general relativity - Derivation of the energy-momentum tensor for an imperfect fluid

In chapter 7 of the "Physical Foundations of Cosmology" Mukhanov uses this energy-momentum tensor for an imperfect fluid:

$$T^\mu_\nu = (\rho + p)u^\mu u_\nu - p\delta^\mu_\nu - \eta(P^\mu_\gamma u^{;\gamma}_\nu+P^\gamma_\nu u^\mu_{;\gamma}-\frac{2}{3}P^\mu_\nu u^{\gamma}_{;\gamma}) $$

where $\eta$ is the shear viscosity coefficient and $P \equiv \delta^\mu_\nu - u^\mu u_\nu$ is the projection operator.

Where does this relation come from? can you introduce some references for derivation of this energy-momentum tensor?

Answer

Here is a sketch of where it comes from. First just consider the perfect fluid terms and note the thermodynamic relation $$ \rho + p = \mu n + T s, $$ where $T$ and $s$ are temperature and entropy, $\mu$ and $n$ are a chemical potential and number density.

We also have a relation for derivatives of $p$ $$ dp = n d\mu + s dT. $$

Now if you take the divergence and dot with $u$

$$0=u^\nu \nabla_\mu T^\mu_\nu = \nabla_\mu (\rho+p)u^\mu - u^\nu\nabla_\nu p$$ Now use the thermodynamic relations and the fact that the number density is conserved and you get $$ 0= T \nabla_\mu s u^\mu.$$ so this expresses that the entropy current is conserved in the perfect fluid.

The new viscosity terms will modify this expression but they are chosen in such a way that the divergence of the entropy current will be strictly positive in order to satisfy the second law.

You can work it out yourself if you rewrite the viscosity terms as a four index symmetric traceless tensor contracted with $\nabla_\gamma u_\delta$. After you take the divergence and dot with $u$ as above, you end up with both $\nabla_\mu u_\nu$ and $\nabla_\gamma u_\delta$ contracted into this tensor.

In general there can be more terms, for instance the bulk viscosity. The details can be found in this review paper