ฝัง
- เผยแพร่เมื่อ 24 มิ.ย. 2024
- What are the Christoffel Symbols, and why do we need them? Our exploration into the world of differential geometry continues, as we strive to lay bare the complex mathematical machinery behind General Relativity. Here we calculate the Christoffel symbols for the most common of curvilinear coordinates system - polar coordinates - in order to build a preliminary intuition for what these symbols represent and how they relate to the metric tensor. As a bonus, we unearth the meaning of concepts like the Levi-Civita connection and parallel transport along the way. Plus: bears!
Thank you for watching, and please consider supporting us on Patreon -- it makes a difference. / dialect_philosophy
Contents:
00:00 - Intro
00:54 - Cartesian vs. Polar Land
03:31 - The Metric Tensor
10:16 - The Levi-Civita Connection
14:15 - The Christoffel Symbols
21:14 - A Polar Geodesic
22:27 - Future Agenda - วิทยาศาสตร์และเทคโนโลยี
There are two whoopsies at the 21:47 mark -- first the correct Christoffel symbol there is Gamma - r - theta - theta, as we are considering the change in the r-component of the theta basis vector transported in the theta direction. It is NOT Gamma - r - theta - r, as depicted. (Moral of the story: indices are confusing, so make sure to pay extra careful attention to them!)
The second whoopsie is a subtler, but much bigger and more important one. When we shift the our path radially to the right in Polar Land, the radial and theta components of the new diagonally-directed basis vector both pick up a 1/r acceleration component. This means we have to shrink the theta component by 2/r in Polar land to continue on the path of the Cartesian Land geodesic. We failed to show and account for these extra Christoffel components coming into play in that scene (as we showed only the -r Christoffel component corrections), and so technically the purple geodesic path depicted in that scene is not the truthful one, though the 1/r components do become more and more negligible the further out from the radius, so the path shown is still a close approximation.
The animations are super well done, and the "anthropomorphization" of different coordinate systems is surprisingly effective at teaching even this complicated Riemannian geometry. Teachers of young people are taught to keep in mind that the human mind responds best to human stories (stories about animals, like bears trapped in a matrix, still count as human stories). I'm glad that you are demonstrating that this principle remains true even for more advanced subjects.
Thanks for watching and for the kind review!
I've watched most of the dialekt videos. This is the first one that left me stumped within the first 2 minutes. I'm baffled by the kudos the comments seem to universally bestow.
For "anthropomorphization" I think it would be better called zoomorphization, but in any case I find it a genuine distraction
This is truly the quality content we need. Why the hell does this kind of stuff have only a few hundred likes. It should be in millions.
I think it will be in the millions, hopefully sooner rather than later
Millions of people trying to understand the advanced math behind general relativity? Unrealistic expectations.
dude, it's been uploaded two hours ago
@@chrimony exactly! One can hope 😁 but I'd say surreal rather than unreal. ☺️
I’m at 9 mins and I don’t see the point/usefulness yet. Lots of nonsensical math variables…
I’m just gonna take a moment to appreciate the humor in this video. When I had to pause for a laugh break once I heard “polar bear” as a counterpart to “cartesian bear”
Fantastic video! I don't understand how you produce these so fast and so well!
beaucoup de nuits blanches, mon ami...
Je t'aime dude@@dialectphilosophy
@@dialectphilosophy :O T'es francophone? Québec?
No way its scienceclic??????????????????????
One of the most visually pleasing illustrations of what metric-compatible connections are that I've ever seen.
Imo grad students should be using stuff like this. You're doing a great service by putting videos like this out there.
Amount of work the author has put into this is amazing.
AI did most of it
Riemannian Geometry was one of the hardest subject I studied in grad school. This is an amazing introduction to many important concepts
Thank you! 😊
Riemannian Geometry in grad school?
@@veil6666I learned it in 2nd!
it follows that i knew it when i wasn't even born into this coordinate space
That red pill was dry and painful, but such is the way of the Cartesian Bear. He is a friend nonetheless.
Cartesian Bear and Polar Bear literally kills me. Oh my god. I live for this. My life once again has purpose.
I am studying differential geometry for GR and your videos, especially with the animations, are invaluable for arriving at an intuitive and clear understanding of these concepts. Many physics and math textbooks offer symbolic or proof explanations that are rather stiff and don't promote the intuition as clearly and easily as your videos. This is a serious contribution to higher math and physics and helps so much. Thank you!
Staying tuned to see how the Christoffel symbols lead to the Riemann Tensor!
Which has 256 ( two hundred fifty six) components. Yes, many of them can canceled, but you still will enjoy the ‚big picture‘. 😂
@@lowersaxon In 2D it's just 16 components, which is a bit more reasonable.
That's the plan! However, to temper your expectations, the Christoffel symbols and geodesic equation will require at least a couple more videos, so it'll be some time.
