Einstein simultaneity: just a convention?

[dx]

I can certainly evaluate the usefulness of the guess, and am called on to do so all the time.

You cannot evaluate the usefulness. You can only evaluate the degree to which you believe in it. Whether the guess is close enough to reality to be useful or not cannot be evaluated before the experiment. You may have a lot of evidence to support your claim that "It is unlikely to be useful", but that is also a guess based on information. You are only assigning a degree to your belief.

 

[Ken G]

i never said otherwise.

OK, just making sure.

 

[Ken G]

You may have a lot of evidence to support your claim that "It is unlikely to be useful", but that is also a guess based on information. You are only assigning a degree to your belief.

Yes, except I would replace your "guess" with "useful prediction", and remove your "only".

 

[Hurkyl]

In other words, who needs such a criterion more,

Huh? "Needing a criterion more"? What the heck are you talking about?

(which is what I presume you mean by "wave nature", because it was obvious that light exhibits wave properties)

"Obvious"? What does 'obvious' mean? Is it anything other than an expression of the biases in how we interpret what we see?

that light is fundamentally a wave and not a particle

(Classical) waves, of course, cannot be (classical) particles.

My criterion for judging that is the absence of a justification for concluding the inverse.

So how do you justify dismissing all of the evidence for light being a wave? Keep in mind that your argument is heavily dependant upon you being able to devise a justification that is not also applicable to things like the obviousness of light exhibiting wave properties, or whether or not the sun will rise tomorrow.

Pick any word you like

Fine; I pick the phrase "caused by invisible pink unicorns". So are you seriously criticizing classical mechanics because it doesn't permit gravity to be caused by invisible pink unicorns?

 

[Ken G]

Huh? "Needing a criterion more"? What the heck are you talking about?

Simple enough. You are claiming I need a criterion to claim that physicists should not have, for example, used the success of Newton's laws as evidence that the universe was fundamentally deterministic. I am saying, no, I do not need such a criterion-- not nearly as much as physicist need a criterion to make that undemonstrated leap of faith. Ergo, it is your position that "needs a criterion more", so it is you who need to explain said criterion.

"Obvious"? What does 'obvious' mean? Is it anything other than an expression of the biases in how we interpret what we see?

Obvious means that if you use wave mechanics successfully to understand some aspects of light behavior, then wave mechanics can be used successfully to understand some of light's behavior. That's obvious, yes, and that is just what I said. The point is, if that's all you mean by "wave nature of light", then it was obvious. So I presume you instead mean, "light is fundamentally a wave, in contrast to, say, a particle". If you do not explain what you mean by phrases like "wave nature", it forces me to fill in the blanks in your argument, and I am merely explaining how I'm doing that.

(Classical) waves, of course, cannot be (classical) particles.

This is a definition of the word "classical", I hardly see what this tells us about the concepts of waves and particles.

So how do you justify dismissing all of the evidence for light being a wave?

Your own words belie your argument. "Light being a wave"? What does that mean? Light is light, last I checked.

Keep in mind that your argument is heavily dependant upon you being able to devise a justification that is not also applicable to things like the obviousness of light exhibiting wave properties, or whether or not the sun will rise tomorrow.

I see no challenge there at all. Yes, we can analyze much of the behavior of light using wave mechanics. That's what we do in physics, we make mathematical idealizations and use them to create models that describe certain aspects of what we observe. The entire leap that I am arguing against is that we even use language like "ilght being a wave". Science has no idea how to define the meaning of that sentence, yet we persist in using it, like it said something more than "we successfully apply wave models to unify the results of the following list of experiments, which we assume extends to analogous situations but we have no idea how hard we can push it into new regimes." Now, that didn't hurt, did it?

Fine; I pick the phrase "caused by invisible pink unicorns". So are you seriously criticizing classical mechanics because it doesn't permit gravity to be caused by invisible pink unicorns?

If you see a parallel between the word "mediate" and the words "caused by invisible pink unicorns", then I suppose you can go ahead and imagine that if you like. As for my (and Newton's own) criticism of classical mechanics, it had nothing to do with the agent we ascribe to action at a distance, it had merely to do with the necessity that the agent act instantaneously with no description of how the influence got from point A to point B. It is the issue of causality that is at stake here, a deeply held principle given its vast unifying potential, even before the advent of relativity.

 

[Hurkyl]

Simple enough. You are claiming I need a criterion to claim that physicists should not have, for example, used the success of Newton's laws as evidence that the universe was fundamentally deterministic.

Well, yeah. That's how rational discussion works -- rather than expect us to accept all of your claims as a matter of faith, you attempt to provide rational justification for them. And since you are apparently denying empiricism, one of the cornerstones of the scientific process, your justification needs to be pretty darn good.

I see no challenge there at all. ... "we successfully apply wave models to unify the results of the following list of experiments, which we assume extends to analogous situations but we have no idea how hard we can push it into new regimes." Now, that didn't hurt, did it?

