# Circular definitions in Special Relativity?

Standard textbooks introduce Special Relativity in this way:

1. They introduce two postulates, the second being something like that

The speed of light in a vacuum is the same for all observers, regardless of the motion of the light source [consenus definition on the english wikipedia page ]

or

"... light is always propagated in empty space with a definite velocity [speed] c which is independent of the state of motion of the emitting body" [A. Einstein]

1. They define Einstein synchronisation which relies on the knowledge/assumption that the speed of light is everywere constant

The problem that I see is this one:

• The postulate is speaking about any measure of "speed" and the meaning of (mean) "speed" is supposed to be $\Delta s/\Delta t$ (as far as the reader knows) where $\Delta t=t_1-t_2$, $t_1$ and $t_2$ being two measured times. If the times $t_1$ and $t_2$ are measured in two different positions we need to know in advance how to syncronise distant clocks but we don't know yet and we will know how to do it only afterwards: with Einstein synchronisation. But Einstein syncronisation relies on the postulate itself: it assumes that the "speed" of light is constant. This seems to produce a logical circularity: speed of light is (obviously) constant just because we define synchronisation in order to make it "artificially" constant? (*)

• And what is the postulate actually telling us? We don't know what is "speed" yet since we don't know how to define time for spatially distant places. Maybe the postulate is supposed to be linked with Einstein synchronisation in order to define what is "speed"?

• What actually is speed/velocity on a general (non circular) path when we drop absolute time? Before postulating anything about speed of light we need in the first place to know what speed is, and the definition shouldn't rely on the use of the notion of velocity itself that is the case with Einstein's synchronisation!

Can anyone clarify this issue?

(*) I know the M-M experiment proved that speed of light is constant when measured as an average in a closed path with mirrors but I would say it doesn't say anything about the speed of light in any non-closed interval of the path.

• It is worth keeping in mind that physics is full of constructs that are internally consistent but wholly self-referential after some boot-strapping step that looks arbitrary. The whole framework of ideas surrounding inertial frames and that surrounding energy are both example of this kind of thing. Neither the apparent circularity of the definitions, nor the arbitrariness of the bootstrap is a problem because the results are tested against nature in an objective way. Commented Aug 12, 2016 at 17:32
• The constance of the speed of light can be derived from Maxwells equations. Commented Aug 12, 2016 at 22:03
• @PhysicsGuy Well, sure. And you can derive the speed of sound in a diffuse gas through kinetic theory. But in one case that speed is relative a medium and in the other it is relative all observers which came as a bit of a surprise for all that there is some clarity to it in retrospect. Commented Aug 12, 2016 at 23:24
• If 2 different observers have 2 different views of what events are synchronized, and what events are not, then a bigger picture than either observer can see, is at play. To see the bigger picture, one must observe those events in play, 4 dimensionally. Doing so, removes the mystery/bizarreness of SR. goo.gl/fz4R0I
– Sean
Commented Aug 13, 2016 at 1:30

A synchronisation procedure of ideal clocks at mutual rest must be transitive, symmetric, reflexive and it must remain valid in time once one has adjusted the clocks to impose it. There is no evident a priori reasons why Einstein's procedure should satisfy these constraints. The fact that it instead happens is the physical content of both postulates you quoted.

Actually there is a third physical constraint: The value of the velocity of light must be constantly $c$ when measured along a closed path. This measurement does not need a synchronisation procedure, since just one clock is exploited.

A natural issue show up at this juncture: whether there are synchronisation procedures different from Einstein's one which however fulfill all requirements.

The answer is positive (without imposing other constraints like isotropy and homogeneity) and they give rise to other formulations of special relativity, which are physically equivalent to Einstein's one. (Geometrically speaking, it turns out that the geometry of the rest spaces is not induced by the metric of the spacetime by means of the standard procedure of induction of a metric on a submanifold.)

There are well-known physical situations regarding clocks with non-inertial motion (at rest with respect to each other), where Einstein's procedure cannot be used and other synchronisation procedures must be adopted. The most relevant is the one regarding a rotating platform. If I remember well, the first correct analysis of the problem was proposed by Born.

• Very interesting, thanks. I have a question: what happens to invariance of $c$ when you use alternative synchronisations? Does it still hold true for non-closed path? Commented Aug 12, 2016 at 14:17
• No, it does not. It does not matter however, since it is not a completely convention-independent feature. Commented Aug 12, 2016 at 14:41
• The wikipedia page you quoted actually includes a little discussion on the subject The link toward the page on the so-called Sagnac effect in particular en.wikipedia.org/wiki/Sagnac_effect Commented Aug 12, 2016 at 14:49
• Your name sounds Italian, if it is the case you can have a look at my lecture notes on mathematical foundations of Relativity, where I spent some pages on these issues. Unfortunately I did not have time yet to translate those notes in English. Commented Aug 12, 2016 at 14:58
• Is it "Teoria della Relatività Speciale: formulazione matematica"? I will see it, thanks Commented Aug 12, 2016 at 15:01

Velocity is $dx/dt$ not $\Delta x/\Delta t$ so we aren't measuring the speed between two different places and two different times. The velocity is defined at a point. We don't need any form of Einstein synchronisation to define a velocity.

