Junction conditions for generalized hybrid metric-Palatini gravity with applications

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

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