Einstein simultaneity: just a convention?

[Ken G]

I actually find it quite ridiculous that these postulates are always presented as if they are mathematical axioms from which you can derive everything else, when they are in fact ill-defined. The biggest problem is that the concept of an "inertial frame" hasn't been defined in advance. I gave this some thought a few months ago, and I came to the conclusion that any reasonable definition must actually include these "postulates" in some way. This is what makes them so ill-defined. They are a part of a definition of a concept they depend on!

Yes, I'm coming to a similar conclusion, the usual description of SR is more like a "how to" recipe than an effort to understand reality at its most general level consistent with observations. I find that ironic, because the core concept of relativity is the recognition that certain concepts we tended to associate with reality, such as absolute time, are actually just the conveniences of a particular coordinatization that only work in a particular regime. When that is the message, shouldn't we be trying harder to distinguish the new conveniences we are introducing from the underlying structure that we have actually constrained?

There is of course nothing wrong with this way of finding a theory. Once the theory has been found, it can be tested in experiments, and if the experiments fail to disprove the theory, it doesn't matter how we found it. It's still a good scientific theory.

There is nothing "wrong" with Newtonian mechanics either, which is why it still gets used. It isn't exact, but no theories are intended to be exact, because they are all idealizations of some kind. What was "wrong" was thinking that if we understood Newtonian mechanics, we understood "how reality works". I caution against making the same mistake again, especially in terms of statements like "the speed of light is constant". It is part of the theory that c is a constant, for to say otherwise is to add unnecessary complexity, and it is part of the theory to say what kinds of experimental assumptions will generate a result that light propagates at that speed c. Other descriptions of the situation will not reach that conclusion, yet they can be just as valid. It seems the same to me as saying whether a Doppler shift is a stretching of a wavelength or a lagging of a frequency.

I do however have a problem with the traditional presentation, because it gives the student the impression that Einstein's postulates are sufficient to define the theory, that they are the axioms of a theory, and that all those calculations that the book and their teacher goes through is part of an actual derivation of time dilation, the Lorentz transformation, and so on, when in fact those calculations are just there to help us guess what the real axioms of the theory are (and to improve our general understanding of relativistic effects).

Yes, that seems a valid complaint to me.

I would have thought that an author who really understands this would be inclined to actually say these things, but they never do, so I sometimes wonder if any of them really understand it. Maybe the smart ones do, and just assume that this is obvious to everyone.

Actually, I suspect it is more that they fear they will confuse the reader, who will prefer a more cut-and-dried (yet misleading) approach. It is similar to how cosmology is explained, in terms of space that physically expands and so forth.

Define the velocity associated with a curve C and a point p on the curve, in an inertial frame, as the 3-vector we get by taking the spatial components of the tangent vector of C at p and dividing them by the magnitude of the temporal component. Define the speed as the magnitude of the velocity. These definitions and the properties of Minkowski space imply that the speed associated with any null line, at any point on the line, and in any inertial frame, is =1.

But if we don't restrict to the 3-vector and just use the whole 4-vector, all we are doing is defining a concept of a unit vector in that space. Then we define the spatial direction to be the direction that light moves in, so of course it becomes the 3-space unit vector. I still see definitions here, I'm not seeing where this is a physical statement. It seems to me a lot of what we are doing in SR is choosing a particular coordinatization because it is convenient, like choosing spherical coordinates to treat the electric forces from a charge. We then express the physics in terms of that convenient coordinatization, but we do it in such a way that tends to confuse the latter for the former. It's very difficult to disentagle what nature put there from what we put there, that's my issue with it.

 

[Fredrik]

Then we define the spatial direction to be the direction that light moves in, so of course it becomes the 3-space unit vector.

I'd prefer not to mention the physical phenomenon of "light" yet. The time direction is singled out by the metric, and the spatial directions are orthogonal to those, and to each other, but are otherwise arbitrary.

I'm not seeing where this is a physical statement.

The statement that space and time can be described by Minkowski space is a physical statement. When we have made that statement and made the appropriate identification of things in the mathematical model with things in the real world, the rest is mathematics.

For example, the Michelson-Morley experiment and the fact that homogeneous Lorentz transformations preserve the light-cone at the origin tell us that light in the real world must be identified with null lines in the mathematical model.

It seems to me a lot of what we are doing in SR is choosing a particular coordinatization because it is convenient, like choosing spherical coordinates to treat the electric forces from a charge. We then express the physics in terms of that convenient coordinatization, but we do it in such a way that tends to confuse the latter for the former. It's very difficult to disentagle what nature put there from what we put there, that's my issue with it.

I don't think of it quite like that. The fact that inertial frames exist (the fact that the metric admits a non-trivial group of isometries) is a physical property of space-time, at least approximately. So it's more than just a convenience. But I think I know what you mean, and I share those feelings sometimes, in particular when the subject of VSL theories comes up. I can't even make sense of what it means to have a variable speed of light. (I haven't studied that subject). We would obviously have to replace Minkowski space with something that looks a lot like Minkowski space, but isn't quite the same.

