In fluid mechanics and astrophysics, the relativistic Euler equations are a generalization of the Euler equations that account for the effects of special relativity.

The equations of motion are contained in the continuity equation of the stress-energy tensor T^μν:

For a fluid,

T^μν = (e + p)u_μu_ν + pg_μν.

Here e is the relativistic rest energy of the fluid, p is the pressure, u is the four-velocity of the fluid, and g_μν 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

These equations reduce to the classical Euler equations if u < < c.

The relativistic Euler equations may be applied to calculate the speed of sound in a fluid with a relativistic equation of state (that is, one in which the pressure is comparable with the internal energy density e, including the rest energy; e = ρc² + ρe^C where e^C is the classical internal energy).

Under these circumstances, the speed of sound S is given by

(note that

e = ρ(c² + e^C)

is the relativistic internal energy density). This formula differs from the classical case in that ρ has been replaced by e / c².