Equivalence of Thermodynamic Potential Equations and Special Relativity: A Gravitational-Thermodynamization Perspective

Abstract

Thermodynamic potentials provide equivalent descriptions of equilibrium systems through Legendre transforms and associated Maxwell relations. Motivated by recent developments in the ``thermodynamization'' of gravitational dynamics and by the RVB--residue method viewpoint (attributed to Wen-Xiang Chen), we propose a unified framework comparing two notions of equivalence: (i) the algebraic equivalence classes of thermodynamic potential equations, and (ii) the covariance constraints of special relativity. We formulate a map between potential-based contact geometry identities in thermodynamics and invariant structures in relativity via a variational ``residue'' extraction on constrained manifolds. Conceptually, we show that the equivalence of different thermodynamic potentials can be viewed as a gauge-like redundancy, while Lorentz covariance in relativity acts as a compatibility condition selecting physically admissible gauges. The framework is illustrated on relativistic ideal fluids and a simple horizon thermodynamics toy model.

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