New energy inequalities for tensorial wave equations on spacetimes that satisfy a one-sided bound

TL;DR

This paper establishes new energy inequalities for tensorial wave equations like Maxwell, Yang-Mills, and Weyl fields on curved spacetimes, under one-sided geometric conditions that generalize previous criteria for Einstein's equations.

Contribution

It introduces a novel approach using one-sided geometric bounds to derive energy inequalities for tensorial wave equations on curved spacetimes.

Findings

Energy inequalities hold under one-sided bounds on lapse and deformation tensor.

Method applies to Bel-Robinson energy for Weyl fields, controlling energy growth.

Conditions generalize previous continuation criteria for Einstein's equations.

Abstract

We consider several tensorial wave equations, specifically the equations of Maxwell, Yang-Mills, and Weyl fields, posed on a curved spacetime, and we establish new energy inequalities under certain one-sided geometric conditions. Our conditions restrict the lapse function and deformation tensor of the spacetime foliation, and turn out to be a one-sided and integral generalization of conditions recently proposed by Klainerman and Rodnianski as providing a continuation criterion for Einstein's field equations of general relativity. As we observe it here for the first time, one-sided conditions are sufficient to derive energy inequalities for certain tensorial equations, provided one takes advantage of some algebraic properties enjoyed by the natural energy functionals associated with the equations under consideration. Our method especially applies to the Bel-Robinson energy for Weyl fields, and our inequalities control the growth of the energy in a uniform way, with implied constants depending on the one-sided geometric bounds, only.