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We discuss a puzzle in relativistic spin hydrodynamics; in the previousformulation the spin source from the antisymmetric part of the canonicalenergy-momentum tensor (EMT) is crucial. The Belinfante improved EMT ispseudo-gauge transformed from the canonical EMT and is usually a physicallysensible choice especially when gauge fields are coupled as inmagnetohydrodynamics, but the Belinfante EMT has no antisymmetric part. We findthat pseudo-transformed entropy currents are physically inequivalent innonequilibrium situations. We also identify a current induced by the spinvorticity read from the Belinfante symmetric EMT.

Spin Hydrodynamics and Symmetric Energy-Momentum Tensors -- A current induced by the spin vorticity --