Asymptotics of Solutions to Silent Wave Equations

We study the asymptotics of solutions to a particular class of systems of linear wave equations, namely, of silent equations. Here, the asymptotics refer to the behavior of the solutions near a cosmological singularity, or near infinity in the expanding direction. Leading-order asymptotics for solutions of silent equations were already obtained by Ringström (Astérisque 420, 2020). Here, we improve upon Ringström’s result, by obtaining asymptotic estimates of all orders for the solutions, and showing 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.