As a lowly physics undergrad who has been chewing on this 2nd postulate of special relativity for a year or more, I simply can't wrap my head around reasons why it is true or how Einstein might have been convinced enough to propose this postulate.
Consider Alfred who is riding in a car travelling at 88 m/s with his headlights on and Bernard who is on the side of the road hitch hiking. Why does the light propagating from Alfred's car move at $c$ relative to both Alfred and Bernard, and not at $c$ + 88 m/s relative to Bernard?
The nifty results of special relativity all kind of hinge on this idea, and asking my professors in class hasnt really yielded an answer much more than "because we have never observed a case otherwise".
Answer
Actually given that the first postulate says that all physical laws are the same in all inertial frames, you could replace the second postulate by the postulate: "Maxwell's equations are the physical laws for electromagnetism".
From Maxwell's laws you can derive that the speed of light in vacuum has a specific, constant value, in SI units $c=1/\sqrt{\epsilon_0\mu_0}$. Now there are three possibilities:
Maxwell's laws are valid only in a specific inertial frame (or rather, in a specific set of inertial frames at rest relative to each other).
That's the essence of the aether hypothesis. It would violate the first postulate. Also, experiments failed to measure that preferred frame.
Maxwell's laws are not the correct description of electromagnetism (that is, they are valid in no inertial frame).
That option would, of course, have been compatible with the first postulate, but not very likely, given the huge experimental support for Maxwell's equations.
Maxwell's laws are valid in all inertial frames.
If that is the case, then all of the consequences of Maxwell's equations have to be valid in all inertial frames. One of the consequences of Maxwell's equations is the value of the speed of light in vacuum.
However, it turns out that the only thing from Maxwell's equations you actually need in order to derive special relativity is the constant speed of light. Therefore it makes sense to postulate that directly; that way even if it turned out that Maxwell's theory had to be revised, you don't need to revise relativity as long as the revised theory still predicts a constant speed of light.
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