A Line Source In Minkowski For The de Sitter Spacetime Scalar Green's Function: Massless Minimally Coupled Case

TL;DR

This paper presents a novel method to derive the scalar Green's function in de Sitter spacetime by embedding it into higher-dimensional Minkowski space, avoiding direct wave equation solutions.

Contribution

It introduces a line source approach in Minkowski space for calculating Green's functions in curved spacetimes like de Sitter, simplifying the computation process.

Findings

Green's function sourced by a line near the origin in Minkowski space

Method applicable to higher-dimensional de Sitter spacetime

Avoids solving the wave equation directly in curved spacetime

Abstract

Motivated by the desire to understand the causal structure of physical signals produced in curved spacetimes -- particularly around black holes -- we show how, for certain classes of geometries, one might obtain its retarded or advanced minimally coupled massless scalar Green's function by using the corresponding Green's functions in the higher dimensional Minkowski spacetime where it is embedded. Analogous statements hold for certain classes of curved Riemannian spaces, with positive definite metrics, which may be embedded in higher dimensional Euclidean spaces. The general formula is applied to $(d \geq 2)$-dimensional de Sitter spacetime, and the scalar Green's function is demonstrated to be sourced by a line emanating infinitesimally close to the origin of the ambient $(d+1)$-dimensional Minkowski spacetime and piercing orthogonally through the de Sitter hyperboloids of all finite sizes. This method does not require solving the de Sitter wave equation directly. Only the zero mode solution to an ordinary differential equation, the "wave equation" perpendicular to the hyperboloid -- followed by a one dimensional integral -- needs to be evaluated. A topological obstruction to the general construction is also discussed by utilizing it to derive a generalized Green's function of the Laplacian on the $(d \geq 2)$-dimensional sphere.