# Asymptotically AdS black hole with a conformally-coupled scalar field in the
first-order formalism of gravity

Luis Avilés<EMAIL_ADDRESS>Instituto de Ciencias Exactas y Naturales,
Universidad Arturo Prat, Playa Brava 3256, 1111346, Iquique, Chile  Facultad
de Ciencias, Universidad Arturo Prat, Avenida Arturo Prat Chacón 2120,
1110939, Iquique, Chile  Cristóbal Corral<EMAIL_ADDRESS>Instituto de
Ciencias Exactas y Naturales, Universidad Arturo Prat, Playa Brava 3256,
1111346, Iquique, Chile  Facultad de Ciencias, Universidad Arturo Prat,
Avenida Arturo Prat Chacón 2120, 1110939, Iquique, Chile  Fernando Izaurieta
<EMAIL_ADDRESS>Faculty of Engineering, Architecture and Design, Universidad
San Sebastián, Lientur 1457, Concepción, Chile.  Omar Valdivia
<EMAIL_ADDRESS>Instituto de Ciencias Exactas y Naturales, Universidad Arturo
Prat, Playa Brava 3256, 1111346, Iquique, Chile  Facultad de Ciencias,
Universidad Arturo Prat, Avenida Arturo Prat Chacón 2120, 1110939, Iquique,
Chile  Carlos Vera<EMAIL_ADDRESS>Instituto de Ciencias Exactas y
Naturales, Universidad Arturo Prat, Playa Brava 3256, 1111346, Iquique, Chile
Facultad de Ciencias, Universidad Arturo Prat, Avenida Arturo Prat Chacón
2120, 1110939, Iquique, Chile

###### Abstract

We present a novel asymptotically anti-de Sitter black hole solution with
conformally-coupled scalar fields in the first-order formalism of gravity in
four dimensions. To do so, we consider a one-parameter extension of conformal
transformations by exploiting the fact that the vielbein and spin connection
are regarded as independent fields. We solve the field equations analytically
and obtain a static black hole solution with nontrivial torsion sourced by the
conformal coupling between the scalar field and geometry. The presence of
torsion renders the scalar field everywhere regular, while the curvature and
torsion singularities coalesce into the origin. We show that this
configuration is continuously connected to previously reported solutions in
the limit of vanishing torsion and analyze its main properties, focusing on
the consequences of the torsional singularity.

##  I Introduction

Black holes are one of the most fascinating objects in nature. They were
predicted by Einstein’s theory of general relativity, and their existence has
been well established through different experimental evidence Kormendy and
Richstone (1995); Celotti  _et al._ (1999); Gillessen  _et al._ (2009); Muñoz
_et al._ (2011); Abbott  _et al._ (2016); Akiyama  _et al._ (2022). In
electrovacuum, they are uniquely characterized by their mass, angular
momentum, and electric charge, that is, the physical quantities that can be
measured as asymptotic charges at spatial infinity. Any other type of imprint
that matter fields could leave on black holes is lost once its stationary
phase is achieved; this leads to a uniqueness theorem for the Kerr-Newman
black hole in Einstein-Maxwell theory.111Later, it was shown that this can be
circumvented in Einstein-Yang-Mills theory Volkov and Galtsov (1989); Bizon
(1990); Kuenzle and Masood-ul Alam (1990); Baltsov and Volkov (1992). This is
known as the no-hair theorem Israel (1967, 1968); Carter (1971); Ruffini and
Wheeler (1971); Bekenstein (1995); Chrusciel  _et al._ (2012), where hair is
referred to as any other asymptotic charge apart from the aforementioned
quantities that can be measured by a Gauss law at infinity.

The no-hair theorem is a theory-dependent statement and it relies critically
on the minimal coupling of matter to gravity. For instance, in asymptotically
flat spacetimes, it is known that minimally-coupled scalars lead to a solution
that is not a black hole but, rather, represents a naked singularity Janis
_et al._ (1968, 1969). One could, in turn, allow for nonminimal-scalar
coupling to circumvent this issue. A natural possibility would be to consider
conformally-coupled scalar fields. Nevertheless, the static and spherically
symmetric solution found by Bekenstein Bekenstein (1974) and independently by
Bocharova-Bronnikov-Melnikov Bocharova  _et al._ (1970) reveals that this
class of coupling leads to a scalar field that becomes singular at the
horizon. Even though it was argued that this singularity is not necessarily
problematic from the viewpoint of an asymptotic observer Bekenstein (1975), it
was shown that it renders an ill-defined stress-energy tensor, whose
evaluation through suitable regularization methods yields ambiguous results
Sudarsky and Zannias (1998). Remarkably, this problem can be solved by
introducing a cosmological constant and a conformal potential for the scalar
field, pushing the scalar-field singularity behind the event horizon Martinez
_et al._ (2003, 2006). After this resolution was proposed, different
configurations with self-gravitating scalar fields have been found in four
dimensions Anabalon and Maeda (2010); Charmousis  _et al._ (2009); Astorino
(2015); Anabalon and Cisterna (2012); Bardoux  _et al._ (2012); Astorino
(2013); Bardoux  _et al._ (2014); Cisterna  _et al._ (2021); Barrientos  _et
al._ (2022a, b); Babichev  _et al._ (2022). In higher dimensions, however, the
standard conformal coupling does not admit black hole solutions Xanthopoulos
and Dialynas (1992); Klimcik (1993) and one needs to introduce conformal
higher-curvature corrections to circumvent this no-go theorem Giribet  _et
al._ (2014); Chernicoff  _et al._ (2016); Babichev  _et al._ (2023).

A natural question is whether relaxing some assumptions on the geometry could
help to find a way out to the no-hair theorem. In the first-order formalism of
gravity, i.e. where the metric and connection are treated as independent
dynamical fields, it is known that nonminimally coupled scalar fields act as a
source of torsion (See Barrientos  _et al._ (2017) and references therein).
This class of interaction appears in different contexts. For instance, the
dimensional reduction of low-energy limits of string theory, e.g. Einstein-
Gauss-Bonnet gravity Zwiebach (1985), induces a nonminimal coupling between
the dilaton and topological terms of the Euler class Charmousis (2015); Lu and
Pang (2020); Hennigar  _et al._ (2020); Ma and Lu (2020). In the context of
cosmology, the torsion-induced interaction might explain the accelerated
expansion of the Universe from a geometrical viewpoint Toloza and Zanelli
(2013); Espiro and Vásquez (2016); Castillo-Felisola  _et al._ (2016); Cid
_et al._ (2018). On the other hand, the shift symmetry in the linear pseudo-
scalar-Nieh-Yan coupling Hojman  _et al._ (1980); Nelson (1980); Nieh and Yan
(1982a) allows one to renormalize the divergent piece coming from the
torsional contribution to the axial anomaly Chandia and Zanelli (1997);
Mercuri (2009). The latter generates an axion-like particle that could provide
a solution to the strong CP problem from a gravitational perspective Lattanzi
and Mercuri (2010); Castillo-Felisola  _et al._ (2015). In this context,
axionic hair on slowly-rotating black holes Campbell  _et al._ (1990); Duncan
_et al._ (1992) and locally $\text{AdS}_{3}\times\mathbb{R}$ black string
solutions have been found Cisterna  _et al._ (2019); Corral  _et al._ (2022);
as well as in the first-order formulation of Chern-Simons modified gravity
Jackiw and Pi (2003); Alexander and Yunes (2008, 2009); Yunes and Pretorius
(2009); Konno  _et al._ (2009); Amarilla  _et al._ (2010); Nashed and Nojiri
(2023). Finally, from a phenomenological viewpoint, (pseudo-)scalar-induced
torsion leads to luminal propagation of gravitational waves, although their
polarization can experience birefringence in vacuum Barrientos  _et al._
(2019a); Sulantay  _et al._ (2023). All of this suggests that torsional
scalar-tensor theories could provide a novel way to circumvent the no-hair
theorem by relaxing assumptions on the geometry.

