A Swift Introduction to Spacetime Algebra
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- čas přidán 14. 06. 2024
- 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: • A Swift Introduction t...
I also have an important addendum that I made to that video as well: • Addendum to A Swift In...
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 geocalc.clas.asu.edu/html/GAi... 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
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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
As a physicist: Mind=blown.
I am so used to the other set of mathematical tools that it is hard for me to do anything but simple problems with geometric algebra, but I can appreciate how amazing and nice of a tool it is.
To my viewers that are wanting more videos in From Zero to Geo: Good news! I'll be getting back to it right away. I just wanted to get this video out there that I think is a better SoME2 submission than the videos in From Zero to Geo.
EDIT: Since people keep asking me and I realize I should have said it in the video, I said "the zitterbewegung interpretation of quantum mechanics" near the end.
I wanted to say, you should submit this for SoME. I guess that's on me for not reading the description
You might want to put the #SoME2 tag on this video... 🙃
Great work! If it’s for SoME2, may I suggest to put the hashtag in the description and/or the title, to help referencing?
Edit: … just seen the other answer above ^^
Great video! Also "zitterbewegung" is pronounced more like "tsitter-beh-veh-goong"
Wonderful you are back ! Thank you for your time producing this videos sir.
I feel bad, I had never heard of David Hestenes until watching this video, and after looking him up I found out he works in the physics department of my university! I'll have to pay him a visit it seems.
Edit: It seems he retired before I started but this was not conveyed on the university's website :/
If you can convince him to make online videos or have an interview if you can that would be great,
did you?
Did you???
Can you film an interview with him?
@@umbraemilitos thank you for reminding me of this! He actually retired as a professor a few years before I started :/ but his profile on the university website just wasn't updated, so I found out when one of my professors told me he had retired. And unfortunately I was not able to talk with him.
1. I want you to know that your intro to geo algebra was the greatest math video I’ve ever seen and I was so excited to see this
2. What was the name of the interpretation of QM, and where can I learn more?
3. I would LOVE similar videos for PGA and CGA!
4. No seriously, thank you so dearly much for the effort you put into this niche but stunning topic
I realize now that I should have written the word "zitterbewegung" on the screen or something. You can find some information on it here: geocalc.clas.asu.edu/html/GAinQM.html
I took notes during my second watch and omg, I didn't realised how much information there was. Everything just came up so naturally that I just took it in at first. I hope you continue to do videos on more complex notions in GA, It's so engaging !
I would like to quote David Hestenes: "I have been pursuing the theme of this talk for 25 years, but the road has been a lonely one where I have not met anyone travelling very far in the same direction.'" Looks like his theory is becoming more popular. Thanks for this video it is fantastic! May be some more details how we can do more calculas in STA. Just one note: Your style is very similar to 3Blue1Brown as a talk. I would recommend to select your own style of presentation.
The fact that time-space split can be modeled by a geometric product blew my mind. Geometric algebra will definitely become the mainstream tool to do and teach physics in a close future. keep the good work your videos are an asset for humanity !
This really helps me see how "hyperbolic rotations" are just like the notations I know, and why there's good intuition in doing geometric algebra stuff with the Minkowski basis/quadratic form. Thanks for such clear explanations!
hell yeah 5hr video outlining Hestenes' Zitterbewegung structure in electrons and photons when?
seriously through, i love your work in making these concepts surrounding both relativity and STA easy to follow and visually intuitive whenever possible. keep up the good work :)
I’ve been working on this topic for one year now and I can’t get over how beautifully simple the algebra is. It should be standard in physics.
Its so cool how simple some of these equations can get when viewed in the right lens of relativity!
This is some god-tier math channel. it does not have this appearance because the audio is behind others, but really good microphones are expensive and the audio is good enough.
this is mind boggling.
geometric algebra is the most exciting branch of mathematics i have ever encountered.
Actually underrated... ur zero to geo video textbook series are the works of a good samaritan... keep up the good work!
For me, watching your last two videos has liberated much of the known laws of physics from the rubbish bin. Much I have yet to learn and relearn. Thank you so much. I will be back.
After watching this video, I feel like my brain has been trapped in a cage for my whole life, and this video broke me out and gave me wings. ❤
Thanks for the clear and very welcome explanations! In order to propagate the fundamentals and applications of Geometric Algebra, such more advanced videos are desperately needed. Great work and looking forward to more videos on this topic!
