Quasi-local mass for small surfaces

Penrose's quasi-local mass for a small sphere is calculated by expanding in powers of the affine parameter along the generators of a light cone. The result is checked and extended to small surfaces of more general shape by techniques involving more direct use of spinor calculus. In a non-vacuum spacetime, the mass inside a small sphere is obtained from a 4-momentum Psub(a) which is the product of the volume enclosed by the surface and the 4-vector Tsub(ab)tsup(b), where tsup(a) is a unit timelike vector orthogonal to a spanning 3-surface. In the vacuum case, the mass vanishes at the fifth order in the diameter of the surface, both for small spheres and for small surfaces of more general shape. (author)