Relativistic Euler equations

Relativistic Euler equations
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "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 $T μν$ :

$\nabla_\mu T^{\mu\nu}=0$

where the right hand side is the zero tensor. For a perfect fluid,

$T_{\mu\nu} \, = (e+p)u_\mu u_\nu+p \eta_{\mu\nu}$ .

Here $e$ is the relativistic rest energy of the fluid, $p$ is the fluid pressure, $u$ is the four-velocity of the fluid, and $η μν$ 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

$\nabla_\mu (nu^\mu)=0.$

These equations reduce to the classical Euler equations if $u\ll c$ .

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; $e = ρ c 2 + ρ e C$ where $e C$ is the classical internal energy per unit mass).

Under these circumstances, the speed of sound $S$ is given by

$S^2=c^2 \left. \frac{\partial p}{\partial e} \right|_{\rm adiabatic}.$

(note that

e = ρ(c 2 + e C)

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 This formula differs from the classical case in that $ρ$ has been replaced by $e / c 2$ .

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