Formulation and Proof of the Gravitational Entropy Bound

Abstract

We provide a formulation and proof of the gravitational entropy bound. We use a recently given framework which expresses the measurable quantities of a quantum theory as a weighted sum over paths in the theory's phase space. If this framework is applied to a field theory on a spacetime foliated by a hypersurface $\hat{I}\pounds ,$ the choice of a codimension-2 surface $B$ without boundary contained in $\hat{I}\pounds$ specifies a submanifold in the phase space. We show here that this submanifold is naturally restricted to obey an entropy bound if the field theory is diffeomorphism-invariant. We prove this restriction to arise by considering the quantum-mechanical sum of paths in phase space and exploiting the interplay of the commutativity of the sum with diffeomorphism-invariance. The formulation of the entropy bound, which we state and derive in detail, involves a functional $K$ on the submanifold associated to $B.$ We give an explicit construction of $K$ in terms of the Lagrangian. The gravitational entropy bound then states: For any real $\frac{\hat{I}}{»}\hslash ,$ consider the set of states where $K$ takes a value not bigger than $\hat{I}»$ and let $V$ denote the phase space volume of this set. One has then $\ln (V)\le \frac{\hat{I}}{»}\hslash .$ Especially, we show for the Einstein-Hilbert Lagrangian in any dimension with cosmological constant and arbitrary minimally coupled matter, one has $K=\frac{A}{4G}.$ Hereby, $A$ denotes the area of $B$ in a particular state.