Junction Conditions for General Gravitational Theories

The junction conditions for general theories of gravity based on actions that depend on arbitrary functions of the curvature scalar invariants (including differential invariants) are obtained using the distributional formalism. In case of the existence of thin shells, a general expression for the shell energy-momentum tensor is presented. Generalized Israel equations are also obtained. The conditions for a proper matching, without shells, are derived. The main results are: (i) shells arise if the $m$th-covariant derivative of the Riemann tensor is continuous at the matching hypersurface, where $m$ is the maximum order of differentiation appearing in the Lagrangian density; (ii) a proper junction without thin shells requires further that the $(m+1)$-th derivative be also continuous, (iii) theories with $m=0$ that are quadratic in the scalar curvature invariants are special and unique for they allow for discontinuities of the Riemann tensor resulting in the existence of thin shells and {\em gravitational double layers} and (iv) General Relativity and $F(R)$ theories are extraordinary theories that admit shells of curvature (i.e. impulsive gravitational waves) because other theories require the absence of jumps of the second fundamental form across the matching hypersurface. For proper junctions, the continuity across the matching hypersurface of the normal components of the energy-momentum tensor is proven to be a {\em universal} property, independently of the field equations, thereby providing important necessary conditions for any matching in any gravitational theory. All results are derived for a minimal coupling with the matter, but the strategy would be analogous for more general couplings.