Editing Relativistic Euler equations

{{More citations needed|date=April 2020}}
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 [[general relativity]]. They have applications in [[high-energy astrophysics]] and [[numerical relativity]], where they are commonly used for describing phenomena such as [[gamma-ray burst]]s, [[Accretion disk|accretion phenomena]], and [[neutron star]]s, often with the addition of a [[Magnetohydrodynamics|magnetic field]].<ref name=":0">{{Cite book|last=Rezzolla, L. (Luciano)|title=Relativistic hydrodynamics|others=Zanotti, Olindo|date=14 June 2018|isbn=978-0-19-880759-9|location=Oxford|oclc=1044938862}}</ref> ''Note: for consistency with the literature, this article makes use of [[natural units]], namely the speed of light'' <math>c=1</math> ''and the [[Einstein notation|Einstein summation convention]].''

== Motivation ==
For most fluids observable on Earth, traditional fluid mechanics based on Newtonian mechanics is sufficient. However, as the fluid velocity approaches the speed of light or moves through strong gravitational fields, or the pressure approaches the energy density (<math>P\sim\rho</math>), these equations are no longer valid.<ref name=":1">{{Cite book|last1=Thorne|first1=Kip S.|title=Modern Classical Physics|last2=Blandford|first2=Roger D.|publisher=Princeton University Press|year=2017|isbn=9780691159027|location=Princeton, New Jersey|pages=719–720}}</ref> Such situations occur frequently in astrophysical applications. For example, gamma-ray bursts often feature speeds only <math>0.01%</math> less than the speed of light,<ref>{{Cite journal|last1=Lithwick|first1=Yoram|last2=Sari|first2=Re'em|date=July 2001|title=Lower limits on Lorentz factors in gamma-ray bursts|journal=The Astrophysical Journal|volume=555|issue=1|pages=540–545|doi=10.1086/321455|arxiv=astro-ph/0011508|bibcode=2001ApJ...555..540L|s2cid=228707}}</ref> and neutron stars feature gravitational fields that are more than <math>10^{11}</math> times stronger than the Earth's.<ref>{{Cite book|title=An introduction to the sun and stars|date=2004|publisher=Open University|others=Green, S. F., Jones, Mark H. (Mark Henry), Burnell, S. Jocelyn.|isbn=0-521-83737-5|edition=Co-published|location=Cambridge|oclc=54663723}}</ref> Under these extreme circumstances, only a relativistic treatment of fluids will suffice.

== 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" />

==Equations of motion in flat space==
In the case of flat space, that is <math>\nabla_{\mu} = \partial_{\mu}</math> and using a [[metric signature]] of <math>(-,+,+,+)</math>, the equations of motion are,<ref>{{cite book |last1= Lifshitz|first1= L.D.|last2= Landau|first2= E.M.|title= Fluid Mechanics|edition= 2nd|publisher= Elsevier |date= 1987 |page= 508|isbn= 0-7506-2767-0}}</ref>
<math display="block">
\left(e + p\right) u^{\mu} \partial_{\mu} u^{\nu} = -\partial^{\nu}p - u^{\nu} u^{\mu} \partial_{\mu}p
</math>

Where <math>e = \gamma \rho c^2 + \rho \varepsilon</math> is the energy density of the system, with <math>p</math> being the pressure, and <math>u^{\mu} = \gamma(1, \mathbf{v}/{c})</math> being the [[four-velocity]] of the system.

Expanding out the sums and equations, we have, (using <math>\frac{d}{dt}</math> as the [[material derivative]])
<math display="block">
\left(e + p\right) \frac{\gamma}{c} \frac{du^{\mu}}{dt} = -\partial^{\mu} p - \frac{\gamma}{c} \frac{dp}{dt} u^{\mu}
</math>

Then, picking <math>u^{\nu} = u^i = \frac{\gamma}{c}v_i</math> to observe the behavior of the velocity itself, we see that the equations of motion become
<math display="block">
\left(e + p\right) \frac{\gamma}{c^2} \frac{d}{dt} {\left(\gamma v_i\right)} = -\partial_i p -\frac{\gamma^2}{c^2} \frac{dp}{dt} v_i
</math>

Note that taking the non-relativistic limit, we have <math display="inline">\frac{1}{c^2} \left(e + p\right) = \gamma \rho + \frac{1}{c^2} \rho \varepsilon + \frac{1}{c^2} p \approx \rho</math>. This says that the energy of the fluid is dominated by its [[rest energy]].

