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

Area of Science:

• General Relativity and Cosmology
• Partial Differential Equations
• Mathematical Physics

Background:

• The wave equation describes fundamental wave phenomena in spacetime.
• Friedmann-Robertson-Walker (FRW) spacetimes are standard models for a homogeneous and isotropic universe.
• FRW spacetimes exhibit a metric singularity at t=0, posing challenges for initial value problems.

Purpose of the Study:

• To analyze the wave propagator for the Cauchy problem in FRW spacetimes with K=0 and K=-1.
• To derive new results concerning the behavior of wave solutions near the spacetime singularity.
• To construct solutions to the wave equation that are well-defined across the t=0 singularity.

Main Methods:

• Utilizing the spherical means representation for the general solution of the wave equation on FRW backgrounds.
• Deriving properties of the wave propagator from this representation.
• Analyzing the behavior of solutions for initial data given at t=t0 > 0 and taking the limit t0 -> 0.

Main Results:

• An elementary proof is provided for the sharp rate of time decay of wave solutions with compactly supported initial data.
• The sharp Huygens principle is shown to be violated in these spacetimes, although finite propagation speed is maintained.
• A limit of the wave propagator is shown to exist as t0 approaches 0, yielding well-defined solutions for all t > 0, continuous through the t=0 singularity.

Conclusions:

• Solutions to the wave equation can be constructed in FRW spacetimes that are continuous across the t=0 singularity.
• These solutions satisfy the wave equation for all t, with specified initial data at the singular time t=0.
• The study extends the understanding of wave propagation in cosmological spacetimes and the behavior of solutions near singularities.