In this work, we address this question by focusing on conformally-coupled
scalar fields in the first-order formalism of gravity. To this end, we study a
one-parameter extension of conformal transformations by considering that the
metric and affine properties of the manifold are independent Nieh and Yan
(1982b); Chakrabarty and Lahiri (2018); Izaurieta  _et al._ (2020)—see Eq. (5)
below. Conformal symmetry is relevant in this context because, in four
dimensions, conformally invariant metric theories are known to be finite for
any asymptotically AdS solutions Grumiller  _et al._ (2014); a result that can
be extended for Einstein-AdS spaces in six dimensions Anastasiou  _et al._
(2021, 2022, 2023a) and in the presence of conformally-coupled scalar fields
in four dimensions Anastasiou  _et al._ (2023b).222Indeed, one can embed
Einstein-AdS gravity in conformal gravity by imposing Neumann boundary
conditions in the Fefferman-Graham expansion Maldacena (2011); Anastasiou and
Olea (2016). We construct a scalar-tensor theory in the first-order formalism
of gravity that remains invariant under these transformations and obtain the
field equations. By assuming a static ansatz, we solve the field equations
analytically and obtain an asymptotically AdS black hole solution with
nontrivial torsion sourced by the scalar field. We analyze different
properties of the solution putting particular attention on the torsional
singularities. Remarkably, the scalar field becomes fully regular, while the
curvature and torsion singularities coalesce at the origin of the spacetime.
The solution is continuously connected to previous results reported in the
literature in the limit where torsion vanishes. In contrast to previous
findings, this limit can be achieved for a particular value of the conformal
parameter without trivializing the scalar field. Finally, we analyze the
asymptotic behavior of the solution and provide evidence that the value of the
conserved charges will change due to the presence of torsion.

The manuscript is organized as follows: In Sec. II, the one-parameter
extension of conformal transformations is discussed and a scalar-tensor theory
that remains invariant under the latter is proposed. In Sec. III, the solution
is presented by assuming a static ansatz with a compact constant-curvature
transverse section. Section IV is devoted to analyzing the physical properties
of the black hole solution, focusing on the torsional effects on the geometry.
Finally, in Sec. V we present a summary and discussion. Appendix A is included
for the sake of comparison where we rewrite the action and field equations
with tensor components.

##  II Conformally coupled scalar fields in first-order gravity

Here, we discuss the dynamics and the symmetries of the theory we are
interested in. To this end, we focus on a scalar-tensor theory that remains
invariant under a one-parameter family of conformal transformations. In
particular, we consider the metric and connection as independent fields; this
is usually regarded as the first-order formalism of gravity, where torsion is
not assumed to vanish beforehand. This allows one to extend the typical
conformal scalar-tensor couplings in (pseudo)-Riemannian geometries to
theories in which torsional degrees of freedom are present.

Let $\mathcal{M}_{4}$ be a four-dimensional Lorentzian manifold endowed with a
metric tensor $g_{\mu\nu}$. Henceforth, Greek indices denote tensor components
in the coordinate basis while lowercase Latin indices are used for the Lorentz
orthonormal basis. We denote by $\Omega^{p}(\mathcal{M}_{4})$ the set of
$p$-forms defined over $\mathcal{M}_{4}$. Change of frame matrix components
$e^{a}{}_{\mu}$ help us to define the vielbein $1$-form
$e^{a}=e^{a}{}_{\mu}\mathrm{d}x^{\mu}$.333From these definitions, one can
define a Lorentz-vector basis $E_{a}=E^{\mu}{}_{a}\partial_{\mu}$, such that
it is dual to the vielbein $1$-form, that is,
$e^{a}{}_{\mu}E^{\mu}{}_{b}=\delta^{a}_{b}$ and
$e^{a}{}_{\mu}E^{\nu}{}_{a}=\delta^{\nu}_{\mu}$. The line element is given by

$\mathrm{d}s^{2}=g_{\mu\nu}\mathrm{d}x^{\mu}\otimes\mathrm{d}x^{\nu}=\eta_{ab}e^{a}\otimes
e^{b}\,,$ (1)

where $\eta_{ab}=\mathrm{diag}(-1,1,1,1)$ denotes the Minkowski metric and,
consequently, we have the local mapping
$g_{\mu\nu}(x)=e^{a}{}_{\mu}(x)e^{b}{}_{\nu}(x)\eta_{ab}$. Since we are
considering a Riemann-Cartan geometry, we introduce the Lorentz connection
$1$-form, $\omega^{ab}=\omega^{ab}{}_{\mu}\mathrm{d}x^{\mu}$, as an
independent field. From these quantities, one defines the curvature and
torsion $2$-forms which are given by

$\displaystyle R^{ab}$
$\displaystyle=\mathrm{d}\omega^{ab}+\omega^{a}{}_{c}\wedge\omega^{cb}\,,$ (2)
$\displaystyle T^{a}$ $\displaystyle=\mathrm{d}e^{a}+\omega^{a}{}_{b}\wedge
e^{b}\,,$ (3)

respectively. They satisfy the Bianchi identities $\text{D}R^{ab}=0$ and
$\text{D}T^{a}=R^{a}{}_{b}\wedge e^{b}$. Additionally, the Lorentz connection
can always be decomposed into their torsion-free and contorsional pieces, that
is, $\omega^{ab}=\mathring{\omega}^{ab}+K^{ab}$, where the Levi-Civita
connection satisfies the torsion-free condition
$\text{d}e^{a}+\mathring{\omega}^{a}{}_{b}\wedge e^{b}=0$ and
$K^{ab}=K^{ab}{}_{\mu}\text{d}x^{\mu}$ is the contorsion $1$-form defined via
$T^{a}=K^{a}{}_{b}\wedge e^{b}$.

We consider a scalar-tensor theory where the gravitational dynamics for the
vierbein, Lorentz connection, and scalar field is dictated by the action
functional

$\displaystyle I[e,\omega,\phi]$
$\displaystyle=\int_{\mathcal{M}_{4}}\Bigg{[}\frac{1}{4\kappa}\epsilon_{abcd}\left(R^{ab}-\frac{\Lambda}{6}e^{a}\wedge
e^{b}\right)\wedge e^{c}\wedge e^{d}$
$\displaystyle-\frac{1}{24}\epsilon_{abcd}\left(\phi^{2}R^{ab}+\left\\{\lambda\left[1-\frac{\lambda}{2}\right]Z^{2}+V(\phi)\right\\}e^{a}\wedge
e^{b}+4\lambda\phi Z^{a}T^{b}\right)\wedge e^{c}\wedge e^{d}\Bigg{]}\,.$ (4)

where $Z^{a}=e^{a}{}_{\mu}\nabla^{\mu}\phi$. Here, $\kappa=8\pi G_{N}$ is the
gravitational constant, $\Lambda$ is the cosmological constant, $V(\phi)$ is a
potential for the scalar field $\phi$, and $\lambda$ is a dimensionless
paremeter that characterizes the extended Weyl transformation for the Lorentz
connection. Additionally, the kinetic term of the scalar field is constructed
in terms of $Z^{2}=Z_{a}Z^{a}$. Neglecting the potential $V(\phi)$, the
scalar-tensor sector of the action (4) remains invariant under the one-
parameter family of Weyl transformations

