Spontaneous scalarization of black holes in Gauss-Bonnet teleparallel gravity

In this paper, we find new scalarized black holes by coupling a scalar field with the Gauss-Bonnet invariant in teleparallel gravity. The teleparallel formulation of this theory uses torsion instead of curvature to describe the gravitational interaction, and it turns out that, in this language, the usual Gauss-Bonnet term in four dimensions decays in two distinct boundary terms, the teleparallel Gauss-Bonnet invariants. Both can be coupled individually or in any combination to a scalar field, to obtain a teleparallel Gauss-Bonnet extension of the teleparallel equivalent of general relativity. The theory we study contains the familiar Riemannian Einstein-Gauss-Bonnet gravity theory as a particular limit and offers a natural extension, in which scalarization is triggered by torsion and with new interesting phenomenology. We demonstrate numerically the existence of asymptotically flat scalarized black hole solutions and show that, depending on the choice of coupling of the boundary terms, they can have a distinct behavior compared to the ones known from the usual Einstein-Gauss-Bonnet case. More specifically, nonmonotonicity of the metric functions and the scalar field can be present, a feature that was not observed until now for static scalarized black hole solutions.

Figure 1

The radius of the horizon (left panel) and the scalar charge (right panel) as functions of mass for a fixed ${\alpha }_2=-1$ and varying ${\alpha }_3$. The bald Schwarzschild black hole is depicted with a solid black line, the scalarized branches with ${\alpha }_3{\dot{\mathcal{G}}}_3({\psi }_H)\ne {\alpha }_2{\dot{\mathcal{G}}}_2({\psi }_H)$ and using BC1 with solid lines of different colors, while the solutions with ${\alpha }_3{\dot{\mathcal{G}}}_3({\psi }_H)={\alpha }_2{\dot{\mathcal{G}}}_2({\psi }_H)$ and BC2 are marked by a dashed line. The pure sGB case (when ${\alpha }_3=0$) is depicted with a solid dark green line.

Figure 2

The radius of the horizon (left panel) and the scalar charge (right panel) as functions of mass for a fixed ${\alpha }_3=1$ and varying ${\alpha }_2$. The conventions for solution branches are the same as in Fig. 1.

Figure 3

The tetrad functions $A(r)$ and $B^{-1}(r)$, and the scalar field $\psi$ as functions of the normalized to the horizon radius radial coordinate $r/r_H$. The lines corresponding to different functions are marked with different colors—black, red, and blue for $A(r)$, $B^{-1}(r)$, and $\psi (r)$, respectively. The left $y$ axis corresponds to the $\psi$ function, while the right $y$ axis sets the scale for $A(r)$ and $B^{-1}(r)$. Different patterns of the lines correspond to solutions with varying $r_H$. The theory parameters are fixed to ${\alpha }_2=-1,a_3=-0.5$ (left panel) and ${\alpha }_2=-1,a_3=0.5$ (right panel). Since ${\alpha }_2\ne a_3$ in both cases, BC1 is used. These solutions are a subset of the ones presented in Fig. 1.

Figure 4

The tetrad functions $A(r)$ and $B^{-1}(r)$, and the scalar field $\psi$ as functions of the normalized to the horizon radius radial coordinate $r/r_H$. Notations are the same as for Fig. 3, while the theory parameters are set to ${\alpha }_2={\alpha }_3=1$ (left panel) and ${\alpha }_2={\alpha }_3=-1$ (right panel). Since ${\alpha }_2={\alpha }_3$ in both cases, BC2 is used. These solutions are a subset of the ones presented in Figs. 1 and 2.

Figure 5

The tetrad functions $A(r)$ and $B^{-1}(r)$ and the scalar field $\psi$ as functions of the normalized to the horizon radius radial coordinate $r/r_H$. Notations are the same as for Fig. 3. The two figures correspond to the two pure teleparallel cases with ${\alpha }_2=0$ and ${\alpha }_3=1$ (left panel) and ${\alpha }_2=-1$ and ${\alpha }_3=1$ (right panel). These solutions are a subset of the ones presented in Figs. 1 and 2, respectively.

Figure 6

The teleparallel terms $T_G(r)$ and $B_G(r)$, as well as the teleparallel Gauss-Bonnet invariant equal to the summation of both [see Eq. (8)] as functions of radial coordinate $r$. The coupling parameters are chosen to be ${\alpha }_2=-1$ and ${\alpha }_3=-0.5$. These solutions are a subset of the ones presented in Fig. 1.