Relativistic Euler equations

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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 <math>T^{\mu\nu}</math>:

<math>

\nabla_\mu T^{\mu\nu}=0 </math> For a fluid,

<math>T^{\mu\nu}=(e+p)u_\mu u_\nu+pg_{\mu\nu}</math>.

Here <math>e</math> is the relativistic rest energy of the fluid, <math>p</math> is the pressure, <math>u</math> is the four-velocity of the fluid, and <math>g_{\mu\nu}</math> is the metric tensor.

To the above equations, a statement of conservation is usually added, usually conservation of baryon number. If <math>n</math> is the number density of baryons this may be stated

<math>

\nabla_\mu (nu_\mu)=0.</math>

These equations reduce to the classical Euler equations if <math>u<< c</math>.

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 <math>e</math>, including the rest energy; <math>e=\rho c^2+\rho e^C</math> where <math>e^C</math> is the classical internal energy).

Under these circumstances, the speed of sound <math>S</math> is given by

<math>

S^2=c^2 \left. \frac{\partial p}{\partial e} \right|_{\rm adiabatic}.</math>

(note that

<math>e=\rho (c^2+e^C)</math>

is the relativistic internal energy density). This formula differs from the classical case in that <math>\rho</math> has been replaced by <math>e/c^2</math>.