Sunday, 6 September 2009

The Moon, considered as a Flat Disc

The Moon considered as a flat disc gives Lorentz relationships
Mathematics doesn't always translate directly to physics.
That statement might sound odd to a mathematician, but consider this: even if you believe that physics is nothing but mathematics, that makes physics a subset of mathematics ... which means that there'll be other mathematics that lies outside that subset, that doesn't correspond cleanly to real-world physical theory. The key (for a physicist) is to know which is which.

That's not to say that "beauty equals truth" isn't a good working assumption in mathematical physics – it is – the problem is that the aesthetics of the two subjects are different, and mathematical beauty doesn't necessarily correspond well to physical truth. The physicist's concept of beauty is often different to that of the mathematician.

The "beauty equals truth" idea is often used as an argument for special relativity. SR uses the Lorentz relationships, and to a mathematician, it can sometimes seem that these are such beautiful equations that a system of physics that incorporates them has to be correct.

But the Lorentz relationships can also appear in bad theories, as a consequence of rotten initial starting assumptions:
Our Moon is tidally locked to the rotation of the Earth, so that it always shows the same face to us, and we always see the same circular image, with the same mappable features. Now suppose that a 1600's mathematician has a funny turn and decides that it's so outrageously statistically improbable that the moon would just coincidentally just happen to have an orbit that results in it presenting the same face to us at all times, that something else is going on. Our hypothetical "crazy mathematician" might decide that since we always see the same disc-image of the Moon, that perhaps, (mis)applying Occam's Razor, it really IS a flat disc.

Our mathematician could start examining the features on the Moon's surface, and discover a trend whereby circular craters appear progressively more squashed towards the disc's perimeter. We'd say that this shows that we're looking at one half of a sphere, but our mathematician could analyse the shapes and come up with another explanation. It turns out that, in "disc-world" the distortion corresponds to an apparent radial coordinate-system contraction within the disc surface. For any feature placed at a distance r from the disc centre, where R is the disc radius, this radial contraction comes out as a ratio of 1 : SQRT[1 - rr/RR ] .

In other words, by treating the Moon as a flat disc, we'd have derived the equivalent of the Lorentz factor as a ruler-contraction effect! :)
Our crazy mathematician could then go on and use that Lorentz relationship as the basis of a slew of good results in group theory and so on. They could argue that local physics works the same way at all points on the disc surface, because the disc's inhabitants can't "see" their own contraction, because their own local reference-rulers are contracted, too. Our mathematician could arguably have advanced faster and made better progress by starting with a bad theory! So "bad physics" sometimes generates "good" math, and sometimes the worse the physics is, the prettier the results.

The reason for this is that, sometimes, real physics is a bit ... boring. If we screw physics up, the dancing pattern of recursive error corrections sometimes generates more fascinating structures than the more mundane results that we'd have gotten if we simply got the physics right in the first place.

Sometimes these errors are self-correcting and sometimes they aren't.
If we considered the Earth as flat, then, because it's possible to map a flat surface onto a sphere (the Riemann projection), it'd still be theoretically possible to come up with a complete description of physics that worked correctly in the context of an infinite rescaled Flat Earth. We'd lose the inverse square law for gravity, but we'd gain some truly beautiful results, that would allow, say, a lightbeam aimed parallel to one part of the surface to appear to veer away. We'd end up with a more subtle, more sophisticated concept of gravitation than we'd tend to get using more "sane" approaches, and all of those new insights would have to be correct. In fact, studying flat-Earth gravity might be a good idea! We'd eventually end up deriving a mathematical description that was functionally identical to the physics that we'd get by assuming a sphericial(ish) Earth ... it'd just take us longer. Once our description was sufficiently advanced, the decision whether to treat the Earth as "really" flat or "really" spherical would simply be a matter of convenience.

