A Generalized Representation Formula for Systems of Tensor Wave Equations

TL;DR

This paper extends the Kirchhoff-Sobolev parametrix to tensor wave systems with first-order terms, enabling broader applications in Einstein-related spacetimes by emphasizing null cone support.

Contribution

It introduces a generalized representation formula for tensor wave equations, supporting systems with first-order terms, crucial for advancing Einstein spacetime analyses.

Findings

Generalized representation formula for tensor wave systems.

Supports Einstein-Maxwell and Einstein-Yang-Mills spacetimes.

Highlights null cone support of formulas.

Abstract

In this paper, we generalize the Kirchhoff-Sobolev parametrix of Klainerman and Rodnianski to systems of tensor wave equations with additional first-order terms. We also present a different derivation, which better highlights that such representation formulas are supported entirely on past null cones. This generalization is a key component for extending Klainerman and Rodnianski's breakdown criterion result for Einstein-vacuum spacetimes to Einstein-Maxwell and Einstein-Yang-Mills spacetimes.