Spontaneous Scalarization of Black Holes in Gauss-Bonnet Teleparallel Gravity

General Relativity (GR) predicts that in vacuum, there is a unique spherically symmetric solution: the Schwarzschild metric, which is characterised only by the mass $M$ of the gravitational object. Adding electromagnetic fields to the theory, i.e., considering the Einstein-Maxwell case, then the unique spherically symmetric and asymptotically flat black hole solutions are characterized by two parameters, the mass and the charge $Q$ of the gravitational object. Further, by considering a rotating scenario, the angular momentum $J$ appears as an additional parameter in Einstein-Maxwell black hole solutions and measures the amount of rotation of the body. Thus, when one describes stationary and asymptotically flat black holes in GR with minimally coupled electromagnetism, the solutions are uniquely characterised by three parameters ($M,Q,J$) only. This fact is known as Israel’s no-hair theorem [1, 2].

One of the well-studied ways of violating the no-hair theorem is by considering a theory consisting of a scalar field nonminimally coupled to the Gauss-Bonnet invariant [3, 4, 5], which in the context of scalarized black holes is labeled as scalar Gauss-Bonnet (sGB) gravity. It has been shown that for some coupling functions, there are scalarized black hole solutions where the black hole scalar charge emerges from a spontaneous scalarization process. In this process, the predictions of the theory match the predictions of GR in the weak field regime but differ in the strong field regime. This mechanism was first demonstrated for black holes  [6, 7, 8] and it shares many similarities with the neutron star scalarization observed in scalar-tensor theories [9, 10]. The process for black holes works as follows: The described black hole coincides with Schwarzschild one for weak gravitational fields when the spacetime curvature near the horizon is too weak to source the scalar field. For strong gravitational fields at the horizon, that realizes when the black hole mass $M$ falls below a certain threshold, the Schwarzschild solution becomes unstable, and a non-trivial scalar field emerges. Thus, the solution bifurcates to a different black hole solution with scalar hair. This transition is usually smooth in sGB and shares similarities with second order phase transitions. This idea is analog for charged black holes in, e.g., Einstein-Maxwell-scalar gravity  [11] or black holes in multi-scalar Gauss-Bonnet theories [12, 13]. After these findings, several studies have suggested that those scalarized black holes could be important to describe astrophysical scenarios in the strong regime [14, 15, 16, 17, 18, 19].

All the studies mentioned so far assumed from the beginning that the geometrical description of gravity is based on Riemannian geometry, i.e. that the gravitational field is encoded in a spacetime metric and its canonical torsion free and metric compatible Levi-Civita connection. When one considers a more general geometry as it is done in the metric-affine gravity formalism, one can obtain black hole solutions beyond Schwarzschild (and Kerr in axial symmetry) [20, 21, 22, 23]. Another alternative and interesting description of gravity can be formulated in terms of tetrads and an independent flat ($R^{\lambda}{}_{\mu\nu\beta}=0$) metric compatible ($\nabla_{\mu}g_{\alpha\beta}=0$) spin connection which thus possesses only torsion¹¹1Note that, if the curvature $R^{\lambda}{}_{\mu\nu\beta}$ of the independent connection is vanishing, the curvature $\accentset{\circ}{R}^{\lambda}{}_{\mu\nu\beta}$ of the Levi-Civita connection associated to the metric by the tetrads is not.. Those theories are labeled as torsional Teleparallel theories of gravity (or metric Teleparallel theories) [24, 25]. In that framework, one can formulate an equivalent theory to GR which is known as the Teleparallel equivalent of GR (TEGR) where the action is constructed solely by quadratic contractions of the torsion summed with specific coefficients forming the so-called torsion scalar $T$ [24, 26]. Since GR and TEGR are dynamically equivalent, their predictions are equivalent and then, one cannot distinguish them by making any classical experiment.

Motivated by the fact that GR might not be the final theory of gravity due to its theoretical and observational challenges (see [27, 28, 29, 30, 31, 32, 33] for some of them), one can further follow a similar route as it is done in the Riemannian case and consider modified gravity in the Teleparallel framework. A very interesting aspect of such theories is that they can be interpreted as a gauge theory of translations [34], which is a subcase of Poincaré gauge theory [35, 36]. The mathematical details of modified Teleparallel theories are still under intense investigation since not all details have been understood from the theoretical point of view yet. The theories retain local Lorentz invariance as symmetry when one transforms simultaneously all the fundamental fields involved, i.e. the tetrad and the spin connection [37, 38, 39, 40]. However, there are suggestions that they also predict strong couplings, i.e., there are some gravitational modes that are strongly coupled in different backgrounds [41, 42, 43, 44, 45, 46].

