Formulation and Proof of the Gravitational Entropy Bound
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916505936986112 |
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| author | Averin, Artem |
| author_facet | Averin, Artem |
| contents | 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 $\fracλ{\hbar},$ 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 $\ln (V) \le \fracλ{\hbar}.$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_02470 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Formulation and Proof of the Gravitational Entropy Bound Averin, Artem High Energy Physics - Theory General Relativity and Quantum Cosmology High Energy Physics - Phenomenology 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 $\fracλ{\hbar},$ 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 $\ln (V) \le \fracλ{\hbar}.$ 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. |
| title | Formulation and Proof of the Gravitational Entropy Bound |
| topic | High Energy Physics - Theory General Relativity and Quantum Cosmology High Energy Physics - Phenomenology |
| url | https://arxiv.org/abs/2412.02470 |