A Swift Introduction to Spacetime Algebra

Video Description

This video is a fast-paced introduction to Spacetime Algebra (STA), which is the geometric algebra of Minkowski space. In it, we figure out what the problems are with the way introductory textbooks usually describe special relativity and how we can solve those problems by using spacetime and redefining length in terms of the spacetime interval. This creates Spacetime Algebra. We then study STA and learn about spacetime splits and Lorentz transformations. This video is my submission for SoME2, and just in time too! It was probably the hardest video to make that I've ever done because of the length of the video and the difficulty of the material being presented. In the end, I hope it can be of use to you. Here's a link to my original video on geometric algebra: https://www.youtube.com/watch?v=60z_hpEAtD8 I also have an important addendum that I made to that video as well: https://www.youtube.com/watch?v=0bOiy0HVMqA When it comes to resources for applications of STA, you can't go wrong with Doran and Lasenby's Geometric Algebra for Physicists. It includes chapters on every topic discussed here. Here's a few other resources: Relativistic Mechanics: New Foundations for Classical Mechanics by Hestenes, although he actually uses VGA instead of STA Electrodynamics: Understanding Geometric Algebra for Electromagnetic Theory (starts out in VGA but switches to STA later on) Relativistic Quantum Mechanics: Spacetime Algebra by Hestenes. Also, everything on https://geocalc.clas.asu.edu/html/GAinQM.html is good, and it includes some information on the zitterbewegung interpretation of quantum mechanics. Gauge Theory Gravity: The original paper describing GTG is "Gravity, Gauge Theories and Geometric Algebra" which can be found online on arxiv: https://arxiv.org/abs/gr-qc/0405033 Discord: https://discord.gg/3Zj59zA2Rg Patreon: https://www.patreon.com/sudgylacmoe Patreon Supporters: AxisAngles David Johnston p11 Richard Penner Rosario Sections: 00:00 Introduction 00:54 Prerequisites 02:05 Outline 03:18 Symmetry 05:36 Lorentz Boosts 07:03 Problems With Lorentz Boosts 08:02 Lorentz Boosts Mix Space and Time 08:34 Making Time a Vector 10:08 Visualizing Spacetime 12:27 Lorentz Boosts Change Lengths 12:54 Length vs. Square 13:33 Finding an Invariant Square 14:11 Spacetime Vectors as Reference Frames 14:59 Measuring Length in a Vector's Reference Frame 15:29 Derivation of the Spacetime Interval 17:38 Examples of the Square of a Vector 19:37 Negative Length? 20:26 Spacetime Algebra 22:22 Correspondence Between Space and Spacetime 23:38 Converting Between Spacetime and Space 24:51 Spacetime Splits 25:57 Algebraic View of Spacetime Splits 29:00 Return to Lorentz Boosts 29:51 2D Lorentz Boosts 32:17 Lorentz Boosts = Rotations 33:52 Higher-Dimensional Lorentz Boosts 35:06 Lorentz Transformations 35:40 Various Applications