Excellent work. Love how you connected VGA to STA. Very clear as always. I know you plan to push on from zero to geo but I must say I found this type of video even more important. GA is so powerful that it takes a lot of time to explore it and the books from Hestenes or Lasenby are quite dense (for good reasons) which makes it hard (time consuming) for people to go through it. With a video like this it is now much easier for me to have people watch it and then have a discussion about the powers of GA. I could see three additional videos on the 3 connections you mention in the end: mechanics, electrodynamics and quantum mechanics. One main issue I found in talking to people about GA is it takes very long to get to powerful applications and many see the elegance but ask: what do I do with this that I can’t do already? There are videos about the details of GA already, they could be improved with Manim but they exist. The three above don’t (to my knowledge). Well done. Congrats.
I agree, 30-40 min videos like this are good for exploring the more advanced topics and getting a better bigger picture, while the Zero2Geo ones are more educational and slow paced.
Your videos are a godsend! Simply amazing clarity but I shall still have to listen to the presentation several times in order to fully comprehend it all. This is so exciting and I hope to see many more on Geometric Algebra, Symmetries and Groups in the near future. Please keep up the excellent expositions!!!
This chapter on GA4P has always left me quite stumped. so stoked for this video
Well done! The whole essence conveyed in 1 accessible lecture.
Dude im drunk as fuck and i understood pretty miuch all of this video... probably a testament to how good ur video making skills are. Congrqts brother ☝️
Real nice video. Thanks for that! I have a somewhat related request for future video content: I would love to see a Swift introduction to geometric calculus with some concrete examples! You are a great teacher and I think that video would be awesome!
While I agree that that would be a great video, the issue is that I'm actually not the greatest at geometric calculus. Definitely not enough to be able to teach it. Maybe someday in the distant future.
this video is incredible!! geometric algebra gives such good intuition for understanding this
I've been a physicist for 11 years now and I've never heard someone call it the "Lore Ints" boost before. xD This pronunciation is killing me!
This is fantastic!! Back in 2016 I made a 3-part video introduction to multivectors (using an axiomatic approach centered on the geometric product), and I had a lot of ideas for a series of follow-up videos, but then life got in the way and those plans had to be set aside. Now that I see talented creators like you making GA videos, I don't feel so bad for dropping the ball on my plans.
BTW, are you planning to attend ICACGA in Denver this October? Prof. Hestenes (who is 89 years old this year) will be the keynote speaker. I'm sad I can't be there in-person, but fortunately there's also an option to attend virtually.
I thought that I know special relativity well. But you shattered my confidence. I will look into it. Thanks for sharing.
Very very instructive, your explanation is excellent! Keep on going please …
That is an excellent video, I can’t thank you enough for all of this !
I hope I will one day understand all of this. Thank you for making these great videos! :)
I am about to watch and I’m so pumped!!!
Amazing, this is Amazing... Clifford's algrebras seem to have promising potential but ignored. Thank you !!. More videos like this!!
A video about APS would be great! A fell in love with Geometric Algebra. I come from an engineering background but the math always caught my atention.
Simply Fantastic!
Thank you so much, great video, would love to see more
Loved every second of this :)
Ty
This is video I needed so bad, thanks a lot!
Wow this is almost beautiful in a weirdly complex but organized way
Great explanation! Many thanks again …
Another great video, thank you
I did a postdoc in geometric algebra, it's awesome.
Very insightful >> 36:53
It changed my perspective about electromagnetism entirely!!! Thank you.
Zen like simplicity and elegance. Thank you very much.
By the way, has anybody read "Dichronauts" by Greg Egan which is set on a world with 2 normal space dimensions and one hyperbolic dimension? I never realized that some of the "physical" rotations there were mathematically equivalent to Lorenz Boosts.
Woohoo! Love this channel!
This is mind blowing! Why is this not thought in physics courses?!? 🤯
Top tier content!
After this and a few other videos you've made (about a year ago and more -- including the series that abruptly ends a few months back), I'd very much like to support your further work through regular payments to Patreon. I want to see you making a reasonable living out of learning more (yourself) and furthering your ability to teach/share what you develop in your mind in the process. I'll be discussing this with my wife. But slightly more than a hundred bucks a year is cheap if it allows you to progress and then share with us your mind. Best wishes, regardless!
Why is Geometric Algebra not standard in physics? I can't stand all these matrices in QFT etc. and love how neat everything looks in your notation. Is it because it has not caught on yet, or are there shortcomings?
