  It is now well-understood that a Lorentz boost of a spatially coherent
monochromatic optical beam yields a so-called space-time wave packet (STWP): a
propagation-invariant pulsed beam whose group velocity is determined by the
relative velocity between the source and observer. Moreover, the Lorentz boost
of an STWP is another STWP, whose group velocities are related by the
relativistic law for addition of velocities typically associated with massive
particles. We present an experimental procedure for testing this prediction in
both the subluminal and superluminal regimes that makes use of spatio-temporal
Fourier synthesis via a spatial light modulator. Our approach enables realizing
the change in temporal bandwidth, the invariance of the spatial bandwidth, the
concomitant change in the spatio-temporal wave-packet envelope, and the change
in group velocity that all accompany a Lorentz boost of a monochromatic optical
beam. The only consequence of the Lorentz boost not captured by this
methodology is the Doppler shift in the optical carrier. This work may provide
an avenue for further table-top demonstration of relativistic transformations
of optical fields.
