Even parity junction conditions for perturbations on most general spherically symmetric space–times

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Abstract
A new highly efficient and versatile general relativistic perturbational formalism for general matter occupied spherically symmetric space–times is developed. The perturbations are geometrical objects on the two dimensional totally geodesic submanifold spanned by the radial and time coordinates. The geometrical objects are ’’gauge invariant’’ scalars, vectors, and tensors which are independent of infinitesimal coordinate transformations on the background space–time. This article gives the even parity gauge invariant perturbation objects for arbitrary background scalars, vectors, and symmetric tensors on a spherically symmetric space–time. In particular, metric, matter, first and second fundamental forms, as well as vacuum-matter interface gauge invariant perturbations for a collapsing star are given. In addition four even parity continuity conditions across discontinuous timelike hypersurfaces are given. Two are conditions on the metric gauge invariants, one is a condition on the perturbation away from the spherical contour of the interface, and the fourth couples that contour perturbation to the metric gauge invariants.