@@dialectphilosophy Sounds good, and as always when watching Dialect, I'm always staying tuned.
Subscribed. Beautiful job.
How is it possible I've found this channel only now? I was working on a thesis in General relativity and you explain all of the concepts incredibly well. You're doing an excellent work of passing the information to the viewer as well as keeping their attention. The presentation, the information, all of it is just magnificent. As one educator to another, I tip my hat to you. Amazing content.
Thank you very much! 😊
Impressed by the amount of effort going to the animation
When I heard "Cartesian Bear" I thought "That sure is a seemingly random choice of animal, but alright" - but then came "Polar Bear" and I was like "AHH I GET IT" 🤣 Well played!
Very clear and concise. In just two opening sentences you described what others can't in a book. 👏👏👏
This video is INSANELY well animated and explained. I remember struggling to visualise christoffel symbols in college, this would have been a massive boon. Hoping this blows up soon⚡
I just finished a course in differential geometry and was frustrated because it’s such an awesome subject but the course moved so fast that I couldn’t really understand what was actually being done. Your videos are incredible! I feel like I’m finally understanding the things I have learnt. Thank you for your hard work :)
Best, most thorough (and patient!) explanation of this content!! So great! And also something I very much plan to point to when I get to trying to explain how small/infinitesimal games relate to the Levi-Civita connection, etc. 🙏💗 Thank you! ☺️
Hey I was just talking to you earlier, didn't expect to see you here.
At twenty-four minutes, this video definitely tried our patience, surely you can relate 😂 but thank you and looking forward to checking out your podcasts soon!
Dialect is back with a bang, thank you
this is turning a sphere inside out level content here. and i mean that in full appreciation. the draw is the memeability, but the educational content is legit. here’s hoping this flourishes in the ytp space.
thank you!
This is the first of your videos I've seen, and I have to say, it is excellent. Thank you for your carefully explained and wonderfully illustrated video. 🙌🏻🤩
Thank you for watching!
Just outstanding! E.B. Christoffel, G. Ricchi-Curbastro & T. Levi-Civita would be very proud of their enormous legacy! 💖
I’m a PhD physicist and struggled with GR but now thanks to this patient walkthrough I finally grasp Christoffel symbols! Thank you!
Now if I could only get an intuition for a “one-form”…
A one-form is just an oriented line segment. A two form is an oriented patch. A three form is an oriented volume.
@@keithdow8327 that's very helpful, thanks!
@@keithdow8327 I think they're joking
@@oni8337
Dude,
For a physicist, forms end up in integrals where they represent a small patch of something. The orientation of the patch is important. For Gauss' law we care is the integral over the surface is about a vector pointing in or out, for each patch.
@@keithdow8327 Gauge Theory Gravity exists and it's written in geometric algebra. Im pretty sure multivectors and k-blades are nothing new to physicists
Dang finally understand the metric tensor and it was thanks to a bear pun animation. Bravo sir!
😅
I discovered your channel last year and only realized yesterday that you've been stepping up the pace of uploading. I stayed up late last night and caught up with six of your last videos - I'm so hooked. Cannot wait to see your next video!
That's very encouraging to hear :-) Thanks for your support, and for binge-watching 🤪
This is the best video to visualize the metric tensor ive seen so far
This and ScienceClic's excellent series on "The Mathematics of General Relativity" are my favorite intuitive explanations of Christoffel Symbols, Geodesics and the Metric Tensor. Can't wait to watch the next few videos in this series!
This is damn good. Nothing less expected from dialect. 👌👌👌👌 and polar bear in the matrix is just amazing😂😂
Wow this was fantastic! Your visuals did a great job making the math feel intuitive. Looking forward to the subsequent videos!
it's such a pity that this video didn't make it in time to attend SoME3, it will definitely win a prize!😢
Your videos are seriously top tier
Man watching more of it, you really must have spent weeks on this. Thank you for your service
This is incredible work. Both your explanation and animations are so well done and make this challenging topic approachable. Bravo!
I can see the love that's been put into this. Truly inspiring!
I am learning so much from these videos, the world that we occupy is absolutely beutiful and you are proving this to me in every single video! Keep it up :)
only a person that truly and deeply understands a subject can present it with such elegance and clarity. very well done! it was a joy to watch and listen. i walked away with much better understanding of why these symbols exist in the first place. happily subscribed and looking forward to more excellent content.
Even having already learnt this before, this makes it so much more visual! Kudos!
One of the easiest explanation of Christoffel symbol I ever had seen in the history of maths or physics............ Piece of mind... 🎉
This is one of the hardest concepts I came across and this is explained in the easiest way ever. Hatts of to you for providing this quality content on TH-cam ❤ extreme respect for you man ❤️❤️
Great visualization. Thank you. Looking forward to your next installment
By far the greatest explanation I've encountered. Bravo!!!
