Albert Einstein, 1950 (Scientific American):
"[re: the general principle of relativity] ... without this ... it would be practically impossible for anybody to hit on the gravitational equations, not even by using the principle of special relativity ...... This is why all attempts to obtain a deeper knowledge of the foundations of physics seem doomed to me unless the basic concepts are in accordance with general relativity from the beginning. This situation makes it difficult for us to use our empirical knowledge, however comprehensive, in looking for the fundamental concepts and relations of physics, and it forces us to apply free speculation to a much greater extent than is presently assumed by most physicists.
I do not see any reason to assume that the heuristic significance of the principle of general relativity is restricted to gravitation and that the rest of physics can be dealt with separately on the basis of special relativity, with the hope that later on the whole may be fitted consistently into a general relativistic scheme. I do not think that such an attitude, although historically understandable, can be objectively justified. The comparative smallness of what we know today as gravitational effects is not a conclusive reason for ignoring the principle of relativity in theoretical investigations of a fundamental character. In other words, I do not believe that it is justifiable to ask: what would physics look like without gravitation? "
Some people really don't like the implications of what Einstein seemed to be saying here about his own theories. Some online physics people in the past have insisted that Einstein couldn't possibly have said such a thing, but I've checked paperback reprints and the original 1950 SciAm publication and there it is, in black and white. That's what he said.
Einstein's point is entirely logical, and supported by the historical record. If we look at the background to the development of relativity theory, notably the Newtonian Catastrophe, we find that the version of general relativity that we ended up with, with special relativity providing an underlying flat-spacetime layer, was not just not inevitable, it actually seemed to depend on a rather unlikely-looking chain of historical accidents. Probablistically, it really shouldn't have happened like this.
We should have understood that gravity slowed time a century before Einstein eventually noticed, shortly after John Michell had predicted gravitational shifts way back in 1783. With the benefit of that missing piece of information, Gauss and Riemann and Clifford and friends ought to have been able to complete their curved-space projects (allowing curvature in four dimensions rather than just three) and should have been able to produce a general theory of relativity in the Nineteenth Century, before special relativity had been thought of. Einstein's special theory was partly a reaction against the aether models that dominated in the absence of a proper curvature-based description, and when Einstein went on to try derive his more ambitious general theory a few years later, he naturally wanted it to incorporate his earlier theory.
But special relativity assumed inertia without gravity, and energy-concentration without curvature. Its founding geometrical principles are fundamentally incompatible with key results that arise from the general principle of relativity. Like Einstein said, historically understandable, but not objectively justifiable with the benefit of hindsight.
You have badly mis-read Einstein's remarks in that Scientific American article. He was not suggesting that general relativity can or should be separated from special relativity, he was saying just the opposite, that they should not be separated. Quantum physicists try to separate them, by adopting special relativity, but truncating it at that point, neglecting the completion of the theory that constitutes the general theory. This is clearly what Einstein said, and what he was understood to be saying. The idea that a curved spacetime theory of gravity could exist without the Minkowski pseudo-metric is simply ridiculous, since the geodesic paths represent the longest - not the shortest - between two events. Only in the context of special relativity (Minkowski spacetime locally at each point) does this make sense, which is why no one who understands the first thing about relativity would ever make the mistake of thinking general relativity could be separated from special relativity.
ReplyDeleteAnonymous, you might have a slight mental block on this subject yourself.
ReplyDeleteSome problems have multiple solutions, with the choice of solution depending on what further auxiliary assumptions we make.
If you want an example of the basics of general relativity derived independently of SR, you could look at
Wolfgang Rindler, "General relativity before special relativity: an unconventional view of relativity theory" American Journal of Physics 62 887-8939.
At the end of Rindler's paper, he tidies things up by pointing out that of course, having derived the basics of GR, one can derive special relativity from it as a flat-spacetime limiting case.
But of course, that final reduction doesn't automatically mean a reduction to flat-spacetime physics, unless one also believes that spacetime geometry must be totally unaffected by the presence and relative motion of particulate matter, and that a flat-spacetime description of high-energy physics is actually valid. And from our experimental data (and from more general logical arguments), that seems to be an unsafe assumption.
