  Within the general theory of relativity, the curvature of spacetime is
related to the energy and momentum of the present matter and radiation. One of
the more specific predictions of general relativity is the deflection of light
and particle trajectories in the gravitational field of massive objects.
Bending angles for electromagnetic waves and light in particular were measured
with a high precision. However, the effect of gravity on relativistic massive
particles was never studied experimentally. Here we propose and analyze
experiments devoted to that purpose. We demonstrate a high sensitivity of the
laser Compton scattering at high energy accelerators to the effects of gravity.
The main observable -- maximal energy of the scattered photons -- would
experience a significant shift in the ambient gravitational field even for
otherwise negligible violation of the equivalence principle. We confirm
predictions of general relativity for ultrarelativistic electrons of energy of
tens of GeV at a current level of resolution and expect our work to be a
starting point of further high-precision studies on current and future
accelerators, such as PETRA, European XFEL and ILC.
