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In discussing relativity with a (somewhat mathematical) friend the other day, I ran into a problem showing why special and/or general relativity could be considered as exact descriptions of reality rather than just approximations that are working okay so far. Here's his argument:

Special relativity applies to a situation where there is no curvature of spacetime. Since both matter and energy cause curvature, the only situation where special relativity would apply would be one which has no matter and energy, and therefore one which is not achievable in any experiment. And even as a thought experiment, a clock has to have mass and energy and so there can be no exact predictions in such a theory.

He went on to say that therefore the general theory of relativity rests on a very shaky foundation. This is as compared, for example, to thermodynamics which rests comfortably on statistical mechanics and quantum mechanics.

A related question: Special Relativistic approximation to GR

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Can't you consider that fact that SR is valid as a limiting case of vanishing curvature and inertial framing, as a "non-approximation"? If your friend is "mathematical", ask him if he rejects calculus as an exact mathematical tool.. (anyway, the term "exact description of reality" is by itself a bit philosophically shaky isn't it) –  BjornW Apr 15 '11 at 1:07
    
@Bjorn; I'm going to guess on his argument; I'll likely seem him tonight at chess club and will update if needed. We could treat SR as the limit, but that makes SR a part of mathematics rather than a part of reality; that is, it reduces SR to just a description of our mathematical approximations. It makes the argument for GR somewhat circular. By contrast, calculus is defined from the ground up. One begins with the function, the limit comes later. –  Carl Brannen Apr 15 '11 at 1:31
    
you are delving deep into philosophy here. Of course SR is just a mathematical approximation to reality. So what? Which physical theory isn't? Can you point out at least one? –  Marek Apr 15 '11 at 3:28
    
No of course not! Neither is Newtonian mechanics or GR for that matter. If you want an exact descriptions of reality you'll have to speak to Hare Krishna's or some other philosophical cults who have all the answers. Barring that I don't think your question makes much sense. –  user346 Apr 15 '11 at 4:35
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"Layman's perspective"? LOL, I won an honorable mention at the prestigious annual gravitation essay contest and published a paper: arxiv.org/abs/0907.0660 The reason a lot of people are so certain of the foundations of special and general relativity is because they don't know anything about the foundations of these subjects. That's why they're still being argued in the literature. –  Carl Brannen Feb 23 '13 at 21:57
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5 Answers 5

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From an experimentalist's point of view, data trumps theory and mathematics too :) .

Special relativity is a measured fact, within the accuracy of our measuring systems, and has not been falsified. It rests on data, as all physical theories.

Edit: Your friends

"the general theory of relativity rests on a very shaky foundation

reflects the necessity of special relativity in any theory of gravity. His argument that

the only situation where special relativity would apply would be one which has no matter and energy, and therefore one which is not achievable in any experiment

is at fault, because he is not including experimental limits and data. Any mathematical curvature is orders of magnitude smaller than our measuring instruments can measure and validate, and physical theories are judged on that.

Physical theories go mathematically as far as the data can take them. Before the observation of particles special relativity was an interesting observation of Lorenz in the Maxwell electromagnetic theory. Before the photoelectric effect classical mechanics sufficed. If ever experiments can go down to the curvatures of gravity for elementary particles, maybe special relativity will be modified to fit the data. It is theories that change to fit the data, not vice versa.

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To a mathematician, "has not been falsified" is ridiculous. Of course it cannot be falsified. It applies exactly in the context of an absence of matter and energy that one might use to falsify it. –  Carl Brannen Apr 15 '11 at 21:21
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Data trumps mathematics in the sense that if a mathematical system used in a physical theory leads to conclusions against the data, it should be discarded from the theory and a different modified mathematical formulation used. –  anna v Apr 16 '11 at 6:23
    
Great answer @annav. –  Sklivvz Dec 23 '12 at 14:37
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The general theory of relativity does not rest on special relativity, so it does not make sense to say the foundation of GR is shaky because SR is shaky. One merely requires consistency, i.e., we want general relativity to reduce to special relativity in the limit.

