Scalar, vector and tensor fields on $dS_3$ with arbitrary sources: harmonic analysis and antipodal maps

TL;DR

This paper analyzes scalar, vector, and tensor harmonics on three-dimensional de Sitter space, exploring their asymptotic data, antipodal relationships, and decomposition theorems, which are crucial for understanding fields at spatial infinity.

Contribution

It provides explicit harmonic definitions, antipodal data relationships, and decomposition theorems for fields on de Sitter space, advancing the understanding of asymptotic structures.

Findings

Explicit harmonic definitions for scalar, vector, and tensor fields on de Sitter.

Antipodal relationships between past and future asymptotic data.

Decomposition theorems for tensors obeying wave equations.

Abstract

The scalar, vector and tensor spherical harmonics on three-dimensional de Sitter spacetime are defined and analyzed. Each harmonic defines two sets of asymptotic data on the two sphere in the asymptotic expansion close to both the past and the future of de Sitter spacetime. For each case, we make explicit the antipodal relationship of both sets of asymptotic data between past and future infinity, which can be non-local. A procedure is defined to extract these asymptotic data in the presence of sources. This provides for each class of propagating field on de Sitter the relationship between two independent sets of data defined on the sphere in the asymptotic future with the corresponding data defined in the asymptotic past. We also provide several theorems on the decomposition of vector and tensors on de Sitter such as one proving that a large class of tensors obeying an inhomogeneous wave equation can be expressed locally in terms of a symmetric transverse traceless tensor. These results are instrumental in the description of interacting four-dimensional asymptotically flat fields at spatial infinity.