First law and Smarr formula of black hole mechanics in nonlinear gauge theories

Abstract

Motivated by the fact that Bardeen black holes do not satisfy the usual first\nlaw and Smarr formula, we derive a generalized first law from the Lagrangian of\nnonlinear gauge field coupled to gravity. In our treatment, the Lagrangian is a\nfunction of the electromagnetic invariant as well as some additional\nparameters. Consequently, we obtain new terms in the first law. With our\nformula, we find the correct forms of the first law for Bardeen black holes and\nBorn-Infeld black holes. By scaling arguments, we also derive a general Smarr\nformula from the first law. Our results apply to a wide class of black holes\nwith nonlinear gauge fields.\n