Scalar, vector and tensor fields on $dS_3$ with arbitrary sources: harmonic analysis and antipodal maps
Geoffrey Compère, Sébastien Robert
TL;DR
This paper provides a comprehensive construction and analysis of scalar, vector, and tensor harmonics on $dS_3$, with careful attention to their antipodal data and how these data map between past and future infinity. It introduces a unified framework for both homogeneous and inhomogeneous equations, including explicit procedures to extract asymptotic data in the presence of sources and to define conserved charges across spatial infinity. A key advance is the detailed decomposition of vector and tensor fields into longitudinal, transverse, and SDT sectors, along with new lemmas extending prior work and enabling rigorous control of asymptotic behavior. The results have potential implications for describing interacting four-dimensional asymptotically flat fields at spatial infinity and suggest avenues for extending the analysis to higher dimensions and fermionic fields.
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 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.