aainstitutetext: Sede Esmeralda, Universidad de Tarapacá, Avenida Luis Emilio
Recabarren 2477, Iquique, Chilebbinstitutetext: Institute of Theoretical
Physics, Faculty of Mathematics and Physics, Charles University, V
Holešovičkách 2, 180 00 Prague 8, Czech Republicccinstitutetext: Instituto de
Matemática, Universidad de Talca, Casilla 747, Talca, Chileddinstitutetext:
Departamento de Física, Universidad de Concepción, Casilla, 160-C, Concepción,
Chile

# Revisiting Buchdahl transformations: New static and rotating black holes in
vacuum, double copy, and hairy extensions

José Barrientos a,b Adolfo Cisterna c Mokhtar Hassaine d Julio Oliva
<EMAIL_ADDRESS><EMAIL_ADDRESS>hassaine@inst-
mat.utalca.cl<EMAIL_ADDRESS>
###### Abstract

This paper investigates Buchdahl transformations within the framework of
Einstein and Einstein-Scalar theories. Specifically, we establish that the
recently proposed Schwarzschild-Levi-Civita spacetime can be obtained by means
of a Buchdahl transformation of the Schwarschild metric along the spacelike
Killing vector. The study extends Buchdahl’s original theorem by combining it
with the Kerr-Schild representation. In doing so, we construct new vacuum-
rotating black holes in higher dimensions which can be viewed as the Levi-
Civita extensions of the Myers-Perry geometries. Furthermore, it demonstrates
that the double copy scheme within these new generated geometries still holds,
providing an example of an algebraically general double copy framework. In the
context of the Einstein-Scalar system, the paper extends the corresponding
Buchdahl theorem to scenarios where a static vacuum seed configuration,
transformed with respect to a spacelike Killing vector, generates a hairy
black hole spacetime. We analyze the geometrical features of these spacetimes
and investigate how a change of frame, via conformal transformations, leads to
a new family of black hole spacetimes within the Einstein-Conformal-Scalar
system.

##  1 Introduction

Einstein field equations represent a set of second-order, nonlinear coupled
partial differential equations, of a mixed type since the momentum and
Hamiltonian constraints are of elliptic type, while the evolution of the
initial data is given by a hyperbolic system. Due to this inherent complexity,
direct brute force integration of these equations proves challenging, with
successful integration typically limited to cases exhibiting a high degree of
symmetry. Consequently, the pursuit of exact solutions necessitates the
utilization of specialized techniques, with Lie point symmetries serving as a
prominent approach Stephani:2003tm . These symmetries refer to a set of local
transformations that consistently map each solution of the system to another
solution within the same system. Hence, possessing a solution to the field
equations and identifying a Lie point symmetry within the framework enables
the discovery of additional solutions to the same equations. Although this
seems to be a sort of recipe, there is a highly nontrivial effort behind the
finding of a new Lie point symmetry, as they are usually hidden in the tensor
language. Moreover, recognizing and accurately assessing the applicability of
a particular Lie point symmetry is of paramount importance, as it may occur
that the entirety of the symmetry’s potential may go unexplored due to a lack
of physical intuition. In the realm of the electro-vacuum, the pioneering
contributions of Ernst Ernst:1967wx ; Ernst:1967by , Geroch Geroch:1970nt ,
and Kinnersley Kinn1 , building upon earlier works by Ehlers Ehlers:1957zz ;
Ehlers:1959aug , and Harrison harrison1968new , have played a pivotal role in
advancing our understanding of exact solutions to the Einstein-Maxwell
equations. Particularly significant is the Ernst scheme, which formulates the
Einstein-Maxwell field equations in a space characterized by two complex
potentials, one gravitational and the other electromagnetic, that enormously
facilitates the detection of certain Lie point symmetries otherwise hidden in
the Einstein-Maxwell theory, the so-called Ehlers and Harrison transformations
Ehlers:1957zz ; Ehlers:1959aug ; harrison1968new .

In this study, our objective is to reexamine and extend the discussion on
arguably two of the earliest Lie point symmetries pertinent to the Einstein
field equations. Specifically, we focus on the transformations introduced by
Buchdahl Buchdahl:1956zz ; Buchdahl:1959nk  and Janis-Robinson-Winicour (JRW)
Janis:1969ivo , which represent straightforward symmetries within the contexts
of the Einstein vacuum, the Einstein-Scalar, and the Einstein-Scalar-Maxwell
theories. These transformations facilitate the generation of new solutions
from existing static vacuum solutions of the Einstein equations and have been
very useful in the identification of novel spacetimes. Buchdahl
transformations are categorized into two types: first Buchdahl:1956zz  and
second kind Buchdahl:1959nk .111The categorization we have introduced here is
new and serves the purpose of distinguishing between different types of
transformations. The former type converts any static vacuum solutions of the
Einstein equations into a new “reciprocal” static vacuum solution, while the
latter type transforms any static vacuum solution into a static solution
augmented with a nontrivial scalar field, specifically, a massless scalar
field relevant to the Einstein-Scalar system. On the other hand, the JRW
transformation Janis:1969ivo  generates an electrically charged solution from
a known Buchdahl scalar solution, maintaining the scalar profile from the
uncharged seed. A Buchdahl transformation of the first kind may alter
Minkowski spacetime, yielding a reciprocal spacetime that may not correspond
to Minkowski spacetime in disguise. However, when applied to the Schwarzschild
spacetime, it reproduces the Schwarzschild geometry in a different gauge, a
result consistent with Birkhoff’s theorem. In fact, Buchdahl transformations
of the first kind (as well as Buchdahl transformations of the second kind), as
typically employed, do not alter the spherical symmetry of the reciprocal
solution, and in consequence, the latter cannot depart from the Schwarzschild
spacetime. Conversely, Buchdahl transformations of the second kind
Buchdahl:1959nk  are recognized for their inability to introduce a scalar
field onto Minkowski spacetime, but they do so onto the Schwarzschild
geometry, producing the well-known Fisher-Janis-Newman-Winicour (FJNW)
Fisher:1948yn ; Janis:1968zz  spacetime. This spacetime, a solution of
Einstein gravity enhanced with a massless minimally coupled scalar field,
becomes electrically charged in the context of Maxwell theory through JRW
transformations Janis:1969ivo . While the scalar field profile remains
unaltered from the uncharged seed, the electric configuration adopts a
Coulombian nature.

In its original formulation Buchdahl:1956zz , Buchdahl theorem (of the first
kind) can be presented as follows: Consider any $d-$dimensional vacuum
solution of the Einstein field equations

$ds_{0}^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=g_{aa}(x^{k})(dx^{a})^{2}+g_{ij}(x^{k})dx^{i}dx^{j},$
(1)

which is said to be static with respect to a coordinate “$a$”, namely, that it
satisfies $g_{ai}=0=\partial_{a}g_{\mu\nu}=0$. Here, $a$ represents a cyclic
coordinate, and $i,j$ run from $1$ to $d-1$.222Note that the index $a$ is
fixed, no summation is assumed even when repeated. It is proven that, knowing
(1), a new static and vacuum solution ($d\geq 4$) is automatically given by

$ds^{2}=(g_{aa})^{-1}(dx^{a})^{2}+(g_{aa})^{\frac{2}{d-3}}g_{ij}dx^{i}dx^{j},$
(2)

which is said to be reciprocal to the original seed. In addition, it is proven
Buchdahl:1959nk  (Buchdahl transformation of the second kind) that given a
static vacuum solution of the form (1), the spacetime configuration ($d\geq
4$)

$\displaystyle ds^{2}_{ES}$
$\displaystyle=(g_{aa})^{\beta}(dx^{a})^{2}+(g_{aa})^{1-\beta}g_{ij}dx^{i}dx^{j},$
(3) $\displaystyle\Phi$
$\displaystyle=\sqrt{\frac{1-\beta^{2}}{2\kappa}}\ln{(g_{aa})},$

will correspond to a solution of the Einstein-Scalar field equations

$R_{\mu\nu}=\kappa\partial_{\mu}\Phi\partial_{\nu}\Phi,\qquad\Box\Phi=0,$ (4)

where $\kappa=8\pi G$ represents Newton’s constant, and $|\beta|<1$ a scalar
hair parameter.

The conventional understanding of staticity is tied to the metric’s time
independence and the absence of stationary terms in the metric tensor.
Consequently, investigations into the Buchdahl theorem have primarily occurred
within the framework where the cyclic coordinate $a$ represents the time
coordinate. In this context, this is what we previously called “the typical”
manner of applying the Buchdahl transformations (2) and (3). However, this
limitation is not inherent in the theorem itself; rather, $a$ could denote any
cyclic coordinate associated with any other Killing vector of the geometry,
not necessarily the timelike Killing vector $\partial_{t}$. An intriguing
avenue of inquiry involves employing Buchdahl transformations concerning
spacelike Killing vectors. Therefore, a logical progression of exploration
involves examining the effects of a Buchdahl transformation when applied along
the Killing vector $\partial_{\phi}$.

In a recent publication Mazharimousavi:2024hrg , a vacuum solution to the
Einstein field equations has been introduced, termed the Schwarzschild-Levi-
Civita black hole. This geometry depicts a Schwarzschild black hole situated
within a Levi-Civita spacetime, as its geometric analysis can testify. The
geometry corresponds to the Levi-Civita spacetime in the massless limit, while
it asymptotically approaches a Levi-Civita spacetime at radial infinity, or in
terms of cylindrical coordinates, asymptotically far from the symmetry axis.
Although not advertised in Mazharimousavi:2024hrg , further analysis reveals
that this solution is algebraically general, namely, of Petrov type I,
indicating that its classification departs from the general expanding
Plebanśki-Demiański class Plebanski:1976gy , where most known black holes
belong to. Furthermore, particularly noteworthy reveals to be the fact that
this solution corresponds to a Buchdahl-transformed version of the
Schwarzschild black hole. However, unlike previous transformations, which were
typically applied with respect to the timelike Killing vector $\partial_{t}$,
this transformation occurs concerning the azimuthal spacelike Killing vector
$\partial_{\phi}$.

