Linear algebra in hyperbolic space: Introduction to Gyrovector spaces

[@UliRaudales]

A you tube channel about coding, advance mathematics and physics. I AM 100% IN!!!!

[@Philomatha]

:)

[@angelmendez-rivera351]

I appreciate that you are attempting to give more exposition to gyrovector spaces and gyrogroups. However, there are several problems with this video, which are bound to mislead viewers not familiar with the concepts. 0. There is nothing inherently Euclidean about the mathematical structure we call a vector space. A vector space over a finite field is not a Euclidean vector space. An infinite dimensional vector space over the complex numbers is not a Euclidean vector space. A Euclidean vector space is specifically a vector space over the real nunbers, with an inner product on the space. 1. While it is true that gyrovector spaces can be used to study hyperbolic geometry, vector spaces also enable the study of hyperbolic geometry via Lorentzian and ultra-Lorentzian manifolds. 2. Vector spaces are still the more natural setting for studying the special theory of relativity. The vector space is R^1,3 and the defining feature is that we use not an inner product, but a generalization, a nondegenerate indefinite symmetric bilinear form, where (e0) • (e0) = 1, but (e1) • (e1) = -1, and similarly for e2 and 3, where e0 is the timelike basis vector and e1, e2, and e3 are the spacelike basis vectors. As for velocity addition, we forego the concept of velocity altogether and instead work with a quantity called the rapidity ζ, defined as artanh(v/c). Rapidities do form a vector space, and add in the intuitive way one would expect. You can use gyrovector spaces, and there is nothing wrong with this, but very few physicists do this in practice, and there are many reasons why this is the case. 3. There is no such a thing as a Lorentz acceleration. The transformations you were talking about are Lorentz boosts, for which the velocity is constant, so the acceleration is indeed the zero vector. Therefore, they are not actually accelerations. The Thomas rotation is not a Lorentz boost, but it is still a Lorentz transformation, as all rotations are Lorentz transformations. Lorentz transformations are generated by the Lorentz group, and this includes not only boosts, but rotations as well. In fact, a Lorentz boost is just a hyperbolic rotation, a rotation in the e0e1-plane, e0e2-plane, or e0e3-plane.

[@Philomatha]

0) You're right; in fact I never claimed this, I only claimed that its formalism arose from euclidean geometry, and that it can be used to study euclidean geometry. The same can be said for gyrovector spaces though; both gyrovector spaces and vector spaces are abstract mathematical structures and there is nothing inherently euclidean or hyperbolic about them. 1) Can you expand more on this, perhaps by sending some resources? Lorentzian manifolds (such as Minkowski space time) are pseudo-riemannian manifolds, whereas the models of hyperbolic geometry I mentioned in the video are musually isometric riemannian manifolds; i don't know how a geometrico-analytical setting could be set on them without gyrovector spaces, employing only vector spaces. 2) Yes I know, in fact very few special relativity textbooks mention gyrovector spaces, however it's definitely another nice tool to have in a mathematical toolbox. 3) I apologize for the mistake; it was due to a translation error, since most of my resources on special relativity are not in english. In all the instances where I said that, I meant "Lorentz boosts" like you correctly pointed out. Thanks for the comment and for watching the video.

[@rv706]

​​​@@Philomatha: As for 1), he/she probably meant that Euclidean vector spaces appear in the study of general Riemannian manifolds because the fibers of the tangent bundle have such a structure. In analogy with that, indefinite orthogonal vector spaces (by which I mean real v.s. equipped with a nondegenerate symmetric bilinear form of indefinite signature) appear as fibers of the tangent bundle of Lorentzian manifolds.

[@Tannz0rz]

These geometries can likewise be obtained via the Projective (or Plane-Based) Geometric Algebras: G(n, 0, 1) - Euclidean PGA G(n+1, 0, 0) - Elliptic PGA G(n, 1, 0) - Hyperbolic PGA In Euclidean PGA the plane at infinity squares to 0, in Elliptic PGA the plane at infinity squares to +1, and in Hyperbolic PGA the plane at infinity squares to -1. Likewise, an algebra of relativistic spacetime can be obtained via the Spacetime Algebra G(1, 3, 0), whose basis vectors reflect the Minkowski metric.

[@newwaveinfantry8362]

11:02 - Shouldn't there be a square root around 1/(1+v*u/c^2)? Otherwise, how do you get the velocity time dilation formula t = t0/sqrt(1-v^2/c^2)?

[@ARBB1]

I first learned about gyrovectors in a book about the Relativistic Thomas Precession and uses in General Relativity, always found them a little useless, but you helped me see some of their use

[@umbraemilitos]

I really appreciate this video! Thank you for making it. How would these ideas change in Elliptic, Parabolic, or more abstract spaces?

[@Philomatha]

A generalization of gyrovector spaces to Elliptic spaces (but in general to all riemannian manifolds of constant sectional curvature) exists, it is called the "K-stereographic model". andbloch.github.io/K-Stereographic-Model/

[@umbraemilitos]

@@Philomatha Thank you! Have you looked into geometric algebra? I wonder if that formalism can also generalize this.

[@Philomatha]

@@umbraemilitos This is actually a very good question. To do so, we would need a generalization of the constructions of tensor products (and thus quotients, in order to prove the existence of the latter) on gyrovector spaces. The problem here is the non associativity and non commutativity of gyrovector addition, which does not allow you to define (in straightforward way I should specify) an analogous construction of quotient of vector spaces.

[@mrl9418]

This is fascinating

[@rv706]

There might be a typo in the equation at 11:11 - One of the two u vectors appearing in the ratio inside the parentheses should be a v vector? Otherwise the ratio would be equal to 1 and hence pleonastic...

[@Philomatha]

Yes correct, thanks for pointing it out. I actually noticed there are some other typos too (near the end I forgot to include the \ for the latex symbol \otimes (which looks like the tensor product and the gyroscalar multiplication symbol) and it ended up being displayes just as otimes).

[@davidkeirsey9477]

Very helpful. Now I need to apply these insights I gained to the 27 Finite Sporadic Simple Groups and the 17+1 Finite Simple Group Families. Integration with discrete (Rational) Riemann, Weyl, Ricci geometries using Scott continuity.

[@RalphDratman]

"Let's consider them genetically" -- I think you mean generically.

[@Philomatha]

"genetically" is what I actually wanted to say. This is what I meant by "genetic" in this context: en.wikipedia.org/wiki/Genetic_method

[@RalphDratman]

@@Philomatha Of course, but this is what I thought you meant by "generic" in this context: en.wikipedia.org/wiki/Generic_point --- and I assumed you had made a typographical error.