Demystifying The Metric Tensor in General Relativity 128,267 viewsOct 22, 2021

Demystifying The Metric Tensor in General Relativity 128,267 viewsOct 22, 2021 5.6K DISLIKE SHARE DOWNLOAD CLIP SAVE Dialect 7.58K subscribers The path to understanding General Relativity starts at the Metric Tensor. But this mathematical tool is so deeply entrenched in esoteric symbolism and complex mathematics that its simplistic, intuitive purpose often goes neglected. Here we attempt to expose the basic principles behind the Metric Tensor -- and investigate how they tie into General Relativity -- in as simplistic a manner as possible. Do we succeed? Feel free to let us know in the comments! Support us on Patreon! https://www.patreon.com/dialect_philo... Contents: 00:00 - Intro 00:54 - The Equations of General Relativity 01:33 - The Metric as a Bar Scale 02:45 - Reading Topography on a Map 04:53 - Coordinate Distance vs. Real World Distance 05:35 - Components of the Metric Tensor 08:50 - Mapping the Earth 09:48 - Stretching and Skewing / Law of Cosines 11:37 - Geometrical Interpretation of the Metric Tensor 12:37 - Coordinate Systems vs. Manifolds 13:39 - Conclusions Chapters Intro 0:00 The Equations of General Relativity 0:54 The Metric as a Bar Scale 1:33 Reading Topography on a Map 2:45 Coordinate Distance vs. Real World Distance 4:53 Components of the Metric Tensor 5:35 310 Comments rongmaw lin Add a comment... ScienceClic English ScienceClic English 2 months ago (edited) Fantastic video! Truely clear and very well produced! Just a tiny precision, at the beginning I believe you need two numbers not to describe the altitude, but to take into account the direction of the slope. If the slope of the mountain is along x for example, you need one number for the metric along x, and one number for the metric along y, the perpendicular direction along the map (parallel to the contour lines). 151 Dialect Mikko Haatainen Mikko Haatainen 1 month ago I've followed popularized science for over 20 years, and this video is one of the clearest representations of a difficult idea. Very well done, and deserving more views. 23 Dialect Nilberth Souza Nilberth Souza 1 month ago This subject of tensors has been a very difficult thing for me to learn. However, this is the first time that I have been able to clarify several doubts. Thanks for the excellent video 41 Jae Choi Jae Choi 1 month ago Oh man as an army infantry dude who did a crap ton of land navigation, this analogy really hit the spot! Thanks for the amazing video for helping people like me on the course of learning general relativity please keep more coming! 28 Dialect Saswata Roy Saswata Roy 1 month ago Appreciate the effort you have put into this video to elaborate on such a difficult topic in such a intuitive way. Your analogy using cartography is brilliant and the hard work on animation to back it up is just phenomenal. Best wishes. 7 Dialect Zucc Zucc 1 month ago Really appreciate all the effort that went into this to make it more intuitive. Love how you mapped the g matrix to stretching and skewing. Of course the scale analogy is fantastic. Subscribed! 8 Lucid thinking Lucid thinking 6 months ago Great video! Amazing animations! well done. 41 Dialect Jonathan Gasser Jonathan Gasser 1 month ago I can't imagine how much time this must have taken! Great job, extremely informative! 10 Ricardo Serra Ricardo Serra 1 month ago Congratulations for the great video. I am deeply impressed by the quality of the graphics and the clarity of the explanations using the map scale analogy. 15 Dialect Jim Beaver Jim Beaver 2 months ago Excellent explanation...never heard it described this well before! thanks!! 10 Dialect RMN RMN 1 month ago Dialect is surely going to be one of the top informative channels. There illustrations, pace, visuals, voice over, everything gives a proper understanding, and is perfect. This shows their (team dialect) dedication. 5 81546mot 81546mot 5 months ago Nice video. Looking forward to your future videos following up on this one. Noticed you showed the Riemann tensor--hope you fully explain this in future videos, as well as the Ricci tensor and the Ricci scalar. There is a ton of calculus behind these and that is not a problem if it is explained slowly and succinctly. Would like to hear your explanation of the connection coefficients and exactly what each superscript and subscript mean in their interpretation. You do nice clear explanations and your graphics are helpful. 22 Dialect green_mamba green_mamba 1 month ago Amazing video, never imagined the metric tensor can be understood and visualized in such a fantastic way. Excellent graphics and superb explanation! 3 miguelvb miguelvb 1 month ago This is greatly amazing, really. how much work, how well explained, ... you deserve a medal , really !!! Hope this kind of media is played to general relativity students, and metric-mathematics student ... Thank you !!! 