## effect of acceleration on shape and size of a light wave front

Sometimes I get frustrated with the dialogs in this forum because there are so many misunderstandings and outright half truths being passed around (including from myself). Well, I was just over at SciForums.com warily participating in the thread “According to SR…” and it gave me a whole new perspective on frustration.

The subject of the thread at SciForums.com is quite interesting. I hope I am not violating the rules by introducing it here. Summarized, it goes something like this.

“Suppose that a flash of light is emitted from Earth at the start of the Twins problem when t=t’=0. Since the earth twin is modeled as inertial, that twin would always calculate the geometry of the light wave front to be a sphere of radius ct where t is the elapsed time on the earth twin’s clock. Now the astronaut has been calculating the location and geometry of the light wave front too (which is different during the trip). At the end of the problem, the twins are reunited and must agree on the final location and geometry of the light wave front. But the elapsed time on the astronaut clock is t’ < t. So, how can the astronaut calculate the same final radius for the light wave front?”

P.S. Over at SciForums.com, I referenced a very good and relevent thread from this forum titled “Accelerated Frames in SR” but only one of the many participants got it.
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 Quote by MikeLizzi At the end of the problem, the twins are reunited and must agree on the final location and geometry of the light wave front. But the elapsed time on the astronaut clock is t’ < t. So, how can the astronaut calculate the same final radius for the light wave front?
It is not correct that they must agree on the same final radius. They only need to agree that it is a sphere and that it propagated at c in any inertial frame. They can disagree about the radius just as they can disagree on the time.

Also, note that the second postulate refers to inertial frames. There are three relevant inertial frames in this problem, and the light cone has the appropriate geometry in each, and each matching the elapsed time in their frame but not the elapsed time in the other frames.

The traveling twin's frame is non-inertial, so it is not uniquely defined. However, if you use the definition from Dolby and Gull (http://arxiv.org/abs/gr-qc/0104077) then it will be a sphere of a radius matching the traveller's time in the non-inertial frame also. That is not necessarily the case with other defintions of the frame of a non-inertial observer.

 Quote by DaleSpam It is not correct that they must agree on the same final radius. They only need to agree that it is a sphere and that it propagated at c in any inertial frame. They can disagree about the radius just as they can disagree on the time. Also, note that the second postulate refers to inertial frames. There are three relevant inertial frames in this problem, and the light cone has the appropriate geometry in each, and each matching the elapsed time in their frame but not the elapsed time in the other frames. The traveling twin's frame is non-inertial, so it is not uniquely defined. However, if you use the definition from Dolby and Gull (http://arxiv.org/abs/gr-qc/0104077) then it will be a sphere of a radius matching the traveller's time in the non-inertial frame also. That is not necessarily the case with other defintions of the frame of a non-inertial observer.
Here we go again.
Think about it DaleSpam. At the end of the problem the twins are coincident in the same inertial reference frame. While their clocks don't agree, they must agree on the geometry of everything in the universe from that point forward.

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## effect of acceleration on shape and size of a light wave front

 Quote by MikeLizzi they must agree on the geometry of everything in the universe
For geometrical quantities that are frame-variant, such as the radius, that is only true if they measure those quantities with respect to the same reference frame. And if they do, then they agree, regardless of which reference frame they use, as long as it is the same.

 Quote by MikeLizzi At the end of the problem, the twins are reunited and must agree on the final location and geometry of the light wave front. But the elapsed time on the astronaut clock is t’ < t. So, how can the astronaut calculate the same final radius for the light wave front?”
Easy, he must calculate his differential time dilation due to his acceleration and change in directions. See for instance Minguzzi's papers on differential aging.

 Quote by DaleSpam For geometrical quantities that are frame-variant, such as the radius, that is only true if they measure those quantities with respect to the same reference frame. And if they do, then they agree, regardless of which reference frame they use, as long as it is the same.
That's what I said in the original post. That's the problem. When the twins have reunited they are in the same reference frame. They must now agree on the geometry of the wave front. The earth twin can make the simple calculation that the radius of the wavefront is ct where t is the elapsed time of the earth twin's clock. But if the astronaut tries to do that using the time elapsed on his clock he will get a different answer. His calculation must produce the same answer.

 Quote by Passionflower Easy, he must calculate his differential time dilation due to his acceleration and change in directions. See for instance Minguzzi's papers on differential aging.
I'm not sure you understand the problem I posted. The problem is not how to calculate differential aging.