The Schwarzschild-Levi-Civita spacetime prompts a reevaluation of Buchdahl’s
transformations. Specifically, we aim to extend the Buchdahl theorem to
encompass geometries featuring two Killing vectors in dimension four. However,
an extension with an arbitrary number of commuting Killing vectors and in an
arbitrary number of dimensions is also feasible. We will examine the new
geometries achievable via this generalized Buchdahl theorem when applied to
spacelike Killing vectors. Furthermore, it will be demonstrated that the
solution introduced in Mazharimousavi:2024hrg  admits a Kerr-Schild
representation, with the Kerr-Schild vector being null and geodesic but not
shear-free. This observation aligns with the Goldberg-Sachs theorem GS , which
stipulates that the geometry must belong to algebraically special spacetimes
for a null and shear-free geodesic congruence. This naturally leads to the
question of how Buchdahl transformations can be applied to rotating vacuum
spacetimes. Explicit Buchdahl versions of the Myers-Perry black hole
Myers:1986un  and their Kerr-Schild form will be provided. In addition, it
will be clearly seen why four-dimensional rotating black holes cannot be
subjected to Buchdahl transformations. Additionally, we will elucidate how the
double copy scheme Bern:2008qj ; Bern:2010ue ; Monteiro:2014cda , which
establishes connections between solutions of Einstein-Maxwell theory and
Maxwell theory in flat spacetime, operates in the presence of Buchdahl-
transformed geometries. Lastly, at a static level, we will revisit the
standard Buchdahl theorem within Einstein-Scalar theory Buchdahl:1959nk , in
order to construct new hairy extensions of the Schwarzschild-Levi-Civita black
hole. In addition, we will introduce a framework through which an external
magnetic field can influence these solutions, providing a magnetic
generalization of the JRW transformations Janis:1969ivo . We will analyze the
main geometric properties of these new hairy black hole solutions and provide
several avenues to be explored in the context of black holes in scalar-tensor
theories.

This work is structured as follows: In Section 2, we delve into the main
properties of the Levi-Civita black hole Mazharimousavi:2024hrg , elucidating
its evident connection with the Buchdahl theorem (2) and providing a
magnetized version that regularizes the curvature singularity it presents
along the symmetry axis. Section 3 extends the Buchdahl theorem in vacuum in
several directions. First, to scenarios where two commuting Killing vectors of
a four-dimensional geometry are considered. Next, Buchdahl and Kerr-Schild
transformations are composed, resulting in the Kerr-Schild form of the
Schwarzschild-Levi-Civita black hole and facilitating the construction of new
higher-dimensional rotating black holes. In particular, Buchdahl-transformed
extensions of the Myers-Perry spacetime. In addition, leveraging the Kerr-
Schild form of our solutions, we establish a novel double copy scheme, notable
for its algebraically general nature. Section 4 presents a generalization of
the original Buchdahl theorem within the Einstein-Scalar system, facilitating
the construction of a Levi-Civita version of the Fisher-Janis-Newman-Winicour
black hole Fisher:1948yn ; Janis:1968zz ; Janis:1969ivo . We analyze its key
geometric features and introduce its magnetically charged extension, in an
attempt to generalize the standard JRW electric transformation Janis:1969ivo .
Additionally, we perform the well-known Bekenstein change of frame
Bekenstein:1974sf , obtaining the Levi-Civita extension of the celebrated BBMB
black hole Bocharova:1970skc ; Bekenstein:1974sf . Finally, in Section 5, we
conclude by outlining further avenues for exploration within the realm of
these novel geometries.

##  2 Revisiting the Schwarzschild-Levi-Civita Black Hole

As we have already pointed out the Schwarzschild-Levi-Civita black hole
Mazharimousavi:2024hrg , although it can be easily integrated from the field
equations, corresponds with the Buchdahl-transformed version of the
Schwarzschild geometry, however, not with respect to the standard timelike
Killing vector but with respect to the spacelike Killing vector
$\partial_{\varphi}$. As presented in Mazharimousavi:2024hrg  the geometry
reads

$ds^{2}=\frac{d\varphi^{2}}{\lambda_{0}^{2}r^{2}\sin^{2}\theta}+\lambda_{0}^{2}r^{4}\sin^{4}\theta\left[-\left(1-\frac{2M}{r}\right)dt^{2}+\frac{dr^{2}}{\left(1-\frac{2M}{r}\right)}+r^{2}d\theta^{2}\right],$
(5)

being $\lambda_{0}$ an integration constant and $M$ the mass parameter. In
fact, considering the coordinate $a$ to be the azimuthal angle $\varphi$ and
remaining in four dimensions, the Buchdahl-transformed metric (2) becomes

$ds^{2}=\frac{d\varphi^{2}}{r^{2}\sin^{2}\theta}+r^{4}\sin^{4}\theta
g_{ij}dx^{i}dx^{j}.$ (6)

As a matter of fact, if the seed spacetime is chosen to be the Schwarzschild
geometry, this precisely reproduces the Schwarzschild-Levi-Civita spacetime
(5) for $\lambda_{0}=1$. Contrary to the case in which the timelike Killing
vector has been considered in the transformation, the resulting metric cannot
be turned back to the seed by a large diffeomorphism as the initial spherical
symmetry of the seed has been lost. Direct inspection reveals that the effect
of the aforementioned Buchdahl transformation consists of embedding the given
seed on a Levi-Civita spacetime LC . Actually, considering the Minkowski
spacetime in cylindrical coordinates $(t,\rho,z,\varphi)$

$ds_{0}^{2}=-dt^{2}+d\rho^{2}+dz^{2}+\rho^{2}d\varphi^{2},$ (7)

making use of (2) with respect to the azimuthal coordinate provides us with

$ds^{2}=\rho^{4}(-dt^{2}+d\rho^{2}+dz^{2})+\frac{d\varphi^{2}}{\rho^{2}}.$ (8)

This spacetime is nothing else than the Levi-Civita spacetime

$ds_{LC}^{2}=-\rho^{4\sigma}dt^{2}+k^{2}\rho^{4\sigma(2\sigma-1)}(d\rho^{2}+dz^{2})+\rho^{2(1-2\sigma)}d\varphi^{2},$
(9)

with mass per unit length $\sigma=1$ and $k=1$. Notice that a rescaling of the
noncompact coordinates is necessary. This represents a particular case of
vacuum Weyl spacetimes in which the geometry features an additional spacelike
Killing vector, $\partial_{z}$. Now, the Schwarzschild black hole embedded in
the Levi-Civita spacetime certainly behaves asymptotically as the Levi-Civita
spacetime and it acquires most of its features, besides losing the evident
symmetry with respect to the $z$-coordinate which is lost due to the presence
of the mass term $M$. Therefore, it is of a general algebraic nature, namely,
of Petrov type I. Its background, the Levi-Civita spacetime, is algebraically
special for $\sigma=(-1/2,1/4,1)$ and conformally flat for
$\sigma=(0,1/2,\infty)$. Moreover, since it belongs to a vacuum Weyl solution,
its interpretation in terms of a Newtonian potential is direct to obtain,
representing the gravitational potential of an infinite uniform line source
with a mass per unit length expressed by $\sigma$. Consequently, the
Schwarzschild-Levi-Civita black hole will always feature a curvature
singularity all over the symmetry axis, which will be nothing but the
relativistic analog of the Newtonian line source sourcing the Levi-Civita
spacetime.

The curvature singularity located at the axis of symmetry of the
Schwarzschild-Levi-Civita black hole can be cured by embedding the geometry on
an external magnetic field. In direct analogy with the Schwarzschild-Melvin
spacetime Bonnor_1954 ; Melvin1966 , which is obtained via a magnetic Harrison
transformation harrison1968new , we construct the Melvin extension of the
Schwarzschild-Levi-Civita black hole

$\displaystyle ds^{2}$
$\displaystyle=\Lambda^{2}\left(r^{4}\sin^{4}\theta\right)\left[-\left(1-\frac{2M}{r}\right)dt^{2}+\frac{dr^{2}}{1-\frac{2M}{r}}+r^{2}d\theta^{2}\right]+\frac{1}{(r^{2}\sin^{2}\theta)\Lambda^{2}}d\varphi^{2},$
(10) $\displaystyle A$
$\displaystyle=\frac{B}{\Lambda}\left(\frac{1}{r^{2}\sin^{2}\theta}\right)d\varphi,$

where
$\Lambda=\Lambda\left(r,\theta\right)=1+\frac{B^{2}}{r^{2}\sin^{2}\theta}$.
The metric exhibits the standard curvature singularity at $r=0$ only. As
expected, asymptotically far from the axis of symmetry it behaves as a Levi-
Civita spacetime.

##  3 Extending Buchdahl theorem in vacuum

Thus far, our investigation has yielded a preliminary observation: employing
the Buchdahl transformations relative to the spacelike Killing vector
$\partial_{\varphi}$ presents an intriguing approach for generating novel
spacetimes within the framework of Einstein’s theory. As shown below in Sec.
4, this method also extends to the construction of distinctive spacetimes
within Einstein-Scalar and Einstein-Scalar-Maxwell theories. The primary
consequence appears to be the incorporation of the original seed onto Levi-
Civita spacetime, at least when employing one single transformation. This
significantly modifies the geometric characteristics of these newly derived
solutions compared to the familiar solutions obtained through the application
of Buchdahl transformations according to the standard lore.

###  3.1 Buchdahl theorem and its extension in $d=4$ with two Killing vectors

A generalization of the Buchdahl transformation can be achieved, when the
number of Killing vectors is greater or equal than 2, through compositions
between two different transformations associated with two different Killing
vectors. We will initially limit ourselves to the case in four dimensions with
exactly 2 Killing vectors, however, this construction can be extended in
arbitrary dimensions and with any number of Killing vectors.

