title: On the initial value problem for the wave equation in Friedmann-Robertson-Walker space-times.
abstract: The propagatorW(t0,t1)(g,h) for the wave equation in a given space-time takes initial data (g(x),h(x)) on a Cauchy surface {(t,x) :t=t0} and evaluates the solution (u(t1,x),∂tu(t1,x)) at other timest1. The Friedmann-Robertson-Walker space-times are defined fort0,t1>0, whereas fort0→0, there is a metric singularity. There is a spherical means representation for the general solution of the wave equation with the Friedmann-Robertson-Walker background metric in the three spatial dimensional cases of curvatureK=0 andK=-1 given by S. Klainerman and P. Sarnak. We derive from the expression of their representation three results about the wave propagator for the Cauchy problem in these space-times. First, we give an elementary proof of the sharp rate of time decay of solutions with compactly supported data. Second, we observe that the sharp Huygens principle is not satisfied by solutions, unlike in the case of three-dimensional Minkowski space-time (the usual Huygens principle of finite propagation speed is satisfied, of course). Third, we show that for 0<t0<tthe limit, [Formula: see text] exists, it is independent ofh(x), and for all reasonable initial datag(x), it gives rise to a well-defined solution for allt>0 emanating from the space-time singularity att=0. Under reflectiont→-t, the Friedmann-Robertson-Walker metric gives a space-time metric fort<0 with a singular future att=0, and the same solution formulae hold. We thus have constructed solutionsu(t,x) of the wave equation in Friedmann-Robertson-Walker space-times which exist for all [Formula: see text] and [Formula: see text], where in conformally regularized coordinates, these solutions are continuous through the singularityt=0 of space-time, taking on specified datau(0,⋅)=g(⋅) at the singular time.
authors.affiliation: Department of Mathematics and Statistics , McMaster University , Hamilton, Ontario L8S 4K1, Canada.
authors.forename: Bilal
authors.initials: B
authors.lastname: Abbasi
authors.affiliation: Department of Mathematics and Statistics , McMaster University , Hamilton, Ontario L8S 4K1, Canada ; The Fields Institute , 222 College Street, Toronto, Ontario M5T 3J1, Canada.
authors.forename: Walter
authors.initials: W
authors.lastname: Craig
keywords: general relativity; space–time singularity; wave equation