In my opinion it's because it hasn't caught on yet. I talk about this a bit here: czcams.com/video/2hBWCCAiCzQ/video.html
Excellent
Thank you for video.
Great way of introducing the Minkowski metric, thanks.
I know, I was surprised when I worked it out how much it made sense. I had always just been introduced to it as "Oh look this quantity we dreamed up is invariant under Lorentz transformations so let's base our whole theory on this thing we don't even understand."
@@sudgylacmoeidk the necessity of the minkowski metric follows quite trivially from the speed of light being equal in every reference frame
I'm really curious about the implementation of this to relativistic quantum mechanics. Is there a way to write Lagrangians and such in a way that doesn't reference the coordinates? (And do you know where I could find videos about this?)
I find this pretty abstract. How about making up a special relativity problem (with numbers) and solving it in both the conventional way and with this spacetime algebra approach? So the advantages will become more obvious.
What is pretty powerful about this particular system is that it allows us to talk about quantum problems and relativity problems using the same language.
Maybe in 400 years somebody will see this in college. Until then... youtube.
May I suggest an example problem: muon decay.
Can you make a video on how EM, QM or even QFT can benefit from Geometric Algebra? I really can't stand the contemporary physics notation. It seems so ... ugly in comparison to how beautiful these theories are supposedly are.
incredible
While I was working with those stupid tensors and co-variant and contra-variant indices I always wondered if there was an easier way to do this.
Damn you have some absolute banger content
woah dude nice!
You make this seem way too understandable. Awesome job,
This really makes relativity much better to work with. I wonder if there's a book or some open course teaching general relativity with geometric algebra as well
Near the end of Geometric Algebra for Physicists by Doran and Lasenby there's a few chapters on general relativity.
very interesting
Oh my gosh, yes!!
Most of the content is trivial, but still interesting.
This is going to take a few times through to get into my tiny brain!
wow, very informative-TU
Thank you for the awesome video! What is the interpretation of spacelike rotations and timelike rotations you mentioned at the end of the video? Is a rotation in spacetime equivalent to a translation at a certain velocity in 3D space?
Spacelike rotations are ordinary rotations, and timelike rotations are Lorentz boosts. When considering them applied to the path of a particle, they correspond to rotation and acceleration.
I also could not recognized the mangled German word, the "zit-r-BB-gone" thing was "Zitterbewegung", literally translateable as "shivering movement". The word is stated in the video description, now that I notice.
Very good video! Thank you! Does anyone know what software he uses to animate everything?
This is answered in question two of my FAQ: czcams.com/users/postUgwFByhvEg1_hD_L1Ch4AaABCQ
Love the channel! Feeling like there’s something missing at minute 7:24 where rotation is defined on (a)with R dagger (a) R and an isomorphism between the Lorentz Boost. I’m novice and could be missing something.
The most you'd be missing is how to get R, for which there are a few ways. Technically, there's not even much restriction on what R can be other than the product of any number of vectors or the exponential of a bivector. Once you have R, R†aR is the way to transform a using R.
I guess R† could be confusing. R†, which is also sometimes noted as R̃, is the "reverse" of R, which is similar to the complex conjugate. It's the result of multiplying the original vectors is was made from in reverse order. If R was from an exponential, then it's the result of using the negative exponent.
I almost jumped when I saw the notification
I just realized that special relativistic spacetime is quite quaternionic. (Also, I think it's funny that the best thing my spell-checker can guess I meant by "quaternionic" is "fraternization".)
This is so coooool
I prefer to reserve "i" for the arbitrary units squaring to -1
This is a more general question, but is division an operation which can be done on arbitrary multivectors, whether in only some geometric algebras or all of them?
Division is possible for many, but not all, multivectors. As an example of a noninvertible multivector in 1D VGA, consider 1 + e1. Because (1 + e1)^2 = 2(1 + e1), if 1 + e1 was invertible, we could cancel 1 + e1 and get 1 + e1 = 2, which is clearly false.
Great video, but i really don't see the usual tensor algebra foundation going anywhere sadly. Clifford algebra is Great in certain situations but tensor algebra with vectors etc is just so simple imho.
How comes the Dirac equation you show contains gamma 1 and 2, but not 3? This seems to break some symmetry I would expect to hold.