Hence, let $g_{\mu\nu}$ be the metric tensor in four dimensions with
coordinates defined by $x^{\mu}=(t,x^{i},\varphi)$ and let
$\xi^{\mu}\partial_{\mu}=\partial_{t}$ and
$\chi^{\mu}\partial_{\mu}=\partial_{\varphi}$ be two Killing vector fields,
timelike and spacelike, respectively. In addition, let us assume that
$g_{it}=g_{i\varphi}=g_{t\varphi}=0$, that is, the metric tensor acquires the
form $g=\left(-g_{tt},\,g_{ij},\,g_{\varphi\varphi}\right)$ with all the
metric components being independent of $t$ and $\varphi$.
In order to generalize the original Buchdahl theorem (in four dimensions) to
the case in which the transformation involves the two Killing vectors, we
start by defining the family of applications $T_{(n,m)}$, with
$n,m\in\mathbb{Z}$, that acts on the metric components as follows

$T_{(n,m)}(g)=\left[-\left(g_{tt}\right)^{a_{n,m}}\left(g_{\varphi\varphi}\right)^{a_{m,n}}g_{tt},\,\left(g_{tt}\right)^{b_{n,m}}\left(g_{\varphi\varphi}\right)^{b_{-n,-m}}g_{ij},\,\left(g_{tt}\right)^{-a_{n,m}}\left(g_{\varphi\varphi}\right)^{-a_{m,n}}g_{\varphi\varphi}\right],$
(11)

where we have defined

$\displaystyle a_{n,m}=2(-1)^{n+m}n,\qquad b_{n,m}=(n+m)(n+m+1).$ (12)

It can be proven that if
$g_{\mu\nu}=\left(-g_{tt},\,g_{ij},\,g_{\varphi\varphi}\right)$ is a vacuum
metric, the reciprocal spacetime metrics defined by

$\displaystyle g^{\prime}=T_{(n,n)}(g):=T^{(1)}_{n}(g),$ (13a) $\displaystyle
g^{\prime\prime}=T_{(n,n-1)}(g):=T^{(2)}_{n}(g),$ (13b)

will also satisfy $R_{\mu\nu}=0$, and this for any integer $n\in\mathbb{Z}$.

Many comments can be made concerning this result. First of all, note that
since $a_{n,m}$ and $b_{n,m}$ (12) are always even, the signature of the
transformed metric $T_{(n,m)}(g)$ is the same as the original one, $g$. One
can also notice that for the Killing vector field $\xi$ (resp. $\chi$), the
standard Buchdahl transformation as defined in Buchdahl:1956zz  will
correspond to (13b) with $n=1$ (resp. $n=0$). For example, for a starting
metric given by the Schwarzschild solution with coordinates
$(t,r,\theta,\varphi)$ and metric components
$g_{tt}=g_{rr}^{-1}=1-\frac{2M}{r}$, $g_{\theta\theta}=r^{2}$ and
$g_{\varphi\varphi}=r^{2}\sin^{2}\theta$, the transformation (13b) with $n=0$
yields the Schwarzschild-Levi-Civita black hole (5), recently presented in
Mazharimousavi:2024hrg . Note that for the transformed Schwarzschild solution
with $n=1$, the resulting metric is nothing but the Schwarzschild spacetime
itself written in a different set of coordinates, see below (19). It is also
interesting to see that the set of all the transformations (13) forms a group
under composition

$\displaystyle
T^{(1)}_{0}=\mbox{id},\qquad(T^{(1)}_{n})^{-1}=T^{(1)}_{-n},\qquad(T^{(2)}_{n})^{-1}=T^{(2)}_{n}$
$\displaystyle T^{(1)}_{m}\circ T^{(1)}_{n}=T^{(1)}_{n+m},\quad
T^{(2)}_{m}\circ T^{(2)}_{n}=T^{(1)}_{n-m},\quad T^{(2)}_{m}\circ
T^{(1)}_{n}=T^{(2)}_{n+m},\qquad T^{(1)}_{m}\circ T^{(2)}_{n}=T^{(2)}_{n-m}.$

Moreover, as previously stated, the transformations (13) are nothing but the
compositions of the standard Buchdahl transformations associated with each of
the two Killing vectors. In other words, the group of generalized Buchdahl
transformations are generated by $T^{(2)}_{1}$ and $T^{(2)}_{0}$ only, which
are exactly the standard Buchdahl transformations associated with $\xi$ and
$\chi$. That is, any transformation (13), can be expressed as an alternated
composition of the transformations $T^{(2)}_{1}$ and $T^{(2)}_{0}$ as

${\cal T}_{1}:=T^{(2)}_{1}\circ T^{(2)}_{0}\circ T^{(2)}_{1}\circ
T^{(2)}_{0}\cdots\qquad\mbox{or}\qquad{\cal T}_{0}:=T^{(2)}_{0}\circ
T^{(2)}_{1}\circ T^{(2)}_{0}\circ T^{(2)}_{1}\cdots.$ (14)

More precisely, ${\cal T}_{1}$ will generate all the transformations
$T^{(1)}_{-n}$ and $T^{(2)}_{n}$ for $n\in\mathbb{N}$ while ${\cal T}_{0}$
will induce the remaining transformations, namely $T^{(1)}_{n}$ and
$T^{(2)}_{-n}$. For example, one can represent the transformation
$T^{(1)}_{3}$ as

$T^{(1)}_{3}=T^{(2)}_{0}\circ T^{(2)}_{1}\circ T^{(2)}_{0}\circ
T^{(2)}_{1}\circ T^{(2)}_{0}\circ T^{(2)}_{1}.$ (15)

In order to be concrete, once again we exemplify with the Schwarzschild-Levi-
Civita black hole of Mazharimousavi:2024hrg

$\displaystyle
ds_{SLC}^{2}=r^{4}\sin^{4}\theta\left[-\left(1-\frac{2M}{r}\right)dt^{2}+\frac{dr^{2}}{1-\frac{2M}{r}}+r^{2}d\theta^{2}\right]+\frac{d\varphi^{2}}{r^{2}\sin^{2}\theta}.$
(16)

As already mentioned, this spacetime can be obtained from the Schwarzschild
metric

$\displaystyle
d{s}_{Schw}^{2}=-\left(1-\frac{2M}{r}\right)dt^{2}+\frac{dr^{2}}{1-\frac{2M}{r}}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\varphi^{2},$ (17)

by means of the transformation (13b) with $n=0$, that is
$g_{SLC}=T^{(2)}_{(0,-1)}g_{Schw}$. Let us now go beyond a single
transformation, and from the vacuum solution (16), operate with a generalized
transformation with the timelike vector field
$\xi^{\mu}\partial_{\mu}=\partial_{t}$. The resulting new vacuum spacetime,
dubbed extended Schwarzschild-Levi-Civita, reads

$\displaystyle ds_{ESLC}^{2}$
$\displaystyle=\frac{-dt^{2}}{r^{4}\sin^{4}\theta\left(1-\frac{2M}{r}\right)}+\left[r^{4}\sin^{4}\theta\left(1-\frac{2M}{r}\right)\right]^{2}\left[r^{4}\sin^{4}\theta\left(\frac{dr^{2}}{1-\frac{2M}{r}}+r^{2}d\theta^{2}\right)\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left.+\frac{d\varphi^{2}}{r^{2}\sin^{2}\theta}\right],$
(18)

and is simply given via the transformations and
$g_{ESLC}=T^{(2)}_{(1,0)}g_{SLC}=T^{(2)}_{(1,0)}\circ
T^{(2)}_{(0,-1)}g_{Schw}=T^{(1)}_{(-1,-1)}g_{Schw}$. In a similar way, we can
start by acting on the Schwarzschild metric with a standard Buchdahl
transformation with respect to the timelike Killing vector $\partial_{t}$.
This metric, which is nothing else than the Schwarzschild metric in a
different set of coordinates, can be obtained via (13b) with $n=1$ as
$\bar{g}_{Schw}=T^{(2)}_{(1,0)}g_{Schw}$. It is given by

$d\bar{s}_{Schw}^{2}=\frac{-dt^{2}}{\left(1-\frac{2M}{r}\right)}+\left(1-\frac{2M}{r}\right)^{2}\Bigg{[}\frac{dr^{2}}{\left(1-\frac{2M}{r}\right)}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\varphi^{2}\Bigg{]}.$ (19)

Acting on this metric with a Buchdahl transformation with respect to the
spacelike Killing vector $\partial_{\varphi}$ provides the spacetime
$\bar{g}_{ESLC}=T^{(2)}_{(0,-1)}\bar{g}_{Schw}=T^{(1)}_{(1,1)}g_{Schw}$ of
which the line element reads

$\displaystyle d\bar{s}_{ESLC}^{2}$
$\displaystyle=\Bigg{[}r^{2}\sin^{2}\theta\left(1-\frac{2M}{r}\right)\Bigg{]}^{2}\Bigg{[}-\frac{dt^{2}}{1-\frac{2M}{r}}+\left(1-\frac{2M}{r}\right)^{2}\left(\frac{dr^{2}}{(1-\frac{2M}{r})}+r^{2}d\theta^{2}\right)\Bigg{]}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\quad\quad\qquad\qquad+\frac{d\varphi^{2}}{r^{2}\sin^{2}\theta\left(1-\frac{2M}{r}\right)}.$
(20)

Note that this metric is not diffeomorphic to (18). Therefore, the different
vacuum metrics generated by means of these transformations (13) are not
diffeomorphic in general, and this is consistent with the non Abelian nature
of the transformations.

Finally, we propose to re-derive the previous results with a static and
spherical seed metric using the mini-superspace formalism. This approach also
presents the advantage of specifying the thermodynamic properties of the
solutions obtained by means of the generating technique method. As usual, we
work in the Euclidean formalism with the Euclidean time $\tau$ related to the
Lorentzian time $t$ by $t=i\tau$. In order to fit with the prescription of
(11), we shall consider an Euclidean ansatz of the form

$\displaystyle ds_{{\text{\tiny{\mbox{E}}}}}^{2}=$
$\displaystyle\left(N^{2}f\right)^{a_{n,n-1}}\Lambda^{a_{n-1,n}}\,d\tau^{2}+\left(N^{2}f\right)^{b_{n,n-1}}\Lambda^{b{-n,1-n}}\left[\frac{dr^{2}}{f}+r^{2}d\theta^{2}\right]$
(21)
$\displaystyle+\left(N^{2}f\right)^{a_{-n,1-n}}\Lambda^{-b_{-n,1-n}}\,r^{2}\sin^{2}\theta
d\varphi^{2},$

where $f=f(r)$, $N=N(r)$ and $\Lambda=\Lambda(r,\theta)$, and where the
factors $a$ and $b$ are defined in (12). The Euclidean Einstein-Hilbert action
evaluated on this ansatz (after some integration by parts) and with a periodic
time $\tau$ of period $\beta=\frac{1}{T}$, with $T$ being the temperature,
reads

$\displaystyle I_{E}=\frac{1}{2\kappa}\int\sqrt{g_{E}}\,d^{4}x\,R$ (22)
$\displaystyle=\frac{\beta}{\kappa}\int_{\theta}\int_{r}N\Bigg{[}\sin(\theta)\left(rf^{\prime}+f-1+\left(\frac{\partial_{\theta}\Lambda}{\Lambda}\right)^{2}+r^{2}f\left(\frac{\Lambda^{\prime}}{\Lambda}\right)^{2}-2rf\frac{\Lambda^{\prime}}{\Lambda}\right)-2\cos(\theta)\frac{\partial_{\theta}\Lambda}{\Lambda}\Bigg{]}.$

