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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 σ_μν. 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 Omega_μνconstructed with the Frenet-Serret tetrad. We recover the proper generalization of the fundamental thermodynamical relation, involving an additional term -(1/2) Ω_μνσ^μν. We also show that the spin density tensor has a non-vanishing projection onto the four-velocity field, i.e. t^μ= sigma_μνu^ν\ne 0, in contrast to the common assumption t^μ= 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.