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Yesterday, I found the time to write a bit further on my SRT FAQ and wanted to give a quantitative analysis of the Bell space-ship paradox on the example of the two rockets accelerating with constant proper acceleration, and I found a problem, I cannot solve. So I took this section out from my FAQ for the time being.
The problem occurs, when one looks at the situation from the point of view of the heading rocket C and if ##\alpha L_A>c^2##, where ##\alpha## is the constant proper acceleration and ##L_A## the constant distance of the space ships in A's reference frame, where the space ships start accelerating from rest simultaneously at time ##t=0##. Perhaps you can help me out here. The problem finally seems to be to find a proper, i.e., invariant definition of the distance of the spaceships. I still have to understand the physical meaning of bcrowell's analysis using the time-like congruence of space-like separated hyperbolic-motion hyperbola. This seems to be the only analysis in terms of a frame-independent quantity in the literature.
I've put my analysis here, because for some reason I cannot upload it as an attachment here:
http://fias.uni-frankfurt.de/~hees/tmp/bell-paradox.pdf
Note that the figure in bcrowell's Insights article is depicted in other form as the left-hand panel of the figure in the text. For this case (point of view of the rear space-ship B) no problem occurs. The problem occurs in the situation depicted in the right panel, but that's explained in detail in the text too.
The problem occurs, when one looks at the situation from the point of view of the heading rocket C and if ##\alpha L_A>c^2##, where ##\alpha## is the constant proper acceleration and ##L_A## the constant distance of the space ships in A's reference frame, where the space ships start accelerating from rest simultaneously at time ##t=0##. Perhaps you can help me out here. The problem finally seems to be to find a proper, i.e., invariant definition of the distance of the spaceships. I still have to understand the physical meaning of bcrowell's analysis using the time-like congruence of space-like separated hyperbolic-motion hyperbola. This seems to be the only analysis in terms of a frame-independent quantity in the literature.
I've put my analysis here, because for some reason I cannot upload it as an attachment here:
http://fias.uni-frankfurt.de/~hees/tmp/bell-paradox.pdf
Note that the figure in bcrowell's Insights article is depicted in other form as the left-hand panel of the figure in the text. For this case (point of view of the rear space-ship B) no problem occurs. The problem occurs in the situation depicted in the right panel, but that's explained in detail in the text too.
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