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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?

Physics_Plasma
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  • I've included the equation of state in my list of equations, though. – Physics_Plasma Jan 17 '18 at 04:44
  • Ah, I just saw that. The only thing that I can see that is missing explicitly is the law of baryon conservation: $dn/dt = -n \nabla \cdot \mathbf{v}$. – Dr. Ikjyot Singh Kohli Jan 17 '18 at 04:50
  • Isn't that just the continuity equation, which I have listed? – Physics_Plasma Jan 17 '18 at 04:52
  • What's your $\gamma$ represent in the continuity equation? – Dr. Ikjyot Singh Kohli Jan 17 '18 at 04:53
  • The Lorentz factor, since this is relativistic hydrodynamics. – Physics_Plasma Jan 17 '18 at 04:53
  • Okay, see answer below. – Dr. Ikjyot Singh Kohli Jan 17 '18 at 04:53
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    I believe he is not explicitly specifying the first law of thermodynamics, which is what you integrate to give the fluid energy density $\rho$. – Dr. Ikjyot Singh Kohli Jan 17 '18 at 05:06
  • I think you are right about the first law being the missing piece. I thought he derived the entropy equation from the first law, so I figured they were equivalent, but perhaps not. I'll look again tomorrow. – Physics_Plasma Jan 17 '18 at 05:08
  • It is best to derive the entropy conservation equation from baryon conservation, as opposed to the 2nd law, the condition that $ds/dt = 0$ falls out naturally. – Dr. Ikjyot Singh Kohli Jan 17 '18 at 05:11
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    I understand it now. You have the equations $\partial_{\mu}n^{\mu}=0$, $\partial_{\mu}T^{\mu\nu}=0$, and the EoS $P = P(\rho)$. Those are 6 equations in terms of 6 variables ${ n, \mathbf{v}, P, \rho}$. Then you can add to that the first law of thermodynamics $d(\rho/n)+Pd(1/n)=Tds$, which gives one equation and one more unknown ($s$), as long as you assume temperature is constant, and imposed from the outside. – Physics_Plasma Jan 17 '18 at 16:39
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    Yes, that is the idea. The main equation of course is $T^{uv}_{,u} = 0$ which is equivalent to local conservation of energy, and comes from the divergence-free property of the Einstein tensor. However, you shouldn't assume that temperature is constant if you want the full picture. In terms of cosmology, you can assume certain epochs where the temperature is almost constant, but you will have to modify your equations to include a temperature variation. But, you have the general idea. – Dr. Ikjyot Singh Kohli Jan 17 '18 at 16:43
  • References and discussion of relativistic hydrodynamics here: https://physics.stackexchange.com/a/730704/226902 – Quillo Mar 06 '23 at 16:11

2 Answers2

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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 $\mu$.)

Dr. Ikjyot Singh Kohli
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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.

Jajati Nayak
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