Junction conditions for generalized hybrid metric-Palatini gravity with applications

Abstract

The generalized hybrid metric-Palatini gravity is a theory of gravitation that has an action composed of a Lagrangian given by $f(R,\mathcal{R})$, where $f$ is a function of the metric Ricci scalar $R$ and a new Ricci scalar $\mathcal{R}$ formed from a Palatini connection, plus a matter Lagrangian. This theory can be rewritten by trading the new geometric degrees of freedom that appear in $f(R,\mathcal{R})$ into two scalar fields, $\ensuremath{\varphi}$ and $\ensuremath{\psi}$, yielding a dynamically equivalent scalar-tensor theory. Given a spacetime theory, the next important step is to find solutions within it. To construct appropriate solutions it is often necessary to know the junction conditions between two spacetime regions at a separation hypersurface $\mathrm{\ensuremath{\Sigma}}$, with each spacetime region being an independent solution of the theory. The junction conditions for the generalized hybrid metric-Palatini gravity are found here, both in the geometric representation and in the scalar-tensor representation, and in addition, for each representation, the junction conditions for a matching with a thin-shell of matter and for a smooth matching at the separation hypersurface are worked out. These junction conditions are then applied to three configurations, namely, a star, a quasistar with a black hole, and a wormhole. The star is made of a Minkowski interior, a thin shell at the interface with all the matter energy conditions being satisfied, and a Schwarzschild exterior with mass $M$, and unlike general relativity where the matching can be performed at any radius ${r}_{\mathrm{\ensuremath{\Sigma}}}$, for this theory the matching can only be performed at a specific value of the shell radius, namely ${r}_{\mathrm{\ensuremath{\Sigma}}}=\frac{9M}{4}$, that corresponds to the general relativistic Buchdahl radius. The quasistar with a black hole is made of an interior Schwarzschild black hole surrounded by a thick shell that matches smoothly to a mass $M$ Schwarzschild exterior at the light ring radius ${r}_{\mathrm{\ensuremath{\Sigma}}e}=3M$, and with the matter energy conditions being satisfied for the whole spacetime. The wormhole is made of some interior with matter that contains the throat, a thin shell at the interface, and a Schwarzschild-AdS exterior with mass $M$ and negative cosmological constant $\mathrm{\ensuremath{\Lambda}}$, with the matter null energy condition being obeyed everywhere within the wormhole.