10 - Penrose's Quasi-local Mass

Summary

Introduction

Penrose's definition of a kinematic twistor associated to an arbitrary topologically spherical 2-surface in an arbitrary space-time [14] was given seven years ago. Since then the definition has been a focus of constant attention. The aim of this review is to collect together what is now known and to say what the successes and failures have been.

The definition was originally made in the hope of identifying within the kinematic twistor, A_αβ, a total momentum and angular momentum for the space-time threading through the 2-surface S. Most subsequent investigations have concentrated on the less ambitious target of deriving just a momentum or just a scalar invariant, the total mass-energy. One notable exception to this trend has been the study of angular momentum for 2-surfaces S at futurenull-infinity or space-like infinity in asymptotically-flat space-times [60, 21].

The dominating fact about the subsequent investigations has been the necessity to distinguish 2-surfaces S in which the data, by which I shall mean the first and second fundamental forms, reveal the presence at S of conformal or Weyl curvature from those in which it does not. Calling the former contorted (and the latter non-contorted) following the suggestion of Penrose [15] we shall see that for non-contorted S there do exist scalar invariants. Furthermore the definitions of total mass-energy which then arise have appealing properties in a large variety of particular cases.