Relativistic Euler equations
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For a perfect fluid,
These equations reduce to the classical Euler equations if the fluid three-velocity is much less than the speed of light, the pressure is much less than the energy density, and the latter is dominated by the rest mass density.
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 , including the rest energy; where is the classical internal energy per unit mass).
Under these circumstances, the speed of sound is given by
is the relativistic internal energy density). This formula differs from the classical case in that has been replaced by .
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