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

TL;DR

This paper develops a new coordinate-based method for analyzing timelike junctions between spherically symmetric solutions of Einstein's equations, with applications to FLRW models and phenomena like vacuum bubbles.

Contribution

It introduces a coordinate system that absorbs junction surface motion, derives evolution equations, and clarifies conditions for matching spherically symmetric spacetime regions, including FLRW models.

Findings

Junction surface can reach the speed of light in finite proper time.

Explicit conditions for matching spherically symmetric regions are provided.

Numerical results demonstrate the behavior of junctions, including vacuum bubbles.

Abstract

We investigate timelike junctions (with surface layer) 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. The evolution equations for all relevant quantities are derived. We discuss the no-surface layer case (boundary surface) and study the behaviour for small surface energies. It is shown that one should expect cases in which the speed of light is reached within a finite proper time. We carefully discuss necessary and sufficient conditions for a possible matching of spherically symmetric sections. For timelike junctions between spherically symmetric space-time sections we show explicitly that the time component of the Lanczos equation always reduces to an identity (independently of the surface equation of state). The results are applied to the matching of FLRW models. We discuss `vacuum bubbles' and closed-open junctions in detail. As illustrations several numerical integration results are presented, some of them indicate that the junction surface can reach the speed of light within a finite time.