Could SR not be built from only one postulate?

Me too (as you probably guessed). But I have not seen Robertson's treatment. Where can I find it?

 

This thread is really interesting and helpful (for me at least). I said the reference to a specific set of phenomena, namely the propagation of light in the empty space, creates an asymmetry in SR foundations because it is peculiar whereas the theory embraces all kinds of phenomena. So it is not the invariance of a parameter named c which is problematic, it is the definition of c as the speed of light. This is why I think that only a more general justification for the existence of an invariant speed could remove this asymmetry. Then experiments would show that the speed of light is close or equal to c.

 

It isn't invariant "because of c". In that context, c is just a dimension-conversion quantity so that the sum makes sense. (You can't meaningfully add apples and oranges.)

There are textbooks that do start from an assumption of Minkowski spacetime. But in doing so, one magically assumes that the interval ##-(ct)^2+x^2+y^2+z^2## is invariant -- which then necessarily implies the Lorentz group. Such an approach is probably preferred by mathematicians, but I find it doesn't give much physical insight into the foundations.

 

I'm not a mathematician, but I'm more math-nerdy than most. I find that approach vastly superior when we only want to define SR and see what it says about the world. Those other things that we like to discuss are still interesting, for at least two reasons:

1. If we allow ourselves to make many assumptions along the way, the argument shows how a person who doesn't know SR already can discover SR through clever guesses, and some input from experiments (including the invariance of c).

2. If we take a fixed set of assumptions as the starting point of a rigorous proof, we see that there's no significant difference between SR and pre-relativistic classical mechanics other than the group of functions that change coordinates from one global inertial coordinate system to another. We also see that these are the only two theories that are consistent with the assumptions.

So 1 tells you something about how the theory was found, and 2 gives you some insight into what theories can be defined. I think this is good stuff, but one can argue that this is "just history and philosophy", and not SR at all.

 

Note that Galilean relativity has two invariant intervals instead of one. Having done a general abstract derivation from POR, isotropy, homogeneity you get that you have a choice of two geometric structures:

1) invariant time + invariant Euclidean distance.
or
2) invariant spacetime interval (equivalent to some invariant speed which would be labeled c).

I still have not seen a single convincing argument for choosing one over the other except experiment. It is, for example, false that (1) requires action at a distance. Newton's gravitation is a specific theory in the framework of (1), and was chosen not because it was forced by this structure but to match the observation that orbits had high stability. GR had an issue of replicating the appearance of instant action at a distance required for orbital stability while having a propagation speed that was too slow by orders of magnitude. This difficulty GR had matching observation (and the solution) are well explained in the classic paper by Steven Carlip: http://arxiv.org/abs/gr-qc/9909087.

[Edit: one statement you could make about this is that gravitational observations falsified SR and were more consistent with Newtonian physics, especially given the modern understanding that a satisfactory SR theory of gravitation that matches observation as well as Newtonian theory is impossible. Thus, you can say EM experiments favored SR, gravitational experiments favored Newtonian physics, and it took GR (general relativity) to successfully encompass all classical observations.]

 

Thank you!

[Edit]... after reading it...

Hmm, it turned out to be rather different than I expected...

Robertson's postulates/assumptions/restrictions seem to be:

P1) There exists a reference frame ##\Sigma## in which light is propagated rectilinearly and isotropically in free space with constant speed ##c##.

P2) The physical geometry of 3-space, as revealed by measuring rods, is Euclidean.

P3) All clocks at rest in ##\Sigma## are synchronized.

P4) The velocity of light in free space is independent of the motion of its source.

P5) There exists a reference frame ##S## which is moving with an arbitrary constant velocity of magnitude ##|v|<c## with respect to ##\Sigma##.

P6) The physical geometry of 3-space in ##S##, revealed by the same meaurement techniques as above, is also Euclidean.

(No assumption is made about the speed of light in ##S## -- that is derived.)

P7) The transformation between coordinates in ##\Sigma## and ##S## has only ##v^i## as its essential parameter, and ##v^i=0## corresponds to the identity.

P8) The considerations are confined to laboratory--scale experiments, hence only in a small spacetime neighborhood, hence only a transformation between infinitesimal differentials is considered, therefore the transformation is taken to be linear.

(Some coefficients of the transformations are reduced using a radar method to perform clock synchronization.)

P9) The 1-way velocity of light is independent of direction. Probably this follows from (P1).

He then appeals to the Michelson-Morley and Kennedy-Thordike experiments to reduce the possibilities further, followed by a further appeal to the Ives-Stilwell experiment.

He doesn't seem to make much (any?) use of the group multiplication property.

All in all, at the point before he starts appealing to experiments he seems to be quite a long way from the most-general set of possibilities.

 

Thanks for challenging my views. In this thread we are looking at the simplest set of preliminary hypotheses/statements/postulates which necessarily impose the Lorentz transformation of the R4-coordinates of a physical event in response to a change of the inertial frame of reference into which this event gets described/recorded.

First I can't see in which way the “focus on theories that use R4 as a model of space and time” could be seen as a reduction in generality or a “limit” to our attention: it is intrinsic to the pattern of the problem at stake.

Second, if the set of preliminary statements leaves open two potential solutions (the Lorentz or the galilean transformation), this shows that further constraints must be added at the forefront in order to reduce the range of potential solutions and derive the Lorentz transformation as the sole but necessary outcome. Invoking some “experiments to distinguish between the two possibilities” is just the same as stating that we haven't so far produced a satisfactory answer. Indeed we know that some experiments (e.g. the decay of muons across the atmosphere) confirm that SR leads to better predictions than the Newtonian mechanics, but this only underlines the importance of resolving the problem at stake.

