# Thermodynamic formulation of pefect fluid with spin

The spin phenomena in an asymmetric spin-related Einstein-Cartan candidate solution is due to microscopic spin of small volumes of a globally "ordinary" perfect fluid.

The only description I have seen is:

$$S_{\mu \nu} + S_{\nu \mu}=0$$

And a constraint relating the spin to the fluid acceleration.

So I can just choose anything for the non vanishing axial vector, to represent the spin vector distribution? I don't fully get it.

How does one go about giving a statistical formulation of this strictly microscopic angular degree of freedom?

• You should add that the spin tensor should be orthogonal to the local 4-velocity of the fluid : $S_{\mu \nu} \, u^{\nu} = 0$. That constraint says that $S_{i 0} = 0$ in the rest frame of a fluid element (spin is a spacelike property).
– Cham
Commented Jun 29, 2018 at 14:30
• OP prefers explicit formulae, before putting Dirac things in there. :) Commented Jun 29, 2018 at 16:19
• It's not clear what you want to know. You should make your question more precise.
– Cham
Commented Jun 29, 2018 at 17:42
• @Someone I have edited with the punch line Commented Jun 29, 2018 at 19:47
• If you want to see how to arrive from microscopic description of point particles with spin to a continuous fluid medium by averaging, have a look at a book "Relativistic theory of spinning fluids", by Halbwachs (pdf here), but it is for a flat space. For equations of spinning fluid in Einstein–Cartan theory have a look, for example, here Commented Jun 30, 2018 at 17:47