As for what the postulate is telling us: while the postulate is of historical significance it is not a great way to understand special relativity. The fundamental principle behind special relativity is that the line element defined by:

$$ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2$$

where $c$ is a constant, is an invariant i.e. it has the same value for all observers regardless of their motion. The invariance of the line element results in all the weird effects associated with relativity like time dilation and length contraction, and it also tells us that the constant $c$ is the speed of light i.e. the first postulate.

• Thanks for your reply. If we consider the "instantaneous velocity" we should compute a limit of $\Delta x / \Delta t$ with $\Delta t \to 0$ and this would involve a sequence of computations of "mean" velocities with "distant" (even if close) positions involved, which would require a synchronization. Wouldn't it? Commented Aug 12, 2016 at 11:35
• All the references I've seen recently refer to the constancy of the speed of light as the second postulate, not the first. E.g. en.wikipedia.org/wiki/Postulates_of_special_relativity and this question itself. Did you mean to say "the second postulate" instead of "the first postulate" in your last sentence? Commented Sep 19, 2021 at 11:06

> If the times $t_1$ and $t_2$ are measured in two different positions we need to know in advance how to syncronise distant clocks but we don't know yet and we will know how to do it only afterwards: with Einstein synchronisation

But we do know how to synchronize distant clocks. Just take one clock (2) to the other (1), synchronize it with 1, then return it to its original position.

> And what is the postulate actually telling us? We don't know what is "speed" yet since we don't know how to define time for spatially distant places.

Speed of light in the postulate is a one-way speed of wavefront of light ray. We know how to measure this one-way speed of light.

One way needs two distant synchronized clocks coupled with switch turning light on and detector detecting that light is present. I'm not sure this was ever done though. The postulate is an assertion of how things are thought to work, not necessarily an experimental result.

The other (indirect) way is to measure independently frequency $f$ and wavelength $\lambda$ of harmonic EM wave of distant source at various places. The speed of light is then calculated as $\lambda f$. This seems quite possible to do with microwaves or radiowaves with MHz-GHz frequency and wavelengths in the realm of centimeters - meters. Again, I do not know if there in fact was such an experiment.

• "But we do know how to synchronize distant clocks. Just take one clock (2) to the other (1), synchronize it with 1, then return it to its original position" Wouldn't relativistic effects destroy the synchronisation in this case? Commented Aug 12, 2016 at 17:53
• Well in the special theory of relativity it is true that the above sync process would be imperfect, as the moving clock would be subject to time dilation. However, in practice this results in very small shift from synchronized state, which does not prevent measuring speed of light in any substantial way. In theory, it can be shown that the error of synchronization can be made arbitrarily close to 0 by moving the clock slowly enough. Commented Aug 12, 2016 at 18:07

"I know the M-M experiment proved that speed of light is constant"

Originally it proved the opposite - the speed of light is variable (depends on the speed of the light source). Then FitzGerald and Lorentz procrusteanized moving objects (introduced, ad hoc, length contraction) and the experiment started to support a tenet of the ether theory - the speed of light is independent of the speed of the source. Finally, Einstein made the tenet his 1905 second postulate:

http://books.google.com/books?id=JokgnS1JtmMC Banesh Hoffmann, Relativity and Its Roots, p.92: "There are various remarks to be made about this second principle. For instance, if it is so obvious, how could it turn out to be part of a revolution - especially when the first principle is also a natural one? Moreover, if light consists of particles, as Einstein had suggested in his paper submitted just thirteen weeks before this one, the second principle seems absurd: A stone thrown from a speeding train can do far more damage than one thrown from a train at rest; the speed of the particle is not independent of the motion of the object emitting it. And if we take light to consist of particles and assume that these particles obey Newton's laws, they will conform to Newtonian relativity and thus automatically account for the null result of the Michelson-Morley experiment without recourse to contracting lengths, local time, or Lorentz transformations. Yet, as we have seen, Einstein resisted the temptation to account for the null result in terms of particles of light and simple, familiar Newtonian ideas, and introduced as his second postulate something that was more or less obvious when thought of in terms of waves in an ether. If it was so obvious, though, why did he need to state it as a principle? Because, having taken from the idea of light waves in the ether the one aspect that he needed, he declared early in his paper, to quote his own words, that "the introduction of a 'luminiferous ether' will prove to be superfluous."

• "Originally it proved the opposite - the speed of light is variable (depends on the speed of the light source)" Interesting, can you link something or drive me somewere to better undertand this point? Commented Aug 14, 2016 at 15:43
• I don't have enough space here to develop the argument (it is simple). The falsehood ("M-M has shown that the speed of light is constant") has been taught for so long that it is regarded as absolute truth now. A few authors admit that M-M is compatible with the variable speed of light predicted by Newton's emission theory of light but discuss the problem in a way not allowing the reader to reach even partial understanding. Commented Aug 14, 2016 at 17:25
• Saying "it proved the opposite" is quite different from " is compatible with the variable speed" Commented Aug 14, 2016 at 17:31
• "It proved the opposite" is what I'm saying; "is compatible with the variable speed" is what a few authors say. Commented Aug 14, 2016 at 17:38
• "It proved the opposite" is your personal thesis? Commented Aug 14, 2016 at 17:58