Holy shit bro your channel is godsend, clear animation and just enough amount of math for me to finally intuitively understand the physics rather than just the tedious algebra
Completely astonishing! Thats new level content.
What a great step by step explanation. Thank you!
great stuff... great ending to this round!
Thank you and stay tuned!
Your educational videos are the best I've ever had the pleasure of learning. I constantly share your vids.
Absolutely hilarious and beautiful video! Amazing presentation of fundamental differential geometry!
When I watch, listen and read this over and over again I get so much new information each time.
My emotions are a little of inadequacy on my part, but mainly of amazement and feeling very lucky to have come across this, knowing I could never have understood it otherwise. Such quality, ingenuity and exposition.
That's just my long-winded way of expressing my thanks to you and acknowledging your incredible work.
I wish I could have thanked other you-tubers for similar reasons, but I write to you in the moment of emotion and particular opportunity.
Thank you for sharing 😌
This is the best explanation of the metric tensor I've ever seen.
great video! good job! Honestly, this platform is the best!
This is such high quality content. Thank you for spending so much time putting this together!
Awesome video, please continue with your great work!! Many young and early scientist will be very grateful.
Fantastic animations. It really helps the description.
What a wonderful explanation of parallel transport. This work is truly a wonder
Where have you been this whole time? The best videos
This is a fantastic explanation!
Thanks for taking the time to explain this so clearly.
Hats off, I would really like this kind of video to be produced more frequently.
Other explanation are burried in math without a visual understanding. But your explanation are very intuitive. Many thanks
Amazing! :) You make it so simple ❤
And I love the matrix spin :) Patiently waiting for the next videos, I want to see the real world! :)
Simply the best and the greatest channel on TH-cam.
This is such good intuition for metric compatibility! Thanks
crazy high quality! great job
Fantastic as always, thanks so much for those videos !
Thank you for watching 😌
Holy shit. Now THIS is some insane production value. Incredible video.
I absolutely love the content of this creator. Who else get happy/excited when they see he has uploaded a new video? 🥰 it's like a gift 🤗🥰
Yup! Though, this particular video seems kind of useless so far (I’m at minute 16)
@@PhysicsWithoutMagic This is useful for General Relativity and calculating geodesic in spacetime
@@PhysicsWithoutMagic This video is far from useless to those who have just learned something new. ;)
@@greenappleisspicy I’m pretty sure everyone who needs to know how to do that could already do that, no?
@@---Lola--- to what use will you put what you’ve learned, if any?
As someone that likes concepts explained geometrically, this was extremely helpful! I wish I had something like this earlier!
This video is amazing. Great explanation coupled with great visualizations.
Excelent video as usual. Thank you for your great work.
This is gold for people who are somewhat interested in this stuff
the animations are absolute next level wow
Holy shit how have I never foud this channel before. This is golden content right here!
You are the masters of visualisation!
Tnank you so much!. It’s really a great work. Please more video like this!! I enjoy and appreciate your job
The content quality is great and of course I am really grateful for explaining such a complex topic in a clear way but what really impressed my is the kinda old-school animation style, it fits so well and just makes me want to sit and watch... So relaxing. The cartesian bear and polar bear idea is 10/10 :)
Thank you for watching 😌
Such great content, thanks for posting ‼️
Holy cow this is a good watch. Thanks :)
Great video, thank you!
Well done. I wrote an article over spherical coordinates and Christoffel symbols and it’s on Wikipedia cited to me.
That's awesome, we've probably read it...
Wow, amazing video. I'm not amazingly versed at math and despite of that I understood this! You have talent
Hype! Knowing the topics ahead, you will be excited, too!!! I like how he started with the most feared topic, which is the Christoffel Symbols, its entry barrier to many who study Tensor calculus. ( this why some dub them chirs-woful symbols 😅)
Never had I thought that a polar bear with an existencial crisis would be the gateway to understanding relativity
Great video, thanks.
I didn't watch this one before. And the L With a top to bottom rotation for christoffel's..
It's very good so far 👋👋👋👋👋 excellent graphical works of yours 👋👋👋👋
If this is going where it is most likely, with these great animations this will be a great way for ppl to learn!!!! 👋👋👋👋👋
Very useful video!❤
Dude, this had to have taken forever to make. Thanks
This video is amazing!
Fantastic, thank you. Making more sense than my dimly remembered uni lectures...
Amazing, thank you.
Superb clarity
AMAZING EXPLANATION 👏🏻
I'm happy that polar beard could find its way into the real world the same way I finally understood this concept: Thanks tou you!
Great video!
your videos before were already priceless. But the amount of work in this one is impressive
Let's just appreciate what a smart bear Polar Bear is.
cutest video on general relativity 🐻❄️🐻 I want to give these bears a non euclidean hug
Wtf the production quality on this is insane
Excellent content
This was the most sophisticated polar bear I ever encountered.
Nice presentation
So beautiful!
Somehow I needed this video