In fluid mechanics and astrophysics, the relativistic Euler equations are a generalization of the 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 bursts, accretion phenomena, and neutron stars, often with the addition of a magnetic field. 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 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), these equations are no longer valid. 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, and neutron stars feature gravitational fields that are more than <math>10^{11}</math> times stronger than the Earth's. 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. 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.
<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 = \rho (c^2 + \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) = \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)