I'm trying to learn special relativity by myself. I've been following this series of videos, plus some other articles I've managed to find online. At this point I'm already quite far into the theory, but I have a question which might be trivial (I thought about it only as soon I started dealing with non-inertial frames of reference).
The two postulate of special relativity are the following:
- The laws of physics take the same form in all inertial frames of reference.
- The speed of light in the vacuum is the same in all inertial frames of reference and its value is $c$
Now my question is, what is that has brought Einstein to conclude that the speed of light is constant only for inertial reference frames and not for non-inertial ones? The math behind this concept does, in fact, make sense: if the frame is inertial, then Lorentz transformations are used to change basis, and the hyperbolic geometry of these transformations doesn't produce a change in the steepness of the light's worldline in the new reference frame; on the other hand, Rindler transformations for accelerating observers do not preserve the steepness of the light's worldline in the new frame. Nonetheless, the fact that Lorentz transformations are used for inertial frames and Rindler transformations for non-inertial ones is derived by that postulate.
The thing that remains obscure to me is how it was possible to limit the applicability of constancy of light only to non accelerating observers: if we take Maxwell's equations as a starting point, it is possible to see that the speed of light in the vacuum is $c$, however it is not clear with respect to which reference frame(s), then why not consider non-inertial frames too?
Best Answer
We know that non-inertial frames violate the first postulate. In a non-inertial frame a non-interacting object does not travel in a straight line at constant speed. This violates Newton's 1st law and Newton's 2nd law. We can add a fictitious force to fix those, but then Newton's 3rd law is violated.
Since non-inertial frames do not obey the first postulate, the immediate initial guess would be that non-inertial frames also do not obey the second postulate. So the usual approach would be to see if you can find an example of a non-inertial frame which does not obey the second postulate, thereby confirming the initial guess.
An easy one to consider is a rotating reference frame. In a rotating frame at sufficiently large distances objects exceed $c$. For example, if you spin around at ordinary speed, then in your frame the sun is moving faster than $c$. So light leaving its surface on the "forward" side is going even faster than that.
Thus, the restriction to inertial frames is confirmed by a simple example.