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In the derivation of the relativistic formula for adding velocities, the Lorentz factor drops out. Mathematically, the formula works for inertial frames with relative velocity c and even gives an answer to Einstein's famous question about what happens if you drive at the speed of light and turn your headlights on.

However, is there a physical meaning to this? Massive light sources can't reach c because the required energy would be infinite. But is there some framework (QFT maybe?) where a photon decays into multiple photons such that one of them can be regarded as "headlights" and another as "light from the headlights"?