relativistic Euler equations

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fields, or the pressure approaches the energy density ($P\backslash sim\backslash rho$), these equations are no longer valid.{{Cite book|last1=Thorne|first1=Kip S.|title=Modern Classical Physics|last2=Blandford|first2=Roger D.|publisher=Princeton University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}} Such situations occur frequently in astrophysical applications. For example, gamma-ray bursts often feature speeds only $0.01\%$ less than the speed of light,{{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}} and neutron stars feature gravitational fields that are more than $10^\{11\}$ times stronger than the Earth's.{{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}} Under these extreme circumstances, only a relativistic treatment of fluids will suffice.

The equations of motion are contained in the continuity equation of the stress–energy tensor $T^\{\backslash mu\backslash nu\}$:

$$\backslash nabla\_\backslash mu\; T^\{\backslash mu\backslash nu\}\; =\; 0,$$

where $\backslash nabla\_\backslash mu$ is the covariant derivative.{{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}} For a perfect fluid,

$$T^\{\backslash mu\backslash nu\}\; \backslash ,\; =\; (e+p)u^\backslash mu\; u^\backslash nu+p\; g^\{\backslash mu\backslash nu\}.$$

Here $e$ is the total mass-energy density (including both rest mass and internal energy density) of the fluid, $p$ is the fluid pressure, $u^\backslash mu$ is the four-velocity of the fluid, and $g^\{\backslash mu\backslash nu\}$ is the 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

$$\backslash nabla\_\backslash mu\; (nu^\backslash mu)\; =\; 0.$$

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. To close this system, an equation of state, such as an ideal gas or a Fermi gas, is also added.

In the case of flat space, that is $\backslash nabla\_\{\backslash mu\}\; =\; \backslash partial\_\{\backslash mu\}$ and using a metric signature of $(-,+,+,+)$, the equations of motion are,{{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}}

\left(e + p\right) u^{\mu} \partial_{\mu} u^{\nu} = -\partial^{\nu}p - u^{\nu} u^{\mu} \partial_{\mu}p

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

Expanding out the sums and equations, we have, (using $\backslash frac\{d\}\{dt\}$ as the material derivative)

\left(e + p\right) \frac{\gamma}{c} \frac{du^{\mu}}{dt} = -\partial^{\mu} p - \frac{\gamma}{c} \frac{dp}{dt} u^{\mu}

Then, picking $u^\{\backslash nu\}\; =\; u^i\; =\; \backslash frac\{\backslash gamma\}\{c\}v\_i$ to observe the behavior of the velocity itself, we see that the equations of motion become

\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

Note that taking the non-relativistic limit, we have $\backslash frac\{1\}\{c^2\}\; \backslash left(e\; +\; p\backslash right)\; =\; \backslash gamma\; \backslash rho\; +\; \backslash frac\{1\}\{c^2\}\; \backslash rho\; \backslash varepsilon\; +\; \backslash frac\{1\}\{c^2\}\; p\; \backslash approx\; \backslash rho$. This says that the energy of the fluid is dominated by its rest energy.

In this limit, we have $\backslash gamma\; \backslash to\; 1$ and $c\; \backslash to\; \backslash infty$, and can see that we return the Euler Equation of $\backslash rho\; \backslash frac\{dv\_i\}\{dt\}\; =\; -\backslash partial\_i\; p$.

In order to determine the equations of motion, we take advantage of the following spatial projection tensor condition:

\partial_{\mu}T^{\mu\nu} + u_{\alpha}u^{\nu}\partial_{\mu}T^{\mu\alpha} = 0

We prove this by looking at $\backslash partial\_\{\backslash mu\}T^\{\backslash mu\backslash nu\}\; +\; u\_\{\backslash alpha\}u^\{\backslash nu\}\backslash partial\_\{\backslash mu\}T^\{\backslash mu\backslash alpha\}$ and then multiplying each side by $u\_\{\backslash nu\}$. Upon doing this, and noting that $u^\{\backslash mu\}u\_\{\backslash mu\}\; =\; -1$, we have $u\_\{\backslash nu\}\backslash partial\_\{\backslash mu\}T^\{\backslash mu\backslash nu\}\; -\; u\_\{\backslash alpha\}\backslash partial\_\{\backslash mu\}T^\{\backslash mu\backslash alpha\}$. Relabeling the indices $\backslash alpha$ as $\backslash nu$ 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

T^{\mu\nu} = wu^{\mu}u^{\nu} + pg^{\mu\nu}

where we have implicitly defined that $w\; \backslash equiv\; e+p$, we can calculate that

$$\backslash begin\{align\}$$

\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}

and thus

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

Then, let's note the fact that $u^\{\backslash alpha\}u\_\{\backslash alpha\}\; =\; -1$ and $u^\{\backslash alpha\}\backslash partial\_\{\backslash nu\}u\_\{\backslash alpha\}\; =\; 0$. Note that the second identity follows from the first. Under these simplifications, we find that

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

and thus by $\backslash partial\_\{\backslash mu\}T^\{\backslash mu\backslash nu\}\; +\; u\_\{\backslash alpha\}u^\{\backslash nu\}\backslash partial\_\{\backslash mu\}T^\{\backslash mu\backslash alpha\}\; =\; 0$, we have

(\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

We have two cancellations, and are thus left with

(e+p)u^{\mu}\partial_{\mu}u^{\nu} = - \partial^{\nu}p - u^{\nu}u^{\alpha}\partial_{\alpha}p

• Relativistic heat conduction
• Equation of state (cosmology)

{{Reflist}}

Euler equations

Category:Equations of fluid dynamics

{{relativity-stub}}