It is interesting to note that the reduced action (22) is generic and does not
depend on the coefficients $a$ and $b$, and its field equations obtained by
varying with respect to $f$, $N$, and $\Lambda$ are easily integrated. Indeed,
one can show that the equation we obtain via variation with respect to $f$ can
be re-written as

$\frac{N^{\prime}}{N}=r\left(\frac{\Lambda^{\prime}}{\Lambda}\right)^{2}-\frac{2\Lambda^{\prime}}{\Lambda},$

which automatically implies that $\Lambda$ is separable in product as
$\Lambda=r^{2}H(\theta)$, and consequently $N=\mbox{cte}$. Finally, the
integration of the remaining equations yields

$\displaystyle f(r)=1-\frac{2M}{r},\qquad\qquad
N=\mbox{cst},\qquad\qquad\Lambda(r,\theta)=r^{2}\sin^{2}\theta.$ (23)

This result is in complete accordance with the extended Buchdahl
transformations (11) together with the Birkhoff theorem. The thermodynamics
can be carried out by adding a boundary term $B_{E}$ to the Euclidean action
such that the full variation vanishes $\delta(I_{E}+B_{E})=0$. In our case,
the variation of the appropriate boundary term is given by

$\delta
B_{E}=-\frac{\beta}{\kappa}\int_{\theta=0}^{\pi}\int_{r=r_{h}}^{\infty}N\Big{[}\sin(\theta)\Big{(}r\delta
f+2\delta\ln\Lambda(1+r^{2}f-2rf)\Big{)}-2\cos(\theta)\delta\ln\Lambda\Big{]},$

where $r_{h}=2M$ is the location of the horizon. For the class of solutions
(23), since $\delta\ln\Lambda=0$, we end up with the standard Schwarzschild
boundary term

$B_{E}=\beta\left(\frac{4M}{\kappa}\right)-\frac{4\pi r_{h}^{2}}{\kappa},$

from which one can infer the mass ${\cal M}=\frac{4M}{\kappa}$, and that the
entropy satisfies the one-quarter area law ${\cal S}=\frac{4\pi
r_{h}^{2}}{\kappa}$.

As a last comment before closing this section, we recall that the Buchdahl
transformation of the Schwarzschild solution along the timelike Killing vector
yields, after an appropriate re-definition of the radial coordinate, the
Schwarzschild spacetime. There is nevertheless an example where this
transformation on a vacuum black hole solution generates a new solution which
is not diffeomorphic to the starting seed. Indeed, in Hassaine:2015ifa , a
five-dimensional vacuum solution with a three-dimensional Nil geometry horizon
was constructed, of which the line element is given by

$ds^{2}=-\left(1-\frac{M}{r}\right)dt^{2}+\frac{dr^{2}}{r^{5}\left(1-\frac{M}{r}\right)}+\frac{1}{r}(dx_{1}^{2}+dx_{2}^{2})+r(dx_{3}-x_{1}dx_{2})^{2}.$

It is clear that the Buchdahl transformation can only be done along the
timelike Killing vector field yielding to a new vacuum solution with a Nil
geometry horizon

$ds^{2}=-\frac{dt^{2}}{\left(1-\frac{M}{r}\right)}+\left(1-\frac{M}{r}\right)\Bigg{[}\frac{dr^{2}}{r^{5}\left(1-\frac{M}{r}\right)}+\frac{1}{r}(dx_{1}^{2}+dx_{2}^{2})+r(dx_{3}-x_{1}dx_{2})^{2}\Bigg{]},$

which after a redefinition of the radial coordinate can be rewritten as

$ds^{2}=-\left(1-\frac{M}{r}\right)dt^{2}+\frac{dr^{2}}{r^{5}\left(1-\frac{M}{r}\right)^{5}}+\frac{r}{(r-M)^{2}}(dx_{1}^{2}+dx_{2}^{2})+r(dx_{3}-x_{1}dx_{2})^{2}.$

###  3.2 Composition of Buchdahl and Kerr-Schild transformations

In order to explore whether any of the spacetimes acquired through Buchdahl
transformations can be expressed in a Kerr-Schild form, we delve into the
composition of Buchdahl and Kerr-Schild transformations. Here, we will only
use the Buchdahl transformation with respect to the spacelike Killing vector
$\partial_{\varphi}$. This is due to the fact that the effect of the
transformation with respect to the timelike Killing vector is pure gauge if
the seed is either standard Minkowski or the Schwarzschild spacetime. Starting
from this observation, we will then split the $d$-dimensional coordinates
appropriately as $x^{\mu}=(x^{i},\varphi)$, where $i=1,\ldots,d-1$. Our
objective is to fix the conditions for which a metric written in the Kerr-
Schild form maintains a Kerr-Schild decomposition under the action of a
Buchdahl transformation with respect to the spacetime Killing vector
$\partial_{\varphi}$. Let us assume that the starting vacuum metric admits a
Kerr-Schild representation

$ds^{2}=ds_{0}^{2}+H(x^{\mu})l\otimes l,$ (24)

where the seed metric is $g_{0}$, and $l$ is a null and geodesic vector field
for the seed and the full metrics

$g^{{(0)}\mu\nu}l_{\mu}l_{\nu}=l^{\mu}(\nabla^{(0)}_{\mu}l_{\nu})=0=g^{\mu\nu}l_{\mu}l_{\nu}=l^{\mu}(\nabla_{\mu}l_{\nu})=0.$
(25)

Now, in order for the full metric to satisfy the Buchdahl conditions along the
spacelike Killing vector $\partial_{\varphi}$, the seed metric $g_{0}$ must
have the following decomposition

$ds_{0}^{2}=g_{ij}^{(0)}dx^{i}dx^{j}+g_{\varphi\varphi}^{(0)}d\varphi^{2},$
(26)

while $H=H(x^{i})$ and the component of the null $1-$form $l$ along the basis
element $d\varphi$ is vanishing, i.e., $l=a_{i}(x^{j})dx^{i}$. Under these
conditions, it is clear that using the Buchdahl transformation, the metric
defined by

$d\tilde{s}^{2}=\left(g_{\varphi\varphi}^{(0)}\right)^{\frac{2}{d-3}}\left[g_{ij}^{(0)}dx^{i}dx^{j}\right]+\frac{d\varphi^{2}}{g_{\varphi\varphi}^{(0)}}+\left(g_{\varphi\varphi}^{(0)}\right)^{\frac{2}{d-3}}H(x^{i})l\otimes
l,$ (27)

is also a vacuum metric. It remains to show that (27) has a Kerr-Schild
representation, that is $l$ is null and geodesic with respect to seed and the
full metric (27). In fact, under our assumption, particularly that
$l=a_{i}(x^{j})dx^{i}$, the null character of $l$ is trivial while the
geodesic equation simplifies to

$l^{\mu}(\tilde{\nabla}_{\mu}l_{\nu})=l^{i}\left[\Gamma^{(0)j}_{ik}-\tilde{\Gamma}^{j}_{ik}\right]l_{j}=-\frac{1}{\Omega}\left[(\partial_{i}\Omega)\,\delta^{j}_{k}+(\partial_{k}\Omega)\,\delta^{j}_{i}-(\partial^{j}\Omega)\,g_{ik}\right]l^{i}l_{j},$
(28)

where $\Omega=\left(g_{\varphi\varphi}^{(0)}\right)^{\frac{1}{d-3}}$. By
utilizing the null nature of the congruence
$\tilde{g}^{\mu\nu}l_{\mu}l_{\nu}=l_{i}l^{i}=0$, it becomes evident that $l$
is geodesic as well.

Let us exemplify this outcome with two cases. We will first show that the
Schwarzschild-Levi-Civita black hole can fit in the Kerr-Schild
representation, and that this is related to the fact that the Schwarzschild
metric itself has a Kerr-Schild form respecting the previous conditions.
Unfortunately, although Kerr’s solution has a Kerr-Schild type representation,
the latter is not compatible with the above-mentioned conditions unless the
rotation parameter vanishes. However, in higher dimensions where there is more
than one rotation parameter, one can show that the general Myers-Perry
rotating black hole solutions with one vanishing rotation are in complete
adequacy with the previous hypotheses and thus new spinning solutions can be
generated.

####  3.2.1 Kerr-Schild form of the Schwarzschild-Levi-Civita Black Hole

Firstly, observe that the Schwarzschild-Levi-Civita black hole metric (16) can
be expressed through a Kerr-Schild transformation of the form

$\displaystyle
ds_{SLC}^{2}=r^{4}\sin^{4}\theta\left[-dt^{2}+dr^{2}+r^{2}d\theta^{2}\right]+\frac{d\varphi^{2}}{r^{2}\sin^{2}\theta}+\left(2Mr^{3}\sin^{4}\theta\right)(dt+dr)^{2},$
(29)

where $l=dt+dr$ defines a null and geodesic but not shearfree congruence, and
that its standard form (16) is recovered by implementing the following change
of coordinates

$\displaystyle dt\to dt+\frac{2M}{r-2M}dr.$ (30)

On the other hand, it is known that the Kerr-Schild representation of the
Schwarzschild metric is given by

$\displaystyle
ds^{2}_{Schw}=-dt^{2}+dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\varphi^{2}+\left(\frac{2M}{r}\right)(dt+dr)^{2},$ (31)

and that its standard form is recovered using the same change of variables
(30). It is direct to observe that the Kerr-Schild representation of the
Schwarzschild-Levi-Civita black hole (29) can be obtained from (31) using the
transformation (27). Although this observation proceeds straightforwardly, it
suggests an interesting fact. Indeed, it seems that the Schwarzschild-Levi-
Civita black hole constitutes the first example of an algebraically general
spacetime in vacuum admitting a Kerr-Schild representation. Note that in
contrast to the Schwarzschild-Levi-Civita black hole (29), the metric (19)
cannot be obtained from a standard Kerr-Schild transformation but instead from
a Kerr-Schild transformation in the $(t,r)-$sector and a conformal
transformation in the orthogonal sector $(\theta,\varphi)$. Indeed, the metric
defined by

$d\bar{s}_{Schw}^{2}=-dt^{2}+dr^{2}-\left(\frac{2M}{r-2M}\right)\left(dt+dr\right)^{2}+\left(1-\frac{2M}{r}\right)^{2}\left[r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\varphi^{2}\right],$ (32)

following the change of variable, $dt\to dt-\frac{2M}{r}dr$, yields (19).

