The Thermodynamic Sky

The Unruh effect predicts that an accelerating observer in vacuum perceives a thermal bath at a temperature proportional to the acceleration. The connection between acceleration and temperature is usually derived from quantum field theory on curved spacetime — it requires the machinery of Bogoliubov transformations and the KMS condition. The thermodynamic analogy is treated as a deep result that emerges from the quantum structure of the vacuum.

Polterovich (arXiv:2603.28039, March 2026) shows that the connection is not an analogy but a literal coordinate transformation. The contact-geometric formulation of general relativity, when restricted to the Minkowski hyperboloid, admits a hodograph transform that maps the sky of an accelerating observer directly into a thermodynamic phase space. The generating functions that describe the evolution of the observer's sky — how the positions of stars change as the observer accelerates — become reduced free energies in the thermodynamic coordinates. An effective temperature emerges proportional to acceleration, consistent with Unruh scaling.

The hodograph transform is a classical change of variables from fluid mechanics, where it swaps the roles of dependent and independent variables. Applied here, it exchanges the geometric variables (positions on the sky, proper time) with thermodynamic variables (entropy, temperature, free energy). The exchange is exact — it is not an approximation or a limit but a mathematical identification. The two descriptions are the same object in different coordinates.

The numerical constant relating temperature to acceleration differs from the standard Unruh result (which includes a factor of 2π from the quantum vacuum). The hodograph transform captures the geometric/classical structure of the relationship; the quantum correction supplies the factor. The classical skeleton of the Unruh effect — the proportionality, the functional form, the thermodynamic structure — is purely geometric, not quantum.

The structural observation: the thermodynamic structure of accelerating observers is not a quantum result dressed in thermal language. It is a geometric result — a coordinate transformation — that quantum mechanics makes physical by supplying the numerical coefficient. The deep connection between acceleration and temperature is not that acceleration creates particles. It is that the geometry of an accelerating sky is literally the same mathematical object as a thermodynamic phase space, related by a classical change of variables.