On the initial value problem for the wave equation in Friedmann -- Robertson -- Walker space-times

TL;DR

This paper analyzes the wave equation in Friedmann-Robertson-Walker space-times, revealing decay properties, the failure of sharp Huygens property, and well-posedness of initial data at the singularity, showing smooth information propagation through the singularity.

Contribution

It introduces a spherical means formulation to study wave solutions in singular space-times, establishing decay, non-sharp Huygens behavior, and well-posed initial data at the singularity.

Findings

Wave solutions exhibit sharp time decay.

Wave equation does not satisfy sharp Huygens property.

Initial value problem at the singularity is well-posed.

Abstract

We study the wave propagator for a Friedmann - Robertson - Walker background space-time, which is singular at time t=0. Using a spherical means formulation for the solution of the wave equation that is due to Klainerman and Sarnak, we derive three properties of solutions. the first is sharp time decay properties, using ideas of Fritz John. the second is that the wave equation in Friedmann - Robertson - Walker space-time does not satisfy the sharp Huygens property. The third is that the initial value problem for a class of data posed at the tingular time is well defined. Since the wave equation is reversible, this represents a class of data which propagates information smoothly from the past to the future passing through the space-time singularity.