Third, it is clear that the addition of a constraint like “no instantaneous action at a distance”, because it implies the existence of a maximum speed limit, resolves the problem at stake. Einstein made his second postulate specific to the propagation of light, but his formal derivation would of course work as well with the more upstream constraint I proposed.

Finally, is this “an improvement over the simple idea that we can use experiments to distinguish between the two possibilities”? I can understand your doubts insofar the replacement of a postulate about the world (the speed of light is finite and invariant) with another postulate about the world (there exists an invariant finite speed for any action inside the world) does not bring much. Eventually we don't know anything about the world (how it is, how it works, what happens there) and it is illusory to believe that experiments will ever bring any knowledge of that kind. Any postulate about the world is merely speculative, its content cannot be certified. Postulates about the world cannot root SR (or any other physics theory) into solid ground.

Conversely the key added value of my approach is that the existence of a maximum invariant speed is NOT derived from a statement about the world and moreover it is NOT a postulate. It is a true statement reflecting a fact that all physicists agree upon: our physics theories are based on a causal paradigm which excludes instantaneous actions at a distance. It is this change of perspective which makes the difference, replacing the usual metaphysical approach based on postulates about the world with a pragmatic approach based on true factual statements about the concept of causality which guides the development of our physics theories. Because the existence of the maximum speed limit is integrated (upstream to the consideration of any external phenomenon) into the very structure of the formal space-time framework used to record physical events, SR ensures that the exclusion of instantaneous actions at a distance is enshrined into the theory.

My (somewhat challenging) conclusion is therefore that the second postulate of SR, dealing with the invariance of the speed of light, should be dropped and replaced with a true factual statement: our physics theories exclude a priori any kind of instantaneous action at a distance. Overall I suggest that both SR postulates could and should be dropped because they propagate the illusion that we know something about the world, to the benefit of true, factual statements about the pragmatic constraints that must be met by any proper physics theory.

 

You keep ignoring the fact that Galilean relativity, per se, does not require instant action at a distance. Instead, to reject it you need the much more specific postulate of an invariant speed, which seems no more natural than the invariant times that comes with Galilean relativity.

You also haven't commented on the point I made (with references) that Gravity (a specific theory, not a required feature of Galilean relativity) made the global symmetries of SR somewhat stillborn, in that it provided apparent experimental support for essentially instant action at a distance (> 10¹⁰ c ). Thus, in fact, the global symmetries of SR do not exist in our world. This makes it particularly hard to sustain that it is uniquely natural, but also false (in the precise sense that its symmetries are local, or exist only in the tangent space at an event).

 

There are theories that don't use ##\mathbb R^4## as the model of space and time. GR uses a smooth manifold, which bears a number of relationships with ##\mathbb R^4## (in particular, it's locally homeomorphic to ##\mathbb R^4##), but it isn't ##\mathbb R^4##. And GR is a better theory than any of the ones we find.

You could also loosen up the constraints and find a vastly superior theory: general relativity.

It seems that what you would consider a satisfactory answer is a perfect theory that can be found just by thinking.

The pre-relativistic spacetime doesn't imply that there's action at a distance. It just allows us to define such theories. SR doesn't allow it (unless we drop the principle of relativity and introduce a preferred coordinate system). But if this is a reason to dismiss the pre-relativistic framework, then we have gone from having a "not so great" theory of gravity (Newton's) to having no theory of gravity*. It's far from obvious that this is a step in the right direction.

*) I know almost nothing about attempts to define a theory of gravity in Minkowski spacetime. This probably means that such attempts haven't been very successful. I have a feeling that if there was a good theory of gravity in this framework, then it would be taught in SR classes and books.

I don't agree that it can't bring us any knowledge of that kind. It already has. We know that Earth is in an approximately elliptical orbit around the Sun for example. But I would say that in many situations, in particular most situations where quantum mechanics is needed, it would be naive to think that the theory (or the experiments that test the accuracy of its predictions) is telling us what's "actually happening" to the system between state preparation and measurement.

However, the fact that the scientific method can't tell us everything that we would want to know, doesn't change the fact that it's the best we've got, and ever will have. It's vastly superior to the method of "just thinking".

I agree with the stuff before the colon. But the only things that should be regarded as facts in physics are experimental results. So what I would have said after the colon is that SR makes better predictions about the results of experiments (when gravity is irrelevant).

I don't see a reason to consider "no instantaneous action at a distance" anything more than an assumption.

 

If I remember correctly, Jackson explicitly shows an alternative relativistic force law that reduces to Coulomb's Law in the nonrelativistic limit, but isn't the same as Maxwell's equations. I don't remember what that was.

I guess that charge could be a Lorentz scalar, instead of a component of a 4-vector.

 

Well, the point of Einstein's original derivation was not really to come up with the most general derivation. Instead, he had a number of things that had plenty of experimental support, but couldn't all be true simultaneously:

1. Newton's equations of motion.
2. Maxwell's equations (specifically, the prediction of the constancy of the speed of light).
3. The principle of relativity (that there is no preferred reference frame).

Einstein's goal was to find a replacement for Newton's laws that was compatible with relativity and the constancy of the speed of light.

 

Charge density is the time-like component of the current 4-vector and charge is a Lorentz scalar. Charge is defined as the integral of the charge density over a space-like hypersurface. The proof that charge is a Lorentz scalar is actually quite simple, given the 4-divergence of the current 4-vector. It's basically just an application of Stokes' theorem.

 

Yes, you're right. I can't find my copy of Jackson, so I'm not sure what the alternative theory is that reduces to Coulomb's law in the nonrelativistic limit.

 

The 4D divergence theorem