A new approach to spherically symmetric junction surfaces and the matching of FLRW regions

We investigate timelike junctions between spherically symmetric solutions of the Einstein-field equation. In contrast to previous investigations this is done in a coordinate system in which the junction surface motion is absorbed in the metric, while all coordinates are continuous at the junction surface. Furthermore, there is no ambiguity in the direction of the normal vectors at the junction hypersurface. We show that non-trivial timelike junction surfaces without surface layer (boundary surfaces) are generally not possible, except the metric components satisfy certain conditions. We study the behaviour of the junction surface for small values of the surface energy density and show that in many cases the junction surface appears to reach the speed of light in a finite proper time. In other cases the junction surface approaches the speed of light asymptotically with respect to proper time on each side, but within a finite proper time measured along the junction. For the junction between spherically symmetric space-time sections we show explicitly that the time component of the Lanczos equation always reduces to an identity. We carefully discuss necessary and sufficient conditions for a possible matching of spherically symmetric space-time sections and apply the results to the matching of FLRW models. Several junctions between FLRW models are studied numerically. Examples are given for junction surfaces which reach the speed of light within a finite and infinite time, respectively.