The Cauchy problem for Lorentzian Dirac operators under non-local boundary conditions

TL;DR

This paper introduces a class of Lorentzian boundary conditions, including APS conditions, that ensure well-posedness of the Dirac operator's Cauchy problem on spacetimes with timelike boundaries.

Contribution

It extends the understanding of non-local boundary conditions for Dirac operators from Riemannian to Lorentzian manifolds with timelike boundaries.

Findings

Defined Lorentzian boundary conditions that are local in time and non-local in space.

Proved well-posedness of the Cauchy problem for the Dirac operator under these conditions.

Applied to APS conditions on level sets of Cauchy temporal functions.

Abstract

Non-local boundary conditions, such as the Atiyah-Patodi-Singer (APS) conditions, for Dirac operators on Riemannian manifolds are well understood while not much is known for such operators on spacetimes with timelike boundary. We define a class of Lorentzian boundary conditions that are local in time and non-local in the spatial directions and show that they lead to a well-posed Cauchy problem for the Dirac operator. This applies in particular to the APS conditions imposed on each level set of a given Cauchy temporal function.