Relativistic Euler equations

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 T^{mu u}:

: abla_muT^{mu u}=0where the right hand side is the zero tensor. For a perfect fluid,

:T_{mu u} , = (e+p)u_mu u_ u+p eta_{mu u}.

Here e is the relativistic rest energy of the fluid, p is the fluid pressure, u is the four-velocity of the fluid, and eta_{mu u} 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: abla_mu(nu^mu)=0.

These equations reduce to the classical Euler equations if ull 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= ho c^2+ ho e^C where e^C is the classical internal energy per unit mass).

Under these circumstances, the speed of sound S is given by:S^2=c^2left.frac{partial p}{partial e} ight|_{ m adiabatic}.

(note that :e= ho (c^2+e^C)

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 This formula differs from the classical case in that ho has been replaced by e/c^2.


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