# Speed of light postulate in special relativity in inertial vs. non-inertial frames

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:

1. The laws of physics take the same form in all inertial frames of reference.
1. 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?

• I'm interested to know too. As I see it, choosing c constant from Maxwell is a starting point. We then develop the theory in inertial frames, as we know that physics is simpler there. Then, having got our theory, we find that in accelerated frames c is not constant. So, by this "argument" non-constant c is a result, not a postulate. I suppose you could try to derive SR in an accelerated frame (how?), and either you would find it very hard or impossible to come up with a consistent theory. I don't know if there is anything more to it than that, let's see! Sep 8 at 14:59
• If you are interested in non-inertial frames and acceleration in the context of relativistic physics, you can move on from special relativity and study general relativity.
– hft
Sep 8 at 15:12
• Your approach is wrong. Even if in the case that even non-inertial observers would agree that the speed of light is constant, a person who is trying to derive a new theory is still justified to first consider the case of constancy only for inertial frames, because then it will be easier. It is only after establishing a good theory on the simplest cases, do we get the confidence to derive things for the more complicated cases, because the solutions to the more complicated cases must reduce to the simpler in appropriate limits. Sometimes the simpler fixes the complicated, as happened here. Sep 8 at 15:21
• @Luke__ Unfortunately, all or almost all texts on Special Relativity (SR) address only inertial frames, leaving noninertial frames for GR -- which is wrong, since acceleration in a flat space-time still belongs in the realm of SR. A notable exception from this tradition is this masterfully written text, which discusses at length both linear acceleration and rotating frames. Strongly recommended. Sep 8 at 16:35
• @Michael_1812 Thanks for the recommendation: I'll give it a look! Sep 8 at 17:21

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?

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.

• Oops, I should have thought about rotation straight away! Sep 8 at 17:13

Seems to me that the second postulate of Special Relativity is redundant.

If "the laws of physics take the same form in all inertial frames of reference", then the quantitative parameters in those laws of physics would be the same in all inertial frames of reference. One of those parameters is the speed of light.

• As far as I can tell this is presupposing that the speed of light is constant. Suppose instead that the various aether based theories were true, then light could have multiple different velocities without violating the laws of physics. Of course Maxwell's equations make it natural to assume that the speed of light is fixed for all reference frames but it is not a priori required. This is where the second postulate is required. Sep 9 at 3:27
• Constant with respect to what? The first postulate states that the speed of light is invariant between the two inertial observers that are moving relative to each other. Doesn't say anything about constant with respect to time. Sep 9 at 3:31
• Constant with respect to observer. The analagous case would be the speed of sound. Clearly that is not constant with respect to observers, so what makes light special? It might be possible to justify but there can certainly be models of physics without a constant (relative to inertial frames) speed of light. (Also the first postulate does not state that the speed of light is invariant, you are simply interpreting it to do so) Sep 9 at 3:33
• Seems to me that the first postulate clearly implies that, among all other physics, that this speed of light is the same (or "constant") between observers in different inertial frames. About what makes the speed of light "special", it's not just about EM. It's every instantaneous interaction. Gravity. Strong force. $c$ is the speed of causality. Sep 9 at 3:37
• @Luke__ lookie here: PBS Space Time: The Speed of Light is NOT About Light Sep 10 at 16:11