Black Holes, Global Monopole Charge and Quasi-Local Energy

Abstract

Black holes are described by solutions of the Einstein vacuum (or electrovac) equation and these solutions are unique. We wish to consider the question: can we get the same solutions from a slightly different and more general equation, which is not the derivative of the original equation? The answer is yes. Take the case of the Schwarzshild solution. It is the unique spherically symmetric solution of the Einstein vacuum equation, R _ik = 0. It can also be obtained as the solution of a slightly more general equation than that of vacuum. This happens because the vacuum equation ultimately reduces to two equations: one the good old Laplace equation and the other its first integral [1]. It is therefore sufficient to have only the first integral equation leaving the other free because it would always be implied by the former. This raises an interesting question: what happens when we have the Laplace equation but not its first integral? We shall show that the two sets of equations are dual of each other in a certain sense [2]. In the former case the unique solution is the Schwarzschild black hole while the dual set provides a black hole with a global monopole charge [3]. The latter solution is also unique.