$\displaystyle e^{a}\to$ $\displaystyle\bar{e}^{a}$
$\displaystyle=\exp[\sigma(x)]e^{a}\,,$ (5a) $\displaystyle\omega^{ab}\to$
$\displaystyle\bar{\omega}^{ab}$
$\displaystyle=\omega^{ab}+\lambda\,\theta^{ab}\,,$ (5b)
$\displaystyle\phi\to$ $\displaystyle\bar{\phi}$
$\displaystyle=\exp[-\sigma(x)]\phi\,,$ (5c)

where $\theta^{ab}=2e^{[a}e^{b]}{}_{\mu}\nabla^{\mu}\sigma$ and $0<\lambda<1$.
This, in turn, implies that the contorsion $1$-form transforms as
$K^{ab}\to\bar{K}^{ab}=K^{ab}+(\lambda-1)\theta^{ab}$. Notice that, if
$\lambda\to 0$, the Lorentz connection remains invariant under Weyl rescalings
Nieh and Yan (1982b) while, if $\lambda\to 1$, the contorsion does Chakrabarty
and Lahiri (2018); Izaurieta  _et al._ (2020). We refer to as these two limits
as the exotic and canonical Weyl rescalings, respectively.

The field equations are obtained by performing arbitrary variations with
respect to the vierbein, Lorentz connection, and the scalar field, giving

$\displaystyle 0$ $\displaystyle=\frac{1}{2}\epsilon_{abcd}R^{ab}\wedge
e^{c}-\frac{\Lambda}{3!}\epsilon_{abcd}e^{a}\wedge e^{b}\wedge
e^{c}-\kappa\tau_{d}\,,$ (6a) $\displaystyle 0$
$\displaystyle=\epsilon_{abcd}T^{c}\wedge e^{d}-\kappa\sigma_{ab}\,,$ (6b)
$\displaystyle 0$
$\displaystyle=\epsilon_{abcd}\left[\lambda(2-\lambda)\text{D}Z^{a}\wedge
e^{b}-\frac{1}{2}\phi R^{ab}-\lambda(3\lambda-5)Z^{a}\wedge
T^{b}+\lambda\text{d}\phi\wedge\text{I}^{a}(T^{b})\right]\wedge e^{c}\wedge
e^{d}$
$\displaystyle+\epsilon_{abcd}\left[\lambda\phi\text{D}(\text{I}^{a}T^{b})\wedge
e^{c}-2\lambda\phi\text{I}^{a}(T^{b})\wedge
T^{c}-\frac{1}{4}\frac{\mathrm{d}V(\phi)}{\mathrm{d}\phi}e^{a}\wedge
e^{b}\wedge e^{c}\right]\wedge e^{d}\,,$ (6c)

respectively. Throughout this manuscript, $\text{I}^{a}$ denotes the inner
contraction operator along the dual vector basis to the vierbein
$1$-form444See Ref. (Barrientos  _et al._ , 2019b, Section 3.2) for further
details.. Additionally, we have defined the stress-energy and spin density
$3$-forms as $\tau_{a}$ and $\sigma_{ab}$, respectively; they are

$\displaystyle\tau_{d}$
$\displaystyle=\frac{1}{3}\epsilon_{abcd}\left(\frac{\phi^{2}}{4}R^{ab}+\lambda\phi
Z^{a}T^{b}\right)\wedge
e^{c}+\frac{1}{6}\epsilon_{abcd}\left[\frac{\lambda(\lambda-2)}{2}Z^{2}+V(\phi)\right]e^{a}\wedge
e^{b}\wedge e^{c}$
$\displaystyle-\frac{\lambda}{6}\epsilon_{abcd}\left[\phi\text{D}Z^{a}+(3\lambda-5)Z^{a}\,\text{d}\phi+\phi
Z_{n}\text{I}^{n}T^{a}\right]\wedge e^{b}\wedge e^{c}\,,$ (7)
$\displaystyle\sigma_{ab}$
$\displaystyle=\frac{\left(1-\lambda\right)}{6-\kappa\phi^{2}}\epsilon_{abcd}\,\phi\,\text{d}\phi\wedge
e^{c}\wedge e^{d}\,.$ (8)

The field equation for the connection can be solved algebraically for the
torsion in terms of the scalar field and derivatives thereof. The solution is
given by

$T^{a}=\frac{\kappa\left(1-\lambda\right)}{\left(6-\kappa\phi^{2}\right)}\,\phi\,\text{d}\phi\wedge
e^{a}\,,$ (9)

where $\phi^{2}\neq\tfrac{6}{\kappa}$. Thus, we conclude that the nonminimal
coupling of the scalar field sources the nontrivial torsion in this theory.
This has been observed in Refs. Toloza and Zanelli (2013); Cid  _et al._
(2018); Barrientos  _et al._ (2019a, 2017) as well. Moreover, in the limit
$\lambda\to 1$, the torsion vanishes independent of the value of the scalar
field. In contrast, if $\lambda\neq 1$, the torsion is nontrivial as long as
the scalar field is not constant. In what follows, we solve the remaining
field equations by assuming a static ansatz and show that the system admits a
black hole solution with nontrivial torsion.

##  III Torsional black holes dressed with scalar fields

In this section, we explore the space of solutions of the theory by solving
the field equations (6). To do so, we consider a one-parameter extension of
the standard quartic potential, that is,

$V(\phi)=\frac{\Lambda\pi\,G\,J^{2}(\phi)}{9}\left[\left(\frac{3+\sqrt{12\pi
G}\phi}{3-\sqrt{12\pi
G}\phi}\right)^{-\sqrt{\lambda(2-\lambda)}}+\left(\frac{3+\sqrt{12\pi
G}\phi}{3-\sqrt{12\pi
G}\phi}\right)^{\sqrt{\lambda(2-\lambda)}}\right]-\frac{\Lambda}{8\pi G}\,,$
(10)

where $J(\phi)=\phi^{2}-\frac{3}{4\pi G}$. The conformal quartic potential is
obtained in the vanishing-torsion limit, that is, $\lambda\to 1$. It is worth
noticing that Eq. (10) has extrema at $\phi_{0}=0$ and at
$\phi_{0}=\pm\sqrt{\frac{3}{4\pi G}}$. The latter, however, represents a
torsional singularity as one can check from Eq. (9). Indeed, we will see that
these points correspond to limiting values of the scalar field. Stability of
the latter will depend on the value of the cosmological constant at these
points. The potential satisfies
$V^{\prime\prime}(0)=-\tfrac{2\Lambda}{3}(\lambda-1)^{2}$.555For $\lambda\neq
1$ this potential contributes to the mass term of the scalar field. Thus, if
$\Lambda<0$ ($\Lambda>0$), the extremum at $\phi_{0}=0$ represents a global
minimum (maximum). It should be noticed that, for $\lambda=1$, the torsion
vanishes and we recover the results obtained in Ref. Martinez  _et al._
(2004).

