Relativity 'controversy' 2 “The actual tick rates of clocks in relative inertial motion in free space are necessarily different”. The correct interpretation of the facts is that for clocks moving inertially in free space, it is impossible to determine any tick rate difference, because they can only meet once. In order to compare tick rates, one of the two clock readings must necessarily be done at a distance and then it depends on the convention for clock synchronization that has been adopted.

In the historical “twin paradox”

^{[1]}, one of the twins is sent on a round trip journey, accelerating quickly to some constant relative speed, maintains that speed for a rather long time, then makes a quick turnaround, flies back at the same relative speed, before decelerating to stop at her brother. This makes it possible to compare their clocks a second time and find any differences in the elapsed times. In the conventional scenario, the ‘away-twin’ will always record less elapsed time on her clock than the elapsed time on her brother’s clock. She is not inertially moving all the time; she changes inertial frames three times and uses energy to do so.

The controversy surfaces when we ask the question: during the trip, when has the away-twin ‘lost’ that time relative to her brother? There is a school of thought saying that it happens progressively as the away-twin travels, but this positions has its problems, as can be shown by very simple logic. Say just before sister ignites here engines to decelerate for the turnaround, we decide that she must rather stay inertial and that her brother must ignite his engines to accelerate and catch up with her, stopping by her side. Now his clock would be the one that has lost time between the two meetings, even if the relative speeds were exactly the same during the two inertial phases of the test. So, whose clock lost what time during which phase? It is impossible to say. All that we can say is that at the end of the test, we can measure whose clock has lost time and how much. If we know the exact test setup in advance, we can also predict the outcome.

There is a vague similarity to quantum mechanics here. Suppose we do not decide in advance which of the twins will do the acceleration, but we ask sister to toss a coin when she nears the turnaround point. Heads and she turns around, tails and she just continuous on inertially. Whatever happens, she lets brother know by radio and when he gets the message, he does the appropriate thing to join his sister – he waits for her, or he fires up his engines. When the test begins, the chances are fifty-fifty and the toss of the coin later decides whose clock will be behind at the end of the test. So for the first part of the test, each clock is running both ‘slower’ and ‘faster’ than the other one, a la quantum particles in a state of superposition. When the result of the toss is known and the choice of test is made, only one state is predicted and subsequently observed.

Since our two clocks are not really quantum-mechanical in nature, the better view is that the proper “relative clock rate” can only be determined between clocks in the same location (collocated) when observed. Any discussion about relative clock rates when they are in relative motion depends on the convention for simultaneity that has been adopted by each clock's frame of reference. There are scientific limits to the range of reasonable simultaneity conventions, but as we have seen in the prior post, there are many of them. Einstein’s convention just proved to be the wisest choice.

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Regards

Jorrie

[1] AFAIK, the “twin paradox” was originally formulated in 1911 by Paul Langevin.