Junction conditions for odd-parity perturbations on most general spherically symmetric space-times

Perturbations of a general background space-time with a hypersurface of discontinuity, such as the history of a collapsing star, are considered. The junction conditions that these perturbations obey are expressed in terms of the perturbed first and second fundamental forms (intrinsic metric and extrinsic curvature) of the (perturbed) set of hypersurfaces one of which is the discontinuous one. These junction conditions are applied to the odd-parity metric and hydrodynamical asymmetries of a slightly aspherical but otherwise general and realistic spherically collapsing star. The junction conditions are stated in terms of those metric and matter perturbational objects that are the most natural, economic, and versatile: gauge-invariant geometrical objects. For odd-parity perturbations these are scalars and vectors on the totally geodesic submanifold spanned by the time and radial coordinates. The end result is simple: The junction conditions amount to the continuity of the gradient of a master gauge-invariant scalar wave function from which all other perturbational quantities can be derived.