Conventionality of simultaneity and velocity measurement

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Dated: 15 November 2023 and 19 November 2023

One interpretation of the Sagnac effect in the rotating frame is based on different speeds of light in the corotating and counterrotating directions, arising from central synchronization. The claim that this points to a preference of some kind of "absolute simultaneity" kept creeping up, suggesting that its proponent(s) had not appreciated the logical link between the simultaneity relation and one-way velocities.

Therefore I took part of an old draft of my paper on Ways to resolve Selleri's paradox [1] and extended it to a detailed description of how one-way velocities are defined, using a synchronized time in the definition, what synchronization and simultaneity mean, and why causal ordering is insufficient in the relativistic spacetime to uniquely define simultaneity. Since this leads to the possibility of several different definitions of simultaneity (in a single frame of reference), simultaneity is conventional, i.e., it is not fixed by physics alone.

I emphasize the logical priority of the simultaneity definition over the definition of time derivatives, meaning that we can (and must!) define simultaneity before one-way velocities. (To define simultaneity, we need a local time definition and a way to extend it to finite distances. To define one-way velocities, we need a time definition that is already extended from the starting point to the end point of the spatial interval, on which we wish to define the velocity.) One-way velocities can only be defined when we already have a notion of simultaneity. That this is not understood well by the majority of people, even if they have a scientific background, is probably due to the fact that historically velocities were defined before even the thought came up that there may be more than one definition of simultaneity. Before Einstein, simultaneity was implicitly defined (for example by reference to Newton's absolute time) and those who think one-way velocities can be defined or even measured independent of synchronization, fail to realize that invariably a notion of simultaneity is used in the definition and/or the measurement of these velocities.

Three examples are discussed that might be invoked as cases in point for a velocity measurement that is synchronization independent. They include the Sagnac effect that Quattrini claimed to produce the rotation velocity (of the circumference) independent of any synchronization, because it is essentially determined by the Sagnac phase shift, which certainly does not depend on synchronization.

I show by explicit calculation in different synchronizations (Einstein synchronization and a variant of Reichenbach's ε synchronization) that these arguments against synchronization dependence of velocities overlook a subtlety: the formal expressions by which the velocity is determined from measured quantities that may well be synchronization independent, will depend on the synchronization themselves.¹ In claiming that these expressions are well-known and independent of synchronization, Quattrini fails to realize that the well-known expressions were all derived under the (implicit and possibly inadvertent) assumption of Einstein synchronization. In fact, my calculations demonstrate explicitly that different expressions are valid for Reichenbach's ε synchronization with ε≠½. The calculations are given in the pdf file below and everyone can check them!

Conventionality of simultaneity and velocity measurement, 15.11.2023

Quattrini claimed that there was an error in my first example, dealing with a velocity measurement using the Doppler effect. This error was that I had used formulas that do not apply to the Doppler radar experiment. Of course, this is not the kind of error that could invalidate my arguments, because it would not change the fact that the expressions are different for different synchronizations. But Quattrini was also wrong in assuming that I was describing the Doppler radar experiment. For the experiment that I was referring to, velocity measurement (for example of stars) by measuring the Doppler shift of the (known) frequency of (e.g., spectral lines of) an emitter (i.e. the star), the results I gave are correct.

Anyway, I took the opportunity to treat the case of the Doppler radar experiment, too, and showed that in this experiment the measured Doppler shift does not allow to determine a synchronization independent velocity either. The same Doppler shift corresponds to different velocities in different synchronies.

Doppler radar and conventionality of simultaneity, 19.11.2023

[1] K. Kassner, Ways to resolve Selleri’s paradox, Am. J. Phys. 80, 1061 (2012).

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