You lose; your form of argument also denies the claim "the sun will rise tomorrow".

If you see a parallel between the word "mediate" and the words "caused by invisible pink unicorns", then I suppose you can go ahead and imagine that if you like.

Please clarify -- do you criticize classical mechanics on the grounds that it does not allow for the possibility that all observed phenomena are simply the actions of invisible pink unicorns?

If you do make this criticism, then do you really expect me to take you seriously?
If you do not make this criticism, then why not?

As for my (and Newton's own) criticism of classical mechanics, it had nothing to do with the agent we ascribe to action at a distance, it had merely to do with the necessity that the agent act instantaneously with no description of how the influence got from point A to point B.

You assume, a priori, that influences have to "get" from one point to another. You also assume, a priori, that the process is not 'fundamental', but instead can be described in terms of some other processes that you find more palatable.

Why should I make such a prior assumptions? More importantly, why should a scientist?

It is the issue of causality that is at stake here, a deeply held principle given its vast unifying potential, even before the advent of relativity.

Causality has nothing to do with whether or not 'action at a distance' is possible. (That is a consequence of the principles of special relativity, but not something inherent in the notion of causality itself)

 

[Fredrik]

I'm curious about how people here view Einstein's prescription for determining simultaneity in an inertial frame, and...

I don't really understand what it is that you're concerned about, or what sort of answer you're looking for, but I'm going to explain how I think about these things:

SR is just the claim that space and time can be represented mathematically by Minkowski space. That's the one and only "axiom" of the theory. The Minkowski metric has a non-trivial group of isometries called the Poincaré group. We call the members of the Poincaré group Lorentz transformations, or inertial frames. (They are diffeomorphisms that map the manifold onto itself, and since the manifold is R⁴, they are also global coordinate systems on the manifold).

Pick any two events that are space-like separated. There's always an inertial frame x that assigns the same time coordinate to those two events. The two events are said to be simultaneous in any such frame.

You asked if the invariance of the speed of light is a law of nature. The word "law" usually refers to some small part of a theory that can be expressed in one sentence or one equation. In this case we're definitely talking about a part of a theory that can be expressed succinctly, so I think the world "law" is appropriate.

 

[Ken G]

Well, yeah. That's how rational discussion works -- rather than expect us to accept all of your claims as a matter of faith, you attempt to provide rational justification for them.

I fear you are rather missing the definition of "skepticism". Let me try another approach. When Newton's laws were discovered, what would you describe as the basic for the subsequent prevailing view by scientists that the universe was fundamentally deterministic? On what basis was that view held? Finally, on what basis can it be challenged? You see, it is you who need to answer these questions, for mine are quite simple-- there was no sufficient basis, and the basis can be challenged from any angle you choose-- it must meet all challenges. Did it?

And since you are apparently denying empiricism, one of the cornerstones of the scientific process, your justification needs to be pretty darn good.

I have no idea what you mean by the word "deny" in that sentence, but I certainly did not deny the usefulness or importance of empiricism in science. Indeed, all I did is require that we keep track of what empiricism is really about-- and what it is not about (to wit, it is not about forming philosophical expectations about the fundamental nature of the universe outside of what we have ever actually tested, that would have to be counted as a tremendous miscarriage of empiricism-- indeed that is pretty much my whole point).

You lose; your form of argument also denies the claim "the sun will rise tomorrow".

Only if you are following it badly. Perhaps my statement above has clarified that empirical statements about what we have found to be true already are valid ways of predicting what will continue to be true, insofar as past testing is relevant to future similar tests (a core assumption of all science). I thought I had been quite clear on that when I distinguished "usefully reliable predictions" made "inside the box" of past experience from philosophical expectations "outside the box"-- the latter being an example of mistaking a simple hypothesis for a reliable prediction.

Please clarify -- do you criticize classical mechanics on the grounds that it does not allow for the possibility that all observed phenomena are simply the actions of invisible pink unicorns?

Classical mechanics certainly does "allow" that possibility, why would you claim it doesn't? All it claims is that whatever is the agent, your unicorns or my "mediators", they act at a distance and instantaneously. That's all it requires, but of course this is a limited model that is not going to stand up as a fundamental truth of reality, even Newton suspected that strongly. There was never any reason to think it would stand up, that's my point, and the fact that we still use action at a distance even after we know it doesn't stand up proves that this aspect never even needed to stand up as a fundamental truth. Ergo, it was never a necessary part of the theory, this is the point.

Indeed, I have not criticized classical physics at all, it's a lovely model for what it's good at. I criticized the foolish tendency to think any theory is something more than what it is, so that we get "surprised" when we find out that the philosophical scaffolding we imagined was supporting that theory turns out to not be a required element of the theory's usefulness. That's been the theme of the whole thread, in fact.

If you do make this criticism, then do you really expect me to take you seriously?
If you do not make this criticism, then why not?

