Tuesday, November 24, 2015

A Traveling Twin's Rest Frame?

Okay, let's play with this idea of the traveling twin's  "rest frame"  One trouble with the idea of any noninertial frame is that there is no unique way of prescribing its construction. 

But I think I understand how you are thinking of your construction...  Here is the link to WWoods diagram on Wikipedia: File:Twin 5.png

My impression of your "rest frame" is that you want to take these diagrams, and chop them along the lines of simultaneity and stitch together the result like so:

One reason I would object to that is that in this “rest frame of the traveling twin” all of the events that happen on earth between Event A and Event C are unrepresented.  Also, if the traveling twin were to look in the opposite direction, some events would cross the line of simultaneity in the other direction, and happen twice.

Let me suggest another version of this "traveling twin's rest frame."

As a solution, I would offer the following modification. 

Instead of stitching together the rest-frame along the line of simultaneity, what if we stitched it together along the line of x=-ct (e.g. the locus of events that are actually observed by the traveling twin at the instant of his acceleration when he looks toward earth.)

For that, we'll start with another diagram

File:Three frames.JPG

This diagram does not share the same labels as the earlier one.  Instead, event B here is the event-(stream) on earth observed by the traveling twin at the time of the turnaround.

Instead of chopping the two diagram along the line of simultaneity, I chop it along the line EB and combine them like this.

We still have this line from beginning-to end of the trip for the traveling twin represented as a straight line. But now it shows all of the events on earth observed by the traveling twin instead of skipping a bunch.

Also it has the advantage of actually showing, rather than hiding the discontinuity in the motion of the earth in the reference frame(s) of the accelerating observer.

By using t=0 as your stitching boundary, you have some observed events repeated twice, and some observed events not represented.  So your concept of an accelerating rest frame would be much improved if you used the locus of events t=r/c of the past light-cone as our stitching boundary. 

Now, just to be clear, though.  Just because I can stitch together parts of two diagrams to make the world-line of an accelerating observer look straight, I still think it is probably better to just say he is "switching inertial reference frames" because even those events that are far in the observer's past are still in the locus of events affected by the Lorentz Transformation.

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