Integral Formula for the Characteristic Cauchy Problem on a curved Background

TL;DR

This paper presents a local integral formula for solving the characteristic Cauchy problem for the Dirac equation on general curved spacetimes, extending previous flat spacetime results and enabling potential sharp estimates.

Contribution

It introduces a new integral formula applicable to curved backgrounds for the Dirac equation, generalizing Penrose's flat spacetime results using Friedlander's method.

Findings

Valid on general curved spacetimes

Recovers Penrose's flat spacetime results

Potential for sharp estimates in the characteristic Cauchy problem

Abstract

We give a local integral formula, valid on general curved space-times, for the characteristic Cauchy problem for the Dirac equation with arbitrary spin using the method developed by Friedlander in his book "the wave equation on a curved spacetime" (1975). The results obtained by Penrose in the flat case in "Null hypersurface initial data for classical fields of arbitrary spin for general relativity" (Gen. Rel. Grav 1980) are recovered directly. It is expected that this method can be used to obtain sharp estimates for the characteristic Cauchy problem for the Dirac equation.