Is 2nd postulate of SR necessary?

[meopemuk]

clem,

thanks for the references. I didn't read all of them, but I am familiar with some of them. They suffer from a common unjustified assumption, which, I think, is important.

For example, take Levy-Leblond's paper. In his eq. (1) he assumes that event's space-time coordinates (x', t') in the moving frame depend ONLY on its coordinates (x, t) in the frame at rest and on parameters {a}, which determine the relative velocity of the two frames. He doesn't entertain the possibility that x' and t' may also depend on the physical nature of the event.

Let me explain what I mean on the following example. Suppose that we are interested in the event of collision of two particles. This event is rather well localized in both space and time for both observers. So, coordinates (x, t) and (x', t') have clear meaning, and it makes sense to ask which transformation connects these two pairs of numbers. Consider two possibilities. In one case the particles are non-interacting. In the other case the particles interact with each other (for example, these are two opposite charges). Is there a good reason to assume a priori that the transformation laws (x, t) -> (x', t') will be exactly the same in these two cases? I don't think so.

I would be much happier if Levy-Leblond (and others) explicitly formulated their tacit postulate, which sounds roughly like this: "Transformations of space-time coordinates of events do not depend (i) on the physical nature of events; (ii) on the composition of the system in which the events occur; (iii) on the nature and strength of interactions in the system." This assumption lies in the foundation of all special relativity, but it is never (or rarely) mentioned.

 

[Fredrik]

My opinion is that people should care a lot less on Einstein's postulates (and possible alternatives to them) and a lot more about the details of how to interpret the mathematics of Minkwski space as predictions about the results of experiments. But I probably say that too often in this forum.

I decided to look up the original version of the postulates. I found a translation here (page 140) and the original german http://www.pro-physik.de/Phy/pdfs/ger_890_921.pdf . They are a bit different from what I said in #23. This is a better summary of what they say:

1. The laws of electrodynamics and optics are valid in any coordinate system where the equations of mechanics are valid.

2. In empty space, light always propagates at a definite velocity V which is independent of the motion of the light source.

These aren't exactly well-defined mathematical axioms either. Does the specification of what coordinate systems he's talking about in postulate 1 even make sense? (That's not a rhetorical question. I don't know what the answer is, but then I'm always a bit dumber the last hour before I go to sleep). Also, note that postulate 2 doesn't even mention coordinates.

 

[Meir Achuz]

Fred: I agree with your first paragraph, so I probably should stop right here but...
Neither postulate, as stated in that translation make much sense today.
For (1), the "equations of mechanics" as Einstein knew them at the time were mostly not valid.
For (2), isn't that completely compatible with an aether? That postulate holds for sound in air.
If anyone does want one or two basic postulates for SR, it can't be either one of those.

 

[robphy]

Be fair. The only filled 15 pages with words that say very little. The rest are references. And besides, they are philosophers. It's their job to fill pages with words that say very little.

In defense of some "philosophers [of science]",
there are some more-technical ones...
like Malament ( http://www.lps.uci.edu/home/fac-staff/faculty/malament/ )
who is interested in foundational aspects that most physicists gloss over in their treatment of the subject. (Foundational aspects are important, of course, in any search for a more fundamental theory.)

a review article "Classical General Relativity" http://arxiv.org/abs/gr-qc/0506065
and this famous paper: http://link.aip.org/link/?JMAPAQ/18/1399/1

 

[DrGreg]

I decided to look up the original version of the postulates. I found a translation here (page 140) and the original german http://www.pro-physik.de/Phy/pdfs/ger_890_921.pdf . They are a bit different from what I said in #23. This is a better summary of what they say:

1. The laws of electrodynamics and optics are valid in any coordinate system where the equations of mechanics are valid.

2. In empty space, light always propagates at a definite velocity V which is independent of the motion of the light source.

