# Boundary value problems for the Lorentzian Dirac operator

## TL;DR

This paper explores boundary conditions for the Lorentzian Dirac operator on globally hyperbolic spacetimes, examining how more general conditions affect the operator's index compared to classical elliptic cases.

## Contribution

It investigates the replacement of Atiyah-Patodi-Singer boundary conditions with more general ones for the Lorentzian Dirac operator and analyzes the resulting index changes.

## Key findings

- Boundary conditions influence the index of the Lorentzian Dirac operator.
- Differences arise compared to the classical elliptic case.
- General boundary conditions can alter the Fredholm property.

## Abstract

On a compact globally hyperbolic Lorentzian spin manifold with smooth spacelike Cauchy boundary the (hyperbolic) Dirac operator is known to be Fredholm when Atiyah-Patodi-Singer boundary conditions are imposed. In this paper we investigate to what extent these boundary conditions can be replaced by more general ones and how the index then changes. There are some differences to the classical case of the elliptic Dirac operator on a Riemannian manifold with boundary.