We assume a static metric whose codimension-2 hypersurfaces of constant $t-r$
represent locally a constant curvature space. In particular, we consider

$\displaystyle\text{d}s^{2}=h(r)\left(-f(r)\text{d}t^{2}+\frac{\text{d}r^{2}}{f(r)}+r^{2}\text{d}\Sigma_{(k)}^{2}\right)\,,\;\;\;\;\;\mbox{where}\;\;\;\;\;\text{d}\Sigma_{(k)}^{2}=\frac{\text{d}\vec{x}\cdot\text{d}\vec{x}}{\left(1+\frac{k}{4}\vec{x}\cdot\vec{x}\right)^{2}}$
(11)

represents the line element of a compact transverse section of constant
curvature $k$ and local coordinates $\vec{x}=(x^{1},x^{2})$, with $k=\pm 1,0$.
Additionally, the scalar field compatible with the isometries of this metric
depends on the radial coordinate only, namely, $\phi=\phi(r)$.

As we mentioned above, the field equation for the connection can be solved for
the torsion in terms of the scalar field and derivatives thereof, whose
solution is given in Eq. (9). This, in turn, implies that the functions
$\omega_{I}(r)$, with $I=1,\ldots,8$, can be solved algebraically in terms of
the scalar field and the metric functions. The nontrivial pieces of the
connection are found to be

$\displaystyle\omega_{1}(r)$
$\displaystyle=\frac{1}{h(r)}\left[\sqrt{h(r)f(r)}\right]^{\prime}+\frac{(1-\lambda)\sqrt{f(r)}\,\phi(r)\phi^{\prime}(r)}{J(\phi)\sqrt{h(r)}}\,,$
(12a) $\displaystyle\omega_{5}(r)$
$\displaystyle=-\frac{\sqrt{f(r)}}{2r^{2}h^{3/2}(r)}\left[h(r)\,r^{2}\right]^{\prime}-\frac{(1-\lambda)\sqrt{f(r)}\,\phi(r)\phi^{\prime}(r)}{J(\phi)\sqrt{h(r)}}\,,$
(12b)

where prime denotes differentiation with respect to the radial coordinate,
i.e. ${}^{\prime}=\text{d}/\text{d}r$, and the other components of the
connection vanish on shell. The field equations for the vierbein and the
scalar field are solved by

$\displaystyle f(r)$
$\displaystyle=k\left(1+\frac{\mu\,G}{r}\right)^{2}-\frac{\Lambda
r^{2}}{3}\,,$ (13a) $\displaystyle h(r)$ $\displaystyle=\frac{r(r+2\mu
G)}{4(r+\mu G)^{2}}\left[2+\left(1+\frac{2\mu
G}{r}\right)^{\frac{1}{\sqrt{\lambda(2-\lambda)}}}+\left(1+\frac{2\mu
G}{r}\right)^{-\frac{1}{\sqrt{\lambda(2-\lambda)}}}\right]\,,$ (13b)
$\displaystyle\phi(r)$ $\displaystyle=-\sqrt{\frac{3}{4\pi
G}}\left[\frac{1-\left(1+\frac{2\mu
G}{r}\right)^{\frac{1}{\sqrt{\lambda(2-\lambda)}}}}{1+\left(1+\frac{2\mu
G}{r}\right)^{\frac{1}{\sqrt{\lambda(2-\lambda)}}}}\right]\,,$ (13c)

where $\mu$ is an integration constant. Notice that the solution in Eq. (13)
is continuously connected to that of Refs. Martinez  _et al._ (2004, 2006) in
the limit $\lambda\to 1$. Indeed, the causal structure of this configuration
is the same as the one in those references. If $\mu>0$ for
$r\in\mathbb{R}_{>0}$, then the scalar field is bounded as
$0<\phi<\sqrt{\frac{3}{4\pi G}}$. Conversely, if $\mu<0$, reality on the
scalar field implies that the range of the radial coordinate is $r>-2\mu G$
while the scalar field is bounded according to $-\sqrt{\frac{3}{4\pi
G}}<\phi<0$. Thus, since the scalar field is constant as $r\to 0$, we conclude
that it is real and everywhere regular if
$(2\lambda-\lambda^{2})^{-1/2}\in\mathbb{N}_{>1}$.

##  IV Properties of the solution

Let us discuss the most relevant properties of the solution. First, the
asymptotic behavior of the metric is modified by the presence of torsion when
$\lambda\neq 1$, since

$\displaystyle g_{tt}$ $\displaystyle\approx\frac{\Lambda
r^{2}}{3}-k-\frac{\Lambda\mu^{2}G^{2}(\lambda-1)^{2}}{3\lambda(\lambda-2)}-\frac{2\mu
G}{r}\left(k-\frac{\Lambda\mu^{2}G^{2}(\lambda-1)^{2}}{3\lambda(\lambda-2)}\right)+\mathcal{O}(r^{-2})\,,$
(14) $\displaystyle g_{rr}$ $\displaystyle\approx-\frac{3}{\Lambda
r^{2}}-\frac{9}{\Lambda^{2}r^{4}}\left(k-\frac{\Lambda\mu^{2}G^{2}(\lambda-1)^{2}}{3\lambda(\lambda-2)}\right)-\frac{18\mu
G}{\Lambda^{2}r^{5}}\left(k+\frac{\Lambda\mu^{2}G^{2}(\lambda-1)^{2}}{3\lambda(\lambda-2)}\right)+\mathcal{O}(r^{-6})\,.$
(15)

Notice that the presence of torsion generates an effective curvature of the
transverse section as one can see from the zeroth-order term in the radial
asymptotic expansion of $g_{tt}$. Additionally, we expect that this behavior
will change the value of the conserved charges as the torsion modifies the
value of the $\mathcal{O}(r^{-1})$ term as $r\to\infty$. Since $0<\lambda<1$,
one can see that the parameter $\mu$ can be negative (positive) for certain
values of $\Lambda>0$ ($\Lambda<0$).

This solution has a curvature singularity when $r\to 0$ as it can be seen by
computing its Kretschmann invariant. However, it is hidden behind a horizon at
$r=r_{h}$ defined by the largest positive root of the polynomial $f(r_{h})=0$.
Remarkably, if $\mu>0$, the presence of torsion renders the scalar field fully
regular, in contrast to their Riemannian counterpart which develops a scalar’s
singularity for a finite value of the radial coordinate. Additionally, there
exists a torsional invariant that can be computed from
$T=T_{\mu\nu\lambda}T^{\mu\nu\lambda}$. Evaluating the latter on the solution
of Eq. (13), we obtain

$\displaystyle T$
$\displaystyle=\frac{8G\mu^{2}(a^{2}-1)\left(\frac{G\mu}{r}+1\right)^{2}\left(\frac{2G\mu}{r}+1\right)^{a}\left[\left(\frac{2G\mu}{r}+1\right)^{a}-1\right]^{2}\left[3k\left(\frac{G\mu}{r}+1\right)^{2}-\Lambda
r^{2}\right]}{\pi
r^{4}\left(\frac{2G\mu}{r}+1\right)^{3}\left[\left(\frac{2G\mu}{r}+1\right)^{a}+1\right]^{4}}\,,$
(16)

where $a=(2\lambda-\lambda^{2})^{-1/2}$ and, since $\lambda\in(0,1]$, we have
that $a>1$.