####  3.2.2 A rotating example: Myers-Perry black holes à la Buchdahl

The next example is provided by the rotating Myers-Perry vacuum solutions
Myers:1986un  in $d=5$, and later generalized in any higher dimension $d\geq
5$. In a $d-$dimensional Myers-Perry solution, the rotation group is
$\mbox{SO}(d-2)$ which has $[\frac{d-1}{2}]$ independent Casimir invariants
corresponding to the rotations in the distinct planes (each of them associated
to a different rotation parameter). One can show that for the Myers-Perry
solutions the hypothesis allowing the Buchdahl transformation is not respected
unless (and at least) one of the rotation parameters vanishes. We shall also
take advantage from the fact that the rotating Myers-Perry solutions admit a
Kerr-Schild representation to present the new generated spinning solutions in
the Kerr-Schild form.

In five dimensions, the Myers-Perry solution with two rotations has the
following Kerr-Schild representation

$ds^{2}_{\mbox{\tiny{MP}}}=-dt^{2}+\frac{r^{2}\rho^{2}}{(r^{2}+a_{1}^{2})(r^{2}+a_{2}^{2})}dr^{2}+\rho^{2}d\theta^{2}+(r^{2}+a_{1}^{2})\sin^{2}\theta
d\varphi_{1}^{2}+(r^{2}+a_{2}^{2})\cos^{2}\theta
d\varphi_{2}^{2}-\frac{M}{\rho^{2}}\,\,l\otimes l,$ (33)

where $\rho^{2}=r^{2}+a_{1}^{2}\cos^{2}\theta+a_{2}^{2}\sin^{2}\theta$, and
the null and geodesic vector $l$ is defined as

$l=dt+\frac{r^{2}\rho^{2}}{(r^{2}+a_{1}^{2})(r^{2}+a_{2}^{2})}dr-
a_{1}\sin^{2}\theta d\varphi_{1}-a_{2}\cos^{2}\theta d\varphi_{2}.$ (34)

In order for our Buchdahl-Kerr-Schild composition to be valid in the presence
of rotation, it is mandatory to turn off at least one of the angular momenta,
let us say $a_{2}=0$ (the case $a_{1}=0$ can be treated identically).
Therefore, taking $\xi^{\mu}\partial_{\mu}=\partial_{\varphi_{2}}$ as the
Killing vector of our transformation, our new spinning vacuum solution
($R_{\mu\nu}=0$) in the Kerr-Schild form reads

$\displaystyle ds^{2}_{\mbox{\tiny{MPLC}}}$
$\displaystyle=\left(r^{2}\cos^{2}\theta\right)\left[-dt^{2}+\frac{\rho_{1}^{2}}{r^{2}+a_{1}^{2}}dr^{2}+\rho_{1}^{2}d\theta^{2}+(r^{2}+a_{1}^{2})\sin^{2}\theta
d\varphi_{1}^{2}\right]+\frac{d\varphi_{2}^{2}}{r^{2}\cos^{2}\theta}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad-\frac{Mr^{2}\cos^{2}\theta}{\rho_{1}^{2}}l_{1}\otimes
l_{1},$ (35)

where now $\rho_{1}^{2}=r^{2}+a_{1}^{2}\cos^{2}\theta$ and where the null and
geodesic vector field $l_{1}$ is given by

$l_{1}=dt+\frac{\rho_{1}^{2}}{(r^{2}+a_{1}^{2})}dr-a_{1}\sin^{2}\theta
d\varphi_{1}.$ (36)

Notice that the presence of a cosmic string localized at
$\theta=\frac{\pi}{2}$ is unavoidable. Similarly, the construction could have
been performed by turning off the angular momentum $a_{1}=0$, in such a case
ending up with a cosmic string localized at $\theta=0$.

In the more intuitive Boyer-Lindquist set of coordinates the line element (35)
reads

$\displaystyle ds^{2}_{MPLC}$
$\displaystyle=\left(r^{2}\cos^{2}\theta\right)\left[-dt^{2}+\frac{M}{\rho_{1}^{2}}\left(dt-
a_{1}\sin^{2}\theta
d\varphi_{1}\right)^{2}+\rho_{1}^{2}\left(\frac{dr^{2}}{r^{2}+a_{1}^{2}-M}+d\theta^{2}\right)\right.$
(37)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+(r^{2}+a_{1}^{2})\sin^{2}\theta
d\varphi_{1}^{2}\bigg{]}+\frac{d\varphi_{2}^{2}}{r^{2}\cos^{2}\theta}.$

A straightforward generalization of this result to the generic $d-$dimensional
case proceeds directly by taking to zero one of the angular momenta. In the
case of odd dimensions, $d=2n+1$, the Buchdahl-transformed Myers-Perry metric
with $(n-1)$ rotations reads

$\displaystyle ds^{2}$
$\displaystyle=\left(r^{2}\mu_{n}^{2}\right)^{\frac{2}{d-3}}\Bigg{[}-dt^{2}+\frac{Mr^{2}}{\Pi
F}\left(dt+\sum_{i=1}^{n-1}a_{i}\mu_{i}^{2}d\varphi_{i}\right)^{2}+\frac{\Pi
F}{\Pi-Mr^{2}}dr^{2}+\sum_{i=1}^{n-1}(r^{2}+a_{i}^{2})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left(d\mu_{i}^{2}+\mu_{i}^{2}d\varphi_{i}^{2}\right)+r^{2}d\mu_{n}^{2}\Bigg{]}+\frac{d\varphi_{n}^{2}}{r^{2}\mu_{n}^{2}},$
(38)

where

$\sum_{i=1}^{n}\mu_{i}^{2}=1,\qquad
F=1-\sum_{i=1}^{n-1}\frac{a_{i}^{2}\mu_{i}^{2}}{r^{2}+a_{i}^{2}},\qquad\Pi=\prod_{i=1}^{n-1}(r^{2}+a_{i}^{2})r^{2}.$
(39)

Similarly, for even dimensions $d=2n+2$ with $(n-1)$ rotations we obtain

$\displaystyle ds^{2}$
$\displaystyle=\left(r^{2}\mu_{n}^{2}\right)^{\frac{2}{d-3}}\Bigg{[}-dt^{2}+\frac{Mr}{\Pi
F}\left(dt+\sum_{i=1}^{n-1}a_{i}\mu_{i}^{2}d\varphi_{i}\right)^{2}+\frac{\Pi
F}{\Pi-Mr}dr^{2}+\sum_{i=1}^{n-1}(r^{2}+a_{i}^{2})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left(d\mu_{i}^{2}+\mu_{i}^{2}d\varphi_{i}^{2}\right)+r^{2}d\mu_{n}^{2}+r^{2}d\alpha^{2}\Bigg{]}+\frac{d\varphi_{n}^{2}}{r^{2}\mu_{n}^{2}},$
(40)

where the variables $(\mu_{i},\alpha)$ are subject to the constraint
$\alpha^{2}+\sum_{i=1}^{n}\mu_{i}^{2}=1$.

It is interesting to remark that, it is natural to expect that these solutions
are going to be of the general algebraic type according to the higher
dimensional Petrov classification Coley:2004jv ; Ortaggio:2012jd . Again, the
attractiveness of these solutions relies on the fact that they are vacuum
algebraically general solutions admitting a Kerr-Schild decomposition. The
Kerr-Schild ansatz can also be useful regarding what is called the double copy
framework. This latter posits that scattering amplitudes in gauge theories,
such as quantum chromodynamics, are intimately connected to those in gravity.
On the other hand, while the Kerr-Schild ansatz is not directly applicable to
the double copy in its usual formulation, there have been explorations of
connections between exact solutions in gravity and their counterparts in gauge
theories within the framework of the double copy. For example, certain
properties of black hole solutions obtained through the Kerr-Schild ansatz may
have analogs or implications in the context of the double copy (see e.g.
Bahjat-Abbas:2017htu  and references therein and thereof). This is precisely
what we are going to investigate in the next subsection.

###  3.3 Buchdahl-transformed Kerr-Schild metrics and double copy

In the double copy scheme, the Kerr-Schild function $H$, employed for
expressing the black hole metric as a perturbation of the seed metric along a
null and geodesic direction $l$, becomes associated with the Maxwell gauge
potential via $A\propto Hl$. This relationship embodies the canonical
correspondence inherent in the double copy formalism utilizing the Kerr-Schild
ansatz. The underlying reasoning is to establish a framework for elucidating
the interrelation between gauge theory amplitudes and gravitational
counterparts, thereby facilitating the examination of scattering phenomena
involving gravitons and their interactions with matter fields within specific
spacetime configurations, notably those of rotating black holes.

In elucidating this correspondence, consider $g$ to represent a metric
formulated within the Kerr-Schild paradigm

$\displaystyle g_{\mu\nu}=g^{(0)}_{\mu\nu}+H\ l_{\mu}l_{\nu},$ (41)

where $l$ is a null and geodesic vector field with respect to both metrics. In
this case, the Ricci tensor transforms as

$\displaystyle
R^{\mu}_{\,\nu}=R^{(0)\mu}_{\quad\,\,\nu}-H\,l^{\mu}l^{\sigma}R^{(0)}_{\sigma\nu}+\frac{1}{2}\nabla^{(0)}_{\sigma}\Big{[}\nabla^{(0)\mu}\left(Hl^{\sigma}l_{\nu}\right)+\nabla^{(0)}_{\nu}\left(Hl^{\sigma}l^{\mu}\right)-\nabla^{(0)\sigma}\left(Hl^{\mu}l_{\nu}\right)\Big{]}.$
(42)

Let us see what could be some hypotheses to ensure that the Kerr-Schild
function $H$ of the Kerr-Schild metric (41) can generate a Maxwell solution
with gauge potential $A\propto Hl$. We also remind that the null and geodesic
vector fields can always be chosen such that $l_{t}=1$. For example, if the
following conditions hold

$\displaystyle R_{\mu t}=R^{(0)}_{\mu
t}=0,\qquad\Gamma^{(0)\mu}_{t\nu}=0,\qquad\partial_{t}g_{\mu\nu}=\partial_{t}H=\partial_{t}l_{\mu}=0,$
(43)

the gauge potential $A\propto H\,l$ will satisfy the Maxwell equations.
Indeed, taking the relation (42) for $\nu=t$, that would imply that
$\nabla^{(0)}_{t}\left(Hl^{\sigma}l^{\mu}\right)=0$, and hence

