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 :
For a fluid, - .
Here is the relativistic rest energy of the fluid, is the pressure, is the four-velocity of the fluid, and is the metric tensor. To the above equations, a statement of conservation is usually added, usually conservation of baryon number. If is the number density of baryons this may be stated
: These equations reduce to the classical Euler equations if . 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). Under these circumstances, the speed of sound is given by
: (note that
: is the relativistic internal energy density). This formula differs from the classical case in that has been replaced by .
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