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 Ω μν constructed with the Frenet–Serret tetrad. We recover the proper generalization of the fundamental thermodynamical relation, involving an additional term −(1/2)Ω μν σ μν . We also show that the spin density tensor has a non-vanishing projection onto the four-velocity field, i.e. t μ = σ μν u ν ≠ 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.