  We introduce a natural notion of incompressibility for fluids governed by the
relativistic Euler equations on a fixed background spacetime, and show that the
resulting equations reduce to the incompressible Euler equations in the
classical limit as $c\rightarrow \infty$. As our main result, we prove that the
fluid pressure of solutions of these incompressible "relativistic" Euler
equations satisfies an elliptic equation on each of the hypersurfaces
orthogonal to the fluid four-velocity, which indicates infinite speed of
propagation.
