An index theorem on asymptotically static spacetimes with compact Cauchy surface

TL;DR

This paper extends index theory for the Dirac operator to asymptotically static spacetimes with compact Cauchy surfaces, establishing Fredholm properties and index formulas under Atiyah-Patodi-Singer boundary conditions at infinite times.

Contribution

It generalizes previous finite-time results to infinite-time settings and constructs a Feynman parametrix as a Fredholm inverse using advanced scattering and pseudo-differential methods.

Findings

Dirac operator is Fredholm with APS boundary conditions at infinite times

Index formula extends to asymptotically static spacetimes

Existence of a Feynman parametrix as a Fredholm inverse

Abstract

We consider the Dirac operator on asymptotically static Lorentzian manifolds with an odd-dimensional compact Cauchy surface. We prove that if Atiyah-Patodi-Singer boundary conditions are imposed at infinite times then the Dirac operator is Fredholm. This generalizes a theorem due to B\"ar-Strohmaier in the case of finite times, and we also show that the corresponding index formula extends to the infinite setting. Furthermore, we demonstrate the existence of a Fredholm inverse which is at the same time a Feynman parametrix in the sense of Duistermaat-H\"ormander. The proof combines methods from time-dependent scattering theory with a variant of Egorov's theorem for pseudo-differential hyperbolic systems.