The ideal relativistic rotating gas as a perfect fluid with spin

Abstract: 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.