Relativistic Euler equations

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.[1] Note: for consistency with the literature, this article makes use of natural units, namely the speed of light ${\displaystyle c=1}$ 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fields, or the pressure approaches the energy density (${\displaystyle P\sim \rho }$), these equations are no longer valid.[2] Such situations occur frequently in astrophysical applications. For example, gamma-ray bursts often feature speeds only ${\displaystyle 0.01\%}$ less than the speed of light,[3] and neutron stars feature gravitational fields that are more than ${\displaystyle 10^{11}}$ times stronger than the Earth's.[4] Under these extreme circumstances, only a relativistic treatment of fluids will suffice.

Derivation

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

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

where ${\displaystyle \nabla _{\mu }}$ is the covariant derivative.[5] For a perfect fluid,

${\displaystyle T^{\mu \nu }\,=(\rho +p)u^{\mu }u^{\nu }+pg^{\mu \nu }.}$

Here ${\displaystyle \rho }$ is the total mass-energy density (including both rest mass and internal energy density) of the fluid, ${\displaystyle p}$ is the fluid pressure, ${\displaystyle u^{\mu }}$ is the four-velocity of the fluid, and ${\displaystyle g^{\mu \nu }}$ is the metric tensor.[2] To the above equations, a statement of conservation is usually added, usually conservation of baryon number. If ${\displaystyle n}$ is the number density of baryons this may be stated

${\displaystyle \nabla _{\mu }(nu^{\mu })=0.}$

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.[1]

Sound speed

The relativistic Euler equations may be applied to calculate the speed of sound in a fluid with a relativistic equation of state. Under these circumstances, the speed of sound ${\displaystyle S}$ is given by

${\displaystyle S^{2}={\frac {\partial p}{\partial \rho }}.}$[5][6]

This equation is superficially similar to the Newtonian sound speed, but ${\displaystyle \rho }$ is a measure of energy density, not rest-mass density (which is typically notated as ${\displaystyle \rho _{0}}$).

See also

• Relativistic heat conduction
• Equation of state (cosmology)

References

1. Rezzolla, L. (Luciano) (14 June 2018). Relativistic hydrodynamics. Zanotti, Olindo. Oxford. ISBN 978-0-19-880759-9. OCLC 1044938862.
2. Thorne, Kip S.; Blandford, Roger D. (2017). Modern Classical Physics. Princeton, New Jersey: Princeton University Press. pp. 719–720. ISBN 9780691159027.
3. Lithwick, Yoram; Sari, Re'em (July 2001). "Lower limits on Lorentz factors in gamma-ray bursts". The Astrophysical Journal. 555 (1): 540–545. arXiv:astro-ph/0011508. Bibcode:2001ApJ...555..540L. doi:10.1086/321455.
4. An introduction to the sun and stars. Green, S. F., Jones, Mark H. (Mark Henry), Burnell, S. Jocelyn. (Co-published ed.). Cambridge: Open University. 2004. ISBN 0-521-83737-5. OCLC 54663723.CS1 maint: others (link)
5. Schutz, Bernard (2009). A First Course in General Relativity. Cambridge University Press. ISBN 978-0521887052.
6. Donat, R.; Font, J.A.; Ibáñez, J.Ma; Marquina, A. (October 1998). "A Flux-Split Algorithm Applied to Relativistic Flows". Journal of Computational Physics. 146 (1): 58–81. Bibcode:1998JCoPh.146...58D. doi:10.1006/jcph.1998.5955.