If I recall correctly, this comes from the spin of the electron. We have to pick some direction that the electron is spinning, and the usual convention is to say that the spin vector is in the z direction. In GA we like to use bivectors for spin, which means that the spin bivector is γ₁γ₂.
This is something I 've been thinking of as I rewatch the video: is there a physical interpretation for STA trivectors? Scalars are scalars, vectors are inertial frames, bivectors are vanilla GA's vectors and bivectors, and the pseudoscalar is the pseudoscalar. But do the trivectors have an interpretation?
They are pseudovectors
Wow.
I hope this is taken as constructive feedback, but I think you need to EQ your audio. Right now it sounds very grating. I think cutting some of the higher frequencies will help out a lot. It would also help with whistly/hissing sounds with words that have s's in them. Love the videos by the way!
Wow, that really does make a difference! Thanks for the tip!
Excellent. I had realized that Lorentz boost should be a hyperbolic rotation because it is described quite simply by introducing imaginary angle. I could not go further. Now I can for the sake of GA.
I wonder why only square of gamma zero is positive one while others being negative. Should this asymmetry have some fundamental reasons?
I would say the reason is as simple as the fact that space and time are not the same.
I tried to run through this with mostly-plus convention to learn more about GA/STA, but ended up with timelike pseudoscalar. Is this something that is supposed to happen?
Dear profesor please could you recommend a math book to learn more about vector
Is there a way to represent the spacetime algebra entirely in terms of vanilla geometric algebra? I think just writing each gamma1, gamma2, gamma3 as basis bivectors would work if you invoked 7 dimensions, but is there a more elegant way? I personally think it's more consistent to have it such that every basis vector squares to one, rather than the weird mixing of 1 and -1 like in spacetime algebra.
I'm sure there's a way, but honestly I wouldn't suggest it. Having basis vectors square to various things is an incredibly useful and powerful idea. Furthermore, space and time are not the same, and this is reflected in the fact that they square to different values.
Whenever I feel smart, I watch this video to put my arrogance in place.
Thank you for this. I thought I had left a comment earlier but I don't see it now. I stumbled upon this looking for a description of Minkowski space-time, and have been blown away! You have an amazing talent for "swiftly" making the complex understandable!
Now, contentwise, I'm wondering whether it would be fair to say STA is not just an alternative "algebra" of space-time but in fact introduces an alternative approach to "geometry" of space-time?
Next, I'm going to look at Zero to Geo and review the Lagrangian.
I would say that the geometry of spacetime was already known before STA. STA is just another algebraic way of representing the same geometry.
Also, if you understood this video, From Zero to Geo is going to be way too low-level with the stuff it currently has covered. It's currently still in the linear algebra review section.
@@sudgylacmoe I got the impression that STA was a significant departure from traditional Minkowski space-time: changing the time axis to a vector and measuring it in units of distance. But maybe I'm just not that familiar with the way Minkowski space is used. As I mentioned, I stumbled upon your video while looking for an explanation of Minkowski space-time.
In any event, in terms of trying to make "math" more understandable, I learned about vectors in physics class. In high school, I was mystified by the idea of vectors (magnitude and direction) until I encountered them in physics--e.g. a boat rowing across a river current. So perhaps you might try introducing physical applications sooner. I'd love to see the "rowboat crossing the river" in STA. One could even do relativity of the boat vs. the shore vs. the river. I'd also like to see the path of a decaying muon from space in STA. Perhaps it's been done--can you suggest a text or paper that does mechanics in STA?
I had earlier attempted to suggest another approach to GA. It seems to me the physical "realities" are not points and vectors but the 4D "enduring objects". These are the real things conserved or transformed (rearranged). The volumes, surfaces, lines, and points are just abstract boundaries or projections of these real things.
Sir, a quick question. I started working with Gauge Theory Gravity recently, do you know any resources for people quantizing the theory? Because it is cool to rewrite GR and all, but is this theory a decent quantum description, i.e, is it renormalizable?
I myself haven't studied this, but I know that some work has been done in this area. I think some of it is discussed near the end of Geometric Algebra for Physicists by Doran and Lasenby, and I'm sure there's a few papers out there somewhere about it.
