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Physicists studying the grounds of physics and some mathematicians often come to a theories which are like a general-relativity, but do not coincide with it. Often these theories contradict the real world. What should one do in order to check one of such theories? Which papers/books should one read in order to be able to understand about some mathematical theories, that "the mathematical theory is nice, but this is not our world"?

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What theories are you talking about ? You should be more specific. –  Cedric H. Nov 2 '10 at 21:03
    
Yes, it could just be me, but I'm not sure what you're asking. You could list a few theories as examples. –  Vortico Nov 2 '10 at 21:22
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Einstein's Special Theory of Relativity could be tested in less than a minute. Turn on a GPS. If the reading is within a few thousand miles accuracy, the theory holds. Without relativity, GPS systems would not work. –  Vortico Nov 2 '10 at 21:26
    
For example cosmologists study Einstein equation with a lambda-term. Using some notions of which objects are natural in Riemannian geometry (see for example page 10 of arxiv.org/abs/math/0609287) another "natural" terms can be written. I don't think, that experiment with turning on GPS checks the lambda term of Einstein equations. Actually I don't know even to which extent does it check it (because I think some adaptive algorithms are used by GPS system). –  Fiktor Nov 2 '10 at 22:19
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@Vortico: no one is going to be convinced by a correct reading on a very complex device involving software, RF electronics, satellites and so on, except the engineers involved in its design. –  DarenW Nov 16 '10 at 4:47

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Considering all the possible low-order corrections to Newtonian motion that there might be to a particle moving in a gravitation field of any kind, or rather, in comparing relative motions of neighboring particles, there are defined some paramaters according to the Parameterized Post Newtonian (PPN) formalism. This is described nicely in the famous book by Misner Thorne & Wheeler. (My other three books on GTR don't mention PPN.)

GTR can be analyzed in terms of PPN parameters - beta=1, eta=0, etc. Alternative theories work out to having different PPN values. Any experimental observations, after being calibrated and curves fitted to and so on, can be judged in terms of PPN parameters, at least some of them, or combinations of some of them. So it's a matter of seeing what fits within error bars.

http://en.wikipedia.org/wiki/Parameterized_post-Newtonian_formalism tells more detail.

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Physical theories are rarely constructed as a purely theoretical exercise - they usually are based on empirical data which is fitted a mathematical solution. The physical theory then tries to explain the data and corresponding math.

The basis of scientific theory is that it must be verified by repeatable tests and data. Sometimes theories that explain some data well are disproven when applied to other situations - and sometimes amendments are made in order to account for the difference. That is the nature of every scientific theory - it is accepted until real world data says it is wrong. There is no way to know whether any theory truly describes our world or not - it holds until it no longer matches observations.

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