Group-theoretic approach to the new conserved quantities in general relativity

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 T_3m^j 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 T_3m^j. It is shown that the infinitedimensional representation is not irreducible; its decomposition into irreducible parts leads to the expansion of η in the T_3m ^j, the latter providing invariant subspaces in which irreducible representations act.