  We report the experimental study of a harmonic oscillator in the relativistic
regime. The oscillator is composed of Bose-condensed lithium atoms in the third
band of an optical lattice, which have an energy-momentum relation nearly
identical to that of a massive relativistic particle, with an effective mass
reduced below the bare value and a greatly reduced effective speed of light.
Imaging the shape of oscillator trajectories at velocities up to 98% of the
effective speed of light reveals a crossover from sinusoidal to nearly
photon-like propagation. The existence of a maximum velocity causes the
measured period of oscillations to increase with energy; our measurements
reveal beyond-leading-order contributions to this relativistic anharmonicity.
We observe an intrinsic relativistic dephasing of oscillator ensembles, and a
monopole oscillation with exactly the opposite phase of that predicted for
non-relativistic harmonic motion. All observed dynamics are in quantitative
agreement with longstanding but hitherto-untested relativistic predictions.
