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If, in vacuum, light is emitted from a source, would it accelerate really really fast until it reaches the well accepted speed of light $c$? If so, how small an interval does it take for it to do so?

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Light doesn’t accelerate in vacuum. It only moves at one speed, $c$.

This can be understood classically as a consequence of Maxwell’s equations: when one derives the equation for electromagnetic waves from them, there is one unique speed at which the waves propagate.

Alternately, it can be understood based on relativistic quantum theory: massless photons can only move at the speed of light, or else they can’t carry energy and momentum.

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