One of the simplest and famous ones is $f(T)$ gravity where one upgrades the TEGR action $T$ to an arbitrary function [47, 48, 49]. There have been several applications to that theory in the context of cosmology [50, 51, 52, 53, 54, 55] and recently on describing black holes [56, 57, 58, 59, 60, 61, 62, 63]. Other alternative Teleparallel theories have been proposed such as $f(T,B)$ gravity where $B$ is the boundary term relating the Ricci scalar $\mathring{R}$ of the Levi-Civita connection with the torsion scalar $T$ [64, 65, 66], or New General Relativity (NGR) [67, 68], where the specific coupling coefficients in the torsion scalar are left arbitrary [69, 70]. In $f(T,B)$ gravity, $f(\mathring{R})$ is obtained as a particular limit, but still, when one considers the purely Teleparallel framework, it might suffer from strongly coupled modes in cosmology [71].

Another route to generalize TEGR is to couple one scalar field $\Psi$ minimally or nonminimally to torsion. Since Teleparallel gravity comes with a canonical boundary term $B$. The simplest way to do this is by taking a Lagrangian like $L=F(\psi)T+G(\psi)B+\partial_{\mu}\psi\partial^{\mu}\psi+V(\psi)$ from where $f(T,B)$ gravity can be obtained in a particular scalar-tensor representation, and also other more general Teleparallel theories can be obtained [72, 73, 74, 75, 76, 77]. Furthermore, that theory contains the standard Riemannian scalar-tensor $L=F(\psi)\accentset{\circ}{R}+\partial_{\mu}\psi\partial^{\mu}\psi+V(\psi)$ theory by choosing $G(\psi)=-F(\psi)$.

Recently, there has been found that Teleparallel scalar-torsion theories of gravity, for some coupling functions, have scalarized black hole solutions such as the Bocharova–Bronnikov–Melnikov–Bekenstein solution and other new solutions that do not exist in the Riemannian case [78]. However, all those solutions have a scalar hair not being independent of the mass of the black hole, which is similar to the behaviour of scalarized black holes in the Riemannian theory with non-minimal coupling to the Ricci scalar [79, 80]. Further, no spontaneous scalarization mechanism has been found for that theory. Therefore, the question of whether Teleparallel gravity could offer a realistic description of scalarized black holes is still open.

In this paper, our main objective is to obtain the first Teleparallel scalarized black hole solutions with spontaneous scalarization. To do this, we will formulate a Teleparallel sGB (TsGB) theory constructed from nonminimal couplings between the torsional Gauss-Bonnet invariants and a scalar field. Similarly to what happens in the Riemannian case, those invariants are boundary terms in four dimensions [81, 82] and therefore, their dynamics would also be nontrivial only when the scalar field is nonminimally coupled to them [83]. As we will show later, our theory can be recast as a combination of the sGB theory plus some additional Teleparallel terms. We will then show that those new contributions can generate spontaneous scalarized black hole solutions triggered by torsion. These could open a new way to study astrophysical objects from non-Riemannian geometries.

This paper is organised as follows. In Sec. II we briefly introduce torsional Teleparallel gravity and the TEGR description of GR whose equations are equivalent to GR. Then, we also discuss the Teleparallel Gauss-Bonnet invariants and how they are related to the Riemannian framework. In Sec. III we present the Teleparallel sGB theory studied in this paper and present its field equations and also its relationship with respect to the sGB theory. Sec. IV is devoted to explaining how to work in spherical symmetries for Teleparallel gravity and the corresponding field equations for spherical symmetry for our theory. Sec. V contains the most important part of our manuscript where we showed the existence of scalarized black hole solutions. First, in Sec. V.1, we found analytical perturbed solutions around Schwarzschild where one notices a scalar charge appearing as an extra parameter. Secondly, in Sec. V.2, we studied the behaviour of the field equations at infinity and near the horizons where we found some conditions that we will use for the numerical study. The spontaneous scalarization process for this theory is presented in V.3 and then in V.4 we numerically solve the equations and find new and novel scalarized black hole solutions. We conclude our main results in Sec. VI.