$\displaystyle\nabla^{(0)}_{\sigma}\Big{[}\nabla^{(0)\mu}\left(Hl^{\sigma}\right)-\nabla^{(0)\sigma}\left(Hl^{\mu}\right)\Big{]}=\nabla^{(0)}_{\sigma}\,F^{\mu\sigma}=0.$
(44)

It is a matter of checking to see that the sufficiency conditions (43) hold in
the case of the Schwarzschild metric (31) and its Kerr extension with the
ellipsoidal flat seed metric as well as for its higher-dimensional version
(33). The legitimate question is what happens for the Schwarzschild-Levi-
Civita (29) or the Buchdahl-transformed Myers-Perry solution (35). A quick
inspection reveals that in both cases, the condition
$\Gamma^{(0)\mu}_{t\nu}=0$ is not fulfilled, and hence the sufficient
conditions given in (43) do not apply. In fact, in the Schwarzschild-Levi-
Civita black hole (29), the Kerr-Schild function and the null geodesic vector
are $H_{\text{{\tiny\mbox{SLC}}}}(r,\theta)\propto r^{3}\,\sin^{4}\theta$ and
$l=dt+dr$, respectively. Then one can directly see that
$A=H_{\text{{\tiny\mbox{SLC}}}}(r,\theta)(dt+dr)\propto
r^{3}\,\sin^{4}\theta(dt+dr)$ will not satisfy the Maxwell equations for the
Schwarzschild-Levi-Civita metric (29). But what is surprising is that by
ignoring the term that generates the singularity along the axis, namely, the
term $\sin^{4}\theta$, one can show that the gauge potential defined by

${\cal
A}=\frac{H_{\text{{\tiny\mbox{SLC}}}}(r,\theta)}{\sin^{4}\theta}(dt+dr)\propto
r^{3}\ (dt+dr),$

will be a solution of the Maxwell equations. Our intuition is re-enforced with
the example of the Myers-Perry solution transformed through the Buchdahl
transformation (35), where
$H_{\text{{\tiny\mbox{MP}}}}(r,\theta)\propto\frac{r^{2}\cos^{2}\theta}{\rho_{1}^{2}}$
and the null geodesic vector is $l_{1}$. In this case again, one can see that
the $1-$form gauge defined by

${\cal
A}=\frac{H_{\text{{\tiny\mbox{MP}}}}(r,\theta)}{\cos^{2}\theta}\,l_{1}\propto\frac{r^{2}}{\rho_{1}^{2}}\left(dt+\frac{\rho_{1}^{2}}{(r^{2}+a_{1}^{2})}dr-
a_{1}\sin^{2}\theta d\varphi_{1}\right),$

will satisfy the Maxwell equations for the Myers-Perry Buchdahl-transformed
metric (35).

We end this section by showing that although it is a modification of the
simple copy gauge field $A\to{\cal A}$ that satisfies Maxwell’s equations, the
zeroth copy scalar field $H$ will satisfy itself the massless Klein-Gordon
equation for the Buchdahl-transformed seed metric along the spacelike Killing
vector field. Let us see how it works. For a vacuum Kerr-Schild metric of the
form (41), it was shown Monteiro:2014cda  that the Kerr-Schild function $H$
would satisfy the massless Klein-Gordon equation for the seed metric
$\Box_{0}H=0$, where $\Box_{0}$ means the d’Alembertian operator with respect
to the seed metric. In our case, the Buchdahl-transformed Kerr-Schild metric
is given by (27) with a seed metric given by

$\displaystyle
d\tilde{s}_{0}^{2}=\left(g_{\varphi\varphi}^{(0)}\right)^{\frac{2}{d-3}}\left[g_{ij}^{(0)}dx^{i}dx^{j}\right]+\frac{d\varphi^{2}}{g_{\varphi\varphi}^{(0)}}.$
(45)

It is now simple to see that since $\Box_{0}H=0$ together with the fact that
$\partial_{\varphi}H=0$, one also has that $\tilde{\Box}_{0}H=0$, with
$\tilde{\Box}_{0}$ being the d’Alembertian operator with respect to (45).
Indeed, this is a direct consequence of the fact that

$\sqrt{-\tilde{g}^{(0)}}=\sqrt{-{g}^{(0)}}\left(g_{\varphi\varphi}^{(0)}\right)^{\frac{2}{d-3}},\qquad\qquad\tilde{g}^{(0)ij}=g^{(0)ij}\left(g_{\varphi\varphi}^{(0)}\right)^{\frac{-2}{d-3}}.$

##  4 Extending Buchdahl theorem in Einstein-Scalar theory

Let us now shift our focus to the Buchdahl theorem of the second kind (3),
which pertains to the Einstein-Scalar system in four dimensions
Buchdahl:1959nk . This theorem follows a similar line of argumentation as
observed in the vacuum case. Initially, it is noted that due to the specific
characteristics of the scalar field, the transformation fails to dress
Minkowski spacetime with a nontrivial scalar field. Conversely, by employing
the Schwarzschild black hole as an initial configuration and employing (as
usual) the transformation along the timelike Killing vector $\partial_{t}$,
the theorem yields what is commonly referred to as the FJNW black hole
Fisher:1948yn ; Janis:1968zz ; Janis:1969ivo . Various facets of this geometry
have been studied Abdolrahimi:2009dc , including its extension to higher
dimensions within the Kaluza-Klein framework Lu:1995yn ; Lu:1995sh ;
Bogush:2022qxl . This, as elucidated in the subsequent section, furnishes an
intriguing framework for applying a geometrical understanding of such
solutions.

The FJNW spacetime is recognized for containing a curvature singularity
precisely coinciding with the location of the would-be horizon at $r_{h}=2M$.
Notably, irrespective of the parameter $\beta$ value, it yields Minkowski
spacetime in the limit where the mass approaches zero. The presence of a naked
singularity stems from the divergence of the associated energy-momentum tensor
at $r_{h}=2M$, primarily attributed to the logarithmic behavior exhibited by
the scalar field profile. However, it has been demonstrated that upon a frame
transformation, specifically to a scalar theory characterized by a conformal
coupling with the curvature, the divergence in the scalar field profile does
not necessarily manifest as a divergence in the energy-momentum tensor.
Consequently, the geometry remains free from naked singularities
Bekenstein:1974sf . For a specific choice of the hair parameter, $\beta=1/2$,
this formulation reproduces the renowned BBMB solution Bocharova:1970skc ;
Bekenstein:1974sf .

###  4.1 The Fisher-Janis-Newman-Winicour-Levi-Civita spacetime

As anticipated, investigations into the scalar-augmented version of the
Buchdahl theorem have predominantly focused on scenarios where the
transformation is applied along the timelike Killing vector $\partial_{t}$.
Consequently, in this context, we proceed to formulate the hairy extension of
the Schwarzschild-Levi-Civita black hole Mazharimousavi:2024hrg . Its
expression is given by333It is pertinent to note that the scalar field profile
remains defined up to a sign. Consequently, this sign can be adjusted in
accordance with the sign originating from its logarithmic behavior, thereby
directing the term $r^{2}\sin^{2}\theta$ towards the numerator of the
expression (46).

$\displaystyle ds^{2}_{\text{{\tiny\mbox{FJNWLC}}}}$
$\displaystyle=\left(r^{2}\sin^{2}\theta\right)^{\beta+1}\left[-\left(1-\frac{2M}{r}\right)dt^{2}+\frac{dr^{2}}{1-\frac{2M}{r}}+r^{2}d\theta^{2}\right]+\frac{d\varphi^{2}}{\left(r^{2}\sin^{2}\theta\right)^{\beta}},$
(46) $\displaystyle\Phi$
$\displaystyle=-\sqrt{\frac{1-\beta^{2}}{2\kappa}}\ln\left(r^{2}\sin^{2}\theta\right).$

This configuration serves as an extension of the FJNW solution Fisher:1948yn ;
Janis:1968zz ; Janis:1969ivo , derived through a Buchdahl transformation of
the second kind applied to the spacelike Killing vector $\partial_{\varphi}$.
Reaching the Schwarzschild-Levi-Civita black hole (16) in the limit $\beta=1$,
we denote (46) as the FJNW-Levi-Civita spacetime. However, it is noteworthy
that in the presence of a nontrivial scalar field profile, the metric (46)
does not exhibit asymptotic behavior akin to that of a Levi-Civita spacetime,
as there exists no value of $\beta$ yielding a Levi-Civita metric. As
previously noted, the Levi-Civita spacetime manifests a curvature singularity
along the entirety of its symmetry axis, thereby inheriting the same
characteristic in the Schwarzschild-Levi-Civita solution (16). This same
feature is preserved in the hairy extension (46); an analysis of the
Kretschmann scalar unveils

$\displaystyle R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$
$\displaystyle=\frac{1}{r^{4}\left(r\sin\theta\right)^{4+4\beta}}\left[4(\beta+1)\left(\frac{(\beta+1)(7\beta^{2}+2\beta+3)}{\sin^{4}\theta}-\frac{12M\beta(2\beta+1)}{r}\right)\right.$
(47)
$\displaystyle\quad\left.-\frac{16M}{r}\left(\frac{(\beta+1)(7\beta^{3}+5\beta^{2}+3\beta+3)}{\sin^{2}\theta}-\frac{M(7\beta^{4}+20\beta^{3}+23\beta^{2}+10\beta+3)}{r}\right)\right],$

a divergent behavior for all possible values of $\beta$. The scalar field
profile exhibits regular behavior throughout the spacetime, except at the
symmetry axis and at locations asymptotically distant from it. While the
standard FJNW solution presents a curvature singularity at $r_{h}=2M$, such a
singularity is absent in this formulation. Of significance is the observation
that, asymptotically distant from the symmetry axis, the kinetic term of the
scalar field remains finite, which, owing to the shift symmetry of the action,
emerges as the pertinent quantity. Moreover, the energy density also retains
finiteness in this regime. It is noteworthy that unlike the FJNW solution,
wherein the massless limit yields Minkowski spacetime, the corresponding limit
in the context of the FJNW-Levi-Civita solution results in a nontrivial
geometry characterized by a nontrivial scalar field. Upon examination in
cylindrical coordinates, it becomes apparent that this geometry cannot be
reconciled with the Levi-Civita spacetime, as there exists no value of $\beta$
that sustains a nontrivial scalar field while simultaneously adhering to the
Levi-Civita form for the geometry. The expression read

$\displaystyle ds^{2}_{\text{{\tiny\mbox{FJNWLC}}}}$
$\displaystyle=\rho^{2(\beta+1)}(-dt^{2}+d\rho^{2}+dz^{2})+\frac{d\varphi^{2}}{\rho^{2\beta}}$
(48) $\displaystyle\Phi$
$\displaystyle=-\sqrt{\frac{1-\beta^{2}}{2\kappa}}\ln\left(\rho^{2}\right).$

It turns out that the solution exhibits a special algebraic nature, classified
as type D, thereby belonging to the broader Plebanśki-Demiański class.
Conversely, for $\beta=-1/2$, it manifests as a conformally flat spacetime
featuring a nontrivial scalar field, akin to the Levi-Civita spacetime with
$\sigma=1$ if $\beta=1$. When $\beta=-1$, it simplifies to Minkowski
spacetime.

