THE 1ST LAW OF BLACK-HOLE PHYSICS FOR A CLASS OF NONLINEAR MATTER MODELS

The discovery of new black hole solutions and other surprises prompted us to study the following topics related to stationary black holes for non-linear matter models, such as Yang-Mills fields or general sigma models: (i) The staticity problem for non-rotating stationary black holes, (ii) the circularity and Frobenius conditions for rotating black holes and (iii) the first law of black hole physics. Definitive and satisfactory results concerning these issues are derived for arbitrary minimally coupled scalar field (non-linear sigma) models. For general Yang-Mills theories we show that, contrary to the Abelian case, the proof of the circularity theorem requires additional assumptions on the Yang-Mills field tensor. Concerning the first law, we derive an expression for the variation of the mass, involving only global quantities and surface terms. This relation generalizes the Bardeen-Carter-Hawking formula to black hole solutions of Einstein-Yang-Mills theories with arbitrary gauge groups.