Could we have assumed the speed of light to be different in different reference frames?

Question

Ptolemy's model of universe assumes that our earth is the static center of universe and everything else move relative to it (ref: The grand design ch:3). This model would give us a consistent picture of universe the only complication would be that the trajectories of other heavenly bodies would be fairly complicated with our earth on center. So assuming any center doesn't contradict fundamental property of nature.

Similarly could we have assumed that the speed of light too was not the same on different frames and had similar properties as speed of sound (for example) which is different on different frame and still not contradict fundamental property of nature only giving some other complicated (may be) equations describing nature?

Answer 1

First, one thing to be clear on is that relativity assumes merely that there is a speed which is the same in all reference frames. The fact that light (or anything else) travels at that speed is mostly irrelevant.

Now, can you formulate a consistent theory in which there is no speed that is the same in all reference frames? Yes, you can, and this is what people did before relativity, or what we still do (as a simplifying approximation) when relativistic effects can be neglected. However, such a theory would contradict the fact that the speed of light is measured to be the same in all reference frames, and the fact that Lorentz contraction and time dilation are measured to precisely agree with the predictions of relativity. So it would clearly not be the best available description of reality.

Answer 2

The given answer is in fact not complete. While aether construction is trivially knocked down, yet another construction must be used to get rid of the remaining theory in which this set up works, that is light takes some convenient reference frame relative to its emitter or collector.

The problem must be solved (again) in two dimensions. The calculation for the angle that light is observed at due to being in a different frame as its source (this is known as Aberration of light) yields the right answer only using relativistic velocity addition.

There are a couple of choices for "convenient reference frame". One is taking the reference frame of the emitter so that all others must add the emitter's velocity. I can't find a name for this one anywhere despite it being the most obvious choice to me. The other is taking the reference frame of the observer (known as aether dragging). The main point is neither work.