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# 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 Tμν:

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

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 ημν 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 = ρc2 + ρeC where eC 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(c2 + eC)

is the relativistic internal energy density). This formula differs from the classical case in that ρ has been replaced by e / c2.