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 B\"ohmer 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 $V(\phi,\xi) \ne 0$, 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.