More on Penrose's quasi-local mass

The author defines the 'data' of a 2-surface S in spacetime in terms of its first and second fundamental forms and show how Penrose's kinematic twistor for S is a functional of these data. The author finds necessary and sufficient conditions in terms of the data for S to be 'contorted', i.e. for its data not to be the data of a 2-surface in a conformally flat space, and show that S is non-contorted iff the usual (local) twistor definition of norm is in fact constant on S. They find a large class of non-contorted 2-surfaces in the Schwarzchild solution and show that the Penrose mass M P to whether one of this class is zero or M s , the Schwarzchild mass parameter, according to whether the 2-surface does not or does go round the central hole. Finally, they calculate the 2-surface twistor space for a stationary black hole and prove the 'isoperimetric inequality' for the Penrose mass of a static black hole.