Small Perturbations in General Relativity: Tensor Harmonics of Arbitrary Symmetry

TL;DR

This paper introduces a general method for constructing tensor harmonics in General Relativity, enabling the analysis of small perturbations in space-times with arbitrary symmetries, and applies it to known cases like Schwarzschild and flat space.

Contribution

The paper develops a unified approach to generate tensor harmonics for arbitrary isometry groups, extending previous methods to more general space-times.

Findings

Constructed tensor harmonics for spaces with arbitrary isometry groups.

Applied the method to specific cases: cylindrical and spherical harmonics.

Demonstrated the method's consistency with known solutions.

Abstract

We develop a method for constructing of the basic functions with witch to expand small perturbations of space-time in General Relativity. The method allows to obtain the tensor harmonics for perturbations of the background space-time admitting an arbitrary group of isometry, and to split the linearized Einstein equations into the irreducible combinations. The essential point of the work is the construction of the generalized Casimir operator for the underlying group, which is defined not only on vector but also on tensor fields. It is used to construct the basic functions for spaces of tensor representations of the background metric's group of isometry. The method, being general, is applied here to construction of the basic functions for the case of the three-parameter group of isometry G_3 acting on the two-dimensional non-isotropic surface of transitivity. As quick illustrations of the method we consider the well-known particular cases: cylindrical harmonic for the flat space-time, and Regge-Wheller spherical harmonics for the Schwarzschild metric.