Scalarized Einstein-Born-Infeld black holes

The phenomenon of spontaneous scalarization of Reissner-Nordström (RN) black holes has recently been found in an Einstein-Maxwell-scalar (EMS) model due to a nonminimal coupling between the scalar and Maxwell fields. Nonlinear electrodynamics, e.g., Born-Infeld (BI) electrodynamics, generalizes Maxwell’s theory in the strong field regime. Nonminimally coupling the BI field to the scalar field, we study spontaneous scalarization of an Einstein-Born-Infeld-scalar (EBIS) model in this paper. It shows that there are two types of scalarized black hole solutions, i.e., scalarized RN-like and Schwarzschild-like solutions. Although the behavior of scalarized RN-like solutions in the EBIS model is quite similar to that of scalarized solutions in the EMS model, we find that there exist significant differences between scalarized Schwarzschild-like solutions in the EBIS model and scalarized solutions in the EMS model. In particular, the domain of existence of scalarized Schwarzschild-like solutions possesses a certain region, which is composed of two branches. The branch of larger horizon area is a family of disconnected scalarized solutions, which do not bifurcate from scalar-free black holes. However, the branch of smaller horizon area may or may not bifurcate from scalar-free black holes depending on the parameters. Additionally, these two branches of scalarized solutions can be both entropically disfavored over comparable scalar-free black holes in some parameter region.

Figure 1

Left: plot of the Hawking temperature $T_H$ against the horizon radius $r_{+}$ for a RN black hole with $\tilde{a}=0$, a RN-like BI black hole with $\tilde{a}=3$ and a Schwarzschild-like BI black hole with $\tilde{a}=10$. Here we take $Q=1$. For $\tilde{a}=0$ and $\tilde{a}=3$, the black holes possess an extremal limit, where $T_H=0$. For $\tilde{a}=10$, the black hole temperature behaves like that of a Schwarzschild black hole, in that it approaches infinity as $r_{+}$ goes to zero. Right: plot of the maximum value of the mass-to-charge ratio $q_{\max }$ against $\tilde{a}$. Extremal RN-like BI black holes have the largest mass-to-charge ratio, while $q_{\max }$ occurs at critical Schwarzschild-like BI black holes, which have $r_{+}=0$ while $Q$ remains finite.

Figure 2

Domain of existence, thermodynamic preference and effective potentials for scalarized RN-like BI black hole solutions with $\tilde{a}=3$. Upper left: domain of existence in the $\alpha -q$ plane, which is displayed by a shaded light blue region and bounded by the bifurcation and critical lines. The blue dashed line represents the bifurcation line, where scalarized black holes bifurcate from BI black holes as zero modes. The red line marks critical configurations of scalarized black holes, where the horizon area vanishes with the mass remaining finite. In the light blue region, only one branch of scalarized black hole solutions exists for a given $\alpha$. Upper right: reduced area $a_H$ versus reduced charge $q$ for BI black holes (a gray line) and scalarized BI black holes with various values of $\alpha$ (red lines). For a given $q$, $a_H$ of the scalarized BI black hole is larger than that of the BI black hole, and increases with the growth of $\alpha$. The scalarized solutions are always entropically preferred. Lower: effective potentials for scalarized BI black holes with $\alpha =1$ (left) and $\alpha =10$ (right) for various values of $q$. Dashed red lines represent positive definite effective potentials, whereas effective potentials that have negative regions are exhibited by solid red lines. The $\alpha =10$ scalarized solutions are stable against radial perturbations, while the stability of the $\alpha =1$ scalarized solutions is inconclusive.

Figure 3

Domain of existence for scalarized Schwarzschild-like BI black hole solutions with $\tilde{a}=10$. For a given $\alpha$, scalarized black hole solutions possess two branches in the dark shaded blue region and one branch in the light one. The domain of existence is delimited by the critical (red lines) and existence (cyan lines) lines. While scalarized solutions emerge with vanishing scalar fields on the bifurcation line (a blue dashed line), scalarized solutions on the existence lines have nonvanishing scalar field configurations. When $\alpha$ is large, the bifurcation and existence lines are hardly distinguishable from each other, and the domain of existence is quite similar to that in the $\tilde{a}=3$ case. The right panel highlights the small $\alpha$ region, where the domains of existence in the RN-like and Schwarzschild-like cases are significantly different.

