Assume when 2 origins O and O' in relative motion are common, a light pulse is emitted. Then, the light sphere emerges spherically from the origin of each frame. http://casa.colorado.edu/~ajsh/sr/paradox.html Now, let's add a twist. When the two coordinate systems are common and the light pulse is emitted, another observer O2 is at rest with O and immediately accelerates in the direction of O' at some constant acceleration "a" for some time T such that after that T, O2 is at rest with O'. In the view of O2, where is the center of the light sphere after the acceleration completes? It is at O'. So, the center of the light sphere moved from O to O' as O2 accelerated.
Simple. The light cone of O2 has tilted, as per the second page of your link, during acceleration, changing frames. The light cone of one is skewed relative to the other, and will remain so. This illustrates the relativity of simultaneity. Even if they end up in the same frame, they do not agree about timing of events.
Very good. This means as O' accelerates he sees co-moving inertial frames in which the center of the light sphere is further ahead of him. As you note, once O' stops accelerating, he must adopt the rules of the O2 frame. Thus, the center of the light sphere is in front of the accelerating observer at all times for any co-moving frame including O2. So, how do you accelerate with the center of the light sphere out accelerating you? Oh, and once they end up in the same frame O' and O2, they agree on simultaneity.
Chinglu, it's an interesting idea. I've found that when we use our imagination to determine light spheres through moving frames we often come up with paradoxes like this. If you made a mechanism by which the observers could ACTUALLY observe the light spheres (with reflective mirrors, etc) you will find that O2 cannot possibly reach O' in time to observe the light pulse's return to conclude that the center is at 2 different locations.
The above post is quoted only as it mentions the paradoxes, which emerge from how we choose to apply our imaginations. These light sphere discussions have always seemed a bit off track to me. So much a hypothetical thought experiment that they cannot be proven one way or the other. While all observers can imagine, conceive of, and even observe a light horizon, from where ever they happen to be, no observer can observe any light sphere. At best they can stand at the origin of what might become a light spere, or somewhere within the light horizon of another observer. To observe a light sphere one would have to observe the involved light in more than one location.., simultaneously. Otherwise, how could one know whether they were observing an expanding light sphere or just happen to be in the path of single coherent ray of light that had no spherical counter part? Add to this that depending on how you construct the thought experiment, confined to the conditions established by SR or GR, if one could observe the photons of a light sphere, without destructive interference, they might conclude either that they were in fact moving relative to the sphere or that it was moving with them, without reguard to its real point of origin. Since the exercise is limited to the thought experiment, it is also always limited to the initial assumptions, the thought experiment is based on. In this case always involving imagination, without the benefit of credible experience. There is no absolute conclusion, because there is no way to experimentally test any hypothesis and any logical conclusions are always dependent on what untestable assumptions one begins with.
I'm not sure I agree with you, OnlyMe. You could in theory have observers O and O' as stated in the OP encased in large spherical mirrors (made of, let's say, vertical strips such that they can pass through each other). That way both observers should, according to SR, be local to each other at the initial flash event and also both see their respective light spheres return to them simultaneously from all directions. That part of the experiment has no technical pitfalls. The problem is that O2 cannot observe O's light sphere reflection and then accelerate quickly enough to also experience O-prime's light sphere reflection. An untestable objection is a powerless objection.
I'm not sure I see what everyone here is bothered about: relativity's invariance of c only applies to inertial reference frames. So an observer could start at rest at the centre of an expanding light sphere, accelerate for some time, then find they're no longer at the centre of the light sphere when they stop accelerating, and that's absolutely not a problem for relativity.
Actually this proves my intended point on two fronts. The assumptions are limited to SR in a GR environment, is one. And the second is that even inside a reflective bubble you are not observing the light sphere. You are observing reflected light, which is itself affected by the act of reflection. A third point might be that both observers cannot observe the same light. Observation of a ray of light is a destructive process. If you then add that in any practcal sense, the light sphere would be moving through curved space, the curvature of space itself would distort the sphere sufficiently that it would no longer resemble a sphere. Some of the same issues come up in some of the variations of the twin paradox, when the assumptions are expanded to include GR. The only way these kinds of thought experiments work in any logical fashion, is if one assumes there is nothing but the two observers and the light sphere, IOW no gravity wells or curved space involved. Which was my point to begin with, the thought experiment only works as an imaginary construct, that does not resemble reality. BTW None of what I have posted negates the value of the logical exercise.
Re-read the OP Przyk. If O correctly concludes that he resides at the center of his light sphere indefinitely and O' correctly concludes that he resides at the center of his light sphere indefinitely, AND O2 has the ability to travel between both observers and agree with their conclusions then there does appear to be a problem; this is because either the light sphere's center appears to be moving with O2's acceleration or there are multiple light spheres. My objection is that the existence of "light spheres" without actual observation of them is dubious.
Again, why would this be a problem? In a sense there are multiple light spheres, obtained by taking different space-like slices along the same light cone. You have multiple light spheres in the same way you have multiple 2D cross sections of any 3D object.
Imagine if O2 could accelerate instantly to some pre-chosen speed (relative to O). Further imagine that O2 does so at the instant the source of the light sphere flashes. Now O2 will find himself located at the center of the light sphere regardless of the pre-chosen speed. Similarly, if O2 gradually accelerates, catches up with O', and then decelerates to remain constantly co-located with O', then O2 will again find himself to be located at the center of the light sphere. But O2 would not have been at the center of the light sphere the whole time, in that case.
If you're talking about parallel slices, then my point is that non parallel sections (because different observers don't share the same notion of simultaneity) do not share a common centre. Even parallel sections don't share the same centre if you compare them vertically on a Minkowski diagram, except in the case of horizontal sections. That's essentially what's being seen here and in the OP, and none of it is surprising or in any way problematic for relativity.
I guess you would just have to accelerate at a constant rate, and the center of the light sphere would move ahead of you. Of course you would still remember that you were co-located with the light source when it flashed, but due to your new simultaneity after accelerating, you find the light sphere is centered on a different point. Interesting, but not a contradiction.
Yes, it is not a problem for SR. But to start at the center of the light sphere, then accelerate and after acceleration you find the center out ahead of you, that is a problem for reality.
There is only one problem. There is no logic in the universe in which you accelerate from and find yourself behind that location after acceleration, except for SR. But, it is a fact, that the center of the light sphere is out in front of you after the acceleration under SR and in that part you are correct.
