QED Prerequisites 27: A Lorentz Boost using the Spacetime Algebra

[@BboyKeny]

Wow 😲 welcome back! I'm very happy I had my notifications on for this channel 😄

[@JimRyan-e4h]

Thanks for continuing, Xyly!!! I'm still way back on some earlier videos and your GR stuff, but I'll get here.

[@jurgenanklam3226]

I‘m glad you continue this series!

[@XylyXylyX]

Thank u for saying so. It is very encouraging to read. ❤️

[@AMADEOSAM]

Welcome back …

[@alecchalmers3744]

Good to see you survived the pandemic.

[@Leo-Vacher]

Amazing! Thank you so much for pursuing this topic!

[@XylyXylyX]

My pleasure

[@linuxp00]

Welcome back, though, I haven't even started properly with your first lessons

[@XylyXylyX]

Thank you for your patience.

[@sahhaf1234]

wellcome back.

[@davidhand9721]

Thank you!

[@davidhand9721]

There is a way to do a boost without even knowing how to exponentiate v or doing any hyperbolic sines because we know both vectors that go into the bivector. That is, the bivector is always gamma0 • v where v is the velocity as a run of the mill vector. To get the 1/2 into the angle, you can sum gamma0 and v then normalize. The rest is just algebra; a fifth grader could do it.

[@davidhand9721]

I'm not positive that the dot product is defined as half of the anticommutator when one or both operands is a multivector. That is the case when both are grade 1 only. The half commutator and half anticommutator always correspond to dot and wedge as a pair, but which one is which depends on the grade. To be fully general, you need to split both multivectors into their grades and do the dot or wedge products for each pair separately. Between the half commutator and anticommutator, the dot product is the one with lower grade than its operands, and the wedge product is the other one. I think you happen to have it right in the end, but it won't always work that way.

[@XylyXylyX]

Opps…of course you are correct. I do know the correct form, I will post an errata in the next lesson!

[@davidhand9721]

@@XylyXylyX believe it or not, I've seen some material recently indicating that the dot product is defined _geometrically_ not algebraically. They said it was always something something magnitude something projection. I kinda don't believe it. Same source also said that there is a similar quirk with even the geometric product itself, such that there are circumstances where it isn't the sum of dot and wedge products. That contradicts everything else I've ever heard or read about it, so again, I kinda don't believe them. I'll shoot you a link in a second.

[@davidhand9721]

@@XylyXylyX kzbin.info/www/bejne/aHKupWmXmayShas is the video, and the guy gave me 2 sources: a book I'll eventually buy, and a pdf that I'll link also

[@davidhand9721]

@@XylyXylyX The paper is on arxiv, pdf/1205.5935

[@XylyXylyX]

@@davidhand9721 Well, any geometric product where one of the factors is grade-1 will be a dot times a wedge. In all other cases, it wont be unless there is some coincidental cancelations.