@@sudgylacmoe they say that work is being done to quantize it in this book but don't actually discuss it, and from what I can tell it is non-renormalizable by power counting unfortunately
@sudgylacmoe - here we are :-) Great video! I also followed your link to the works of David Hestenes and read over the article on real spinor fields. Still strange to me is to see the Dirac equation in a form that does not look Lorentz invariant in an obvious way, and it seems like Hestenes also preferring it in a more coordinate dependent form. Still, I also found the coordinate independent form in his article. So, in the end, the Dirac wave function seems to act as a local scale and Lorentz transform on the vierbein...?
Partly answering myself -- I dove deeper into the text. David Hestenes gives us an introduction to the Dirac equation in a coordinate dependent form to relate it to the formulations one already knows from the textbooks who give the Dirac equation in matrix form. In matrix formulation, you early have to choose a representation to get things computed, so it makes sense introducing the equation this way. But the real beauty arises, at least for me, where you have the invariant formulation at hand and take your coordinate perspectives by chosen vierbeins...
since you touched relativity, do you know about channel called EigenChris?
probably the best lectures for SR and GR out here
The Geometric Algebra evangelist has spoken again
"The magnetic field rotates charged particles in a spacial plane, and the electric field rotates charged particles in a temporal plane, which we perceive as acceleration."
My brain: *windows XP error noise*
I am wondering what’s the unit of gamma_0, 1,2,3 when their geo product are the normal time and space
Basis vectors never have units. The units are always attached to vectors separately.
Hi, did you record your own voice, or it was an AI sound?
I recorded my own voice.
At around 9:00, "time" is referred to as a "vector." I think it is more accurate to refer to "time" as a "component of a vector."
It could be thought of as a vector from a vector basis, as in spacetime from classical einsteinian relativity
37:48 "The ciderbibigon interpretation"?
Did I understand it correctly? I couldn't find anything about it. How is it really called?
"Zitterbewegung" means jittery motion it's in German
@@tariq3erwa English pronouciation...
@@porky1118 zitter like jitter, bewegung like be wig oong
@@tariq3erwa I know, I'm German myself
@@porky1118 I 'm Sudanese nice knowing you
At 14:02, I wonder if t' is not equal to 0, how can we calculate the velocity?
sorry, I mean x' is not equal to 0
Really cool video, this channel is a gem ! But I'm a bit confused by one thing :
At 26:20, it is said that the bivector g1g0 correspond to the physical vector x̂ (and g2g0 correspond to ŷ and g3g0 to ẑ) but I tought that the basis vector where g0, g1, g2 and g3 (like said at 9:16). But in the end it seems that the real basis vector are g1g0, g2g0 and g3g0 ??
I must have missed something but the geometric interpretation of geometric algebra is a bit hard to grasp to me ; if I want to plot a 2+1D vector along 1 dimension in time and 2 dimension in space for example, what is the vector that spans time, and what are the vectors than span space ? And moreover, if I want the space to behave "normaly" (like the complex plane with normal rotation) and the time to behave hyperbolically, what should I choose for g0^2, g1^2 and g2^2 ?....
Spacetime and space are separate things. The γn vectors are spacetime vectors, and γ1γ0, γ2γ0, and γ3γ0 are spacetime bivectors that represent the normal space vectors.
@@sudgylacmoe Ok, so if I want to model a 2+1 space-time, should I use γ0, γ1 and γ2 with γ0^2 = 1, γ1^2 = γ2^2 = -1 and consider the set {γ0,γ1,γ2} as being the base of my 2+1 space-time ?
Thanks again ! :)
Yes, that's correct.
@@sudgylacmoe Ok tanks a lot !
@@sudgylacmoe I'm really sorry to bother you, but I don't really get how we can have γ1^2 = -1 ; I thouht that in 2D VGA e_1^2 = e_1.e_1 + e_1^e_1 = 1 + 0 = 1 (the same for e_2), and then e_1*e_2 = e_1.e_2 + e_1^e_2 = 0 + 1 = -1 = i , which produces a space isomorphic to complex plane and to 2D euclidean space.
I thought that for 2+1 D space-time {γ1,γ2} should produce a space isomorphic to euclidean space, and {γ0,γ1} should produce a space isomorphic to a kid of "hyperbolic space"...
So I don"t get why the space spanned by {γ1,γ2} with γ1^2 = γ2^2 = -1 would produce an euclidean space, since in order to have 2D space-like space (the complex plane) we had e_1^2 = e_1^2 = 1...
Just a thought for your Zero to Geo series:
In further consideration of Minkowski space and STA, it occurs to me that there's a significant difference between the time axis and the space axis, which comes up when time is represented as imaginary (ict), etc.
