STATICITY, CIRCULARITY, AND THE 1ST LAW OF BLACK-HOLE PHYSICS

We give a brief summary of some of our ongoing work on stationary blac k holes for non-linear matter models, such as gauge field theories (in cluding Higgs fields) or general sigma models. Our main result is a ge neralization of the Bardeen-Carter-Hawking formula for the variation o f the mass of stationary black holes, which involves only global quant ities and boundary integrals. Results and open questions related to th e staticity problem for non-rotating black holes, and to the circulari ty and Frobenius conditions for rotating ones, are briefly discussed.