Editing Relativistic Euler equations (section)

== Introduction ==
The [[equations of motion]] are contained in the [[continuity equation]] of the [[stress–energy tensor]] <math>T^{\mu\nu}</math>:

<math display="block">\nabla_\mu T^{\mu\nu} = 0,</math>

where <math>\nabla_\mu</math> is the [[covariant derivative]].<ref name=":2">{{Cite book|last=Schutz|first=Bernard|title=A First Course in General Relativity|url=https://archive.org/details/firstcourseingen00bern_0|url-access=registration|publisher=Cambridge University Press|year=2009|isbn=978-0521887052}}</ref> For a [[perfect fluid]],

<math display="block">T^{\mu\nu}  \, =  (e+p)u^\mu u^\nu+p g^{\mu\nu}.</math>

Here <math>e</math> is the total mass-energy density (including both rest mass and internal 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>g^{\mu\nu}</math> is the [[Metric tensor (general relativity)|metric tensor]].<ref name=":1" /> To the above equations, a [[Conservation law (physics)|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 display="block">\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. To close this system, an [[equation of state]], such as an [[ideal gas]] or a [[Fermi gas]], is also added.<ref name=":0" />