In the case when $\Lambda=0$, the potential vanishes identically. From Eq.
(13), one notices that the existence of a horizon demands that $k=1$ and
$\mu<0$. Thus, the topology of the transverse section is that of
$\mathbb{S}^{2}$. In the limit $\lambda\to 1$, this configuration reduces to
the BBMB solution found in Refs. Bekenstein (1974); Bocharova  _et al._
(1970). If $0<\lambda<1$, however, the scalar field is regular at the horizon
located at $r=-\mu G$ due to the presence of torsion, in contrast to the
aforementioned solution. Nevertheless, an inspection of Eq. (16) reveals that
there exists a torsional singularity at $r=-2\mu G$ which lies outside of the
horizon. Therefore, this case represents a naked torsional singularity.

Let us focus on the case with negative cosmological constant
$\Lambda=-3\ell^{-2}$. First, notice that the existence of event horizons
demands that $k=-1$. In this case, the topology of the horizon is
$\mathbb{H}^{2}/\Gamma$, where $\Gamma$ is a discrete subgroup of $SO(2,1)$
such that the transverse section has finite volume. Then, there are two
possible solutions depending on the sign of $\mu$. For $\mu>0$, hairy
torsional black hole possess a single event horizon located at

$r_{+}=\frac{\ell}{2}\left(1+\sqrt{1+\frac{4\mu G}{\ell}}\right).$ (17)

The torsion and scalar field are regular over this hypersurface. However, a
curvature and torsional singularity occur at the origin $r=0$. Since the
connection is metric compatible, the causal nature of this black hole is
equivalent to that of Ref. Martinez  _et al._ (2006). Defining
$F(r):=h(r)f(r)$ and $H(r):=h^{-1}(r)f(r)$, the Hawking temperature in this
case is given

$\displaystyle
T_{H}=\frac{\sqrt{F^{\prime}(r)H^{\prime}(r)}}{4\pi}\Bigg{|}_{r=r_{+}}=\frac{1}{2\pi\ell}\left(\frac{2r_{+}}{\ell}-1\right)\,,$
(18)

which is positive definite by virtue of Eq. (17). For $\mu=0$, the scalar
field vanishes as well as the torsion and the metric becomes global AdS. If
$\mu<0$, the absence of naked singularities implies that the integration
constant must be bounded according to $\mu\geq-\tfrac{\ell}{4G}$. Then, the
solution has three horizons given by

$\displaystyle r_{--}$
$\displaystyle=\frac{\ell}{2}\left(-1+\sqrt{1-\frac{4\mu G}{\ell}}\right)\,,$
(19a) $\displaystyle r_{-}$
$\displaystyle=\frac{\ell}{2}\left(1-\sqrt{1+\frac{4\mu G}{\ell}}\right)\,,$
(19b) $\displaystyle r_{+}$
$\displaystyle=\frac{\ell}{2}\left(1+\sqrt{1+\frac{4\mu G}{\ell}}\right)\,.$
(19c)

Additionally, reality conditions on the torsion and scalar field demand that
$r\geq-2G\mu$. Then, the latter condition can be combined with bound
$\mu\geq-\tfrac{\ell}{4G}$ to give $r\geq\ell/2$. Indeed, one can see that a
torsional singularity occurs at $r=\ell/2$ [cf. Eq. (16)]. This, in turn,
implies that the autoparallels could end at the torsional singularity, before
reaching the curvature singularity. Nevertheless, the torsional singularity
lies behind the event horizon, namely, $r_{+}>\ell/2$, where $r_{+}$ is given
in Eq. (17) and, therefore, it does not represent a naked singularity.

In the case of a positive cosmological constant, say $\Lambda=3/\ell^{2}$,
with $\ell$ being the de Sitter radius, the absence of naked singularities
demands that $k=1$. If $\mu>0$, there is a unique cosmological horizon located
at $r_{+}$ where the anti-de Sitter radius must be replaced by the de Sitter
one in Eq. (17). Therefore, the lack of an event horizon implies that this
case represents a naked singularity. If $\mu=0$, the scalar field vanishes and
the metric is that of global de Sitter space. If $\mu<0$, however, there is a
torsional singularity at $r=\ell/2$ which lies between the event and the
cosmological horizon. Thus, this case is excluded by the cosmic censorship
conjecture. Finally, in the extremal case, namely, if $\mu=-\tfrac{\ell}{4G}$,
the event and cosmological horizons coalesce into a single horizon. However,
the torsion singularity is located at this single horizon, rendering the
latter a naked singularity. Therefore, in the presence of torsion, we conclude
that only the asymptotically anti-de Sitter black hole is admissible.

It is well known that the notion of geodesics and autoparallels do not
necessarily coincide if torsion is present. In the case of geodesics, the
Killing theorem implies that, if $u=u^{\mu}\partial_{\mu}$ is tangent to a
geodesic, i.e. $u^{\mu}\mathring{\nabla}_{\mu}u^{\nu}=0$, and
$\xi=\xi^{\mu}\partial_{\mu}$ is a Killing vector, then the product
$u^{\mu}\xi_{\mu}$ is constant along the geodesic. As discussed in Ref.
Peterson and Bonder (2019), this conservation law can be extended in the case
of autoparallels with tangent $v=v^{\mu}\partial_{\mu}$, i.e.
$v^{\mu}\nabla_{\mu}v^{\nu}=0$, by introducing a new notion of Killing vectors
such that they satisfy $\nabla_{(\mu}\xi_{\nu)}=0$, where $\nabla$ is the
torsionful connection. These are referred to as T-Killing vectors. The
existence of the latter can be analyzed by solving the T-Killing equation,
which can be written explicitly in terms of the contorsion as

$\mathring{\nabla}_{(\mu}\xi_{\nu)}-{K^{\lambda}}_{(\mu\nu)}\xi_{\lambda}=0\,.$
(20)

Using the definition of the contorsion $1$-form $K^{ab}$ given in Sec. II and
the solution of the torsion $2$-form in Eq. (9), we find that, projection onto
the spacetime components of the former, that is,
$K^{\alpha\beta}{}_{\mu}=E^{\alpha}{}_{a}E^{\beta}{}_{b}K^{ab}{}_{\mu}$,
yields

$K_{\rho\mu\nu}=\frac{2\kappa(\lambda-1)\phi}{6-\kappa\phi^{2}}g_{\nu[\rho}\partial_{\mu]}\phi\,.$
(21)

Assuming an ansatz for the time-like T-Killing vector of the form
$\xi=A(r)\partial_{t}$, then Eq. (20) becomes

$A^{\prime}(r)+\frac{\kappa(\lambda-1)\phi\phi^{\prime}}{6-\kappa\phi^{2}}A(r)=0\,,$
(22)

where $\phi=\phi(r)$. The general solution to this equation is

$A(r)=A_{0}\left(6-\kappa\phi^{2}\right){}^{\frac{\lambda-1}{2}}\,,$ (23)

with $A_{0}$ being an integration constant. Notice that, since
$\phi^{2}\neq\tfrac{6}{\kappa}$, this vector is nondegenerate. Moreover, in
the limit of vanishing torsion, i.e. $\lambda\to 1$, the standard time-like
Killing vector that generates the temporal isometries is recovered. Indeed,
the norm of this T-Killing vector is given by

$\displaystyle\xi\cdot\xi=-h(r)f(r)A^{2}(r)\,.$ (24)

Thus, since $\xi$ is nondegenerate, we see that it becomes null at the
horizon. Therefore, we conclude that the horizon defined at Eq. (17) is also a
T-Killing horizon.

