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 has been replaced by 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