First law of black hole mechanics for charged black holes in a class of Horndeski theories

Abstract

The first law of black hole mechanics is derived for a class of Horndeski theories in the presence of an electromagnetic field. It is shown that for this class of theories, the presence of a nontrivial scalar field does not modify the expression of the first law for black hole solutions of the Einstein-Maxwell system. In particular, it is shown that the Bekenstein-Hawking formula for the black hole entropy remains valid.