Wikipedia:Reference desk/Archives/Mathematics/2019 May 7

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May 7

Are there "interesting" theorems, that only use the addition operation, and need induction for their provability?

Besides the well known "interesting" theorem, stating that every natural number - not being any sum of two identical natural numbers - is followed by a successor which is such a sum (along with analogous theorems about a sum of more than two identical natural numbers, e.g. the theorem stating that every natural number not being any sum of three identical natural numbers - is followed by a successor which is either such a sum or followed by a successor which is such a sum). 185.46.78.64 (talk) 20:28, 7 May 2019 (UTC)

The commutative law of addition comes to mind. 67.164.113.165 (talk) 19:22, 8 May 2019 (UTC)

Möbius—Lorentz group isomorphism ?

Several (usually) reliable sources have asserted an isomorphism between the Möbius group and the Lorentz group: they both have six real parameters as Lie groups and they both contain copies of SO(3). But there are parabolic transformations in the Mobius group. The Galilean transformations are parabolic on space-time, but they are not included in the Lorentz group. An editor has exposed the null rotations which should be in the Lorentz group if indeed the isomorphism held with the Mobius group. Such null rotations fail to exist as seen in coverage of Lorentz transformation. Short of a WP:reliable source calling out the absurd null rotations, is there a way to halt the perpetuation of the harmful identification of the Riemann sphere with the celestial sphere ? Details of some findings are listed at Talk: Lorentz group#Null rotations ?. — Rgdboer (talk) 21:35, 7 May 2019 (UTC)

This is more mathematical physics than mathematics proper, so some of the terminology is unfamiliar to me; maybe someone will correct me if me if I go wrong somewhere. You're right that the articles are poorly referenced in places, but the section Lorentz group#Relation to the Möbius group does construct a homomorphism from the Möbius group to the Lorentz group. The image is not the full Lorentz group but the restricted Lorentz group (the connected component containing the identity), however I think your issue is in the other direction, meaning that parabolic Möbius transformations don't seem to be represented in the Lorentz group. The section Lorentz group#Parabolic describes the Lorentz transformations corresponding parabolic and they appear to be a bit more complex than Galilean transformations (which, as you say, are not Lorentz transformations). In any case, while I don't claim to understand everything in the articles, I'm not convinced there is anything factually incorrect in them and the isomorphism does not seem to be controversial in the literature; see for example [1], [2] and [3] (section 3.7.3). --RDBury (talk) 09:00, 8 May 2019 (UTC)