An insightful approach to grasp the intricacies of this geometry involves its
higher dimensional Kaluza-Klein uplift. Following the Kaluza-Klein framework
Duff:1986hr , whereby a solution of the Einstein-Scalar theory in dimension
$d$ can always be extended to a higher dimensional counterpart pertaining to
Einstein theory in dimension $d+1$, the uplifted geometry is given by

$ds^{2}_{d+1}=e^{2\alpha\Phi}ds^{2}_{d}+e^{2\gamma\Phi}dz^{2},$ (49)

being $z$ the extra spacetime coordinate and where

$\alpha^{2}=\frac{1}{2(d-1)(d-2)},\qquad\gamma=-(d-2)\alpha.$ (50)

It is therefore natural to inquire what the higher dimensional extension of
the FJNW-Levi-Civita solution (46) is and discern the potential geometrical
insights stemming from such an extension. To embark upon this endeavor, let us
commence with the FJNW-Levi-Civita solution expressed in cylindrical
coordinates

$\displaystyle ds^{2}_{\text{{\tiny\mbox{FJNWLC}}}}$
$\displaystyle=\rho^{2(\beta+1)}ds^{2}_{3}+\frac{d\varphi^{2}}{\rho^{2\beta}},$
(51) $\displaystyle\Phi$ $\displaystyle=-\sqrt{1-\beta^{2}}\ln(\rho^{2}),$

where we have used

$ds^{2}_{3}=-\frac{\sqrt{\rho^{2}+z^{2}}-2M}{\sqrt{\rho^{2}+z^{2}}}dt^{2}+\frac{\sqrt{\rho^{2}+z^{2}}}{\sqrt{\rho^{2}+z^{2}}-2M}\frac{(\rho
d\rho+zdz)^{2}}{\rho^{2}+z^{2}}+\frac{(zd\rho-\rho dz)^{2}}{\rho^{2}+z^{2}}.$
(52)

To align with the conventions of Duff:1986hr , we set $\kappa=1/2$.
Consequently, the Kaluza-Klein five-dimensional extension of (46) is given by

$ds_{5}^{2}=\rho^{2(\beta+1)-4\alpha\lambda}ds^{2}_{3}+\rho^{-4\alpha\lambda-2\beta}d\varphi^{2}+\frac{dz^{2}}{\rho^{4\gamma\lambda}},$
(53)

where we have introduced $\lambda=\sqrt{1-\beta^{2}}$, while in four
dimensions, we set $\alpha=1/\sqrt{12}$ and $\gamma=-2/\sqrt{12}$. This
geometry reveals particularly insightful characteristics when opting for the
hair parameter $\beta=-1/2$. Specifically, for this choice, we find that
$2(\beta+1)-4\alpha\lambda=0=-4\alpha\lambda-2\beta$ and $4\gamma\lambda=-1$,
thereby resulting in

$ds_{5}^{2}=-\left(1-\frac{2M}{r}\right)dt^{2}+\frac{dr^{2}}{\left(1-\frac{2M}{r}\right)}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
dz^{2}+d\varphi^{2},$ (54)

where for the sake of the physical interpretation of this line element we have
turned back to the use of spherical coordinates. Consequently, it becomes
evident that for $\beta=-1/2$, the Kaluza-Klein reduction from five to four
dimensions of the standard homogeneous Schwarzschild black string, conducted
along the angular direction $\varphi$, naturally yields the FJNW-Levi-Civita
solution (46). In simpler terms, the FJNW-Levi-Civita solution can be entirely
regularized (for $\beta=-1/2$) through its uplift to five dimensions, where it
manifests as the well-known homogeneous Schwarzschild black string.

It is now natural to inquire whether a charged extension of this geometry can
be devised following the framework of the JRW theorem Janis:1969ivo . The JRW
theorem offers a means to extend a nontrivial solution of the Einstein-Scalar
system to incorporate electric charge, while leaving the scalar field profile
unaffected by the electric field. In direct parallel to this theorem, we
introduce the following construction: Let

$ds^{2}=e^{2U}h_{\alpha\beta}dx^{\alpha}dx^{\beta}+e^{-2U}d\varphi^{2},$ (55)

be a four-dimensional metric that is “static” with respect to $\varphi$ and is
a solution to the vacuum Einstein equations. Hence,

  1. (i)

The reciprocal metric

$ds^{2}=e^{-2U}h_{\alpha\beta}dx^{\alpha}dx^{\beta}+e^{2U}d\varphi^{2},$ (56)

is also a solution to the vacuum Einstein equations.

  2. (ii)

If a vacuum solution to the Einstein equations is given by

$ds^{2}=e^{2V}h_{\alpha\beta}dx^{\alpha}dx^{\beta}+e^{-2V}d\varphi^{2},$ (57)

then the metric (55) is enhanced to be a solution to the Einstein-Scalar
equations (4), where

$\Phi=\sqrt{\frac{2}{\kappa}\frac{1-\beta^{2}}{\beta^{2}}}U,\qquad U=\beta V.$
(58)

  3. (iii)

In addition, if a static solution to the Einstein-Scalar equations (4) is
given by (55) and (58), then a solution to the Einstein-Scalar-Maxwell
equations

$R_{\mu\nu}=\kappa\left(\partial_{\mu}\Phi\partial_{\nu}\Phi+F_{\lambda\mu}F^{\lambda}_{\,\,\,\,\nu}-\frac{1}{4}g_{\mu\nu}F_{\lambda\rho}F^{\lambda\rho}\right),\quad\square\Phi=0,\quad\nabla_{\mu}F^{\mu\nu}=0,$
(59)

is given by (58) along with

$\displaystyle ds^{2}$
$\displaystyle=e^{2W}h_{\alpha\beta}dx^{\alpha}dx^{\beta}+e^{-2W}d\varphi^{2},$
(60) $\displaystyle F_{\mu\nu}$
$\displaystyle=\sqrt{\frac{8}{\kappa}}e^{-2W}\left(U_{,\mu}\delta^{\varphi}_{\nu}-U_{,\nu}\delta^{\varphi}_{\mu}\right),$

where $W=\ln\left|2\cosh(U-U_{0})\right|$, being $U_{0}$ a constant relevant
for the determination of the nature of the magnetic configuration.444This has
been noticed in Maeda:2019tqs  for the higher dimensional electric case, as it
has been unnoticed in the original construction Janis:1969ivo . Additionally,
in Maeda:2016ddh  a detailed classification of higher-dimensional Einstein-
Scalar-Maxwell solutions with Einstein base manifolds has been provided.

Thus, from (ii) solution (46) is straightforwardly constructed, while from
(iii) the magnetized extension of (46) is obtained, yielding

$\displaystyle ds^{2}$
$\displaystyle=\Lambda^{2}\left(r^{2}\sin^{2}\theta\right)^{\beta+1}\left[-\left(1-\frac{2M}{r}\right)dt^{2}+\frac{dr^{2}}{1-\frac{2M}{r}}+r^{2}d\theta^{2}\right]+\frac{1}{(r^{2}\sin^{2}\theta)^{\beta}\Lambda^{2}}d\varphi^{2},$
(61) $\displaystyle\Phi$
$\displaystyle=\sqrt{\frac{1-\beta^{2}}{2\kappa}}\ln\left(\frac{1}{r^{2}\sin^{2}\theta}\right),\quad
A=\frac{B}{\Lambda}\left(\frac{1}{r^{2}\sin^{2}\theta}\right)^{\beta}d\varphi.$

Here it has been defined
$\Lambda=\Lambda(r,\theta)=1+B^{2}\left(\frac{1}{r^{2}\sin^{2}\theta}\right)^{\beta}$.
The limit $\beta=1$ enables the recovery of the hairless magnetized
Schwarzschild-Levi-Civita black hole (10), while $\beta=-1$ yields the
Schwarzschild-Melvin black hole. In contrast to the approach in Janis:1969ivo
, our construction (iii) introduces a magnetic field rather than an electric
one. Specifically, it embeds the FJNW-Levi-Civita black hole within a Melvin
universe, consistent with the “magnetic” nature of the Levi-Civita black hole.
However, it’s important to note that this representation is essentially a
concise depiction of the magnetized solution attainable through the
application of magnetic Harrison transformations harrison1968new , which are
valid in the presence of a minimally coupled scalar field. While an electric
Harrison transformation would also introduce a monopolic electric charge, the
compact construction (iii) would not be applicable. Instead, the approach of
Janis:1969ivo  would be employed, which is formulated in an “electric”
context, as the Buchdahl transformations are applied with respect to the
timelike Killing vector $\partial_{t}$. Contrary to the hairless case (10),
the magnetic field does not regularize the axis of symmetry in the geometry
(61). The curvature singularity persists at the locus of the symmetry axis.

###  4.2 Conformal frame

Up to this point, our focus has been on solutions within the Einstein-Scalar
system. However, it is widely recognized that the Einstein-Scalar theory is
conformally related to the Einstein-Conformal-Scalar theory.

