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 fo r non-linear matter models, such as Yang-Mills fields or general sigma models: (i) The staticity problem for non-rotating stationary black h oles, (ii) the circularity and Frobenius conditions for rotating black holes and (iii) the first law of black hole physics. Definitive and s atisfactory 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 pr oof of the circularity theorem requires additional assumptions on the Yang-Mills field tensor. Concerning the first law, we derive an expres sion for the variation of the mass, involving only global quantities a nd surface terms. This relation generalizes the Bardeen-Carter-Hawking formula to black hole solutions of Einstein-Yang-Mills theories with arbitrary gauge groups.