These aren't exactly well-defined mathematical axioms either. Does the specification of what coordinate systems he's talking about in postulate 1 even make sense? (That's not a rhetorical question. I don't know what the answer is, but then I'm always a bit dumber the last hour before I go to sleep). Also, note that postulate 2 doesn't even mention coordinates.

I would first paraphrase (1) as

1a. An inertial coordinate system is defined as one in which the equations of mechanics are valid.
1b. The laws of electrodynamics and optics are valid in any inertial coordinate system.

Then I would simplify (1a) to say "Newton's first law" instead of "the equations of mechanics", that is:

1a. An inertial coordinate system is defined as one in which every free-falling particle has constant coordinate velocity.

I would then generalise (1b) as

1b. Every law of physics takes the same form in every inertial coordinate system.

(In fact, that isn't technically quite good enough, because it makes no distinction between orthogonal and skew coordinates, so a bit more is needed to resolve that ambiguity. There is also an implicit assumption that inertial coordinate systems do actually exist, otherwise (1b) would be trivially true on an empty set of coordinate systems.)

Einstein's first postulate essentially extends the principle of relativity from Newtonian mechanics (where it was already known to be true -- "Galilean relativity") to all branches of physics.

Although (2) doesn't explicitly mention coordinates, velocity is implicitly defined in terms of coordinates. It can be interpreted as valid in any single frame where V might vary from one frame to another. Subsequent application of (1) would imply all frames have the same V.

 

[Phrak]

To those who have been arguing the point, I fail to see how the exclusion of postulate 2, specifying a finite invariant velocity, would obtain anything but Galilean relativity.

 

[Fredrik]

In fact, that isn't technically quite good enough, because it makes no distinction between orthogonal and skew coordinates, so a bit more is needed to resolve that ambiguity.

The funny thing is that what we need to resolve that ambiguity is something like..."a coordinate transformation from one inertial frame to another maps the light cone at the origin onto itself", but that's a statement that sounds a lot like the second postulate. It just isn't possible to define "inertial frame" in a way that's appropriate for SR without including (some version of) both of the postulates in the definition.

Good post by the way. You noticed a couple of things that I hadn't.

 

[Fredrik]

To those who have been arguing the point, I fail to see how the exclusion of postulate 2, specifying a finite invariant velocity, would obtain anything but Galilean relativity.

The first postulate mentions the equations of electrodynamics explicitly, and Maxwell's equations predict that the speed of an electromagnetic wave is c in all inertial frames. So if we completely ignore that the first postulate actually fails to define what an inertial frame is (which actually seems to be what everyone is doing), then the second is implied by the first.

If we choose to interpret the first postulate as if it doesn't actually include Maxwell's equations (i.e. if we interpret "electrodynamics" as an unspecified theory of electricity and magnetism), then we still don't get Galilean relativity. To get Galilean relativity, we must at the very least postulate that the invariant speed is infinite.

 

[xantox]

To those who have been arguing the point, I fail to see how the exclusion of postulate 2, specifying a finite invariant velocity, would obtain anything but Galilean relativity.

One-postulate derivations such as Levy-Leblond's cited earlier will give a generic transformation law, where depending on the value of some undefined parameter you may obtain both Lorentzian, Galilean and Euclidean transformations.

 

[Phrak]

The first postulate mentions the equations of electrodynamics explicitly, and Maxwell's equations predict that the speed of an electromagnetic wave is c in all inertial frames. So if we completely ignore that the first postulate actually fails to define what an inertial frame is (which actually seems to be what everyone is doing), then the second is implied by the first.

If we choose to interpret the first postulate as if it doesn't actually include Maxwell's equations (i.e. if we interpret "electrodynamics" as an unspecified theory of electricity and magnetism), then we still don't get Galilean relativity. To get Galilean relativity, we must at the very least postulate that the invariant speed is infinite.

Then this is all historical nit-picking over axioms better stated. We could easily have Galilean relativity, compatible with Maxwell's equations within an aether where c is the velocity in the aether. But it Einstein relativity without a finite nonzero invariant velocity.