Complete set of equations in relativistic hydrodynamics

Question

I am reading Zee's "Einstein Gravity in a Nutshell", and in Appendix 2 of Section III.6, he covers the equations governing the hydrodynamics of a perfect fluid. He writes:

The set of equations, continuity (22), Euler (24), entropy conservation (27), together with an equation of state relating $P$ and $\rho$ and thus specifying the fluid, allows us to solve for the motion of the fluid.

The equations he specifies are continuity $$ \frac{\partial}{\partial t}\left(\gamma n\right)+\nabla\cdot \left(\gamma n \mathbf{v}\right) =0,$$ Euler $$ \frac{\partial \mathbf{v}}{\partial t} + \mathbf{v}\cdot \nabla \mathbf{v}=-\left(\frac{1-\mathbf{v}^2}{\rho+P}\right)\left(\mathbf{v}\frac{\partial P}{\partial t}+\nabla P\right), $$ entropy conservation $$ \frac{\partial s}{\partial t}+\mathbf{v}\cdot \nabla s=0, $$ and the equation of state $$ P = P(\rho).$$

I count 6 equations and 7 unknowns $\{n,\mathbf{v},\rho,P,s \}$. How can we completely specify the fluid given just these equations?

Answer 1

I am not familiar with Zee's text, since it seems relatively new on the G.R. front, but, for relativistic hydrodynamics and thus, relativistic thermodynamics, you have 8 quantities that characterize the fluid flow, and thus need 8 equations that govern the fluid's motion. The 8 quantities are: $p$ for pressure, $n$ baryon number density, $s$ for entropy, $\rho$ for energy density, and then the 4 components of the fluid velocity, $\mathbf{u}$.

The equations are thus:

1. An equation of state: $p = p(n,s)$

2. Continuity equation: $dn/dt = -n \nabla \cdot u$

3. Conservation of energy: $ds/dt = 0$

4. 3 Euler equations as you have above

5. 4-velocity normalization: $u \cdot u = -1$

6. The first law of Thermodynamics: $d \rho = \frac{(\rho + p)}{n} dn$

This latter equation is integrated to give the fluid energy density as a function of $n$ and $s$.

(Note that, I have neglected full thermodynamics in this description. Namely, if you include the full thermodynamic picture (for a perfect fluid), then, we must add an equation of state for the fluid temperature, $T = T(n,s)$, and then, the first law of thermodynamics has to have added to it a $n T ds$ term. Further, you need to also define a chemical potential, call it $\mu$.)

Answer 2

Yes, there are 6-equations in ideal relativistic hydrodynamics applied in studying quark gluon plasma fluid produced in relativistic heavy ion collisions. There are 7 unknowns. 6 equations are $$ \partial_{\mu}n_B^{\mu}=0~~~ \text{one} $$ $$\partial_{\mu}T^{\mu\nu}=0~~~ \text{four}$$ $$ \partial_{\mu}s^{\mu}=0~~~ \text{one}$$ Baryon number conservation, energy-momentum conservation and entropy conservation. Unknowns are baryon number density, energy density, pressure, entropy density and three velocity components i.e., $$n_B,\epsilon,P, s,\vec{v}$$, total seven. So 6 equations and 7 unknowns. This is compenstated with equation of state $P=f(\epsilon)$ which is an external input to find the solution. We have to provide equation of state while solving these 6 equations. Hence it is consistent.