The ideal relativistic rotating gas as a perfect fluid with spin

We show that the ideal relativistic spinning gas at complete thermodynamical equilibrium is a fluid with a non-vanishing spin density tensor σμν. After having obtained the expression of the local spin-dependent phase-space density f(x, p)στ in the Boltzmann approximation, we derive the spin density tensor and show that it is proportional to the acceleration tensor Ωμν constructed with the Frenet–Serret tetrad. We recover the proper generalization of the fundamental thermodynamical relation, involving an additional term −(1/2)Ωμνσμν. We also show that the spin density tensor has a non-vanishing projection onto the four-velocity field, i.e. tμ = σμνuν ≠ 0, in contrast to the common assumption tμ = 0, known as Frenkel condition, in the thus-far proposed theories of relativistic fluids with spin. We briefly address the viewpoint of the accelerated observer and inertial spin effects.