Small perturbations in general relativity: tensor harmonics of arbitrary symmetry

Abstract

We develop a method for constructing of the basic functions with which 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 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. As a quick illustration of the general method we consider 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.