The paradox of light

The principle of Galilean relativity states that the laws of physics are the same in all inertial reference frames. In Galileo’s and Newton’s day, “the laws of physics” meant the three laws of dynamics and the Galilean composition of velocities: if I observe an object from the ground, its velocity is the sum of the velocity of the vehicle carrying it and its velocity relative to the vehicle,

$\vv{v}_{\text{suolo}} = \vv{v}_{\text{treno}} + \vv{v}_{\text{oggetto rel.\ treno}}$

It is the most natural intuition in the world: someone running on a train moving at $30\;\text{m/s}$ who throws a ball forward at $10\;\text{m/s}$ sees it move away, as measured from the ground, at $40\;\text{m/s}$. But at the start of the twentieth century new fundamental laws had appeared, Maxwell’s equations of electromagnetism. Einstein asked a radical question: must these too hold in every inertial reference frame?

The domain of validity: Galileo guarantees the invariance of Newton’s laws; Einstein extends the requirement to Maxwell’s equations as well.

If the answer is yes, we run into a surprising fact: Maxwell’s equations imply that the speed of light in vacuum is $c = 1/\sqrt{\varepsilon_0\mu_0}$. But $\varepsilon_0$ (the vacuum permittivity) and $\mu_0$ (the vacuum permeability) are constants of nature: they do not depend on the reference frame. So $c$ must be the same in every inertial reference frame. And here the whole edifice starts to creak.

The paradox

According to Galilean composition, if you switch on a torch pointing in the direction of motion on a train moving at $v = 30\;\text{m/s}$, the beam should travel at $c + 30\;\text{m/s}$ relative to the tracks. But Maxwell says it always travels at $c$!

One of two things must be true: either Maxwell’s equations change form in different reference frames, or the Galilean composition of velocities is wrong.

There is no way to reconcile the two while leaving everything else as it is. Either Maxwell’s universality is sacrificed, or the velocity-addition rule that seemed untouchable is sacrificed. The chapter that follows is, at bottom, the story of this choice.

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Topics: Special relativity Concepts: Einstein’s postulates

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