Regularity of curve integrable spacetimes

TL;DR

This paper investigates the regularity of scalar fields in curve integrable spacetimes, proposing a PDE-based approach to defining gravitational singularities as obstructions to field evolution rather than particle trajectories.

Contribution

It introduces the concept of field regularity to assess well-posedness of wave equations in singular spacetimes, providing a new perspective on classical singularities.

Findings

Classical singularities do not prevent the well-posedness of the wave equation in curve integrable spacetimes.

The PDE approach offers an alternative to geodesic-based definitions of singularities.

Field regularity can serve as a criterion for spacetime regularity in gravitational theories.

Abstract

The idea of defining a gravitational singularity as an obstruction to the dynamical evolution of a test field (described by a PDE) rather than the dynamical evolution of a particle (described by a geodesics) is explored. In particular, the concept of field regularity is introduced which serves to describe the well-posedness of the local initial value problem for a given field.In particular this is applied to (classical) scalar fields in the class of curve integrable spacetimes to show that the classical singularities do not interrupt the well-posedness of the wave equation.