Special Relativity and Hyperbolic Numbers
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- Опубликовано: 26 мар 2023
- A brief introduction to Hyperbolic Numbers and their application to SR.
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Notes: www.mediafire.com/file/w98s0l...
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Nice, nice and nice! Hyper nice!
Thank you, great video. It is not so easy to find a good explanation regarding the fact that spacetime interval is time minus space coordinate.
Glad I could be of help :)
@@EccentricTuber Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Sinh is dual to Cosh -- hyperbolic functions.
Space/time symmetries are dual to Mobius maps -- stereographic projection.
Space is dual to time -- Einstein.
"Always two there are" -- Yoda.
Your channel has an interesting transition, but anyway. This was a cool find. Appreciate the time spent and the upload.
It seems like these Hyperbolic Numbers are more fundamental than the complex numbers, especially considering that seemingly arbitrary boundary conditions of 0 and 2 pi radians the latter have. (How do we know physically if we’ve gone “all the way around” a perfect center point?). But with hyperbolic numbers, you can go one-way to infinity, which seems more natural (look off into the distance, and there is a vanishing point at infinity). Why then do complex numbers and analysis have so much more attention?
In reality I think that they're equally fundamental. There's another algebra called Dual numbers, where there is a "dual unit" that squares to zero.
In general, a number that has a scalar + a scalar multiple of some unit e, the number α + βe is a Study Number. When e^2 = -1, then they're complex numbers. When e^2 = 0, they're dual numbers. When e^2 = +1, they're hyperbolic numbers.
But complex numbers are more readily studied because they're more useful in an everyday context than hyperbolic numbers.
As for " (How do we know physically if we’ve gone “all the way around” a perfect center point?)" there's an easy answer: Math is just a tool! In real life, there might not be a "perfect" center. But there are many things that behave like they have such a center. Likewise, the idea of a point at infinity isn't more natural because infinity is also a concept, just like the idea of a "perfect" center.
@@EccentricTuber thank you, I had no idea about Study or Dual numbers! I read a relatively recent article somewhere saying that they proved quantum mechanics "needs" complex numbers... but after seeing all these alternative formulations you mention, I wonder if one could say more generally that QM just needs some sort of Study Number system? Or is there something special about the complex?
I'm still trying to understand complex numbers... is it correct that all these number system sort of 'expand' the space of positivity and negativity? E.g. on the real number line, you can only go left (neg) or right (pos), but on the complex plane at least you can go in a whole circle's worth of other rays of +/- ? Is this true of all Study Numbers including hyperbolic numbers?
nice :D
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Sinh is dual to Cosh -- hyperbolic functions.
Space/time symmetries are dual to Mobius maps -- stereographic projection.
Space is dual to time -- Einstein.
"Always two there are" -- Yoda.
Why instead of adding we multiply? I mean I can make sense c^2t^2-x^2 from Einstein light mirror experiment but I can't visualize it on hyperbola like Phytagoras on Circle
If you could be more specific in your question I'd be happy to answer it!
I meant in complex numbers modulus is x^2+y^2 why in hyperbolic numbers it is x^2-y^2 I was asking. But it is because of general equation of hyperbolaa right?@@EccentricTuber
Sorry I didn't reply sooner. Sometimes RUclips notifications are bad.@@mustafaercumen3187 Yeah, you can think of the modulus being generated by the fact we need a hyperbola for Minkowski space. The process that I used in my video, which produces the same result, started with using the hyperbolic unit to generate the modulus then look at the hyperbola it creates!
too much title for a 6 minutes video
lol I'll keep that in mind