SR in 3D Geometric Algebra: Frames and Rotors
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- Опубліковано 28 тра 2024
- This, the second video in the series, goes over the basic rotors and how they're related to frames of reference.
Wikipedia page: en.wikipedia.org/wiki/Algebra...
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Correction: The definition of a boost and Lorentz transformation should NOT have a negative exponent, and so
' Ω = w + B ' for ' |B| = -θ ' and ' |w| = φ '. I'll talk about this in the next video.
Man you are awesome!
Thanks dude, you made my day!
At the end, you said that since HR ≠ RH we have that phi v + B ≠ B + phi. But I think addition is indeed always commutative but the rule for exponential are just a bit different in a non commutative algebra.
For two multivectors g1 and g2, under the assumption that g1 and g2 commute (i.e if g1 * g2 = g2 * g1) then exp(g1 + g2) = exp(g1) * exp(g2) = exp(g2) * exp(g1)
L = exp(-1/2 Omega) Omega = phi v + B
L = HR where H is a boost so H = exp(phi' v') for some other phi' and v'
and where L = exp(B') for some other B'
But phi' v' and B' don't commute so we have that exp(phi v + B) = L = H * R = exp(phi' v') * exp(B') ≠ exp(phi v + B)
so phi' ≠ phi or v' ≠ v or B' ≠ B.
Great video I loved it and I'm hyped to watch what's coming !
Glad you liked the video!
That's a good point. I'd been thinking along the lines of the addition within the exponential as being different than the normal addition outside of an exponential. So while they use the same + symbol, being within the exponential implies non-general commutativity additionwise. But now that I look at the definition of a Lie algebra I'm thinking I should've written as you said: exp(g1 + g2) =/= exp(g2 + g1).
Well, at least the audience will understand the concept I'm trying to convey!
The last STA paper I read used the word "paravector" to refer to trivectors in the STA. In APS, I suppose you can sort of fake the XYT, YZT, and ZXT components since you're calling the T basis 1, but it doesn't look like that's what you're doing. I'm sure it's one of the many GA notation inconsistencies out there. What book or paper are you working out of?
You're actually noticing a very important pattern! The APS, the Algebra I'm using, is isomorphic to the even subalgebra of the STA. The section you read is likely on Spacetime Splits, which is a method within the STA and is related to the APS. So it's actually not an inconsistency, but that author was likely referring to the isomorphism between the STA and the APS!
Also, I had included the APS wiki page in the description for, I'll call it, the aggressively curious! The citations and references there go back to Baylis' papers.
What motivated you to come at this without STA? It's like just teaching the equations of SR and removing the part that makes them make sense. Like, you're not ever able to derive any of this stuff, you have to just assert it. What audience is this going to help? It's not me, I don't get it.
The APS is isomorphic to the even subalgebra of the STA. And as for deriving it, I did do so. But I did that using commonly known facts about GA in general, but I applied them in this context. Read up on Baylis' paravector approach to relativity.
As for being more intuitive, I'd argue the average person would find the APS more intuitive that the STA because it works in 3D space, modeling time as a scalar not a vector.
Overall, the STA is better. But it's not the best choice for someone who doesn't need all the bells and whistles of the STA. The audience is then intended for those familiar with the APS, and I hope to show them that the APS can also be used for relativistic problems. Anything presented poorly is more a reflection of my skills in teaching, not the worthiness of the algebra itself.
Space/time symmetries (rotations, boosts) are dual to Mobius maps -- stereographic projection.
It is an equivalent or dual description of space/time.
Space is dual to time -- Einstein.
"Always two there are" -- Yoda.
Sense is dual to nonsense.
@@EccentricTuber What I mean is that, just by deriving the basis you need to describe events in the STA, the entirety of SR can be derived trivially. In the APS, you need to make a lot of assertions that actually come from the STA. For example, why are scalars in APS treated effectively as quantities of time? What motivates the use of the sigma vectors as rotors if they're just vectors when all other rotors are formed from exponentiating bivectors? There are no mysteries like that in the STA. Everything makes perfect sense and couldn't be any other way.
@@hyperduality2838 You have to move on and learn other things. Duality is not the only concept. It's a coincidence that many systems can be described in (sets of) binary distinctions. Compliments and inverses are a common feature necessitated by logic, but it doesn't mean anything deep about nature. It only means that mathematics is logical.
Indeed, not all mathematical operations can be seen as dualities. Projections, ideals, idempotents, modular rings, and countless others have relationships that prohibit a one-to-one mapping that could be called a duality. By obsessively relating everything you learn back to duality, no matter how forced the comparison, you are robbing yourself of the depth of understanding you obviously could have of math and science if you just stopped.
One-to-one relationships exist, but they are not special. They don't underlie the universe. Please think about opening your mind to other concepts. I'm not calling you names, I'm not intending to embarrass or demean you, I'm trying to help you take your interest and intelligence in the right direction. Think about it.