Spherically symmetric static vacuum solutions in hybrid metric-Palatini gravity

Abstract

We consider vacuum static spherically symmetric solutions in the hybrid metric-Palatini gravity theory, which is a combination of the metric and Palatini f (R) formalisms unifying local constraints at the Solar System level and the late-time cosmic acceleration. We adopt the scalar-tensor representation of the hybrid metric-Palatini theory, in which the scalar-tensor definition of the potential can be represented as a Clairaut differential equation. Due to their mathematical complexity, it is difficult to find exact solutions of the vacuum field equations, and therefore we adopt a numerical approach in studying the behavior of the metric functions and of the scalar field. After reformulating the field equations in a dimensionless form, and by introducing a suitable independent radial coordinate, the field equations are solved numerically. We detect the formation of a black hole from the presence of Killing horizon for the time-like Killing vector in the metric tensor components. Several models, corresponding to different functional forms of the scalar field potential are considered. The thermodynamic properties of these black hole solutions (horizon temperature, specific heat, entropy and evaporation time due to Hawking luminosity) are also investigated in detail.

Spherically Symmetric Black Holes and Wormholes in Hybrid Metric-Palatini Gravity

“…Considering exact vacuum solutions of HMPG with zero potential V , we have found that many of them describe space-times with naked singularities, other generic solutions correspond to traversable wormholes, while only solutions of a special twoparameter family contain extremal Killing horizons and describe globally regular black holes with zero Hawking temperature. These results disagree with some of those obtained in [5], where the same set of equations was solved numerically directly in Jordan's frame, and the existence of black holes with finite temperature was stated. The reason for such a discrepancy is yet to be understood.…”

“…One can notice that none of the static, spherically symmetric solutions of the theory (2) with V ≡ 0 possess simple horizons with finite temperature, contrary to the results announced in [5].…”

“…Meanwhile, the transition (6) is also a wellknown efficient method of finding exact or approximate solutions to the equations of the theory (4) due to the comparative simplicity of the action (7). I can hardly agree with the statement from [5] "A crucial mathematical requirement for transformations (28) [(6) in this paper] to be valid is that they must be nonsingular for the considered range of the scalar field variable." In fact, the transformation (6) is, from the viewpoint of solving differential equations, just a substitution which can in principle work in only a part of the relevant range of the variables involved.…”

“…It is also directly confirmed that the scalar curvature is zero for g µν given by (27), as should be the case for V ≡ 0. (According to (3), V = 0 implies R = const · R, therefore in this case R is also zero [5]. )…”

“…and transform the solutions back to g µν and φ according to (6). As is stressed in [5], the HMPG theory is naturally formulated in terms of the Jordanframe metric g µν and the scalar field φ.…”