One thing that I feel is a big problem with the traditional presentation of SR is that it gives students some really strange ideas about the theory, actually about the whole concept of a "theory". For example, there are lots of physicists with Ph.D.s who believe that some of the "paradoxes" of SR can only be resolved by GR. This really is beyond bizarre, for two reasons: 1. They believe that SR contains logical contradictions, and they are OK with that! (If it did, it wouldn't be a theory, so do they really understand what a theory is?) 2. SR consists of real numbers and some functions. If that contains logical contradictions, then all of mathematics would fall with it.

 

[Ken G]

I'd prefer not to mention the physical phenomenon of "light" yet. The time direction is singled out by the metric, and the spatial directions are orthogonal to those, and to each other, but are otherwise arbitrary.

It's another interesting question, what singles out the time direction. I agree the metric says that one direction is different from the other three, but I don't think we can call it time without referencing clocks. So there's something more than just the metric there.

The statement that space and time can be described by Minkowski space is a physical statement. When we have made that statement and made the appropriate identification of things in the mathematical model with things in the real world, the rest is mathematics.

Right, but it is that "identification" that embodies a lot of the physics. Isn't it odd how oftentimes the most "physical" step of all is the one most swept under the rug!

For example, the Michelson-Morley experiment and the fact that homogeneous Lorentz transformations preserve the light-cone at the origin tell us that light in the real world must be identified with null lines in the mathematical model.

indeed, in a particularly convenient version of the mathematical model that uses the Einstein simultaneity convention to define a null line.

The fact that inertial frames exist (the fact that the metric admits a non-trivial group of isometries) is a physical property of space-time, at least approximately. So it's more than just a convenience.

I agree this is an important local property of spacetime, but special relativity, it seems to me, is constructed expressly to extend that local property to a global property. So the specialness of "inertial frames" in SR are not as local isometries (I think that's what survives into GR and appears to be the way you think about it), but as global special frames where the metric integrates trivially. That trivial metric integration is what I mean by a "convenient coordinatization", but it's just that convenience that makes global inertial frames special, not reality. The Einstein simultaneity convention is what dictates that global convenience, so we are only extracting the symmetry that we built right in-- we are finding the coordinate system where the equations simplify the most, like choosing co-rotating coordinates to study the shape of the Earth's surface.

It doesn't lead us to a contradiction, so it isn't wrong, but other coordinatizations that don't respect the symmetry are physically equivalent. It is the symmetry that is real, not the coordinates that respect it, so descriptions of that reality should reference the symmetry not the coordinates (we say the electric force goes inversely with the distance squared in any coordinates, we don't say it goes inversely as radius squared unless the coordinates are clear). As such, I don't think a "global inertial reference frame" has the physical importance SR affords it, it is just a coordinate system like co-rotating coordinates. That's generally not the way SR is taught-- we are led to think that these frames are globally real things in which particular laws of physics apply that don't apply for other observers. It's easy enough to break from that thinking, perhaps, but typically a path one has to find on one's own, as GR is normally reserved for dealing with gravity and has plenty of new issues of its own to grapple with.

But I think I know what you mean, and I share those feelings sometimes, in particular when the subject of VSL theories comes up.

Yes, my suspicion is that it would be easy to come up with a VSL theory that sounds a lot different from SR but is actually equivalent. How confusing would that be for students used to thinking that the constancy of the speed of light is a postulate of SR supported by experiment, and yet VSL theories are also sufficing? What is the key difference in a VSL theory that makes it actually different from SR? And when gravity is put in and SR becomes a purely local theory, what happens to the constancy of the speed of light postulate when you do global integrations or even just when you use nonlocal pictures to describe what is happening? I know for example that gravitational lensing can be understood as a VSL effect just like refraction except involving the "coordinate speed of light", without contradicting Einstein's relativity.

One thing that I feel is a big problem with the traditional presentation of SR is that it gives students some really strange ideas about the theory, actually about the whole concept of a "theory". For example, there are lots of physicists with Ph.D.s who believe that some of the "paradoxes" of SR can only be resolved by GR. This really is beyond bizarre, for two reasons: 1. They believe that SR contains logical contradictions, and they are OK with that! (If it did, it wouldn't be a theory, so do they really understand what a theory is?) 2. SR consists of real numbers and some functions. If that contains logical contradictions, then all of mathematics would fall with it.

I'm not sure what contradictions you are referring to, but I agree that a true contradiction (rather than an esoteric one like a nonconstructive proof of one) would be a big problem for the mathematics that underpins relativity. There's also a deeper question of what we mean by a contradiction-- one might say a "strong" contradiction is when two approaches are both correct in the theory but make different predictions, whereas a "weak" contradiction might be viewed as two observers constructing very different sounding explanations for making the same prediction. The latter is tolerated in relativity, even regarded as a natural consequence of relativity, and I used to accept it as such. Now I'm thinking that the theory should be retooled to eliminate such contradictory sounding explanations-- because when they appear, it means somebody is saying more about what is happening then they really have any right to claim given the data. We often say things like "you can think of it as..." in physics, and if it gets the right answer it doesn't seem so bad-- except when we forget to say the "you can think of it as" part. Maybe it's just better to say, "one way to think of it is this, another is this, but here's the thing we can say that all observers agree is happening that leads to the observed result". Sort of "empiricism plus the minimum theoretical interpretation needed to achieve unification".