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 \sigma_\mu \nu. After having obtained the expression of the local spin-dependent phase space density f(x,p)_(\sigma \tau) in the Boltzmann approximation, we derive the spin density tensor and show that it is proportional to the acceleration tensor Omega_\mu \nu constructed with the Frenet-Serret tetrad. We recover the proper generalization of the fundamental thermodynamical relation, involving an additional term -(1/2) \Omega_\mu \nu \sigma^\mu \nu. We also show that the spin density tensor has a non-vanishing projection onto the four-velocity field, i.e. t^\mu= sigma_\mu \nu u^\nu \ne 0, in contrast to the common assumption t^\mu = 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.