Spherically Symmetric Space-Times in Generalized Hybrid Metric-Palatini Gravity

Abstract

We discuss vacuum static, spherically symmetric asymptotically flat solutions of the generalized hybrid metric-Palatini theory of gravity (generalized HMPG) suggested by B ¨ohmer and Tamanini, involving both a metric g μν and an independent connection Γα μν ; the gravitational field Lagrangian is an arbitrary function f (R, P ) of two Ricci scalars, R obtained from g μν and P obtained from Γα μν . The theory admits a scalar-tensor representation with two scalars φ and ξ and a potential V (φ, ξ) whose form depends on f (R, P ). Solutions are obtained in the Einstein frame and transferred back to the original Jordan frame for a proper interpretation. In the completely studied case V ≡ 0, generic solutions contain naked singularities or describe traversable wormholes, and only some special cases represent black holes with extremal horizons. For V (φ, ξ) = 0, some examples of analytical solutions are obtained and shown to possess naked singularities. Even in the cases where the Einstein-frame metric g E μν is found analytically, the scalar field equations need a numerical study, and if g E μν contains a horizon, in the Jordan frame it turns to a singularity due to the corresponding conformal factor.

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“…It is then straightforward to find the general solution to Eq. ( 41) near z = 0 with any Ω (since any finite Ω 2 is negligible against the infinite negative potential): thus, for the cases (45) we obtain…”

“…If there is no electromagnetic field, q = 0, then the Einstein equation R t t = ... in the coordinates (18) reads simply γ ′′ = 0, which leads to the metric (20) with e 2γ = e −2hu and the relation between the constants k 2 sign k = εC 2 + h 2 instead of (28); the Schwarzschild mass m is equal to h. It is Fisher's well-known solution [7] in the case ε = +1 and its phantom counterpart [25] if ε = −1. These space-times have been studied in detail, including their stability properties, see, e.g., [3,4,6,21,26,31,32], and the stability of their STT counterparts for ε = +1 was recently discussed in [14], so here we will focus on configurations with q ̸ = 0.…”

“…Considering our scalar-electrovacuum solutions with the metric (25) (and even with scalarvacuum solutions studied previously [6,14]), a problem is that z(u) obtained from ( 40) is in all cases quite a long expression depending on the solution parameters and containing combinations of hypergeometric functions. It is impossible to solve it for u(z) and hence to write V eff as an explicit function of z .…”

“…It is therefore reasonable to formulate the boundary conditions unambiguously for the function f (ϕ) characterizing the nonminimal coupling: let us require |δf /f | < ∞. Thus, by analogy with [6,14,32,33], we impose a minimal condition compatible with the perturbation scheme, admitting perturbations diverging not stronger than does the background field.…”

“…In the present paper, we continue the study begun in [14] and consider electrovacuum solutions of the same (Bergmann-Wagoner-Nordtvedt) class of STT, conformally related to Penney's space-time, thus restricting ourselves to STT with massless scalar fields. We can remark that such solutions play in these theories the same role as the Reissner-Nordström solution in GR and, in our opinion, definitely deserve a stability study.…”