Gravity from thermodynamics: optimal transport and negative effective dimensions

Abstract

We prove an equivalence between the classical equations of motion governing vacuum gravity compactifications (and more general warped-product spacetimes) and a concavity property of entropy under time evolution. This is obtained by linking the theory of optimal transport to the Raychaudhuri equation in the internal space, where the warp factor introduces effective notions of curvature and (negative) internal dimension. When the Reduced Energy Condition is satisfied, concavity can be characterized in terms of the cosmological constant $\Lambda$; as a consequence, the masses of the spin-two Kaluza-Klein fields obey bounds in terms of $\Lambda$ alone. We show that some Cheeger bounds on the KK spectrum hold even without assuming synthetic Ricci lower bounds, in the large class of infinitesimally Hilbertian metric measure spaces, which includes D-brane and O-plane singularities. As an application, we show how some approximate string theory solutions in the literature achieve scale separation, and we construct a new explicit parametrically scale-separated AdS solution of M-theory supported by Casimir energy.

arXiv comment57 pages + 4 appendices, 4 figures. v2: expanded introduction and comments on the Casimir vacuum. v3: clarification on the agreement of the Casimir vacuum with conjectured bounds

References in corpus (43)

• Holographic Derivation of Entanglement Entropy from AdS/CFT
• Thermodynamics of Spacetime: The Einstein Equation of State
• On the Origin of Gravity and the Laws of Newton
• Supergravity description of field theories on curved manifolds and a no go theorem
• N=2 Extremal Black Holes
• Metric measure spaces with Riemannian Ricci curvature bounded from below
• Universal Aspects of Gravity Localized on Thick Branes
• Gravitation from Entanglement in Holographic CFTs
• Type IIA Moduli Stabilization
• AdS and the Swampland
• Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below
• Fluxes, moduli fixing and MSSM-like vacua in a simple IIA orientifold
• Riemannian Ricci curvature lower bounds in metric measure spaces with $\sigma$-finite measure
• The Dark Dimension and the Swampland
• A Characterization of Entropy in Terms of Information Loss
• Spin-2 spectrum of defect theories
• Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds
• Quantum Horizons of the Standard Model Landscape
• Remarks on scale separation in flux vacua
• Sharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature bounds
• O-plane Backreaction and Scale Separation in Type IIA Flux Vacua
• On uniqueness theorems for Tsallis entropy and Tsallis relative entropy
• IR/UV Mixing, Towers of Species and Swampland Conjectures
• Compactification on negatively curved manifolds
• Dimensional Reduction and (Anti) de Sitter Bounds
• The Meaning of Einstein's Equation
• Scale-separated AdS${}_4$ vacua of IIA orientifolds and M-theory
• An optimal transport formulation of the Einstein equations of general relativity
• On AdS${}_7$ stability
• Hyperbolic compactification of M-theory and de Sitter quantum gravity
• Causality for nonlocal phenomena
• Renormalization Group Flow as Optimal Transport
• A review of Lorentzian synthetic theory of timelike Ricci curvature bounds
• Cosmological singularity theorems and splitting theorems for N-Bakry-Emery spacetimes
• Curvature-dimension bounds for Lorentzian splitting theorems
• Sharp Cheeger-Buser type inequalities in $\mathsf{RCD}(K,\infty )$ spaces
• An optimal dimension-free upper bound for eigenvalue ratios
• The equality case in Cheeger's and Buser's inequalities on $\mathsf{RCD}$ spaces
• Functional Inequalities on Weighted Riemannian Manifolds Subject to Curvature-Dimension Conditions
• Optimal maps and local-to-global property in negative dimensional spaces with Ricci curvature bounded from below
• Stability of curvature-dimension condition for negative dimensions under concentration topology
• Eigenvalue Estimates on Bakry-Emery Manifolds
• Convergence of metric measure spaces satisfying the CD condition for negative values of the dimension parameter

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