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Hmmm. At the moment the wikipedia article on GR's second sentence is: "General relativity generalises special relativity and Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time, or spacetime." Seems pretty clear to me. –  Carl Brannen Apr 15 '11 at 21:19
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@Carl wikipedia is not the pope on this. If general relativity as it is now formulated mathematically were not consistent with special relativity, it would be rejected (with great praise as we say in Greece). Special relativity encapsulates a huge number of experimental observations and should be a necessary lower energy flat local space limit of any complete description of gravity. Special relativity is a necessary condition, but not sufficient to lead to general relativity. –  anna v Apr 16 '11 at 7:45
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Newton's laws of motion are idealized laws as well. You can't have $0$ force acting on a particle in reality since gravity is a long range force and is present everywhere. So these laws are limiting cases as well.

SR is a good description of reality in the above sense. In the limiting case when gravity may be ignored it is a valid theory of reality.

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But a mathematician can imagine a universe in which all the gravitational forces cancel to zero. –  Carl Brannen Apr 15 '11 at 21:15
    
But this is physics, not mathematics. –  user12345 Jan 14 '13 at 22:07
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Special relativity accurately applies to experiments in a lab or reference frame that is small enough that the tidal force (exactly the same thing as spacetime curvature) in it doesn't alter the outcome of the experiment at the given precision. Search the web for: "For sufficiently small regions, the special theory of relativity is correct!!"

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In discussing relativity with a

Ok.

(somewhat mathematical) friend

Nope, he's not mathematical.

the other day, I ran into a problem showing why special and/or general relativity could be considered as exact descriptions of reality rather than just approximations that are working okay so far.

Both special and general relativity are approximations. Why? They are non-quantised theory. There is a special relativity-quantum mechanics merger called Quantum Field Theory, though, but not yet a universally accepted general relativity-quantum mechanical merger (well, actually there is, semi-classical gravity, but it isn't complete, because gravity is still considered classical, there. String theory happens to solve the problem, along with some other such QG theories, most of which have been proven wrong)

Here's his argument:

You can't argue without mathematics involvedd.

Special relativity applies to a situation where there is no curvature of spacetime.

Yes, this is right.

Since both matter and energy cause curvature,

Or simply, the stress-energy-momentum tensor, is a much better way to say "both mass (not matter) and momentum and pressure and shear stress".

the only situation where special relativity would apply would be one which has no matter and energy,

Yes, but only if you want an exact result (actually, it is still not exact, because it isn't QUANTISED)

and therefore one which is not achievable in any experiment.

How much does a coconut curve spacetime? Not much. Definitely not enough to have any significant (observable) deviation from SR.

And even as a thought experiment, a clock has to have mass and energy and so there can be no exact predictions in such a theory.

A clock is lighter than a coconut (unless it is a grandfather clock, but even that is much lighter than a coconut tree, and how much does a coconut tree curve spacetime?.

He went on to say that therefore the general theory of relativity rests on a very shaky foundation.

You eat a coconut from a coconut tree, and deduce that it is spoilt, unlike what your ancestor (Newton) told you. Also, its flesh is green in colour, you deduce.Then, you eat all the coconuts from the tree and deduce that all are spoilt, not all of them are green. Many are actually very very slightly bluish green (small moons), some are very slightly bluish green (small planets), some are slightly bluish green (planets like the earth), some are bluish green (Stars like the sun), and some are blue (black holes). But you weren't wrong that some coconuts (no-SEM-tensor reigons of spacetime) are actually green in colour.

Ok, let me do a better, more relevant example. Special Relativity is fine as long as gravity is weak. For example, if you are suspended in some reigon of spacetime, then, you are lighter than a coconut tree and you curve spacetime very little, and nothing with a large SEM-tensor is there near your so special relativity can analyse your motion very well. But if you are suspended near the sun, special relativity can't be used, or you will fall into the sun! Call Kerr and Newmann to tell you what to do!

This is as compared, for example, to thermodynamics which rests comfortably on statistical mechanics and quantum mechanics.

Since when did CLASSICAL thermodynamics rest on quantum mechanics? And general relativity improves special relativity, not rest on it. Whereas a lot (but not ALL) of thermodynamic actually uses results from statistical mechanics. I think you're discussing physics with a "non-mainstream" (which is a polite word for saying...) friend?

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