2024/12/03 by Artem Averin, Averin, Artem · 1 citation
Earth and Planetary Sciences · Physics and Astronomy · #Geophysics and Gravity Measurements #Cosmology and Gravitation Theories
paper · pdf · doi:10.48550/arxiv.2412.02470
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 Σ, the choice of a codimension-2 surface B without boundary contained in Σ 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 \fracλℏ, consider the set of states where K takes a value not bigger than λ and let V denote the phase space volume of this set. One has then ln (V) ≤ \fracλℏ. Especially, we show for the Einstein-Hilbert Lagrangian in any dimension with cosmological constant and arbitrary minimally coupled matter, one has K = (A)/(4G). Hereby, A denotes the area of B in a particular state.