Can Kindergarten STR Math Solve Polchinski's Paradox?

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Is it possible to get rid of Polchinski's paradox with kindergarten STR maths?
 
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A more detailed paper would be http://prola.aps.org/abstract/PRD/v44/i4/p1077_1 (but I haven't read it personally, it's on my to-read list).

Reading this paper seems to suggest that the billiard balls analyzed were not relativistic, though. I don't know for sure if anyone has done the relativistic case.
 
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