Light travel times around a closed Universe

Abstract

The travel times for light rays to circumnavigate a closed, spherical Universe, as measured in a frame attached to a moving clock, are determined for Einstein’s static Universe and two cases of the pulsating model. It is shown that the ratio of the difference of the travel times in the forward and rearward directions to the sum of the travel times yields the clock’s velocity relative to space. The implication of these results for Einstein's various views on the ether and motion are discussed, as well as their bearing on the synchronization problem. Lorentzinvariant relations are shown to hold between the respective sum and difference travel times for two clocks traveling on the same path with different velocities. The mathematical similarity of the expressions for the sum and difference travel times to expressions for relativistic energy and momentum is pointed out, and the possible bearing on Mach’s principle is briefly considered. There are two appendices: in the first, synchronization of separated clocks with (hypothetical) faster-than-light signals is critically discussed (including some of the author’s earlier work). In the second, synchronization based on using slowly-moved clocks is analyzed anew.