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 \(T^{\mu\nu}\): \[\nabla_\mu T^{\mu\nu}=0.\] For a perfect fluid, \[T^{\mu\nu} \, = (e+p)u^\mu u^\nu+p \eta^{\mu\nu}.\] Here \(e\) is the relativistic rest energy density of the fluid, \(p\) is the fluid pressure, \(u^\mu\) is the four-velocity of the fluid, and \(\eta^{\mu\nu}\) is the Minkowski metric tensor with signature (-,+,+,+).

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 \[ \nabla_\mu (nu^\mu)=0.\]

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 \(e\), including the rest energy; \(e=\rho c^2+\rho e^C\) where \(e^C\) is the classical internal energy per unit mass).

Under these circumstances, the speed of sound \(S\) is given by \[ S^2=c^2 \left. \frac{\partial p}{\partial e} \right|_{\rm adiabatic}.\]

(note that \[e=\rho (c^2+e^C)\]

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 This formula differs from the classical case in that \(\rho\) has been replaced by \(e/c^2\).