The equations of motion are contained in the continuity equation of the stress–energy tensor :
where is the covariant derivative.[5] For a perfect fluid,
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
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]
In the case of flat space, that is and using a metric signature of , the equations of motion are,[6]
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)
Then, picking to observe the behavior of the velocity itself, we see that the equations of motion become
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 .