13 - Penrose's Quasi-Local Momentum and Angular Momentum

Summary

In general relativity there is a long-standing problem of defining momentum and angular momentum in a general curved space-time. In this chapter, we describe an application of twistor theory which aims to provide a solution to this problem. As we shall see, this aim is not entirely achieved and the programme has both successes and failures.

We recall that in general relativity, all the local matter content is described by the stress–energy tensor T_ab. Gravitational energy, whether in gravitational waves or in the form of gravitational potential energy, is notoriously non-local and one cannot expect to characterise it by a local density. Instead, Penrose (1982) has suggested that one should seek a non-local invariant associated to any two-surface S and representing the total momentum–angular momentum flux through that surface. This non-local invariant will be constructed by twistorial techniques tailored to give the right answer for linearised general relativity where there is a clear right answer.

We begin by reviewing the definition of momentum and angular momentum in special relativity and in linearised general relativity.

A material system in special relativity is defined by its stress–energy tensor T_ab which we may suppose for definiteness to have support within a world tube W in Minkowski space, M. (The case of say electromagnetic fields spread throughout M is a simple generalisation.)