This affects the definition of orthogonality between time and space, and is manifest in the fact that the Lorentz boost is not represented by a simple cartesian rotation of the axes in Minkowski space-time.
Now most Elementary physics texts plot time and distance on orthogonal cartesian real axes, which work fine for Galilean relativity.
But it we are to consider Einstein's relativity, would it not make more sense to treat the time axis as "imaginary" (or the GA equivalent) from the start? For example, in traditional Minkowski space-time graphs, a Lorentz boost does not preserve visual orthogonality of the axes.
But the essence of my question is whether we ought to be teaching "elementary" physics and "elementary" vector / GA analysis using a "complex" relationship / orthogonality between space and time vs. the traditional "real" relationship. This might pave the way for Relativity and perhaps even for the emergence of complex numbers in Quantum Mechanics. Of course, this could represent a significantly different philosophical notion of "velocity", "momentum", "energy", etc.
How might your introduction to vector algebra or STA change if you started with the "complex" (x, ct) plane instead of the cartesian?
It further occurs to me that, just as "gravity" causes everything to "fall" through space,
there is a similar "force" operating in the time dimension that causes "everything" to "fall" continually through time, i.e. what we refer to as the "passage" of time, but what mathematically we would represent as a "worldline". Elsewhere I suggested that "things" are never points; as a minimum, they are "world-x,line segments"" The only "points" are "events" or state-change incidents.
yeh, in the (typical) Minkowskian spacetime formula for the spacetime vector (and "distance", for the time parameter the unit time = i (i^2=-1). In just a (dumb) manipulation with the signs in the formula and the definition of proper time in the graphic representation (geometry) one switches the sign in front of the time parameter (ict)^2 in: (spacetime=(proper time x c-light speed)^2x(or x,y,z)^2 + (ict)^2) into just: (ct)^2- x(and y,z)^2. This algebra of space-time uses the same "trick": changing the meaning of i^2=-1 and using it for the space parameters (x,y,z)and not time (t). Another proof they (physicists do not have any clue about the methodology of science and see the difference between a model (mathematical one) and the physical (sensible and ..smelt, seen, touched) objects, then "space".(I doubt they know the definition of such plain term "space")
@@krzysztofciuba271 I'm really trying to wrap my head around the idea that time and space in relativity are related "hyperbolically" rather than "circularly?", that is, that there is a different notion of "orthogonality" than is assumed in classical physics. One would think that making time (or space) "imaginary" while the other is "real" should profoundly affect the whole of classical physics, well beyond throwing in a miniscule correction factor at "astronomical" speeds, distances, and masses.
@@jjaustin8750 yes. The real unit time = i ((-1)^1/2 in the real space-time because Euclidean space is just an imagery one -Minkowski called it (the relation) a mystical one; a hm, a mysticism entries geometry and physics (his 1908 paper)! Of course, it just only another case to validate the hidden reality of invisible principles that govern this physical (sensible) world- Plato knew it already in 5th cent. BC(his concept of Idea). ps. in court if you can claim the ticket of Police caught you for speeding just arguing all are speeding in the velocity(4-velocity) = c/speed of light; I bet all these lawyers are philosophical idiots and they will give up ( I doubt they know the difference between Euclidean (the fictional one) and Einstein space-time;.i.e the system has failed to educate them but you are a winner in the Court
If you wonder how the interpretation of quantum mechanics mentioned at 37:49 is spelled: It’s called Zitterbewegung (German, roughly translates to “jittery motion”). The subtitles lead nowhere close. The pronunciation is also off quite a bit.
I've been told that this is the English way to pronounce the word, which I know is quite different from the German pronunciation.
@@sudgylacmoe I mainly commented not as a critique to you but because I wanted to read about it and it took me about half an hour to make an input into a search engine that would spit out a correction. And someone else might find my comment.
I haven't encountered the “correct” English pronunciation anywhere. In fact, I added the German pronunciation to the Wikipedia article.
Yeah, that was my bad for not writing it down on the screen. I tried to write it in as many places as I could afterwards (in the description, in the pinned comment, and in an info card in the video), but I can only do so much.
Honestly this seems very useful but it will probably be so much easier to teach if the notation was simpler
28:00 It's a weirdly backwards correspondence, though.
I am now convinced that velocities are timelike bivectors.
Supported by the swift introduction video mentioning that charge can be viewed as a current flowing in the time direction.