Clifford's vision of a system in which curvature is physics is a logical counterexample to the idea that a general theory has to reduce to the physics of SR. A general theory could nominally reduce to SR over pointlike regions, but still not reduce to the physics of SR if real-world descriptions of the properties of objects can't be separated from their associated curvature effects. The spread of real-world interactions between moving bodies don't operate within a dimensionless point, and it might be that the geometry of a region containing two bodies whizzing by each other at high speed is more correctly described by a curved surface. In that case, a nominal reduction to SR at any dimensionless point on that curved surface wouldn't mean that the region containing the two bodies was flat, or that the curved geometry (and resulting physics) of that region was correctly described using special relativity - SR could be "null physics" that fails for regions larger than a point, in which "significant" physics is taking place.
You don't have to agree with that worldview, but you do have to accept that it's at the very least a logical possibility, which means that the logical reduction of any general theory to "SR physics" isn't geometrically proven until someone gets off their backside and shows that an alternative curvature-based system can't work.
The lack of that last result was part of what Einstein was pointing out as an apparent weakness in the current system – that we didn't have a full derivation that generated all the properties of current general relativity from scratch, by applying curved-spacetime principles "all the way down". We were instead stopping when we got to inertial physics and saying that special relativity obviously took over from there. But the general principle of relativity generated results that undermined the idea that SR's geometry was automatically correct, because it also allowed the idea of a Cliffordian model that wouldn't necessarily reduce to SR physics, because the geometry wouldn't be flat. If we could start with the GR-style arguments (as Rindler later did), and prove that the next step from there to special relativity was compulsory, then fair enough … but nobody had managed that by 1950, and nearly sixty years later, in 2009, the GR community still doesn't seem to have found a way to do it.
The more time that passes without theoreticians being able to fix Einstein's problem, the more likely it looks that perhaps the problem simply isn't fixable. Maybe the fundamental concepts of general relativity do operate all the way down to particle physics and quantum theory, and maybe the SR-based, Minkowskian description of physics is simply wrong.
I've made some progress exploring what the consequences might be for physics if this was the case, and I've published what I found, but so far I seem to be the only person who's tried it.
I'm afraid you’re operating under some profound misconceptions. First, you’re mistaken in thinking that no one has shown that a curvature theory of gravity must reduce to special relativity. This can be (and has been) shown quite easily. But before you can even begin to understand how this is shown, you need to understand what it means. Your comments indicate that you don’t. For example, general relativity entails the curvature of space (as well as spacetime), so when you talk about how you think it’s possible that two objects zipping past each other might be best represented in curved space - well, that is what general relativity implies, so you aren’t suggesting anything novel. Likewise when you say that perhaps special relativity applies only for infinitessimal points rather than over extended regions, this again is standard general relativity (it’s a statement of the equivalence principle!), so it’s strange that you make such pronouncements as if you are vouchsafing to us some radical ideas.
ReplyDeleteOn the other hand, you never actually address the specific question you claim to be raising, which is whether the metric of spacetime has a negative signature (i..e, whether spacetime has the Minkowski metric that characterizes special relativity). Your entire thesis amounts to the claim that it would be possible to construct a curvature-based theory of gravity with a positive metric signature. But this is known to be false, and it isn’t terribly difficult to prove it. The negative metric signature of spacetime follows unavoidably from two empirical facts: (1) gravity is an attractive (not repulsive) effect, and (2) light emitted from a region of strong gravitation is red-shifted (not blue-shifted) when viewed from far away. The only way for a curvature theory of gravity to be consistent with those two empirical facts is for the metric signature of spacetime to be negative, meaning that spacetime is Minkowskian in infinitesimal regions, and hence general relativity reduces to special relativity in infinitessimal regions (which, again, is just a statement of the equivalence principle). [continued next post]
[continued from previous]
ReplyDeleteOf course, over extended regions (assuming there is some mass-energy somewhere in the universe) special relativity is not strictly valid, because of the curvature of spacetime, but the metric signature necessarily remains negative, and the deviations from special relativity are extremely slight except in the presence of extremely intense gravitational fields, as described by general relativity. This is completely standard, not at all controversial or novel. And, again, the Minkowskian basis is proveably essential.
Now, it’s certainly possible to question the analytical proof that spacetime must be locally Minkowskian, because all proofs are based on premises, and one would be foolish to assume that we are in possession of the untimate “true” premises as to the workings of nature. Taking this skeptical attitude, we can (and do) subject special relativity to direct empirical tests, to determine whether it is exactly correct (aside from the known deviations in accord with general relativity). This amounts to testing whether local Lorentz invariance is always exactly maintained. This has been examined to incredible levels of precision in the most extreme conditions that can be produced. In every case, strict Lorentz invariance is never violated.
