sudgylacmoe | A Swift Introduction to Spacetime Algebra @sudgylacmoe | Uploaded August 2022 | Updated October 2024, 2 hours ago.
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: youtube.com/watch?v=60z_hpEAtD8
I also have an important addendum that I made to that video as well: 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: arxiv.org/abs/gr-qc/0405033
Discord: discord.gg/3Zj59zA2Rg
Patreon: 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
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: youtube.com/watch?v=60z_hpEAtD8
I also have an important addendum that I made to that video as well: 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: arxiv.org/abs/gr-qc/0405033
Discord: discord.gg/3Zj59zA2Rg
Patreon: 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