##  V Discussion

In this work, we consider a one-parameter family of conformal transformations
in the first-order formalism of gravity. To accomplish this, we exploit the
fact that the vielbein and spin connection are regarded as independent fields
in this setup. Then, by considering the standard conformal weight for a scalar
field, we construct a conformal coupling in the presence of torsion. The
dynamics is dictated by the Einstein-Cartan term and the scalar-tensor
conformal coupling alongside a scalar potential (6). The nonminimal coupling
of the scalar field induces a nontrivial torsion. Remarkably, we find that
there is a particular value for the parameter $\lambda$ in Eq.(5b) that sets
the torsion to zero without trivializing the scalar field. Indeed, this
particular point leaves the torsion invariant under conformal transformations,
while the metric and the scalar field transform in the standard way.

To look for black hole solutions, we assume a static ansatz for the metric,
connection, and scalar field. We solve the field equations analytically in the
presence of a one-parameter extension of the quartic scalar potential that
becomes conformal in the limit $\lambda\to 1$. We obtain an asymptotically AdS
black hole solution with a compact horizon of negative curvature and
nontrivial torsion dressed with scalar fields. In the vanishing torsion limit,
the solution is continuously connected to that of Refs. Martinez  _et al._
(2003, 2006). Remarkably, the torsion renders the scalar field everywhere
regular, in contrast to the black hole solution in which the torsion vanishes.
Nevertheless, there appears a torsional singularity that lies behind the event
horizon, such that there is no violation of the cosmic censorship conjecture.
We solve the T-Killing equation Peterson and Bonder (2019) and conclude that
the event horizon is also a T-Killing horizon.

Interesting questions remain open. First, an asymptotic analysis of the
solution provides evidence that the presence of torsion would modify the
asymptotic charges. The computation of the latter is certainly relevant for
understanding how torsion modifies the global properties of the solution. A
deeper analysis of the latter will provide a starting point to study the
thermodynamic properties of the solution. In fact, there have been some
approaches for the calculation of black hole entropy in other theories with
nontrivial torsion Cvetković and Rakonjac (2023); Blagojević and Cvetković
(2022). Motivated by these results, it would be interesting to obtain the free
energy to first order in the saddle-point approximation and determine whether
the system develops a phase transition between a maximally symmetric space and
a configuration with non-vanishing torsion. This would imply that, above a
certain critical temperature, the torsional configuration would be
thermodynamically preferred. We postpone a detailed analysis of these
questions for a future contribution.

###### Acknowledgements.

We thank Giorgos Anastasiou, Eloy Ayón-Beato, Yuri Bonder, Branislav
Cvetkovic, Daniel Flores-Alfonso, Oscar Fuentealba, Mokhtar Hassaine, Diego
Hidalgo and Marcela Lagos for insightful comments and remarks. The work of
C.C. is partially supported by Agencia Nacional de Investigación y Desarrollo
(ANID) through Fondecyt grants No1240043, 1240048, 1210500, and 1230112. L.A.
is supported by Fondecyt grant No3220805\. O.V. acknowledges Fondecyt grant
No11200742\. F.I. acknowledges financial support from the Chilean government
through Fondecyt grants No1150719, 1180681 and 1211219. F.I. is thankful of
the emotional support of the Netherlands Bach Society. They made freely
available superb quality recordings of the music of Bach, and without them,
this work would have been impossible.

##  Appendix A Action and field equations in tensor components

For the sake of comparison, here we provide the action and field equations in
tensor components to undertake the analysis on a coordinate basis. In this
case, the action principle in Eq. (4) can be rewritten as

$\displaystyle I$
$\displaystyle=\int_{\mathcal{M}_{4}}\text{d}^{4}x\sqrt{|g|}\left[\frac{1}{2\kappa}\left(R-2\Lambda\right)-\frac{1}{12}\phi^{2}R-\lambda\left[1-\frac{\lambda}{2}\right](\nabla\phi)^{2}-\frac{\lambda}{3}\phi
T_{\lambda}\nabla^{\lambda}\phi-V\left(\phi\right)\right]\,,$ (25)

where $T_{\lambda}:=T^{\sigma}{}_{\lambda\sigma}$ is the trace of the torsion
tensor. The field equation for the vierbein can be written as

$G_{\mu\nu}+g_{\mu\nu}\Lambda=\kappa\,\tau_{\mu\nu}\,,$ (26)

where $G_{\mu\nu}$ denotes the Einstein tensor constructed out of the
torsionful connection and

$\displaystyle\tau_{\mu\nu}$
$\displaystyle=\frac{1}{6}\phi^{2}G_{\mu\nu}-g_{\mu\nu}V\left(\phi\right)-\frac{1}{2}\lambda\left(\lambda-2\right)g_{\mu\nu}(\nabla\phi)^{2}+\frac{1}{3}\lambda\phi\nabla_{\mu}\phi
T_{\nu}$
$\displaystyle+\frac{1}{3}\lambda\left(3\lambda-5\right)\left[g_{\mu\nu}(\nabla\phi)^{2}-\nabla_{\mu}\phi\nabla_{\nu}\phi\right]+\frac{1}{3}\lambda\phi\left(g_{\mu\nu}\Box\phi-\nabla_{\nu}\nabla_{\mu}\phi\right)\,,$
(27)

is the stress-energy tensor for the conformally-coupled scalar field. Here,
$\Box=\nabla_{\mu}\nabla^{\mu}$ is constructed out of the torsionful covariant
derivative. It is worth noticing that the field equation Eq. (26) is not
symmetric in general since
$\left[\nabla_{\mu},\nabla_{\nu}\right]\phi=T^{\lambda}{}_{\mu\nu}\nabla_{\lambda}\phi$.
The skew-symmetric piece arises from the fact that the vielbein is not
necessarily symmetric in its two indices; the antisymmetric components are
related to the presence of torsion.

The field equation for the scalar field is given by

$2\lambda\left(1-\frac{\lambda}{2}\right)\Box\phi-\frac{1}{6}\phi
R-\frac{\partial
V}{\partial\phi}-\lambda\left(\lambda-2\right)T^{\mu}\nabla_{\mu}\phi+\frac{1}{3}\lambda\phi\nabla_{\mu}T^{\mu}+\frac{1}{3}\lambda\phi
T_{\mu}T^{\mu}=0\,,$ (28)

Finally, the field equation for the spin connection is given by

$T_{\leavevmode\nobreak\
\alpha\beta}^{\mu}+2\delta^{\mu}_{[\alpha}T_{\beta]}=-\frac{2\kappa\left(\lambda-1\right)}{\left(6-\kappa\phi^{2}\right)}\delta^{\mu}_{[\alpha}\nabla_{\beta]}\phi^{2}\,,$
(29)

whose solution is

$T_{\leavevmode\nobreak\
\mu\nu}^{\rho}=\frac{\kappa\left(\lambda-1\right)}{\left(6-\kappa\phi^{2}\right)}\,\delta^{\rho}_{[\mu}\nabla_{\nu]}\phi^{2}\,.$
(30)

