A variational principle for black holes

Abstract

It is shown that the initial data which gives rise to stationary black hole solutions extremizes the mass for a given angular momentum and area of the horizon. The only extremum of the mass for a given area of the horizon but arbitrary angular momentum is the Schwarzschild solution. In this case, and when the angular momentum is small, the extremum of the mass is a local minimum. This suggests that the initial data for the Schwarzschild solution has a smaller mass than any other initial data with the same area of the horizon. If this is the case, there is no possibility of proving the occurrence of naked singularities by methods suggested by Penrose and Gibbons. Together with Carter's theorem, the fact that the extremum is a local minimum indicates that the Kerr solutions are stable against axisymmetric perturbations.