  The equivalence principle, that is one of the main pillars of general
relativity, is very well tested in the Solar system; however, its validity is
more uncertain on cosmological scales, or when dark matter is concerned. This
article shows that relativistic effects in the large-scale structure can be
used to directly test whether dark matter satisfies Euler's equation, i.e.
whether its free fall is characterised by geodesic motion, just like baryons
and light. After having proposed a general parametrisation for deviations from
Euler's equation, we perform Fisher-matrix forecasts for future surveys like
DESI and the SKA, and show that such deviations can be constrained with a
precision of order 10%. Deviations from Euler's equation cannot be tested
directly with standard methods like redshift-space distortions and
gravitational lensing, since these observables are not sensitive to the time
component of the metric. Our analysis shows therefore that relativistic effects
bring new and complementary constraints to alternative theories of gravity.
