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

The propagator W ( t 0 , t 1 )( 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 = t 0 } and evaluates the solution ( u ( t 1 , x ),∂ t u ( t 1 , x )) at other times t 1 . The Friedmann–Robertson–Walker space–times are defined for t 0 , t 1 >0, whereas for t 0 →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 curvature K =0 and K =−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< t 0 < t the limit, lim t 0 → 0 + W ( t 0 , t ) ( g , h ) = W ( 0 , t ) ( g ) exists, it is independent of h ( x ), and for all reasonable initial data g ( x ), it gives rise to a well-defined solution for all t >0 emanating from the space–time singularity at t =0. Under reflection t →− t , the Friedmann–Robertson–Walker metric gives a space–time metric for t <0 with a singular future at t =0, and the same solution formulae hold. We thus have constructed solutions u ( t , x ) of the wave equation in Friedmann–Robertson–Walker space–times which exist for all − ∞ < t < 0 and 0 < t < + ∞ , where in conformally regularized coordinates, these solutions are continuous through the singularity t =0 of space–time, taking on specified data u (0,⋅)= g (⋅) at the singular time.

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