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Riemannian geometry of the moment manifold in relativistic kinetic theory and the approach to hydrodynamics

Craig S. Wright*

  • *Contact author: cw881@exeter.ac.uk
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Phys. Rev. C 113, 064903 – Published 8 June, 2026

DOI: https://doi.org/10.1103/by8d-d9z2

Abstract

The kinetic-to-hydrodynamic crossover in the quark-gluon plasma is conventionally diagnosed by scale ratios such as the Knudsen and inverse Reynolds numbers, which are not metrical on the space of distributions. I equip the Denicol–Niemi–Molnär–Rischke (DNMR) 14-moment manifold with the Fisher-Rao metric—the local Hessian of the Kullback-Leibler divergence—and study the resulting Riemannian geometry for conformal Bjorken flow in the relaxation-time approximation. The equilibrium Fisher metric fixes the shear viscosity as η=4PτR/5, identifying near-equilibrium transport as an equilibrium statistical quantity. The full geodesic distance exceeds its linearized approximation by 15–35% at moderate shear, quantifying where Navier-Stokes–type expansions under-resolve the state-space distance. The equilibrium Fisher-Rao Ricci scalar of the five-dimensional shear sector is computed exactly within the truncation to be R=−5/21; the negative sign identifies the equilibrium statistical manifold as locally hyperbolic in the Riemannian-geometry sense, which on a statistical model corresponds to diverging geodesic deviation between nearby moment states—a geometric statement about the truncated exponential family, not a prediction for physical relaxation rates. All results are obtained within the DNMR 14-moment truncation for conformal Bjorken flow; the equilibrium relations are closure independent within the truncation, and finite-shear quantities are properties of the Jaynes maximum-entropy completion.

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Phys. Rev. C 113, 064903– Published 8 June, 2026
  • Received 30 March 2026
  • Revised 16 April 2026
  • Accepted 22 May 2026
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