Relativistic Euler equations

In fluid mechanics and astrophysics, the relativistic Euler equations are a generalization of the Euler equations that account for the effects of special relativity.

The equations of motion are contained in the continuity equation of the stress-energy tensor $T^\{mu\; u\}$:

:$abla\_muT^\{mu\; u\}=0$where the right hand side is the zero tensor. For a perfect fluid,

:$T\_\{mu\; u\}\; ,\; =\; (e+p)u\_mu\; u\_\; u+p\; eta\_\{mu\; u\}$.

Here $e$ is the relativistic rest energy of the fluid, $p$ is the fluid pressure, $u$ is the four-velocity of the fluid, and $eta\_\{mu\; u\}$ is the Minkowski metric tensor.

To the above equations, a statement of conservation is usually added, usually conservation of baryon number. If $n$ is the number density of baryons this may be stated:$abla\_mu(nu^mu)=0.$

These equations reduce to the classical Euler equations if $ull\; c$.

The relativistic Euler equations may be applied to calculate the speed of sound in a fluid with a relativistic equation of state (that is, one in which the pressure is comparable with the internal energy density $e$, including the rest energy; $e=\; ho\; c^2+\; ho\; e^C$ where $e^C$ is the classical internal energy per unit mass).

Under these circumstances, the speed of sound $S$ is given by:$S^2=c^2left.frac\{partial\; p\}\{partial\; e\}\; ight|\_\{\; m\; adiabatic\}.$

(note that :$e=\; ho\; (c^2+e^C)$

is the relativistic internal energy density). This formula differs from the classical case in that $ho$ has been replaced by $e/c^2$.