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

We discuss vacuum static, spherically symmetric asymptotically flat solutions of the generalized hybrid metric-Palatini theory of gravity (generalized HMPG) suggested by Böhmer and Tamanini, involving both a metric \(g_{\mu\nu}\) and an independent connection \(\hat{\Gamma}_{\mu\nu}^{\alpha}\); the gravitational field Lagrangian is an arbitrary function \(f(R,P)\) of two Ricci scalars, \(R\) obtained from \(g_{\mu\nu}\) and \(P\) obtained from \({\hat{\Gamma}}_{\mu\nu}^{\alpha}\). The theory admits a scalar-tensor representation with two scalars \(\phi\) and \(\xi\) and a potential \(V(\phi,\xi)\) 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 \(V\equiv 0\), generic solutions contain naked singularities or describe traversable wormholes, and only some special cases represent black holes with extremal horizons. For \(V(\phi,\xi)\neq 0\), some examples of analytical solutions are obtained and shown to possess naked singularities. Even in the cases where the Einstein-frame metric \(g^{E}_{\mu\nu}\) is found analytically, the scalar field equations need a numerical study, and if \(g^{E}_{\mu\nu}\) contains a horizon, in the Jordan frame it turns to a singularity due to the corresponding conformal factor.