In the following the second part of the question Relativistic hydrodynamics is presented:
Consider the hydrodynamic energy-momentum tensor in 3 + 1 dimensions in at space with the Minkowski metric $\eta_{\mu\nu}$. The coordinates are chosen as $u^\mu = (t; x; y; z)$. To first order in derivatives, the energy-momentum tensor can be written as $T^{\mu\nu}=(\epsilon+P)u^\mu u^\nu+P\eta^{\mu\nu}-\eta\Delta^{\mu\alpha} \Delta^{\nu\beta}(\partial_\alpha u_\beta + \partial_\beta u_\alpha -\frac{2}{3}\eta_{\alpha\beta}\partial_\lambda u^\lambda)$
The velocity vector field $u^\mu$ is normalised to $u^\mu u_\mu=-1$ and pressure is a function of energy density, $P(\epsilon)$ (this function is the equation of state). The coeffcient $\eta(\epsilon)$ (also a function of energy density) is known as shear viscosity. The projector $\Delta^{\mu\nu}$ is defined as $\Delta^{\mu\nu}=u^\mu u^\nu + \eta^{\mu\nu}$. Use the facts that trace $ T\equiv \eta_{\mu\nu}T^{\mu\nu}=0$, which yields equation of state $P(\epsilon)=\epsilon/3$, and trace of the bulk viscosity term $T^{\mu\nu}=\dots \zeta\Delta^{\mu\nu}\partial_\lambda u^\lambda$ is also $0$.
(a.) Without deriving the dispersion relation for sound determine the speed of sound for fluids with the equation of state $P(\epsilon)=\epsilon/3$.
(b.) We only look at systems for which the trace is zero: $ T\equiv \eta_{\mu\nu}T^{\mu\nu}=0$. The theory is called conformal hydrodynamics with equation of state $P(\epsilon)=\epsilon/3$ and $\zeta=0$. Perturb $u^\mu (x)$and $\epsilon(x)$ around constant equilibruim values to first order in small $\delta u^\mu$ and $\delta\epsilon$:
$u^\mu=(1,0,0,0) + \delta u^\mu e^{-i\omega t+ikx},\\ \epsilon=\epsilon_0 + \delta \epsilon e^{-i\omega t+ikx}$
This means that $\epsilon_0$, $\delta u^\mu$ and $\delta \epsilon$ are all constant. The only space-time dependence is in the exponentials and it only depends on $t$ and $x$ coordinates (not $y$ or $z$).
Now, consider only transverse fluctuations to the $x$-direction, for example only $\delta u^y$ and set all other fluctuations to zero. By expanding
$\partial_\mu T^{\mu\nu}=0$
to first order in fluctuations, derive the dispersion relation for the mode (solve for $\omega(k)$) and show that the solution has the form
$\omega=-i D k^2$.
Find $D$. This is diffusion. Make sure that your solution solves the complete set of equations $\partial_\mu T^{\mu\nu}=0$.
(c.) Comment on why diffusion looks so non-relativistic. What breaks the Lorentz invariance of $T^{\mu\nu}$?
Could anyone lend me some advice?
