Asymptotics of Solutions to Silent Wave Equations

TL;DR

This paper analyzes the asymptotic behavior of solutions to silent wave equations, providing detailed estimates and applying these results to Maxwell's equations in Kasner spacetimes near singularities, confirming aspects of the BKL conjecture.

Contribution

It offers comprehensive asymptotic estimates for silent wave equations and applies them to electromagnetic fields in Kasner spacetimes, advancing understanding of singularity behavior.

Findings

Solutions are uniquely determined by their asymptotic data.

Electromagnetic potential admits an asymptotic expansion near singularity.

Energy density of solutions diverges along generic timelike geodesics approaching the singularity.

Abstract

We study the asymptotics of solutions to a particular class of systems of linear wave equations, namely, of silent equations. We obtain asymptotic estimates of all orders for the solutions, and show that solutions are uniquely determined by the asymptotic data contained in the estimates. As an application, we then study solutions to the source free Maxwell's equations in Kasner spacetimes near the initial singularity. Our results allow us to obtain an asymptotic expansion for the potential of the electromagnetic field, and to show that the energy density of generic solutions blows up along generic timelike geodesics when approaching the singularity. The asymptotics we study correspond to the heuristics of the BKL conjecture, where the coefficients of the spatial derivative terms of the equations are expected to be small, and thus these terms are neglected in order to obtain the asymptotics.