So, from both a theoretical and an empirical standpoint, there is hardly anything within the realm of human knowledge that is better supported than the proposition that Lorentz invariance is a fundamental symmetry of nature. Special relativity is nothing but the claim that nature is Lorentz invariant, so to argue that there is no support for belief in special relativity – without addressing all these facts – is just plain silly.
Your other claims, such as those regarding Rindler, are simply specious. Rindler’s writing is perfectly consistent with what I’ve said above, and totally contrary to what you claim. Rindler does not make the mistake of thinking a viable theory of gravity can dispense with the negative metric signature, nor does he ever claim that local Lorentz invariance is violated. The same is true for your other references to “19th century general relativity”. They are all consistent with what I’ve explained above, and they all contradict your claim. Also, to re-iterate my original point, you have grossly mis-read and mis-represented Einstein’s views, particularly as expressed in the 1950 Scientific American article. The views he expressed there are exactly opposite what you claim. He did not for one minute suggest that it would be possible to construct some new form of general relativity that was not locally Lorentz invariant. He understood full well that locally Minkowskian spacetime (i.e., special relativity) is an absolutely crucial and indispensible ingredient of general relativity. For you to suggest that he in any way advocated the contrary view is – again – just plain silly.
Anonymous ...
ReplyDeleteThe physics of two objects colliding or exchanging signals is not conducted over infinitesimally-small regions! To study how two particles interact, you need to be using a space large enough to fit in both particles and also any associated curvature. Now, imagine a hypothetical universe in which that interaction is curvature-based ... in such a universe, you could still use your argument for an inevitable reduction to Minkowski spacetime over pointlike regions, even though (in that universe) the rules of how real particles interacted wouldn't be correctly described by Minkowski spacetime and special relativity!
So the apparent mathematical inevitability of the argument doesn't have much significance when it comes to judging how real physics operates, or judging which of these two hypothetical universes - SR-based or curvature-based - we actually live in.
Certainly, general relativity handles the obvious, explicit curvature effects that aren't present in SR, but textbook GR imposes a condition on "credible" gravitational theories that they must reduce to the physics of SR.
That design decision by Einstein alters the syntax and phrasing of certain key definitions, and those subtleties are part of the reason why current GR doesn't have a description of horizon behaviour that agrees with quantum mechanics. If GR was allowed the possibility of reducing to an acoustic metric, then there's no obvious reason why a conflict with QM would have to exist.
But the GR textbooks ruled out the possibility of acoustic metrics by definition when they made SR-compliance compulsory. MTW defined a "metric theory" as being a theory that uses a metric, and reduces to the physics of SR, so if you wanted to study metric theories that didn't reduce to SR physics (but were still "relativistic" in the literal sense), you had a definitional conflict. Even if you did good work, it could be rejected on the grounds that you clearly "hadn't understood" the standard definition of a metric theory.
GR research closed itself off from the other logical possibilities and rewrote its own definitions to make its own way of implementing the GPoR compulsory, even though it wasn't obviously the only possible route if you went back to the original problem.
That's why the recent work on acoustic metric has had to be done "by the back door", by QM guys instead of by the GR guys. The GR guys can't work on these problems without breaking their subject's established defnitions, and shortcircuiting the logical system that they've already learned.
Another good example is your point about the reduction to SR being considered as a statement of the equivalence principle. That's in the textbooks, but its not automatically valid, unless we presuppose that particles have no curvature of their own, and that SR is necessarily correct.
=IF= SR is correct inertial physics, then yes, the equivalence principle will say that "explicitly-gravitational" physics reduces to "non-gravitational" physics over reasonably small regions, which'll be SR. But if SR is wrong physics, and all particles had associated curvature, then the SR-based interpretation of the equivalence principle would be wrong ... over smallish regions, "explicitly-gravitational" physics would reduce to whatever curvature-based version of local physics applied instead of SR. So you can't seriously argue that SR is correct because of the equivalence principle!
contd. ...
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ReplyDeleteBTW, I'm not claiming that Einstein reckoned that the SR equations themselves were definitely wrong ... I don't actually know how far he got with the problem after 1950. What I'm saying is that he correctly identified and described in print a potential weakness in the derivational structure of his own theory, that, if you're striving to be self-critical (as Einstein seemed to be), you'd really want to fix. If you read Einstein's biographical sketch of Isaac Newton, the thing that seemed to impress Einstein the most was Newton's ability to appreciate potential flaws in his own theories better then his contemporaries could, and I think that Einstein was trying to live up to the same ideal, and to identify potential problems in his own models even if his colleagues didn't see them as problems. But over the subsequent fifty or so years, instead of fixing the weakness that Einstein pointed out, we've been ignoring it.