## References

  * Kormendy and Richstone (1995) J. Kormendy and D. Richstone, Ann. Rev. Astron. Astrophys. 33, 581 (1995).
  * Celotti  _et al._ (1999) A. Celotti, J. C. Miller, and D. W. Sciama, Class. Quant. Grav. 16, A3 (1999), arXiv:astro-ph/9912186 .
  * Gillessen  _et al._ (2009) S. Gillessen, F. Eisenhauer, S. Trippe, T. Alexander, R. Genzel, F. Martins, and T. Ott, Astrophys. J. 692, 1075 (2009), arXiv:0810.4674 [astro-ph] .
  * Muñoz  _et al._ (2011) J. A. Muñoz, E. Mediavilla, C. S. Kochanek, E. E. Falco, and A. M. Mosquera, The Astrophysical Journal 742, 67 (2011).
  * Abbott  _et al._ (2016) B. P. Abbott _et al._ (LIGO Scientific, Virgo), Phys. Rev. Lett. 116, 061102 (2016), arXiv:1602.03837 [gr-qc] .
  * Akiyama  _et al._ (2022) K. Akiyama _et al._ (Event Horizon Telescope), Astrophys. J. Lett. 930, L12 (2022), arXiv:2311.08680 [astro-ph.HE] .
  * Volkov and Galtsov (1989) M. S. Volkov and D. V. Galtsov, JETP Lett. 50, 346 (1989).
  * Bizon (1990) P. Bizon, Phys. Rev. Lett. 64, 2844 (1990).
  * Kuenzle and Masood-ul Alam (1990) H. P. Kuenzle and A. K. M. Masood-ul Alam, J. Math. Phys. 31, 928 (1990).
  * Baltsov and Volkov (1992) D. V. Baltsov and M. S. Volkov, Phys. Lett. B 274, 173 (1992).
  * Israel (1967) W. Israel, Phys. Rev. 164, 1776 (1967).
  * Israel (1968) W. Israel, Commun. Math. Phys. 8, 245 (1968).
  * Carter (1971) B. Carter, Phys. Rev. Lett. 26, 331 (1971).
  * Ruffini and Wheeler (1971) R. Ruffini and J. A. Wheeler, Phys. Today 24, 30 (1971).
  * Bekenstein (1995) J. D. Bekenstein, Phys. Rev. D 51, R6608 (1995).
  * Chrusciel  _et al._ (2012) P. T. Chrusciel, J. Lopes Costa, and M. Heusler, Living Rev. Rel. 15, 7 (2012), arXiv:1205.6112 [gr-qc] .
  * Janis  _et al._ (1968) A. I. Janis, E. T. Newman, and J. Winicour, Phys. Rev. Lett. 20, 878 (1968).
  * Janis  _et al._ (1969) A. I. Janis, D. C. Robinson, and J. Winicour, Phys. Rev. 186, 1729 (1969).
  * Bekenstein (1974) J. D. Bekenstein, Annals Phys. 82, 535 (1974).
  * Bocharova  _et al._ (1970) N. M. Bocharova, K. A. Bronnikov, and V. N. Melnikov, Vestn. Mosk. Univ. Fiz. Astron. 6, 706 (1970).
  * Bekenstein (1975) J. D. Bekenstein, Annals Phys. 91, 75 (1975).
  * Sudarsky and Zannias (1998) D. Sudarsky and T. Zannias, Phys. Rev. D 58, 087502 (1998), arXiv:gr-qc/9712083 .
  * Martinez  _et al._ (2003) C. Martinez, R. Troncoso, and J. Zanelli, Phys. Rev. D 67, 024008 (2003), arXiv:hep-th/0205319 .
  * Martinez  _et al._ (2006) C. Martinez, J. P. Staforelli, and R. Troncoso, Phys. Rev. D 74, 044028 (2006), arXiv:hep-th/0512022 .
  * Anabalon and Maeda (2010) A. Anabalon and H. Maeda, Phys. Rev. D 81, 041501 (2010), arXiv:0907.0219 [hep-th] .
  * Charmousis  _et al._ (2009) C. Charmousis, T. Kolyvaris, and E. Papantonopoulos, Class. Quant. Grav. 26, 175012 (2009), arXiv:0906.5568 [gr-qc] .
  * Astorino (2015) M. Astorino, Phys. Rev. D 91, 064066 (2015), arXiv:1412.3539 [gr-qc] .
  * Anabalon and Cisterna (2012) A. Anabalon and A. Cisterna, Phys. Rev. D 85, 084035 (2012), arXiv:1201.2008 [hep-th] .
  * Bardoux  _et al._ (2012) Y. Bardoux, M. M. Caldarelli, and C. Charmousis, JHEP 09, 008 (2012), arXiv:1205.4025 [hep-th] .
  * Astorino (2013) M. Astorino, Phys. Rev. D 88, 104027 (2013), arXiv:1307.4021 [gr-qc] .
  * Bardoux  _et al._ (2014) Y. Bardoux, M. M. Caldarelli, and C. Charmousis, JHEP 05, 039 (2014), arXiv:1311.1192 [hep-th] .
  * Cisterna  _et al._ (2021) A. Cisterna, A. Neira-Gallegos, J. Oliva, and S. C. Rebolledo-Caceres, Phys. Rev. D 103, 064050 (2021), arXiv:2101.03628 [gr-qc] .
  * Barrientos  _et al._ (2022a) J. Barrientos, A. Cisterna, N. Mora, and A. Viganò, Phys. Rev. D 106, 024038 (2022a), arXiv:2202.06706 [hep-th] .
  * Barrientos  _et al._ (2022b) J. Barrientos, A. Cisterna, C. Corral, and M. Oyarzo, JHEP 05, 110 (2022b), arXiv:2202.13854 [hep-th] .
  * Babichev  _et al._ (2022) E. Babichev, C. Charmousis, M. Hassaine, and N. Lecoeur, Phys. Rev. D 106, 064039 (2022), arXiv:2206.11013 [gr-qc] .
  * Xanthopoulos and Dialynas (1992) B. C. Xanthopoulos and T. E. Dialynas, J. Math. Phys. 33, 1463 (1992).
  * Klimcik (1993) C. Klimcik, J. Math. Phys. 34, 1914 (1993).
  * Giribet  _et al._ (2014) G. Giribet, M. Leoni, J. Oliva, and S. Ray, Phys. Rev. D 89, 085040 (2014), arXiv:1401.4987 [hep-th] .
  * Chernicoff  _et al._ (2016) M. Chernicoff, M. Galante, G. Giribet, A. Goya, M. Leoni, J. Oliva, and G. Perez-Nadal, JHEP 06, 159 (2016), arXiv:1604.08203 [hep-th] .
  * Babichev  _et al._ (2023) E. Babichev, C. Charmousis, M. Hassaine, and N. Lecoeur, Phys. Rev. D 107, 084050 (2023), arXiv:2302.02920 [gr-qc] .
  * Barrientos  _et al._ (2017) J. Barrientos, F. Cordonier-Tello, F. Izaurieta, P. Medina, D. Narbona, E. Rodríguez, and O. Valdivia, Phys. Rev. D 96, 084023 (2017), arXiv:1703.09686 [gr-qc] .
  * Zwiebach (1985) B. Zwiebach, Phys. Lett. B 156, 315 (1985).
  * Charmousis (2015) C. Charmousis, Lect. Notes Phys. 892, 25 (2015), arXiv:1405.1612 [gr-qc] .
  * Lu and Pang (2020) H. Lu and Y. Pang, Phys. Lett. B 809, 135717 (2020), arXiv:2003.11552 [gr-qc] .
