

The TwoClock Problem
Storrs McCall (Department of Philosophy, McGill University)
This is analogous to, but not identical with, the twins paradox. In its "threeclock" form, the twins paradox in flat spacetime involves clock 1 following an inertial path from A to C, clock 2 an inertial path from A to B, and clock 3 a third inertial path from B to C. If clocks 1 and 2 are synchronized at A, and 2 and 3 are synchronized at B, clock 3 will record a total elapsed time at C which is less than that of clock 1. This demonstrates that triangles in Minkowski geometry with timelike sides obey a law of triangle inequality opposite to that in Euclidean geometry. The sum of the lengths of AB and BC is less than that of AC.
The twoclock problem also involves two clocks synchronized at A. Clock 1 follows an inertial path from A to C in flat spacetime, far away from massive bodies. Clock 2 starts off on an inertial path at an angle to AC, coasts around a massive body at B without rocket power, using the slingshot effect, and rejoins clock 1 at C (see figure). Both the paths of the two clocks are geodesics. When the clocks are compared at C, will clock 2 record (i) more total elapsed time than clock 1, or (ii) less time (as one would expect from the twins paradox), or (iii) the same amount?
I am sure this result is recorded somewhere in the literature, but I don't know where.