Personally, I think that the reason for this omission is that we can't fix it. I don't think that it's fixable. I think that any time that anybody tries to rederive the full set of relationships for SR-based physics without some artificial cheaty step to enforce flat spacetime, they end up "missing" special relativity and winding up Somewhere Else, with the beginnings of a model that fails the current credibility tests, and then they they end up abandoning the exercise as a failure.
Personally, I find the region of theory-space that you end up in by =not= imposing flat spacetime to be pretty damned interesting. I think that if people just set aside the usual definitions for a moment, went back to basics, and followed what the geometry seemed to be trying to say, they might find themselves in some very interesting territory.
You misunderstand what it means to say that special relativity is only strictly valid for infinitessimal regions of spacetime. This does not mean that special relativity is appreciably wrong for finite regions, and it certainly doesn’t mean that the signature of the metric tensor could be non-negative. According to any empirically viable metric theory of gravity, the metric of spacetime must have a negative signature – a fact which conclusively falsifies your entire thesis. Also, in any extended region, spacetime differs from perfectly Minkowskian spacetime only to a degree that is proportional to the intrinsic curvature in that region, so even for non-zero curvature, spacetime is almost exactly Minkowskian. This is the unavoidable conclusion that was explained to you previously. Nothing you’ve said provides any legitimate justification for you to disregard these fundamental facts and continue clinging to your thoroughly falsified beliefs.
ReplyDeleteYou argue that these things have simply been defined to be true, and that if people were only more open-minded in their definitions, the way would be clear for mankind to recognize the validity of your ideas. But this is false, for the reasons that have already been explained to you in detail. Again, the spacetime of any metrical theory of gravity, if it is to be consistent with the most basic empirical facts, must have a negative signature. This is not an assumption, nor a definition, it is a profoundly meaningful physical necessity, as explained in the previous messages, and it is also abundantly corroborated by experiment. So, whether you are a rationalist or an empiricist, the conclusion is unavoidable. Nothing you’ve said even begins to address this fact, which conclusively falsifies all your stated beliefs.
You then claim that acoustic metrics enable you to evade these obvious facts, but that is nonsense. Of course, one can always postulate hidden mechanisms and substrata that mimic the phenomena, but this doesn’t alter the phenomena, so it doesn’t support your denial of the phenomena of special relativity. Your claim that general relativity and quantum mechanics are rendered mutually compatible by acoustic metrics is self-evident nonsense. Also, as clearly explained previously, you simply mis-read Einstein’s comments in the 1950 Scientific American article. He did not say anything like what you claim he said. When he said we should not try to base physics on special relativity, excluding general relativity, he was saying that we must not separate those two. It is self-evident to anyone who understands general relativity that it subsumes special relativity. This is not an assumption or a definition, it is the only way in which a viable metrical theory of gravity can be constructed, as already explained to you. Einstein was fully aware of this, and he would never have dreamed of suggesting that there was any validity to the idea that the metric of general relativity could have anything other than a negative signature. It is disgracefully dishonest of you to knowlingly mis-represent so egregiously what he said. And it is intellectually dishonest for you to continue to espouse beliefs that have been so thoroughly falsified.
Anonymous, when someone tries to defend a scientific theory by blustering and insisting that its details are "obvious" and "self-evident", then that usually means that the theory sucks.
ReplyDeleteNot always, but usually.
I think that if you look at some of the more extreme fringe guys who don't like SR, you'll find that you're using very similar language to them. They talk about the older models that they've learned and love as being "obvious" and "self-evident", too, and sometimes in their more frothy moments accuse the modern SR guys of "intellectual dishonesty".
They work themselves into a tizzy trying to defend 1880's theory against the encroachments of the Twentieth Century, I think you're doing something similar, except that you're defending 1920's theory.
I don't see a lot of difference between your position and theirs, you and they have simply fixated on and emotionally attached yourselves to different outdated models.
Read my last message carefully. Do you see the explanation of why a negative metric signature (i.e., Minkowski spacetime) is unavoidable for any empirically viable metrical theory of gravity? Do you see the explanation of your flagrant misreading the the 1950 Scientific American article? Do you see the debunking of metaphysical substrata and hidden variables? And so on... Now read your last comment. Do you see any substantive content at all? Honestly, take those two comments, yours and mine, print them out and post them, side by side, on your wall. Take a good look. Can you honestly deny that I have presented a cogent and rational debunking of your stated views? And can you honestly claim to have given any substantive defense of your views? Honestly?
ReplyDelete