  * Hennigar  _et al._ (2020) R. A. Hennigar, D. Kubizňák, R. B. Mann, and C. Pollack, JHEP 07, 027 (2020), arXiv:2004.09472 [gr-qc] .
  * Ma and Lu (2020) L. Ma and H. Lu, Eur. Phys. J. C 80, 1209 (2020), arXiv:2004.14738 [gr-qc] .
  * Toloza and Zanelli (2013) A. Toloza and J. Zanelli, Class. Quant. Grav. 30, 135003 (2013), arXiv:1301.0821 [gr-qc] .
  * Espiro and Vásquez (2016) J. L. Espiro and Y. Vásquez, Gen. Rel. Grav. 48, 117 (2016), arXiv:1410.3152 [gr-qc] .
  * Castillo-Felisola  _et al._ (2016) O. Castillo-Felisola, C. Corral, S. del Pino, and F. Ramírez, Phys. Rev. D94, 124020 (2016), arXiv:1609.09045 [gr-qc] .
  * Cid  _et al._ (2018) A. Cid, F. Izaurieta, G. Leon, P. Medina, and D. Narbona, JCAP 1804, 041 (2018), arXiv:1704.04563 [gr-qc] .
  * Hojman  _et al._ (1980) R. Hojman, C. Mukku, and W. A. Sayed, Phys. Rev. D 22, 1915 (1980).
  * Nelson (1980) P. C. Nelson, Phys. Lett. A 79, 285 (1980).
  * Nieh and Yan (1982a) H. T. Nieh and M. L. Yan, J. Math. Phys. 23, 373 (1982a).
  * Chandia and Zanelli (1997) O. Chandia and J. Zanelli, Phys. Rev. D55, 7580 (1997), arXiv:hep-th/9702025 [hep-th] .
  * Mercuri (2009) S. Mercuri, Phys. Rev. Lett. 103, 081302 (2009), arXiv:0902.2764 [gr-qc] .
  * Lattanzi and Mercuri (2010) M. Lattanzi and S. Mercuri, Phys. Rev. D81, 125015 (2010), arXiv:0911.2698 [gr-qc] .
  * Castillo-Felisola  _et al._ (2015) O. Castillo-Felisola, C. Corral, S. Kovalenko, I. Schmidt, and V. E. Lyubovitskij, Phys. Rev. D91, 085017 (2015), arXiv:1502.03694 [hep-ph] .
  * Campbell  _et al._ (1990) B. A. Campbell, M. J. Duncan, N. Kaloper, and K. A. Olive, Phys. Lett. B 251, 34 (1990).
  * Duncan  _et al._ (1992) M. J. Duncan, N. Kaloper, and K. A. Olive, Nucl. Phys. B 387, 215 (1992).
  * Cisterna  _et al._ (2019) A. Cisterna, C. Corral, and S. del Pino, Eur. Phys. J. C79, 400 (2019), arXiv:1809.02903 [gr-qc] .
  * Corral  _et al._ (2022) C. Corral, C. Erices, D. Flores-Alfonso, and K. Lara, Phys. Rev. D 105, 024050 (2022), arXiv:2111.00912 [hep-th] .
  * Jackiw and Pi (2003) R. Jackiw and S. Y. Pi, Phys. Rev. D68, 104012 (2003), arXiv:gr-qc/0308071 [gr-qc] .
  * Alexander and Yunes (2008) S. Alexander and N. Yunes, Phys. Rev. D77, 124040 (2008), arXiv:0804.1797 [gr-qc] .
  * Alexander and Yunes (2009) S. Alexander and N. Yunes, Phys. Rept. 480, 1 (2009), arXiv:0907.2562 [hep-th] .
  * Yunes and Pretorius (2009) N. Yunes and F. Pretorius, Phys. Rev. D79, 084043 (2009), arXiv:0902.4669 [gr-qc] .
  * Konno  _et al._ (2009) K. Konno, T. Matsuyama, and S. Tanda, Prog. Theor. Phys. 122, 561 (2009), arXiv:0902.4767 [gr-qc] .
  * Amarilla  _et al._ (2010) L. Amarilla, E. F. Eiroa, and G. Giribet, Phys. Rev. D 81, 124045 (2010), arXiv:1005.0607 [gr-qc] .
  * Nashed and Nojiri (2023) G. G. L. Nashed and S. Nojiri, Phys. Rev. D 107, 064069 (2023), arXiv:2303.07349 [gr-qc] .
  * Barrientos  _et al._ (2019a) J. Barrientos, F. Cordonier-Tello, C. Corral, F. Izaurieta, P. Medina, E. Rodríguez, and O. Valdivia, Phys. Rev. D 100, 124039 (2019a), arXiv:1910.00148 [gr-qc] .
  * Sulantay  _et al._ (2023) F. Sulantay, M. Lagos, and M. Bañados, Phys. Rev. D 107, 104025 (2023), arXiv:2211.08925 [gr-qc] .
  * Nieh and Yan (1982b) H. T. Nieh and M. L. Yan, Annals Phys. 138, 237 (1982b).
  * Chakrabarty and Lahiri (2018) S. Chakrabarty and A. Lahiri, Eur. Phys. J. Plus 133, 242 (2018), arXiv:1907.02341 [gr-qc] .
  * Izaurieta  _et al._ (2020) F. Izaurieta, P. Medina, N. Merino, P. Salgado, and O. Valdivia, JHEP 10, 150 (2020), arXiv:2007.07226 [gr-qc] .
  * Grumiller  _et al._ (2014) D. Grumiller, M. Irakleidou, I. Lovrekovic, and R. McNees, Phys. Rev. Lett. 112, 111102 (2014), arXiv:1310.0819 [hep-th] .
  * Anastasiou  _et al._ (2021) G. Anastasiou, I. J. Araya, C. Corral, and R. Olea, JHEP 07, 156 (2021), arXiv:2105.02924 [hep-th] .
  * Anastasiou  _et al._ (2022) G. Anastasiou, I. J. Araya, and R. Olea, JHEP 10, 123 (2022), arXiv:2209.02006 [hep-th] .
  * Anastasiou  _et al._ (2023a) G. Anastasiou, I. J. Araya, C. Corral, and R. Olea, JHEP 11, 036 (2023a), arXiv:2308.09140 [hep-th] .
  * Anastasiou  _et al._ (2023b) G. Anastasiou, I. J. Araya, M. Busnego-Barrientos, C. Corral, and N. Merino, Phys. Rev. D 107, 104049 (2023b), arXiv:2212.04364 [hep-th] .
  * Maldacena (2011) J. Maldacena, (2011), arXiv:1105.5632 [hep-th] .
  * Anastasiou and Olea (2016) G. Anastasiou and R. Olea, Phys. Rev. D 94, 086008 (2016), arXiv:1608.07826 [hep-th] .
  * Barrientos  _et al._ (2019b) J. Barrientos, F. Izaurieta, E. Rodríguez, and O. Valdivia, (2019b), arXiv:1903.04712 [gr-qc] .
  * Martinez  _et al._ (2004) C. Martinez, R. Troncoso, and J. Zanelli, Phys. Rev. D 70, 084035 (2004), arXiv:hep-th/0406111 .
  * Peterson and Bonder (2019) C. Peterson and Y. Bonder, Mod. Phys. Lett. A 35, 2050052 (2019), arXiv:1904.12913 [gr-qc] .
  * Cvetković and Rakonjac (2023) B. Cvetković and D. Rakonjac, Phys. Rev. D 107, 044054 (2023), arXiv:2208.04383 [gr-qc] .
  * Blagojević and Cvetković (2022) M. Blagojević and B. Cvetković, Phys. Rev. D 105, 104014 (2022), arXiv:2203.14696 [gr-qc] .