How to use retarded Green’s functions in de Sitter spacetime

We demonstrate in examples that the covariant retarded Green’s functions in electromagnetism and linearized gravity work as expected in de Sitter spacetime. We first clarify how retarded Green’s functions should be used in spacetimes with spacelike past infinity such as de Sitter spacetime. In particular, we remind the reader of a general formula which gives the field for given initial data on a Cauchy surface and a given source (a charge or stress-energy tensor distribution) in its future. We then apply this formula to three examples: (i) electromagnetism in the future of a Cauchy surface in Minkowski spacetime, (ii) electromagnetism in de Sitter spacetime, and (iii) linearized gravity in de Sitter spacetime. In each example the field is reproduced correctly as predicted by the general argument. In the third example we construct a linearized gravitational field from two equal point masses located at the “North and South Poles” which is nonsingular on the cosmological horizon and satisfies a covariant gauge condition and show that this field is reproduced by the retarded Green’s function with corresponding gauge parameters.

Figure 1

(a) The Carter-Penrose diagram of Minkowski spacetime; (b) Carter-Penrose diagram of de Sitter spacetime [see Eq. (4.1) for the definition of $\chi$]. The field coming from a classical source (its world line represented by a vertical bold line) is nonzero in the shaded region.

Figure 2

A static charge $q$ at the center producing the field in the shaded region.

Figure 3

The spreading of the source field as the time elapses.

Figure 4

The behavior of the initial field as the time elapses.

Figure 5

Coordinates $(\widehat{\lambda },\widehat{r},\widehat{\theta },\widehat{\varphi })$ covering the lower half of the manifold.

Figure 6

The causal past of the point $(\widehat{\lambda },\widehat{r},\widehat{\theta }=0)$ in the case $\widehat{r}>\widehat{\lambda }-{\widehat{\lambda }}^{\prime }$.

Figure 7

The causal past of the point $(\widehat{\lambda },\widehat{r},\widehat{\theta }=0)$ in the case $\widehat{r}<\widehat{\lambda }-{\widehat{\lambda }}^{\prime }$.