Maxwell-Jüttner averaged nonaffine theory of relativistic shear viscosity

We formulate a Maxwell-Jüttner averaged extension of relativistic nonaffine shear-viscosity theory for a classical gas of massive particles. The starting point is the relativistic nonaffine response formula derived from generalized Langevin dynamics, where proper-momentum dissipation produces a Lorentz-factor enhancement of the viscosity. Since a thermal relativistic gas is not characterized by a single Lorentz factor, we replace this scalar enhancement by an equilibrium momentum average over the Maxwell-Jüttner distribution. Within the scalar transverse mean-field closure used in this work, the minimal relativistic correction is the Maxwell-Jüttner mean of the Lorentz factor and becomes a finite-mass enhancement function of ζ=mc²/(kBT). In the general closure, the averaged object is the full product of the Lorentz factor, the zero-frequency memory kernel, and the nonaffine affine-force correlator. This construction separates kinematic relativistic enhancement from dynamical dissipation encoded in the bath memory and collective mode spectrum. We derive the nonrelativistic and ultrarelativistic limits, show that a specified effective hard-sphere scaling reproduces the classical square-root law as a consistency test, and identify the assumptions under which a cubic high-temperature law follows. Comparison with relativistic kinetic-theory examples clarifies that the memory kernel is not equivalent to a relaxation time unless a microscopic closure is specified. Retaining the full frequency dependence instead yields a relaxation spectrum. A single dominant pole reduces this spectrum to one shear relaxation time and gives the usual transient shear equation.