But with the "moon-disc" exercise, we don't have a 1:1 relationship between the physics and the dataset that we're working with, and as a result, although the moon-disc description gets a number of things exactly right, the model fails when we try to extend it, and we have to start applying additional layers of externally-derived theory to bring things back on track.
For instance, the "disc" description breaks down at (and towards) the Moon's apparent horizon. For the disc, the surface stops at a distance R from the centre, and there's a causal cutoff. Events beyond R can't affect the physics of the disk, because there's no more space for those events to happen in. The horizon represents an apparent causal limit to surface physics. But in real life, if the Moon was a busier place, we'd see things happening in the visible region that were the result of events beyond the horizon, and observers wandering about near our horizon would see things that occur outside our map. So if we were to use statistical mechanics to model Moon activity, and were to say that the event-density and event-pressure have to be uniform (after normalisation) at all parts of the surface, then statistical mechanics would force us to put back the missing trans-horizon signals by giving us "virtual" events whose density increased towards the horizon, and whose mathematical purpose was to restore the original event-density equilibrium. In disc-world, we'd have to say that the near-edge observer sees events in all directions, not because information was passing through (or around) the horizon, but because of the disc-world equivalent of Hawking radiation.

So in the disc description, the telltale sign that we're dealing with a bad model is that it generates over-idealised horizon behaviour that can't describe trans-horizon effects, and which needs an additional layer of statistical theory to make things right again. In the "moon-disc" model, we don't have a default agreement with statistical mechanics, and we have to assume that SM is correct, divide physics artificially into "classical" and "quantum" systems, and retrofit the difference between the two predictions back onto the bad classical model – as a separate QM effect, as the result of particle pair-production somewhere in front of the horizon limit – to explain how information seems to appear "from nowhere" just inside the visible edge of the disc.

Clearly, in the Moon-disc exercise this extreme level of retrofitting ought to tell our hypothetical crazy mathematician that things have gone too far, and suggest that the starting assumption of a flat surface was simply bad ...
... but in our physics, based on the early assumption of flat spacetime, and generating the same basic mathematical patterns, we ran into a version of exactly the same problem: Special relativity avoided the subject of signal transfer across velocity-horizons by arguing that the amount of velocity-space within the horizon was effectively infinite (you could never reach v=c), but when we added gravitational and cosmological layers to the theory, the "incompleteness problem" with SR-based physics showed up again. GR1915 horizons were too sharp and clean, and didn't allow outward flow of information, so to force the physics to obey more general rules, we had to reinvent an observable counterpart to old-fashioned transhorizon radiation as a separate quantum-mechanical effect.

So the result of this sanity-check exercise is a little humbling. We can demonstrate to our hypothetical 1600's "crazy mathematician" that the Moon is NOT flat, no matter how much pretty Lorentz math that generates, and we can use the horizon exercise to show them that their approach is incomplete. By assuming that their model is wrong, we correctly anticipate the corrections that they'd have to make from other theories in order to fix things up. That ability to predict where a theory fails and needs outside help is the mark of a superior system, and shows that the "Flat-Moon" exercise isn't just incomplete, it generates results that are physically wrong, and that don't self-correct. It's faulty physics.

But the same characteristic failure-pattern also shows up in our own system, based on special relativity. So have we made a similar mistake?

4 comments:

Anonymous said...