 Quote by MikeLizzi I'm not sure you understand the problem I posted. The problem is not how to calculate differential aging.
It is because the if you know the amount of differential aging you can calculate the actual radius of the wavefront.

 Quote by Passionflower It is because the if you know the amount of differential aging you can calculate the actual radius of the wavefront.
The point of the post is that the twins have not aged the same and if each uses the amount they aged to calculate the radius of the wavefront, it appears to give them different results.

 Quote by MikeLizzi The point of the post is that the twins have not aged the same and if each uses the amount they aged to calculate the radius of the wavefront, it appears to give them different results.
Thee is no difference when we account for the differential aging.

So what is the issue or question here or is this perhaps some attempt to 'disprove' relativity?

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 Quote by MikeLizzi That's what I said in the original post. That's the problem. When the twins have reunited they are in the same reference frame. They must now agree on the geometry of the wave front. The earth twin can make the simple calculation that the radius of the wavefront is ct where t is the elapsed time of the earth twin's clock. But if the astronaut tries to do that using the time elapsed on his clock he will get a different answer. His calculation must produce the same answer.
Ah, back to this misleading concept of necessarily being 'in a frame'. I am on earth, but I can compute (and will compute for planetary problems) in the sun's frame. If I compute planetary motion in an earth centered frame, I am in for a heap of complications (forget GR, and even SR, just imagine Newtonian gravity and Galilean relativity for this analogy).

When the astronaut returns, if he wants to compute using earth's frame, he must use earth's time. If you insist that he use his own watch time, he must use a coherent coordinate chart that maps spacetime (past and ongoing), and has a coordinate time matching his watch time. There is no unique such coordinate system, but there are any number of choices (all much more complex than just choosing one inertial frame). Whichever one he picks, he will continue to label times and distances of events differently than the earthbound twin (as long as he insists on using his watch time, which is influenced by his history of deviation from inertial motion).

 Quote by PAllen Ah, back to this misleading concept of necessarily being 'in a frame'.
Getting close.

 Quote by PAllen When the astronaut returns, if he wants to compute using earth's frame, he must use earth's time.

Of course if the astronaut uses the earth twins data he will get the earth twins answer.

The issue, which I may not have explicitly stated, is how does the astronaut use his own data to get the answer?

He can and if he does it right he will get the same answer as using the earth twins data. Knowing how to do that is the key. I referenced a post in this very forum that explains how that would be done.

 Quote by Passionflower Thee is no difference when we account for the differential aging. So what is the issue or question here or is this perhaps some attempt to 'disprove' relativity?
I'm not trying to disprove relativity. If anything I'm trying to prove that most of the members of this forum don't know as much as they think they know. (And I'm not averse to someone saying that statement also applys to me)

You state there is no difference when we account for differential aging. So show me how each twin would calculate the radius of the light sphere using their own data and get the same answer.

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 Quote by MikeLizzi The issue, which I may not have explicitly stated, is how does the astronaut use his own data to get the answer? He can and if he does it right he will get the same answer as using the earth twins data. Knowing how to do that is the key. I referenced a post in this very forum that explains how that would be done.
You didn't link to anything. I am not going to search for some thread by name, and search for which posts you are referring to.

I insist there is no unique, preferred answer to your question. Are you claiming there is only one coordinate system the astronaut is allowed to use? I have little interest in finding a particular coordinate system in which a particular light sphere has the same radius as for a particular inertial frame (note, different inertial frames may disagree on the radius of a particular light sphere; note, even meeting these constraints, it is trivially provable there are infinite possible such coordinate systems). Why not? Because I think it is very silly in SR to do anything more complex than pick some convenient inertial frame, relate measurements to it, and do all computations in that chosen frame.

[Edit: One very well known feature of non-inertial coordinates is that light speed is not constant. Thus, there is really no mystery in how a well chosen such coordinate system could get the desired agreement : proper time slower, light faster, same radius. I remain uninterested in the details of such an exercise.]

 Quote by MikeLizzi You state there is no difference when we account for differential aging. So show me how each twin would calculate the radius of the light sphere using their own data and get the same answer.
It is trivial as the radius is simply delta t times c for an inertial observer. The traveler can calculate how much his watch ran slower and when he takes that into account he gets the same answer.

I think you ask questions about something that you think is a problem but what is not a problem at all.

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