Generalised hyperbolicity in spacetimes with Lipschitz regularity

TL;DR

This paper establishes conditions for the well-posedness of the wave equation in spacetimes with Lipschitz continuous metrics, enabling analysis of gravitational singularities as obstructions to field dynamics rather than particle trajectories.

Contribution

It introduces a general framework for wave equation well-posedness in Lipschitz regularity spacetimes, applicable to singularities like shell-crossings and matter layers.

Findings

Wave equation is well-posed in Lipschitz spacetimes under certain conditions.

Framework applies to spacetimes with shell-crossing singularities.

Reframes gravitational singularities as obstructions to field dynamics.

Abstract

In this paper, we obtain general conditions under which the wave equation is well-posed in spacetimes with metrics of Lipschitz regularity. In particular, the results can be applied to spacetimes where there is a loss of regularity on a hypersurface such as shell-crossing singularities, thin shells of matter and surface layers. This provides a framework for regarding gravitational singularities, not as obstructions to the world lines of point-particles, but rather as an obstruction to the dynamics of test fields.