Your blog posting is based on several mis-understandings. First, the Lorentz transformation was not developed or adopted based on mathematical beauty, it was painstakingly derived, step by step, from empirical results and purely physical considerations. Second, it is exceedingly well known that special relativity fails because it does not account for the curvature of spacetime associated with any mass or energy, so when you confide to your readers the suspicion that perhaps special relativity doesn’t take the curvature of spacetime into account, you are not exactly making news. Third, the whole concept of anti-particles and hence “pair production” was derived by Dirac as a direct and unavoidable consequence of (wait for it) special relativity. In other words, the very phenomenon that you seem to think undermines special relativity was actually derived FROM special relativity, i.e., from reconciling quantum mechanics with special relativity, the result of which was quantum field theory. Fourth, your appending of “1915” to general relativity is misguided, because the field equations of general relativity are the same today as they were at the end of 1915, and of course flat Minkowski spacetime is still a trivial solution of those equations, and every solution approaches the Minkowski metric over sufficiently small regions. (Have you given any thought to how you are going to change this fact?) Fifth, you claim, on the basis of a patently specious staw analogy, that special relativity is “bad physics” because it leads to wrong results, and yet the result you cite is Hawking radiation, which is a direct consequence of the pair production phenomena involving anti-particles derived from (wait for it) special relativity. So, far from being an example of a failure, the phenomena of anti-matter and pair production are among the greatest triumphs of special relativity. Sixth, when you say the statement that “mathematics doesn’t always translate directly into physics” might sound odd to a mathematician, you are (again) dealing in staw. No mathematician would say that mathematics always translates directly into physics… in fact, I doubt that any mathematician would even know what you mean by that staw assertion. Seventh… well, enough of this…

ErkDemon said...

1. The Lorentz transformation was proposed by George Fitzgerald as an ad-hoc fix for aether theory, and developed independently as a more abstract logical system by Poincare. Most math/physics people I've talked to seem to reckon that the relationship is so "deep" that a theory based on it pretty much has to be right. I don't think they're seeing the bigger picture.

2. I agree that it's not news that SR fails to "do" gravity. That's what GR is for. What I'm saying is that when it comes to gravitational horizons, the embedded SR component causes models like GR1915 to "fail" gravity too. That's why crappy old Newtonian dark stars can successfully predict Hawking radiation classically but the more advanced GR1915 can't handle the effect at all. Newtonian theory, incomplete as it was, still manages to get this part right, because it has the advantage of not having to reduce to SR physics over small regions.

3. Just because pair-production was originally developed in the context of SR doesn't mean that SR "owns" the effect. Historical association is not the same as unique mathematical dependency.

4. I often refer to "GR1915", because I'd like to see Einstein's general theory replaced by an more purist implementation of a general theory that that loses the (IMO troublesome) SR layer, which I think is the root of nearly all the problems with the current system. So when I'm critical of current gravitational theory, I try to be specific and refer to "current GR" or "textbook GR" or "GR1915". "GR1915" is shorter. I used to refer to "GR1916", but someone complained that the essence of GR was published the previous calendar year and that I was being unfair by using a date as late as '16, so I backed up a year.

5. The matter-antimatter thing isn't a great triumph for SR if it's a general result (remember, N.O. has a Lorentz-squared redshift/contraction component in place of SR's single Lorentz component)

6. If nobody competent would say that mathematics always corresponds to physics – then I'd refer you to the anonymous poster here who keeps insisting that the /mathematical/ reduction of curved spacetime to flat-spacetime over sufficiently-small regions means tthat SR has to be correct physics :)

I've pointed out that this mathematical reduction isn't guaranteed to yield a physical result, but they don't seem to understand how it couldn't. Of course, they might not be a mathematician or a physicist, so perhaps they don't count … but the "reduction" argument is also in GR textbooks, which are presumably written by folk who are somewhat representative of the "mathematical physics" community. :)

Anonymous said...

1. Your reply doesn’t address the point at all. You argued that the equations of special relativity were arrived at based on aesthetic criteria of mathematical beauty. I corrected your misconception by reminding you that the Lorentz transformation was not developed or adopted based on mathematical beauty, it was painstakingly derived, step by grudging and somewhat incredulous step, from empirical results and purely physical considerations. Hence your premise is false.