Figure 4

Thermodynamic preference and effective potentials for Schwarzschild-like BI black hole solutions with $\tilde{a}=10$ in the large $\alpha$ regime. Left: reduced area $a_H$ versus reduced charge $q$ for BI black holes (a gray line) and scalarized BI black holes with various values of $\alpha \ge 3.5$ (red lines). The scalarized solutions are always entropically preferred. Right: effective potentials for scalarized solutions with $\alpha =10$ for various values of $q$, which are shown to be positive. The scalarized BI black hole solutions are stable against radial perturbations.

Figure 5

Thermodynamic preference, temperature and effective potentials for the large (red lines), small (blue lines) and tiny (green lines) branches of Schwarzschild-like BI black hole solutions with $\tilde{a}=10$ and $\alpha =2.85$. The large branch coexists with the small and tiny branches, respectively. Upper left: reduced area $a_H$ versus reduced charge $q$ for BI black holes (a gray line) and the three branches of scalarized BI black holes. The difference of the reduced areas of the scalarized and BI black holes $\mathrm{\Delta }{a}_H\equiv a_H^{\text{scalarized BH}}-a_H^{\text{BI BH}}$ is plotted against $q$ for the small and tiny branches in the insets on the right of the panel, which show that the small and tiny branches have smaller area than BI black holes for same $q$. The inset on the left side displays that most of the large branch is entropically preferred. Upper right: reduced temperature $t_H$ versus reduced charge $q$ for BI black holes and the three branches of scalarized BI black holes. For a given $q$, the large branch is hotter than the small branch, while the large branch is colder than the tiny branch when $q$ is large enough. Lower: effective potentials for the three branches of scalarized BI black holes with various values of $q$. The small and tiny branches are stable against radial perturbations. When $q$ is small, the large branch is also stable. However for large $q$, the stability of the large branch is inconclusive.

Figure 6

Thermodynamic preference, temperature and effective potentials for the large (red lines) and small (blue lines) branches of Schwarzschild-like BI black hole solutions with $\tilde{a}=10$ and $\alpha =2.5$. Upper left: reduced area $a_H$ versus reduced charge $q$ for BI black holes (a gray line) and the two branches of scalarized BI black holes. For a given $q$, the small branch and the part of the large branch with small $q$ have smaller area than BI black holes. When $q$ is large enough, the large branch is entropically preferred. Upper right: reduced temperature $t_H$ versus reduced charge $q$ for BI black holes and the two branches of scalarized BI black holes. The large branch is hotter (colder) than the small branch for small (large) values of $q$. Lower: effective potentials for the two branches of scalarized BI black holes with various values of $q$. The small branch and the large branch with small $q$ are stable against radial perturbations. However for large $q$, the stability of the large branch is inconclusive.

Figure 7

Thermodynamic preference, temperature and effective potentials for the large (red lines) and small (blue lines) branches of Schwarzschild-like BI black hole solutions with $\tilde{a}=10$ and $\alpha =2.25$. Upper left: reduced area $a_H$ versus reduced charge $q$ for BI black holes (a gray line) and the two branches of scalarized BI black holes. For a given $q$, the small and large branches both have smaller area than BI black holes, which means that the scalarized solutions are not entropically preferred. Upper right: reduced temperature $t_H$ versus reduced charge $q$ for BI black holes and the two branches of scalarized BI black holes. The scalarized solutions are hotter than BI black holes, and the large branch is hotter (colder) than the small branch for small (large) values of $q$. Lower: effective potentials for the two branches of scalarized BI black holes with various values of $q$. When $q$ is large, the stability of the scalarized solutions is inconclusive. For small enough $q$, the scalarized solutions are stable against radial perturbations.