$I_{ECS}=\int
d^{4}x\sqrt{\bar{g}}\left[\frac{\bar{R}}{2\kappa}-\frac{1}{2}\left(\bar{\partial}\Psi\right)^{2}-\frac{1}{12}\bar{R}\Psi^{2}\right],$
(62)

via the so-called Bekenstein transformations Bekenstein:1974sf

$\bar{g}_{\mu\nu}=\cosh^{2}\left(\sqrt{\frac{\kappa}{6}}\Phi\right)g_{\mu\nu},\quad\Psi=\sqrt{\frac{6}{\kappa}}\tanh\left(\sqrt{\frac{\kappa}{6}}\Phi\right).$
(63)

Hence, given a solution $(g_{\mu\nu},\Phi)$ of the Einstein-Scalar system, the
set of transformations (63) promptly furnishes a solution
$(\bar{g}_{\mu\nu},\Psi)$ for the improved set of field equations

$\displaystyle\bar{G}_{\mu\nu}$
$\displaystyle=\kappa\left[\bar{\partial}_{\mu}\Psi\bar{\partial}_{\nu}\Psi-\frac{1}{2}\bar{g}_{\mu\nu}\left(\bar{\partial}\Psi\right)^{2}+\frac{1}{6}\left(\bar{g}_{\mu\nu}\bar{\square}-\bar{\nabla}_{\mu}\bar{\nabla}_{\nu}+\bar{G}_{\mu\nu}\right)\Psi^{2}\right],$
(64) $\displaystyle\bar{\square}\Psi$ $\displaystyle=\frac{1}{6}\bar{R}\Psi.$

The most iconic solution obtained through this approach is the renowned BBMB
black hole Bocharova:1970skc ; Bekenstein:1974sf , which essentially
corresponds to the FJNW solution translated to the conformal frame,
particularly for the specific value $\beta=1/2$. Following this analogy, let
us execute the change of frame on the FJNW-Levi-Civita spacetime (46),
focusing on the case of $\beta=1/2$. This yields

$\displaystyle d\bar{s}^{2}$ $\displaystyle=\left(1+\alpha
r\sin\theta\right)^{2}\left[r^{2}\sin^{2}\theta\left[-\left(1-\frac{2M}{r}\right)dt^{2}+\frac{dr^{2}}{\left(1-\frac{2M}{r}\right)}+r^{2}d\theta^{2}\right]+\frac{d\varphi^{2}}{r^{2}\sin^{2}\theta}\right],$
(65) $\displaystyle\Psi$ $\displaystyle=\sqrt{\frac{6}{\kappa}}\frac{1-\alpha
r\sin\theta}{1+\alpha r\sin\theta},$

where $\alpha>0$ is a constant that can be introduced after the rescalings
$(t,r,\varphi)\rightarrow(\alpha t,\alpha r,\alpha^{3}\varphi)$ together with
$M\rightarrow\alpha M$. The solution turns out to be of algebraic type I,
retaining a curvature singularity at the axis of symmetry, a characteristic
inherent in the solutions we have considered so far. However, the axis of
symmetry can be regularized by immersing the solution into an exterior
magnetic field, see below (68). At distances asymptotically distant from the
symmetry axis, both the kinetic term and energy density remain finite.
Conversely, the scalar field has been entirely regularized, exhibiting regular
behavior across the entirety of the domain of outer communications, including
the symmetry axis.

An intriguing scenario unfolds when $M=0$. In this instance, the spacetime
diverges from the Minkowski geometry yet retains a nontrivial scalar field,
deviating from the behavior demonstrated by the BBMB solution. The solution in
this case can be represented as

$\displaystyle d\bar{s}^{2}$ $\displaystyle=\left(1+\alpha
r\sin\theta\right)^{2}\left[r^{2}\sin^{2}\theta\left(-dt^{2}+dr^{2}+r^{2}d\theta^{2}\right)+\frac{d\varphi^{2}}{r^{2}\sin^{2}\theta}\right],$
(66) $\displaystyle\Psi$ $\displaystyle=\sqrt{\frac{6}{\kappa}}\frac{1-\alpha
r\sin\theta}{1+\alpha r\sin\theta}.$

This solution also features a singularity along the symmetry axis; however, it
falls into type D in the algebraic classification, unlike (65). Although the
introduction of the parameter $\alpha$ may seem somewhat arbitrary, it proves
instrumental in analyzing the spacetime emerging in the limit as $\alpha$
tends to zero. The solution maintains its type I classification in the Petrov
scheme. Notably, it does not straightforwardly connect with either the
Schwarzschild or the Levi-Civita black hole, partly due to the constant scalar
field present

$\displaystyle d\bar{s}^{2}$
$\displaystyle=r^{2}\sin^{2}\theta\left[-\left(1-\frac{2M}{r}\right)dt^{2}+\frac{dr^{2}}{\left(1-\frac{2M}{r}\right)}+r^{2}d\theta^{2}\right]+\frac{d\varphi^{2}}{r^{2}\sin^{2}\theta},$
(67) $\displaystyle\Psi$ $\displaystyle=\sqrt{\frac{6}{\kappa}}.$

The scalar field constant value is precisely the one that makes the effective
gravitational constant vanish. Therefore, this solution solves the field
equations (64) in the limit where no Einstein term is present in the action.

As previously mentioned, (65) can be regularized along its symmetry axis by
embedding the solution into an external magnetic field. The most direct method
to achieve this is by considering the already magnetized solution (61) and
applying the change of frame provided by (63). The solution that solves the
Einstein-Conformal-Scalar-Maxwell equations then reads

$\displaystyle d\bar{s}^{2}$ $\displaystyle=\left(1+\alpha
r\sin\theta\right)^{2}\left[\Lambda^{2}\left[-\left(1-\frac{2M}{r}\right)dt^{2}+\frac{dr^{2}}{\left(1-\frac{2M}{r}\right)}+r^{2}d\theta^{2}\right]+\frac{d\varphi^{2}}{\Lambda^{2}}\right],$
(68) $\displaystyle\Psi$ $\displaystyle=\sqrt{\frac{6}{\kappa}}\frac{1-\alpha
r\sin\theta}{1+\alpha r\sin\theta},\quad
A=\frac{2\sqrt{\alpha}B}{\Lambda}d\varphi,$

where $\Lambda=r\sin\theta+B^{2}$. The inclusion of the magnetic field
regularizes the symmetry axis and the curvature singularity locates itself at
$r=0$ only. Hence, this geometry remains regular everywhere within the domain
of outer communications, as well as the scalar field profile. This stands in
contrast to the standard BBMB magnetized black hole.

##  5 Further Comments

In this work, we have revisited Buchdahl transformations Buchdahl:1956zz ;
Buchdahl:1959nk , specifically exploring scenarios where the transformations
are applied with respect to the typical azimuthal spacelike Killing vector, as
opposed to the customary timelike Killing vector. This seemingly minor
adjustment has paved the way for the construction and examination of novel
black hole solutions in both vacuum and the Einstein-Scalar system, as well as
its conformally related framework. Indeed, we have elucidated the origin of
the recently introduced spacetime Mazharimousavi:2024hrg  and delved into
Buchdahl transformations of the first kind Buchdahl:1956zz  in cases where the
seed geometry possesses two different Killing vectors. Additionally, we have
composed Buchdahl and Kerr-Schild transformations to unveil the Kerr-Schild
form of these newly discovered algebraically general geometries, which, to the
best of our knowledge, constitute the first vacuum Kerr-Schild geometries of
type I.

This composition has provided valuable insights into the construction of new
vacuum rotating black holes, particularly exemplified by what we have termed
the Levi-Civita extension of the Myers-Perry geometry Myers:1986un . Moreover,
we have identified an algebraically general double copy framework.

Within the realm of Buchdahl transformations of the second kind
Buchdahl:1959nk , we have devised novel black hole solutions in the Einstein-
Scalar and Einstein-Scalar-Maxwell theories. Notably, we have introduced what
we refer to as a Levi-Civita extension of the FJNW solution Fisher:1948yn ;
Janis:1968zz , or alternatively, a hairy extension of the Schwarzschild-Levi-
Civita black hole Mazharimousavi:2024hrg . Furthermore, we have extended the
JRW theorem Janis:1969ivo  to embed this geometry within an external magnetic
field. Finally, these Einstein-Scalar solutions have been uplifted to the
Einstein-Conformal-Scalar framework, yielding a generalization of the well-
known BBMB geometry Bocharova:1970skc ; Bekenstein:1974sf .

Numerous avenues of exploration emerge from these findings. The study of
spacetime with acceleration Kinnersley:1970zw ; Pravda:2000vh  proves to be
particularly promising, offering a highly nontrivial static seed spacetime for
investigation in this context. This could potentially lead to a C-metric
spacetime embedded within a Levi-Civita-like background, possibly altering the
form of the standard conical singularities presented in the metric. Since
these are vacuum solutions, Harrison transformations harrison1968new  can be
readily applied, facilitating the straightforward construction of their
charged extensions with both electric and magnetic monopolic charges.
Furthermore, it would be feasible to embed them within an electromagnetic
background Bonnor_1954 ; Melvin1966 . Similarly, employing Ehlers
transformations Ehlers:1957zz ; Ehlers:1959aug , these solutions could acquire
properties of stationarity, such as NUT charge or vortex-like background
characteristics Barrientos:2024pkt . Investigating the causal structures and
geometric features of these spacetimes would undoubtedly represent a
significant advancement in determining the spectrum of solutions of Einstein
vacuum gravity. Transitioning to scenarios involving rotation, a comprehensive
geometrical analysis of the Levi-Civita extension of the Myers-Perry black
hole appears imperative. Moreover, given that we have access to Buchdahl
transformations of the second kind and that the use of Ehlers and Harrison
transformations is ensured within the Einstein-Scalar framework, all
aforementioned constructions can be executed in the presence of a massless
minimally coupled scalar field. Consequently, within the realm of very
minimalistic scalar-tensor theories, the spectrum of exact solutions seems
notably enriched by these discoveries, particularly when considering the
utilization of frame transformations Bekenstein:1974sf ; Ayon-Beato:2015ada .
A subsequent step could involve augmenting the theory by including a
cosmological constant, as demonstrated in Tangen:2007yn . With this extension,
not only can solutions of the type considered and mentioned here be utilized
as seeds, but also cylindrically symmetric black holes Lemos:1994xp . It would
then be feasible to apply transformations along the translational symmetry
Killing vector $\partial_{z}$, either independently or in combination with the
usual Killing vectors $\partial_{t}$ and $\partial_{\varphi}$.

We anticipate that exploring these additional avenues of research will provide
us with a more comprehensive understanding of the spectrum of exact solutions
in Einstein and Einstein-Scalar theories.

###### Acknowledgements.

We thank Andrés Anabalón for interesting discussions regarding the Kaluza-
Klein uplift of (46). The work of J.B. is supported by FONDECYT Postdoctorado
grant 3230596. A.C. is partially supported by FONDECYT grant 1210500 and by
Primus grant PRIMUS/23/SCI/005 from Charles University. The work of M.H. is
partially supported by FONDECYT grant 1210889. J.O is partially supported by
FONDECYT grant 1221504.

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