The Sagnac Effect revisited
in [Relativity]
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From: Jerry on 24 Sep 2009 11:26 On Sep 24, 5:52 am, Jonah Thomas <jethom...(a)gmail.com> wrote: > Jerry <Cephalobus_alie...(a)comcast.net> wrote: > > Jonah Thomas <jethom...(a)gmail.com> wrote: > > > I have no problem with a zero dot-product, if you're careful with > > > it. > > > > You could get the result you want easily with quaternions. Multiply > > > two four-vectors the quaternion way > > > > (t1,A1)(t2,A2)=(t3,A3) and if t3=0 then you have your minkowski > > > norm=0. But A3 will not equal zero at the same time unless one of > > > those 4-vectors was all zeroes. > > > > And that system gives you easy division, too. You don't have to put > > > up with a broken multiplication. But you do it. > > > You really ought to learn to walk before attempting to run. > > The link between hyperbolic quaternions and spacetime theory > > has long been recognized. > > Yse, but hyperbolic quaternions are bad. They are not positive-definite. Jonah, for the past hundred years, quaternion enthusiasts have touted a quaternion formulation of special relativity as "the next big thing", with a notable lack of success. Meanwhile, relativity moved on. Minkowski space time became subsumed within a far richer mathematical structure in general relativity. The universe does not follow your mathematical intuitions about what is good, bad, or ugly. Planets do not orbit the Earth in perfect circles. Heavier objects do not fall faster than lighter objects. An arrow does not fly because it is loaded with impetus. You need to adapt your intuitions to what the universe tells us about how it works, not the other way around. Jerry
From: Jonah Thomas on 24 Sep 2009 12:06
Jerry <Cephalobus_alienus(a)comcast.net> wrote: > Jonah Thomas <jethom...(a)gmail.com> wrote: > > Jerry <Cephalobus_alie...(a)comcast.net> wrote: > > > Jonah Thomas <jethom...(a)gmail.com> wrote: > > > > > I have no problem with a zero dot-product, if you're careful > > > > with it. > > > > > > You could get the result you want easily with quaternions. > > > > Multiply two four-vectors the quaternion way > > > > > > (t1,A1)(t2,A2)=(t3,A3) and if t3=0 then you have your minkowski > > > > norm=0. But A3 will not equal zero at the same time unless one > > > > of those 4-vectors was all zeroes. > > > > > > And that system gives you easy division, too. You don't have to > > > > put up with a broken multiplication. But you do it. > > > > > You really ought to learn to walk before attempting to run. > > > The link between hyperbolic quaternions and spacetime theory > > > has long been recognized. > > > > Yse, but hyperbolic quaternions are bad. They are not > > positive-definite. > > Jonah, for the past hundred years, quaternion enthusiasts have > touted a quaternion formulation of special relativity as "the > next big thing", with a notable lack of success. Yes. There haven't been very many of them and perhaps they have not understood quaternions well, and not well understood the physics they wanted to explain. It's sad. Maybe there's just no way, either. > Meanwhile, relativity moved on. Minkowski space time became > subsumed within a far richer mathematical structure in general > relativity. Is it positive-definite? I like a system a lot better when you have multiplicative inverses. A*B=C A=C/B That kind of thing. It sounds like physicists have gotten by without that, but it would be really good if it was available. > The universe does not follow your mathematical intuitions about > what is good, bad, or ugly. Planets do not orbit the Earth in > perfect circles. Heavier objects do not fall faster than lighter > objects. An arrow does not fly because it is loaded with impetus. > > You need to adapt your intuitions to what the universe tells us > about how it works, not the other way around. Unfortunately, I don't know enough about it yet to tell whether the universe works without multiplicative inverses or not. How much of that comes from the universe, and how much comes from the theories about the universe that don't have to be that way? When I get it clearer which is which I'll be happier and readier to adapt my intuition to the universe. |