Conserved quantities from pseudotensors and extremum theorems for angular momentum

Abstract

A method for calculating pseudotensor-based conserved quantities for isolated systems in general relativity, independently of the asymptotic behaviour of the coordinate system used, is given. This allows the evaluation of concepts like energy, momentum and angular momentum in any coordinate system. The calculation is carried out for the Schutz-Sorkin gravitational Noether operator, a pseudotensorial vector operator which reduces to the familiar pseudotensors for particular choices of the vector fields; it is illustrated for the Kerr metric using various fields and coordinates. The authors use this to prove a theorem of extremality of angular momentum for vacuum solutions of Einstein's equations, showing that any two of the following imply the third: (i) the metric is axisymmetric; (ii) Einstein's field equations are satisfied; (iii) the total angular momentum is an extremum against all perturbations satisfying a mild (and most reasonable) restriction. This theorem, valid for stationary and non-stationary metrics, is generalised to include matter fields, and, in particular, perfect fluids. A related theorem for extremising the angular momentum flux across a timelike hypersurface is also proved. This theorem provides an alternative way to solve the field equations for axisymmetric gravitational collapse.