Spherical-harmonic tensors

TL;DR

This paper develops a formalism connecting spherical harmonics and symmetric tensors, enabling tensor expansions and analyzing Lorentz invariance violations in gravity.

Contribution

It introduces a novel construction of traceless and trace-extended symmetric tensors linked to spherical harmonics, expanding their application to tensor-valued functions.

Findings

Constructed orthonormal basis for symmetric tensors of any rank

Derived relationship between spherical-harmonic tensors and spin-weighted harmonics

Applied formalism to analyze Lorentz invariance violations in Newtonian gravity

Abstract

The connection between spherical harmonics and symmetric tensors is explored. For each spherical harmonic, a corresponding traceless symmetric tensor is constructed. These tensors are then extended to include nonzero traces, providing an orthonormal angular-momentum eigenbasis for symmetric tensors of any rank. The relationship between the spherical-harmonic tensors and spin-weighted spherical harmonics is derived. The results facilitate the spherical-harmonic expansion of a large class of tensor-valued functions. Several simple illustrative examples are discussed, and the formalism is used to derive the leading-order effects of violations of Lorentz invariance in Newtonian gravity.