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.

Additional recommended knowledge

What is the Sensitivity of my Balance?

Daily Visual Balance Check

How to ensure accurate weighing results every day?

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).