A Line Source in Minkowski for the de Sitter Spacetime Scalar Green's Function: Massive Case

摘要

For certain classes of space(time)s embeddable in a higher dimensional flatspace(time), it appears possible to compute the minimally coupled masslessscalar Green's function in the former by convolving its cousin in the latterwith an appropriate scalar charge density. The physical interpretation is thatbeings residing in the higher dimensional flat space(time) may set up sourcesto fool the observer confined on the lower dimensional curved submanifold thatshe is detecting the field generated by a space(time) point source in her ownworld. In this paper we extend the general formula to include a non-zero mass.We then employ it to derive the Green's function of the massive wave operatorin (d >= 2)-dimensional de Sitter spacetime and that of the Helmholtzdifferential operator -- the Laplacian plus a "mass term" -- on the (d >=2)-sphere. For both cases, the trajectories of the scalar sources are the sameas that of the massless case, while the required scalar charge densities aredetermined by solving an eigenvalue equation. To source these massive Green'sfunctions, we show that the (d+1)-dimensional Minkowski/Euclideanexperimentalists may choose to use either massive or massless scalar linecharges. In de Sitter spacetime, the embedding method employed here leadsdirectly to a manifest separation between the null cone versus tail terms ofthe Green's functions.