Hypersurface in Euclidean 4 space - The thermodynamics of the quermassintegral ladder
Year: 2026
Type: preprint
Abstract: ## Abstract The four quermassintegrals of a convex body in flat four-space are related by a chain rule under normal displacement. We show that this chain rule is the first law of thermodynamics. Taking the area law and the surface-gravity temperature as inputs, the internal energy obtained by integrating $T\,dS$ lands exactly on the Einstein–Hilbert rung; the free energy $E - TS$ equals the intrinsic Einstein–Hilbert action of the boundary metric, so the factor of two between them is a Legendre transform; and the temperature is reproduced independently by the ladder as a ratio of consecutive rungs, with the topological rung in the numerator. The second law is a classification theorem of convex geometry — every rung is non-decreasing under outward displacement, by monotone Hadwiger. The ladder carries an equation of state $w = -1/3$, which is the curvature contribution of a closed universe. Finally, the identification of the free energy with the Einstein–Hilbert rung holds **if and only if** the body is a ball, and the failure is precisely the Alexandrov–Fenchel deficit: thermodynamic consistency selects the round sphere.
Author: Thrussell, Jay
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213616675