A Universal Spinor Bundle and the Einstein-Dirac-Maxwell Equation as a Variational Theory

TL;DR

This paper introduces a universal spinor bundle that unifies metric-dependent spinor bundles, facilitating variational analysis of Einstein-Dirac-Maxwell equations and enabling a coherent definition of maximal Cauchy developments.

Contribution

It constructs a natural finite-dimensional bundle capturing all metric spinor bundles, simplifying variational problems involving multiple metrics.

Findings

Unified spinor bundle for all metrics

Application to Einstein-Dirac-Maxwell variational theory

Defined maximal Cauchy development coherently

Abstract

Not only the Dirac operator, but also the spinor bundle of a pseudo-Riemannian manifold depends on the underlying metric. This leads to technical difficulties in the study of problems where many metrics are involved, for instance in variational theory. We construct a natural finite dimensional bundle, from which all the metric spinor bundles can be recovered including their extra structure. In the Lorentzian case, we also give some applications to Einstein-Dirac-Maxwell theory as a variational theory and show how to coherently define a maximal Cauchy development for this theory.