Scalar, Vector and Tensor Harmonics on the Three-Sphere

TL;DR

This paper reviews scalar, vector, and tensor harmonics on the three-sphere, introduces a new notation for clarity, and presents practical numerical methods with tests demonstrating their accuracy in gravitational physics applications.

Contribution

It provides a comprehensive summary of harmonic properties, a new notation for simplification, and efficient numerical computation techniques with validation.

Findings

Harmonics satisfy eigenfunction and orthogonality conditions

Numerical methods accurately reproduce analytical identities

Test results confirm the effectiveness of the computational approach

Abstract

Scalar, vector and tensor harmonics on the three-sphere were introduced originally to facilitate the study of various problems in gravitational physics. These harmonics are defined as eigenfunctions of the covariant Laplace operator which satisfy certain divergence and trace identities, and ortho-normality conditions. This paper provides a summary of these properties, along with a new notation that simplifies and clarifies some of the key expressions. Practical methods are described for accurately and efficiently computing these harmonics numerically, and test results are given that illustrate how well the analytical identities are satisfied by the harmonics computed numerically in this way.