Group-Theoretic Approach to the New Conserved Quantities in General Relativity

Abstract

The Newman-Penrose formalism for obtaining the recent conserved quantities in general relativity is discussed and a group-theoretic interpretation is given to it. This is done by relating each triad of the orthonormal vectors on the sphere to an orthogonal matrix g. As a result, the spin-weighted quantities η become functions on the group of three-dimensional rotation, η = η(g), where g ∈ O3. An explicit form for the matrix g is given and a prescription for rewriting η(g) as functions of the spherical coordinates is also given. We show that a quantity of spin weight s can be expanded as a series in the matrix elements Tsmj of the irreducible representation of O3, where s is fixed. Infinite- and finite-dimensional representations of the group SU2 are then realized in the spaces of η's and Tsmj. It is shown that the infinite-dimensional representation is not irreducible; its decomposition into irreducible parts leads to the expansion of η in the Tsmj, the latter providing invariant subspaces in which irreducible representations act.