In this paper, we work with the metric signature $(+---)$ and we use Latin indices for tangent spacetime and Greek indices for spacetime indices. Further, an upper $\circ$ symbol will be used to denote quantities computed with the Levi-Civita connection (Riemannian case).

The fundamental variables of metric Teleparallel gravity are and a flat & metric-compatible connection with torsion, with connection coefficients $\omega^{a}{}_{b\mu}$, and tetrad fields $e^{a}=e^{a}{}_{\mu}\differential x^{\mu}$ (i.e. orthonormal frames), see; [24, 25]. Gravity is encoded in the torsion tensor defined as the antisymmetric part of the connection

$\displaystyle T^{a}{}_{\mu\nu}=2\left(\partial_{[\mu}e^{a}{}_{\nu]}+\omega^{a}{}_{b[\mu}e^{b}{}_{\nu]}\right)\,.$

(1)

Because the connection is both flat and metric-compatible, the spin connection coefficients $\omega^{a}{}_{b\mu}$ are purely gauge degrees of freedom, and one can always choose a gauge such that this quantity vanishes (Weitzenböck gauge) [37, 38, 39, 40]. This means that it is sufficient to only consider the tetrads as the dynamical variables of the theory (keeping in mind this gauge choice). The metric and its inverse can be reproduced from the tetrads as

$\displaystyle g_{\mu\nu}=\eta_{ab}e^{a}{}_{\mu}e^{b}{}_{\nu}\quad\text{and}\quad g^{\mu\nu}=\eta^{ab}e_{a}{}^{\mu}e_{b}{}^{\nu}\,,$

(2)

while they satisfy $e^{a}{}_{\mu}e_{b}{}^{\mu}=\delta^{a}_{b}$, and $e^{a}{}_{\mu}e_{a}{}^{\nu}=\delta_{\mu}^{\nu}$ and $\eta_{ab}=\textrm{diag}(1,-1,-1,-1)$ is the Minkowski metric.

It is convenient to define the contortion tensor as

$\displaystyle K^{\rho}{}_{\mu\nu}$

$\displaystyle=$

$\displaystyle\Gamma^{\rho}{}_{\mu\nu}-\accentset{\circ}{\Gamma}^{\rho}{}_{\mu\nu}=\frac{1}{2}\left(T_{\mu}{}^{\rho}{}_{\nu}+T_{\nu}{}^{\rho}{}_{\mu}-T^{\rho}{}_{\mu\nu}\right)\,,$

(3)

which measures the difference between the Levi-Civita connection $\accentset{\circ}{\Gamma}^{\rho}{}_{\mu\nu}$ and the Teleparallel connection $\Gamma^{\rho}{}_{\mu\nu}$. Up to quadratic contraction of the torsion tensor, one can build three possible non-parity violating invariants:

$\displaystyle T_{\alpha\beta\gamma}T^{\alpha\beta\gamma}\,,\quad T_{\alpha\beta\gamma}T^{\beta\alpha\gamma}\,,\quad T^{\lambda}{}_{\lambda\mu}T_{\beta}{}^{\beta\mu}\,,$

(4)

from where one can construct the simplest Teleparallel theories of gravity by considering a Lagrangian formed by a linear combination with arbitrary coefficients $L=a_{1}T_{\alpha\beta\gamma}T^{\alpha\beta\gamma}+a_{2}T_{\alpha\beta\gamma}T^{\beta\alpha\gamma}+a_{3}T^{\lambda}{}_{\lambda\mu}T_{\beta}{}^{\beta\mu}$ ²²2Which is the Lagrangian defining NGR [67].. It turns out that if $(a_{1},a_{2},a_{3})=(\tfrac{1}{4},\tfrac{1}{2},-1)$, one can have a special scalar known as the torsion scalar

$\displaystyle T=\frac{1}{4}T_{\alpha\beta\gamma}T^{\alpha\beta\gamma}+\frac{1}{2}T_{\alpha\beta\gamma}T^{\beta\alpha\gamma}-T^{\lambda}{}_{\lambda\mu}T_{\beta}{}^{\beta\mu}\,,$

(5)

which differs by a boundary term $B$ with respect to the Levi-Civita Ricci scalar, namely,

