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{{Unreferenced stub|auto=yes|date=December 2009}}
In [[fluid mechanics]] and [[astrophysics]], the '''relativistic Euler equations''' are a generalization of the [[Euler equations (fluid dynamics)|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 [[perfect fluid]], 
:<math>T^{\mu\nu}  \, =  (e+p)u^\mu u^\nu+p \eta^{\mu\nu}.</math>
Here <math>e</math> is the relativistic rest energy  density of the fluid, <math>p</math> is the [[fluid pressure]], <math>u^\mu</math> is the [[four-velocity]] of the fluid, and <math>\eta^{\mu\nu}</math> is the [[Minkowski metric|Minkowski metric tensor]] with [[metric signature|signature]] {{math|(-,+,+,+)}}.  

To the above equations, a  [[Conservation law|statement of conservation]] is usually added, usually conservation of [[baryon number]].  If <math>n</math> is the [[number density]] of [[baryon]]s this may be stated
:<math>
\nabla_\mu
(nu^\mu)=0.</math>

These equations reduce to the classical Euler equations if the fluid three-velocity is [[classical mechanics#The Newtonian approximation to special relativity|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 <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 per unit mass).  

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 has been replaced by 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

==See also==
* [[Relativistic heat conduction]]
* [[Equation of  state (cosmology)]]

{{DEFAULTSORT:Relativistic Euler Equations}}
[[Category:Special relativity]]
[[Category:Equations of fluid dynamics]]


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