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.

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