Volume 19: Pages 430-433, 2006
The Self‐Consistency of the Kinematics of Special Relativity, Part VI. Einstein's Argument in Special Relativity Concerning “Time Travel” Is Substantially Correct After All; and Smoothed Polygons
I. J. Good 1
11309 Lynn Drive, Blacksburg, Virginia 24060‐3001 U.S.A.
What I previously called “Einstein's Slip” wasn't a slip after all: “time travel” is a correct inference from KSTR: my slightly subtle mistake about this matter occurred in Part V(A). This correction doesn't imply an inconsistency in KSTR, although it would appear to do so to those who regard time travel as logically impossible. Arthur or A is stationary in an inertial system, and Bertha or B travels relative to Arthur with constant speed v in a closed path Π. According to A she takes time t but according to B she takes less time, the difference in the times (the lag) being (1 − γ−1)t, where γ = (1 − v2/c2)−1/2, at least closely. Einstein, in effect, arrived at this formula precisely for the case where Π is a circle by saying in 1905 in his first article on (special) relativity, “[i]f we may assume that the formula [as cited above] for polygons can be assumed for smooth curves.” But motion along a polygon with constant speed requires infinite acceleration at the vertices and so is physically impossible. We make Einstein's argument closer to physical reality by replacing the polygons by smoothed polygons in a well‐defined sense. (For the sake of generality we allow the smoothed polygons to lie in more than two spatial dimensions.) For the case where Πis a circle Einstein arrived at the same inference as does a limiting argument in which the circle is regarded as the limit of a sequence of regular polygons.
Keywords: challenge to general relativity, Einstein's non‐slip, knotted polygons, KSTR, polygons in three dimensions, smoothed polygons, swings and roundabouts, time lag, time travel in special relativity
Received: March 13, 2006; Published Online: December 15, 2008