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
and the Einstein summation convention.
Introduction
The equations of motion are contained in the continuity equation of the stress–energy tensor
:
![{\displaystyle \nabla _{\mu }T^{\mu \nu }=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7f35bac4e6b4e967bb8942d083f0c689eefc5c2)
where
is the covariant derivative.[5] For a perfect fluid,
![{\displaystyle T^{\mu \nu }\,=(e+p)u^{\mu }u^{\nu }+pg^{\mu \nu }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da301e6b406c33307b5bc9b6accc9eb577fd2b80)
Here
is the total mass-energy density (including both rest mass and internal energy density) of the fluid,
is the fluid pressure,
is the four-velocity of the fluid, and
is the metric tensor.[2] To the above equations, a statement of conservation is usually added, usually conservation of baryon number. If
is the number density of baryons this may be stated
![\nabla _{\mu }(nu^{\mu })=0.](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b4f62f914406adea0600725b1f0ab231dbc5535)
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]
Equations of Motion in Flat Space
In the case of flat space, that is
and using a metric signature of
, the equations of motion are,[6]
![{\displaystyle (e+p)u^{\mu }\partial _{\mu }u^{\nu }=-\partial ^{\nu }p-u^{\nu }u^{\mu }\partial _{\mu }p}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c532fbc6c553d855ddb750bda9c6fcf9829187c)
Where
is the energy density of the system, with
being the pressure, and
being the four-velocity of the system.
Expanding out the sums and equations, we have, (using
as the material derivative)
![{\displaystyle (e+p){\frac {\gamma }{c)){\frac {du^{\mu )){dt))=-\partial ^{\mu }p-{\frac {\gamma }{c)){\frac {dp}{dt))u^{\mu ))](https://wikimedia.org/api/rest_v1/media/math/render/svg/6294ab5df1ee09304894ff6585b423f40025dd87)
Then, picking
to observe the behavior of the velocity itself, we see that the equations of motion become
![{\displaystyle (e+p){\frac {\gamma }{c^{2))}{\frac {d}{dt))(\gamma v_{i})=-\partial _{i}p-{\frac {\gamma ^{2)){c^{2))}{\frac {dp}{dt))v_{i))](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1ad3010ca45c9ad848ae169b6b66debaea20c8a)
Note that taking the non-relativistic limit, we have
. This says that the energy of the fluid is dominated by its rest energy.
In this limit, we have
and
, and can see that we return the Euler Equation of
.
Derivation of the Equations of Motion
In order to determine the equations of motion, we take advantage of the following spatial projection tensor condition:
![{\displaystyle \partial _{\mu }T^{\mu \nu }+u_{\alpha }u^{\nu }\partial _{\mu }T^{\mu \alpha }=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14f4871d045801b89f281ffe3ca1ab551e24ca02)
We prove this by looking at
and then multiplying each side by
. Upon doing this, and noting that
, we have
. Relabeling the indices
as
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
![{\displaystyle T^{\mu \nu }=wu^{\mu }u^{\nu }+pg^{\mu \nu ))](https://wikimedia.org/api/rest_v1/media/math/render/svg/813e4497455c06f9d9ffc6fa6c80cc1080434cd7)
Where we have implicitly defined that
.
We can calculate that
![{\displaystyle {\begin{aligned}\partial _{\mu }T^{\mu \nu }&=(\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 }T^{\mu \alpha }&=(\partial _{\mu }w)u^{\mu }u^{\alpha }+w(\partial _{\mu }u^{\mu })u^{\alpha }+wu^{\mu }\partial _{\mu }u^{\alpha }+\partial ^{\alpha }p\end{aligned))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b3e58cedeff615829430f1db1500e440f2fed46)
And thus
![{\displaystyle 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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/486f9f090610e4db246f8798488a7b79c663e999)
Then, let's note the fact that
and
. Note that the second identity follows from the first. Under these simplifications, we find that
![{\displaystyle 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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9835ef5b78b8bb7fff68472342b277ac098e56c4)
And thus by
, we have
![{\displaystyle (\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e0e27b9065f00fbb3e36fffa3acf2bb4828c3e4)
We have two cancellations, and are thus left with
![{\displaystyle (e+p)u^{\mu }\partial _{\mu }u^{\nu }=-\partial ^{\nu }p-u^{\nu }u^{\alpha }\partial _{\alpha }p=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53b7944158b30445c77a65c0fa785e6ed0e1e190)