3 Dialect Bill McGuinness Bill McGuinness 1 month ago So clearly written, drawn, presented. Thank you 2 Nicolas Kerner Nicolas Kerner 1 month ago Such a great work! You used some basics maths to explain a hard notion, i'm impressed! Finding a video like this one made my day, thank you! 2 Dialect Yousuf Nazir Yousuf Nazir 1 month ago Excellent explanation of the geometry of the coordinate systems and methods of the metric tensors in simple way through the applications of the geophysical maps and the dimensions of the symmetries 3 Science 3D Science 3D 1 month ago What a great video! It is so clear now! I always had difficulty to really understand what a tensor is that now it is so much clearer thank you 2 Arthur Pletcher Arthur Pletcher 1 month ago (edited) "Demystifying" well describes your lesson here. I was intimidated by the term Metric Tensor. However as an artist and spatial thinker, your approach was so easy for me to grasp. No problem, Thanks :) 1 Robert Robinson Robert Robinson 1 month ago Definitely one of the best educational videos... please make more! 2 Victor Toledo Victor Toledo 1 month ago (edited) Thanks for the video! I've been studying the math and this one is an awesome example to explain complex math concepts in geometric way without loss of rigor 1 Phillip Hoffmann Phillip Hoffmann 1 month ago (edited) Absolutely brilliant!! The quality of the animations and voice over at every step is just phenomenal. Bravo and kudos to all concerned. 2 Dialect RxLVND RxLVND 2 weeks ago Absolutely genius how you explained such a complicated concept so clearly. Thank you! Dialect Oh God Not This! Oh God Not This! 1 month ago This is incredibly well done. You really did demystify it. 2 Imusiko Imusiko 3 months ago Really amazing video! When will the follow up on this one be released? Thank you! 5 Dialect Arongil Productions Arongil Productions 1 month ago Wonderful video! I appreciated the excellent visuals you included. Hope word gets out about these videos to more people! (Mathematical note: that kind of matrix is a symmetric, or Hermitian, matrix, which guarantees real eigenvalues for orthogonal eigenvectors. The universe seems to prefer this kind of matrix. I absolutely loved your explanation of why these symmetric matrices describe the kinds of stretches and skews that we'd need to navigate, just like in a map.) 1 Dialect Steven Y Steven Y 6 months ago More people need to watch your stuff 👍 12 jmcsquared jmcsquared 1 month ago Mathematician here. This is by far the best intuitive description of precisely what metrics technically do that I've ever seen. You could in principle teach metrics in this experimental way to a class of high school seniors or college freshman, regardless of major. My wife and I wrote a textbook for a course where we teach general math to liberal arts majors. We have a section for curvature, and we try to teach it at the level of someone who knows zero calculus. Certainly giving credit where it's due, could we use your topography example in lesson plans for teaching metric geometry, perhaps in a future version of our textbook? This has so much potential to be incorporated into meaningful classroom activities that illustrate practical and integrated mathematical understanding. Thank you, and keep up the awesome videos! 3 Dyl WHS Dyl WHS 3 weeks ago This has been the clearest and most relatable introduction to tensors I have ever come across. Thank you for describing it so well. xgozulx xgozulx 1 month ago thank you so much, im in general relativity course right now, and this helped a lot :D 2 Dialect AetheismRules AetheismRules 2 weeks ago Absolutely fantastic !! THANK YOU !! I have seen about a dozen videos on this topic - and all had me completely lost - this was SO CLEAR !! I cant begin to thank you enough . Dialect Umair M. Umair M. 1 month ago You literally outdone it. I have been trying to understand the Mercator projection! But never got it until now. 2 David Schroeder David Schroeder 2 weeks ago I wish these sort of videos had been around when I was attending college in the 60's, it is so wonderfully clear how the metric tensor is build up from simpler geometric concepts. Thank you for this. Magnetron692 Magnetron692 2 months ago Thanks! This is of great help in order to understand what is a metric tensor! 1 mrimatt mrimatt 1 month ago Outstanding explanation! Happy to have found your channel with under 4k subscribers. It will grow rapidly. 1 John Keck John Keck 13 days ago Well explained and very commonsensical! Thanks so much for this! Alberto T. Alberto T. 1 month ago This video was refreshingly good and well explained! 1 Auburn Sandoval Auburn Sandoval 5 months ago Amazing video quality and graphics. Very intuitive presentation. 7 Dialect Philip Robinson Philip Robinson 1 month ago Truely excellent, a mystery revealed. I am a retired electrical engineer and have been trying to understand GR. I was never taught tensor calculus in my physics course long ago. I have also been searching for a book on the same without sucess. So I thank you for this excellent presentation and look foreward to the next episode. 1 Eric Johnson Eric Johnson 11 days ago Kudos. I was plodding through Susskind's lectures of metric tensors. Then I watched this. It just goes to show that you do not need to be a Stanford professor to teach. By focusing on topographic maps and showing the resulting tensor equations everything is crystal clear. Dialect Walter Lampert Walter Lampert 5 months ago Well done! I really enjoyed this. 4 Night Owl Night Owl 3 weeks ago Man, if these videos were around in the 90s when I studied physics, I might have actually understood what I was studying. Excellent video! Dialect klammer75 klammer75 1 month ago Amazing! Best explanation I’ve seen and I’ve seen a lot! Well done sir🤓🎓👏🏼💪🏼 ongzz ongzz 1 month ago this might have been the clearest explanation i've ever seen! instantly subscribed 1 George Evans George Evans 2 months ago Within a couple minutes, I was already like "who is this guy, and does he have any more videos???" Fantastic video, thank you! 3