)

where $U$ and $L$ are the confluent hypergeometric function and the
generalized Laguerre polynomial, respectively, and
$\kappa=\frac{1}{2}\left(1+\sqrt{1+4s_{0}}+\frac{s_{1}}{\sqrt{s_{2}}}\right)$
($c_{1},c_{2}$ arbitrary constants). This solution diverges either at spatial
infinity or at the horizon.

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