Holography, the Second Law and a tilde C-function in higher curvature gravity

Abstract

We analyze the Second Law of black hole mechanics and the generalization of the holographic bound for general theories of gravity. We argue that both the possibility of defining a holographic bound and the existence of a Second Law seem to imply each other via the existence of a certain "c-function" (i.e. a never-decreasing function along outgoing null geodesic flow). We are able to define such a "c-function", that we call \tilde{C}, for general theories of gravity. It has the nontrivial property of being well defined on general spacelike surfaces, rather than just on a spatial cross-section of a black hole horizon. We argue that \tilde{C} is a suitable generalization of the notion of "area" in any extension of the holographic bound for general theories of gravity. Such a function is provided by an algorithm which is similar (although not identical) to that used by Iyer and Wald to define the entropy of a dynamical black hole. In a class of higher curvature gravity theories that we analyze in detail, we are able to prove the monotonicity of \tilde{C} if several physical requirements are satisfied. Apart from the usual ones, these include the cancellation of ghosts in the spectrum of the gravitational Lagrangian. Finally, we point out that our \tilde{C}-function, when evaluated on a black hole horizon, constitutes by itself an alternative candidate for defining the entropy of a dynamical black hole.