Efficient trace-free decomposition of symmetric tensors of arbitrary rank

TL;DR

This paper introduces an improved iterative method for efficiently decomposing symmetric tensors into trace-free parts, applicable in both symbolic and numerical contexts across arbitrary dimensions.

Contribution

The authors develop a closed-form, dimension-independent iterative approach for trace-free decomposition of symmetric tensors of any rank, with practical computations for higher ranks.

Findings

Method works in symbolic and numerical settings

Closed-form representation for arbitrary dimensions

Computed STF mass multipole moments for ranks 5 to 8

Abstract

Symmetric trace-free tensors are used in many areas of physics, including electromagnetism, relativistic celestial mechanics and geodesy, as well as in the study of gravitational radiation and gravitational lensing. Their use allows integration of the relevant wave propagation equations to arbitrary order. We present an improved iterative method for the trace-free decomposition of symmetric tensors of arbitrary rank. The method can be used both in coordinate-free symbolic derivations using a computer algebra system and in numerical modeling. We obtain a closed-form representation of the trace-free decomposition in arbitrary dimensions. To demonstrate the results, we compute the coordinate combinations representing the symmetric trace-free (STF) mass multipole moments for rank 5 through 8, not readily available in the literature.