$\displaystyle\accentset{\circ}{R}=-T+B=-T+\frac{2}{e}\partial_{\mu}(e\,T_{\beta}{}^{\beta\mu})\,,$

(6)

where $e=\sqrt{-g}$ is the determinant of the tetrad. Due to the above equation, one can formulate a theory having the same equations as GR by just considering $T$ linearly in an action:

$\displaystyle S_{\rm TEGR}=-\frac{1}{2\kappa^{2}}\int T\,e\,\differential^{4}x.$

(7)

One can then further consider higher order contractions of the torsion tensor. Some interesting scalars that one can consider are the Teleparallel Gauss-Bonnet invariants that, similarly as (6), can be related to the Riemannian Gauss-Bonnet invariant $\accentset{\circ}{G}$ as

$\displaystyle\accentset{\circ}{G}=T_{G}+B_{G}\,,$

(8)

where the Riemannian part is defined as

$\displaystyle\accentset{\circ}{G}=\accentset{\circ}{R}_{\alpha\beta\mu\nu}\accentset{\circ}{R}^{\alpha\beta\mu\nu}-4\accentset{\circ}{R}_{\alpha\beta}\accentset{\circ}{R}^{\alpha\beta}+\accentset{\circ}{R}^{2}\,,$

(9)

and the Teleparallel Gauss-Bonnet invariants are [81, 82]

	$\displaystyle T_{G}$	$\displaystyle=$	$\displaystyle\delta^{\mu\nu\sigma\lambda}_{\alpha\beta\gamma\epsilon}K^{\alpha}{}_{\chi\mu}K^{\chi\beta}{}_{\nu}K^{\gamma}{}_{\xi\sigma}K^{\xi\epsilon}{}_{\lambda}+2\delta^{\mu\nu\sigma\lambda}_{\alpha\beta\gamma\epsilon}K^{\alpha\beta}{}_{\mu}K^{\gamma}{}_{\chi\nu}K^{\chi\epsilon}{}_{\xi}K^{\xi}{}_{\sigma\lambda}+2\delta^{\mu\nu\sigma\lambda}_{\alpha\beta\gamma\epsilon}K^{\alpha\beta}{}_{\mu}K^{\gamma}{}_{\chi\nu}D_{\lambda}K^{\chi\epsilon}{}_{\sigma}\,,$		(10)
	$\displaystyle B_{G}$	$\displaystyle=$	$\displaystyle\frac{1}{e}\partial_{\mu}\Big{[}e\delta^{\mu\nu\sigma\lambda}_{\alpha\beta\gamma\epsilon}K^{\alpha\beta}{}_{\nu}\Big{(}K^{\gamma}{}_{\xi\sigma}K^{\xi\epsilon}{}_{\lambda}-\frac{1}{2}\accentset{\circ}{R}^{\gamma\epsilon}{}_{\sigma\lambda}\Big{)}\Big{]}\,.$		(11)

Here, $D_{\lambda}K^{\chi\epsilon}{}_{\sigma}=\partial_{\lambda}K^{\chi\epsilon}{}_{\sigma}+(\mathring{\Gamma}+K)^{\chi}{}_{\beta\lambda}K^{\beta\epsilon}{}_{\sigma}+(\mathring{\Gamma}+K)^{\epsilon}{}_{\beta\lambda}K^{\chi\beta}{}_{\sigma}-(\mathring{\Gamma}+K)^{\beta}{}_{\sigma\lambda}K^{\chi\epsilon}{}_{\beta}$ is the covariant derivative of the general connection. Furthermore, $\delta_{ijkl}^{abcd}={\displaystyle{\begin{aligned} 4!\delta_{[i}^{a}\delta_{j}^{b}\delta_{k}^{c}\delta_{l]}^{d}\end{aligned}}}$, and then $\delta^{\mu\nu\sigma\lambda}_{\alpha\beta\gamma\epsilon}=e^{\mu}{}_{a}e^{\nu}{}_{b}e^{\sigma}{}_{c}e^{\lambda}{}_{d}\delta^{abcd}_{ijkl}e^{i}{}_{\alpha}e^{j}{}_{\beta}e^{k}{}_{\gamma}e^{l}{}_{\epsilon}$. It is worth mentioning that the first equation derived for $T_{G}$ (see Eq. (48) in [81]) had four pieces, but the authors did not notice that, due to the symmetries of contortion tensor, one of them is always vanishing. Therefore, the scalar $T_{G}$ only contains three pieces as it is written above. It should be noted that in Riemannian geometry (curvature-based theories), the torsion tensor is identically zero but the total curvature tensor $R$ is not. Therefore, the above equation, as it is displayed here, is no longer valid. The above relationship between the Gauss-Bonnet scalars only exists by assuming that the manifold contains torsion and that the general curvature is vanishing.

