I was trying to solve the following problem from the problem book on relativty:
Problem 5.22. Show that the velocity of sound $v_{s}$ in a relativistic perfect fluid is given by $$v_{s}^{2}=\partial p /\left.\partial \rho\right|_{s=c o n s t a n t}$$
For a high temperature relativistic gas with an equation of state $\rho \approx 3 \mathrm{P}$ (essentially that for a photon gas) show that $v_{s} \approx 1 / \sqrt{3}$
Solution is given as:
Solution $5.22 .$ Assume that the acoustical wave is a perturbation (isentropic) in a uniform static fluid with parameters $\rho_{0}, P_{0},$ and $n_{0}$ Let the perturbations be $\rho_{1}, P_{1},$ and $n_{1}$ and the fluid velocity be $\mathbf{u} \approx\left(1, \underline{v}_{1}\right)$ in the rest frame of the unperturbed fluid. The first order perturbation terms in $\mathrm{T}_{, \nu}^{\mu \nu}=0$ give us
(i) From $\mu=0$ $$\nabla \cdot \underline{v}_{1}=-\frac{\partial \rho_{1}}{\partial t}\left(\rho_{0}+P_{0}\right)^{-1}$$
(ii) From $\mu=1,2,3$
$$\frac{\partial \underline{v}_{1}}{\partial \mathrm{t}}=-\frac{\nabla \mathrm{P}_{1}}{\left(\rho_{0}+\mathrm{P}_{0}\right)}$$ which can be combined into $$\partial^{2} \rho_{1} / \partial t^{2}-\nabla^{2} P_{1}=0$$ since $P_{1}$ and $\rho_{1}$ are related for isentropic flow by
Can anyone explain or derive the equation given in (ii).