2. Please stop and think for just one minute: You say “crappy old Newtonian dark stars can successfully predict Hawking radiation classically… Newtonian theory, incomplete as it was, still manages to get this part right”. What do you mean “right”? Think for a minute. Hawking radiation has obviously never been observed, so we’re not talking about an empirical fact. The reason Hawking radiation is considered to be “right” is because of Hawking’s derivation of it from a combination of quantum field theory – based crucially on special relativity for anti-particles, etc. – and the curved spacetime of general relativity – based crucially on special relativity for the equivalence principle, etc. Do you understand how insane it is to cite the prediction of Hawking radiation as evidence of a failure of special relativity? Only if you could somehow prove that there is no such thing as Hawking radiation would you be in a position to impugn special relativity. As always, your reasoning is completely backwards. Needless to say, the banal fact that analogs of Hawking radiation (not true Hawking radiation) can be found in other physical contexts, like the water in your bathtub, does nothing at all to lessen the silliness of your beliefs.

3. Your answer here makes no sense. The point is that anti-particles make sense only in the context of a spacetime with negative metric signature, i.e., locally Minkowski spacetime. Are you now claiming that anti-particles (or some analog of anti-particles) occur in the context of acoustic analogs of event horizons? I find no reference to “anti-particles” in any paper on that subject. Furthermore, you are basing your claim on “crappy old Newtonian dark stars”, so are you saying that anti-particles are operative in the dark star version of black hole radiation? And still further-more, even if you could find analogs of anti-particles in these models (which you can’t), it still would not be a defense of your original claim, because (again) genuine Hawking radiation is believed to be “right” only because it arises as a consequence of special relativity, so it is utterly silly to cite Hawking radiation as an example of the failure of special relativity. If anything, you might argue that it shows a deficiency in classical GENERAL relativity, which doesn’t mesh perfectly with quantum field theory, but quantum field theory is founded firmly and inextricably on SPECIAL relativity. The part of general relativity that doesn’t mesh with quantum field theory is curvature, the very thing you claim as a secure basis. As always, your reasoning is completely backwards.

continued next post

Anonymous said...

continued from previous post

4. Your comment here makes no sense, because you allude to “all the problems with the present system” being due to special relativity, which you think should be replaced by a fictitious version of general relativity which does not satisfy the equivalence principle (i.e., the principle that we can choose coordinates at any event so that the metric at that event is Minkowskian). In addition to being a contradiction in terms (and completely insane), we’ve just seen that exactly the opposite is the case. Special relativity as embodied fully in quantum field theory is what gets Hawking radiation “right”. The part of relativity that we haven’t been able to reconcile with quantum field theory (which gets Hawking radiation “right”, remember) is precisely the part that you tout as correct, namely, the representation of the universal coupling of gravitational interaction as curvature. THIS is intractable problem with the present system. As to “all the other problerms”, you can’t actually cite any. Look, let’s be honest. When you wrote about “those whose first exposure to special relativity prompted an immediate instinctive loathing”, you were obviously among those people. Your loathing of special relativity dates back to your youth, and is based on simple lack of understanding. All this stuff about acoustic metrics and Hawking radiation came along much later, and really has nothing to do with the source of your loathing for special relativity.

5. Your comment here is a total non-sequitur. Anti-matter is not a general result. It exists only on the context of Minkowski spacetime. See above. Your comment about how pre-relativistic optics gives the wrong redshift in some circumstances is obviously not relevant.

6. You missed the point. No one has said that special relativity (or general relativity) must be correct physics. What has been said is that the only logically self-consistent metrical representation of an attractive effect with redshift near gravitating bodies in terms of curved spacetime must be locally Minkowskian. This is a mathematical fact about the mathematical relationship between two mathematical theories, constrained to be consistent with a small set of fairly evident empirical facts. The very foundation and essence of general relativity is the equivalence principle, which is nothing other than the assertion that spacetime is locally Minkowskian. To suggest that general relativity could be reformulated in a way that is not locally Minkowskian is literally a contradiction in terms, and simply reveals that the person claiming it has not the slightest idea what he is talking about. Of course, Lorentz invariance is among the most thoroughly verified properties of all physical phenomena, but that’s separate from the mathematical fact that nothing remotely resembling general relativity is consistent with the denial of local Lorentz invariance. So, as noted previously, both from a rational and from an empirical standpoint, your claims are absurd.