Relativistic magnetohydrodynamics of a spinful and vortical fluid: Entropy current analysis

We generalize a recently introduced formulation of relativistic spinful and vortical fluid to relativistic magnetohydrodynamics (MHD). We refer to it as the "Spinful-Vortical MHD" (SVMHD). The aim is to scrutinize the interplay between the vorticity, magnetic field, and spin, which is treated as a quantum object, in contrast to other formulations of spin hydrodynamics. To this purpose, we first perform a standard entropy current analysis up to first-order gradient expansion, $\mathcal{O}\left(\partial\right)$ as well as $\mathcal{O}\left(\hbar\partial\right)$, where $\hbar$ is the Planck constant. In contrast to alternative formulations of spin MHD, in the absence of vorticity, the zeroth-order energy-momentum tensor includes an additional magneto-vorticity mixed term and reduces, as expected, to the energy-momentum tensor of MHD. We show that in the first-order of gradient expansion, $36$ dissipative transport coefficients appear. They satisfy certain constraints that guarantee the positive definiteness of the entropy production rate. We then modify the formulation of SVMHD by replacing the magnetic part of the thermal vorticity tensor with its electric part. Carrying out the same analysis as in the standard formulation, we show that in this case, the first-order constitutive relations consist of $11$ nondissipative Hall-like coefficients, apart from $25$ dissipative coefficients. This difference arises from different behavior of the electric and magnetic part of the thermal vorticity under time-reversal transformation.