Spherically symmetric black holes and wormholes in hybrid metric-Palatini gravity

The so-called hybrid metric-Palatini theory of gravity (HMPG), proposed in 2012 by T. Harko et al., is known to successfully describe both local (solar-system) and cosmological observations. This paper gives a complete description of static, spherically vacuum solutions of HMPG in the simplest case where its scalar-tensor representation has a zero scalar field potential $V(\phi)$, and both Riemannian ($R$) and Palatini ($\cal R$) Ricci scalars are zero. Such a scalar-tensor theory coincides with general relativity with a phantom conformally coupled scalar field as a source of gravity. Generic asymptotically flat solutions either contain naked central singularities or describe traversable wormholes, and there is a special two-parameter family of globally regular black hole solutions with extremal horizons. In addition, there is a one-parameter family of solutions with an infinite number of extremal horizons between static regions and a spherical radius monotonically changing from region to region. It is argued that the obtained black hole and wormhole solutions are unstable under monopole perturbations. As a by-product, it is shown that a scalar-tensor theory with $V(\phi)=0$, in which there is at least one nontrivial ($\phi \ne \rm const$) vacuum solution with $R\equiv 0$, necessarily reduces to a theory with a conformal scalar field (the latter may be usual or phantom).