It is well-known that $\accentset{\circ}{G}$ is a topological invariant in four dimensions and it is easy to see that the scalar $T_{G}$ also has the same property. The extra scalar $B_{G}$ is always a boundary term (independently of the number of dimensions). This means that one can add to the TEGR action both $T_{G}$ and $B_{G}$, and the dynamics will be just the same as GR. However, if one introduces a scalar field, by allowing some non-minimal couplings between these two terms, they will become dynamical. This property is analogue to the Riemannian Gauss-Bonnet invariant. However, we have shown that in the Teleparallel formulation, the usual Gauss-Bonnet term actually decays into two distinct boundary terms, which can be coupled separately.

There are many studies concerning black holes in the sGB theory, which is constructed by introducing non-minimal couplings between a scalar field and the Gauss-Bonnet invariant $\accentset{\circ}{G}$, namely

$\displaystyle\mathcal{S}_{\rm sGB}$

$\displaystyle=\frac{1}{2\kappa^{2}}\int\Big{[}\accentset{\circ}{R}-\frac{1}{2}\beta\,\partial_{\mu}\psi\partial^{\mu}\psi+\alpha\mathcal{G}(\psi)\accentset{\circ}{G}\Big{]}\,\sqrt{-g}\,\differential^{4}x\,.$

(12)

Depending on $\mathcal{G}(\psi)$, several papers have found that the above theory has scalarized black hole solutions and spontaneous scalarization exists [6, 7, 8].

Following a similar approach for Teleparallel gravity, one can replace $\accentset{\circ}{R}$ for $-T$, see (6), and introduce non-minimal couplings between the Teleparallel Gauss-Bonnet invariants and the scalar field. By doing this, we have the following Teleparallel sGB theory [83, 82]:

$\displaystyle\mathcal{S}_{\rm TsGB}$

$\displaystyle=\frac{1}{2\kappa^{2}}\int\Big{[}-T-\frac{1}{2}\beta\,\partial_{\mu}\psi\partial^{\mu}\psi+\alpha_{1}\mathcal{G}_{1}(\psi)T_{G}+\alpha_{2}\mathcal{G}_{2}(\psi)B_{G}\Big{]}\,e\,\differential^{4}x\,,$

(13)

where the coupling functions $\mathcal{G}_{1}(\psi)$ and $\mathcal{G}_{2}(\psi)$ depend on the scalar field only. The terms $T_{G}$ and $B_{G}$ are boundary terms if they appear linearly in the action. The classical teleparallel boundary term $B$ could also be non-minimally coupled via a term $\alpha_{0}\mathcal{G}_{0}(\psi)B$; however analysing the spontaneous scalarization properties of this term turns out to be more involved (more details on this will be reported in future work), which is why we do not include it in our study here and focus on the Gauss-Bonnet terms. If $\mathcal{G}_{1}$ and $\mathcal{G}_{2}$ are constants, the theory would be equivalent to the standard Einstein one minimally coupled to a scalar field. From Eq. (8), one can notice that the above action contains (12) since if $\mathcal{G}_{1}=\mathcal{G}_{2}=\mathcal{G}$ and $\alpha_{1}=\alpha_{2}=\alpha$, we recover the last term in (12), which is the coupling between the Riemannian Gauss-Bonnet and the scalar field $\alpha\mathcal{G}(\psi)\mathring{G}$. In several studies [6, 7, 8, 84], spontaneous scalarization of black holes and compact stars have been found for such theories. Since the above action (13), constructed from Teleparallel gravity, is more general than previous theories, as we will show later, the theory we consider has a similar and new scalarization process for black holes.

In this section, we will find the field equations in spherical symmetry. First, we will solve the antisymmetric part of the field equations, and then, the resulting symmetric field equations will be shown.