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It is often said that special relativity is more general than Newtonian mechanics. Is there any precise meaning of what is meant by more "general"?

I would consider a theory A more general than a theory B if the axioms of B can be considered theorems of A. Is that a proper definition?

Consider special relativity vs Newtonian mechanics. We can state that:

For speeds below c, Newtonian mechanics hold.

But this means that Newton's axioms hold, which isn't the case, since the theorem "Newtonian mechanics hold" (i.e. its axioms are true) doesn't hold.

If we had constraint Newtonian mechanics to describe motion for speeds below c, then we could consider special relative as more general (based on the above definition).

Is there a precise meaning of what a more general theory means in physics?

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  • All speeds are below c per relativity. What Newton's laws describe is all speeds where relativistic effects are negligible.
    – Mary
    Apr 17, 2022 at 1:50
  • I don't think one can define "more general" just by comparing the mathematical structure of the two theories--as @JoWehler said the notion depends on experimental results. One could imagine a hypothetical universe where Newtonian gravity described gravitational interactions more accurately than general relativity, so if the inhabitants of that universe discovered general relativity first and Newtonian gravity second, they could call Newtonian gravity "more general" than general relativity.
    – Hypnosifl
    Apr 17, 2022 at 19:41

2 Answers 2

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In physics theory B is more general than theory A, if B explains all results which A explains and some additional results.

According to this definition Special Relativity is more general than Newton’s mechanics, and General Relativity is more general than Special Relativity.

I would not base the definition of ‘more general‘ on comparing or proving axioms. In general, physical theories are not axiomatized like mathematical theories. Nevertheless the mathematical framework of quantum mechanics can be understood as an axiomatization of quantum mechanics. Then the Copenhagen interpretation can be considered one interpretation of these axioms.

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  • Should "some additional results" be interpreted with respect to theory B? Also, why physical theories are not axiomatized? Isn't general relativity or Newtonian mechanics axiomatic? Or do you mean in the "form" of axiomatization?
    – Anton
    Apr 16, 2022 at 21:40
  • @Anton I mean additional results in the same domain of investigation. - Axiomatization is an indicator for mature science. Axiomatization is not done on the front-line of research. - Even classical electrodynamics is not axiomatized, though we have the Maxwell equations. - String theory or quantum loop gravity are far away from being axiomatized. Scientists would be happy about some fundamental equations with possible solutions.
    – Jo Wehler
    Apr 16, 2022 at 21:50
  • In physics theory B is more general than theory A, if B explains all results which A explains and some additional results. This is not necessarily the case. The quantum field theory is more general than quantum mechanics but can't explain things QM can.
    – Pathfinder
    Apr 17, 2022 at 8:04
  • @Felicia The Wikipedia entry on quantum field theory starts "quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity and quantum mechanics." Of course Wikipedia can be wrong. But why do you not agree?
    – Jo Wehler
    Apr 17, 2022 at 11:31
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    @Felicia What you say is simply not true in general. Sure, if you are treating scattering theory in QFT, then you consider asymptotically free fields (not necessarily just two — I don't know why you specify that number). But QFT is a much more general framework that may be used to study strongly-coupled fields beyond perturbation theory. One can also treat bound states in QFT. And again, QFT obeys all the rules of ordinary quantum mechanics; quantum field theories are strictly a subset of all quantum mechanical theories.
    – d_b
    Apr 17, 2022 at 20:00
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Further to @JoWehler's answer, the more general theory recovers the less general one in the latter's regime of applicability. For example, you can show each equation in special relativity behaves, for speeds much smaller than the in vacuo speed of light, like its Newtonian counterpart. (If MathJax worked as well here as it does on some other SE sites, I'd work through an example or two herein.) In this same sense, general relativity is, as the name hints, more general than special relativity.

In some cases, a "more general" theory might not have been shown empirically necessary, so it's worth illustrating how we can apply the above criterion to such cases. For example, here is one generalization of GR, which introduces a function that can easily be chosen to just give GR again. For now, we know of no empirical reason not to just stick with GR; but if we ever did, and this generalization was adopted as more suitable, there would need to be a regime in which its results reduced to GR, and thereby explained why GR had been so successful until now.

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