In this limit, we have <math>\gamma \to 1</math> and <math>c \to \infty</math>, and can see that we return the Euler Equation of <math>\rho \frac{dv_i}{dt} = -\partial_i p</math>.

===Derivation===
In order to determine the equations of motion, we take advantage of the following spatial projection tensor condition:
<math display="block">
\partial_{\mu}T^{\mu\nu} + u_{\alpha}u^{\nu}\partial_{\mu}T^{\mu\alpha} = 0
</math>

We prove this by looking at <math>\partial_{\mu}T^{\mu\nu} + u_{\alpha}u^{\nu}\partial_{\mu}T^{\mu\alpha}</math> and then multiplying each side by <math>u_{\nu}</math>. Upon doing this, and noting that <math>u^{\mu}u_{\mu} = -1</math>, we have <math>u_{\nu}\partial_{\mu}T^{\mu\nu} - u_{\alpha}\partial_{\mu}T^{\mu\alpha}</math>. Relabeling the indices <math>\alpha</math> as <math>\nu</math> shows that the two completely cancel. This cancellation is the expected result of contracting a temporal tensor with a spatial tensor.

Now, when we note that
<math display="block">
T^{\mu\nu} = wu^{\mu}u^{\nu} + pg^{\mu\nu}
</math>

where we have implicitly defined that <math>w \equiv e+p</math>, we can calculate that
<math display="block">\begin{align}
    \partial_{\mu} T^{\mu\nu}    & = \left(\partial_{\mu} w\right) u^{\mu} u^{\nu}    + w \left(\partial_{\mu} u^{\mu}\right) u^{\nu}    + wu^{\mu} \partial_{\mu} u^{\nu}    + \partial^{\nu}   p \\[1ex]
    \partial_{\mu} T^{\mu\alpha} & = \left(\partial_{\mu} w\right) u^{\mu} u^{\alpha} + w \left(\partial_{\mu} u^{\mu}\right) u^{\alpha} + wu^{\mu} \partial_{\mu} u^{\alpha} + \partial^{\alpha}p
\end{align}</math>

and thus
<math display="block">
u^{\nu}u_{\alpha}\partial_{\mu}T^{\mu\alpha} = (\partial_{\mu}w)u^{\mu}u^{\nu}u^{\alpha}u_{\alpha} + w(\partial_{\mu}u^{\mu})u^{\nu} u^{\alpha}u_{\alpha} + wu^{\mu}u^{\nu} u_{\alpha}\partial_{\mu}u^{\alpha} + u^{\nu}u_{\alpha}\partial^{\alpha}p
</math>

Then, let's note the fact that <math>u^{\alpha}u_{\alpha} = -1</math> and <math>u^{\alpha}\partial_{\nu}u_{\alpha} = 0</math>. Note that the second identity follows from the first. Under these simplifications, we find that
<math display="block">
u^{\nu}u_{\alpha}\partial_{\mu}T^{\mu\alpha} = -(\partial_{\mu}w)u^{\mu}u^{\nu} - w(\partial_{\mu}u^{\mu})u^{\nu} + u^{\nu}u^{\alpha}\partial_{\alpha}p
</math>

and thus by <math>\partial_{\mu}T^{\mu\nu} + u_{\alpha}u^{\nu}\partial_{\mu}T^{\mu\alpha} = 0</math>, we have
<math display="block">
(\partial_{\mu}w)u^{\mu}u^{\nu} + w(\partial_{\mu}u^{\mu}) u^{\nu} + wu^{\mu}\partial_{\mu}u^{\nu} + \partial^{\nu}p -(\partial_{\mu}w)u^{\mu}u^{\nu} - w(\partial_{\mu}u^{\mu})u^{\nu} + u^{\nu}u^{\alpha}\partial_{\alpha}p = 0
</math>

We have two cancellations, and are thus left with
<math display="block">
(e+p)u^{\mu}\partial_{\mu}u^{\nu} = - \partial^{\nu}p - u^{\nu}u^{\alpha}\partial_{\alpha}p
</math>

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

==References==
{{Reflist}}

[[Category:Special relativity|Euler equations]]
[[Category:Equations of fluid dynamics]]

{{relativity-stub}}