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

A note on incompressibility of relativistic fluids and the instantaneity of their pressures