# Generalized regularly discontinuous solutions of the Einstein equations

## TL;DR

This paper extends the mathematical framework for matching discontinuous metrics in Einstein's equations, introducing generalized boundary layers, shock waves, and thin shells, with new compatibility conditions and mean-value differential geometry.

## Contribution

It generalizes the theory of gravitational discontinuities to regularly discontinuous metrics, providing a new geometric framework and compatibility conditions.

## Key findings

- Developed a mean-value differential geometry framework on hypersurfaces.
- Derived compatibility conditions for matching discontinuous metrics.
- Analyzed examples including shock waves and thin shells.

## Abstract

The physical consistency of the match of piecewise-$C^0$ metrics is discussed. The mathematical theory of gravitational discontinuity hypersurfaces is generalized to cover the match of regularly discontinuous metrics. The mean-value differential geometry framework on a hypersurface is introduced, and corresponding compatibility conditions are deduced. Examples of generalized boundary layers, gravitational shock waves and thin shells are studied.