A good theory does not include any extraneous and unnecessary elements, like pink unicorns, until they are needed. Nothing I've said in the least contradicts that. What I've said is actually quite the opposite-- that once you have a useful theory that unifies some set of observations, you should never mistake any optional conveniences associated with the application of the theory for a demonstrated part of the theory itself. Instead, you should endeavor to find the minimum theory, that makes the fewest possible assumptions, yet achieves the same unification of the data. Obviously including pink unicorns in classical physics doesn't do that-- and neither does assuming that the universe is fundamentally deterministic. That's what I'm saying, your objections are to things I have not said nor would.

You assume, a priori, that influences have to "get" from one point to another.

On the contrary, I make no such assumption, indeed I've no doubt that picture will suffer from limitations as well. All I have ever said is that simply because classical mechanics met with success by modeling forces as actions at a distance, it gives us no reason to conclude that the influences don't have to get from one point to another. I would also take the same attitude toward any particular model for how the influences get from one point to another-- this is the purpose of skepticism. We need to keep track of what is really necessary in a theory-- and none of Newton's theories required anything beyond a very rapid, possibly instantaneous, propagation of influences (using propagation loosely, as I said specific ideas about what that means will also just be models). When one has a wide range of possibilities like that, it is of course natural to expect that more careful scrutiny will yield a more generic answer than an extreme limiting case. So it is not a surprise that this is just what we find.

You also assume, a priori, that the process is not 'fundamental', but instead can be described in terms of some other processes that you find more palatable.

You are mistaken to think that this is an issue of "palatability". That would be subjective. What I am talking about is simply being true to the ways we define what constitutes the action of science, and the goals of that action-- nothing more. I find it odd that you think I'm breaking from empiricism-- my goal is to restore it from how badly we tend to stray when we become too enamored of our own idealizations. The lure of the "warm fuzzy feeling" of understanding is indeed great, but science is about achieving it without fooling ourselves in the process, not tripping over ourselves in our rush to achieve it by fooling ourselves. As I said before, there are other ways to achieve the latter which require far less educational investment than science.

Why should I make such a prior assumptions? More importantly, why should a scientist?

You, and they, shouldn't, if the data don't require it-- do you not see that this is the whole point?

Causality has nothing to do with whether or not 'action at a distance' is possible.

I can't agree, but that's another discussion entirely.

 

[Ken G]

SR is just the claim that space and time can be represented mathematically by Minkowski space. That's the one and only "axiom" of the theory.

I tend to agree with you, but then the question of the thread becomes, why is that not what you will find in virtually any physics textbook? Instead, you will find two postulates, the first being that the laws of physics are the same in any inertial frame, and the second that the speed of light is constant. Also, you find the Einstein simultaneity convention tacked on as the means for deciding what speed means. So it's really three axioms, or two axioms and a coordinate definition, depending on how you count it. That doesn't sound like your one axiom to me.

The Minkowski metric has a non-trivial group of isometries called the Poincaré group. We call the members of the Poincaré group Lorentz transformations, or inertial frames. (They are diffeomorphisms that map the manifold onto itself, and since the manifold is R⁴, they are also global coordinate systems on the manifold).

Yes, this is also the crucial elements of the picture emerging from my discussion with Hurkyl. Mathematics is often a good way to really see what are the core elements of a theory, and what are unknown philosophical statements. Which brings us to...

You asked if the invariance of the speed of light is a law of nature. The word "law" usually refers to some small part of a theory that can be expressed in one sentence or one equation. In this case we're definitely talking about a part of a theory that can be expressed succinctly, so I think the world "law" is appropriate.

So what part of your "one axiom" requires this "law"? You see, the question is not whether or not this succinct statement is normally used in SR, we know that it is. The question is, is it a necessary part of SR, such that the observations that support SR support this "law"? Or is it purely a statement of philosophical preference, engendering a non-generic aspect to the theory simply because we don't know we can't, not because we need to (examples of that in the past include geocentrism and determinism, to name just a few)? I'm arguing that attaching such non-generic philosophical preferences to our theories is the best way to get "surprised" down the road, leading to yet another unnecessary "revolution".

 

[Fredrik]

I tend to agree with you, but then the question of the thread becomes, why is that not what you will find in virtually any physics textbook?

I suspect that some authors just don't know any better, and that the others are choosing the traditional path because it's easier (for them).

Instead, you will find two postulates, the first being that the laws of physics are the same in any inertial frame, and the second that the speed of light is constant.

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!

So it really doesn't make any sense to think of them as axioms. They should be thought of as items on a check list. Once we have learned about the two "postulates", our next task is to find a theory of space and time that implies some version of them. What the "postulates" are really saying, is that we're only going to consider theories that do.

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.

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).

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.

So what part of your "one axiom" requires this "law"? You see, the question is not whether or not this succinct statement is normally used in SR, we know that it is. The question is, is it a necessary part of SR, such that the observations that support SR support this "law"?

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.

 

[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".