It is of considerable interest to obtain an expression for the entropy of a dynamical black hole in general theories of gravity since it will allow one to study the classical and semiclassical second law. We propose a new formula for the entropy of a dynamical black hole in arbitrary classical Lagrangian theories of gravity. Our formula has the property of being “non-teleological” in that it increases only when matter crosses the event horizon, unlike the Bekenstein—Hawking formula in general relativity and its generalization (arrived at independently) by Dong and by Wall. In general relativity, our formula for the entropy of a dynamical black hole differs from the Bekenstein—Hawking formula A/4 by a term proportional to the integral of the expansion of null generators of the horizon. In this talk, we discuss the first law, classical and semiclassical second law for our entropy formula, and its relation with the Dong—Wall entropy formula.

Based on 2402.00818