  The secular dynamics of a non-relativistic charged particle in an
electromagnetic wave can be described by the ponderomotive potential. Although
ponderomotive electron-laser interactions at relativistic velocities are
important for emerging technologies from laser-based particle accelerators to
laser-enhanced electron microscopy, the effects of special relativity on the
interaction have only been studied theoretically. Here, we use a transmission
electron microscope to measure the position-dependent phase shift imparted to a
relativistic electron wave function when it traverses a standing laser wave.
The kinetic energy of the electrons is varied between $80\,\mathrm{keV}$ and
$300\,\mathrm{keV}$, and the laser standing wave has a continuous-wave
intensity of $175\,\mathrm{GW/cm}^2$. In contrast to the non-relativistic case,
we demonstrate that the phase shift depends on both the electron velocity and
the wave polarization, confirming the predictions of a quasiclassical theory of
the interaction. Remarkably, if the electron's speed is greater than
$1/\sqrt{2}$ of the speed of light, the phase shift at the electric field nodes
of the wave can exceed that at the antinodes. In this case there exists a
polarization such that the phase shifts at the nodes and antinodes are equal,
and the electron does not experience Kapitza-Dirac diffraction. Our results
thus provide new capabilities for coherent electron beam manipulation.
