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- เผยแพร่เมื่อ 27 พ.ค. 2024
- 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!
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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 - วิทยาศาสตร์และเทคโนโลยี
Also mathematician here, I wrote my Masters Thesis in Riemanninan Geometry and this is the best explanation of the metric tensor I've seen here on TH-cam. Keep up the great work.
Thank you!
IM NOT A MATHEMATICIAN.
And it is still totally wrong. Because I don’t understand the Accuracy. They still Don’t understand how to wORK out a CiRCLE.. 3.15… I would have EQUATOR LINES NORTH TO SOUTH..
If you are a mathematican and probably a nacissist then what is the shape of a point in space using the metric tensor?
@@dialectphilosophy The metric tensor is used for - to simplify the meaning - dimensions of space. So, what is the shape of a point in space in metric tensor? Also, how can you apply this tensor to ''time''?
@@user-ky5dy5hl4d Ask a flat earther they will give an accurate answer
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).
Wow -- first off, we absolutely love your videos! They're a complete inspiration, and so thank you very much for gracing our channel, we're very honored!
And secondly, you're absolutely correct to point out that imprecision. In fact, you would only need one ratio/number to describe how your distance element changes in a small region of the map, since as long as you are confined to that small region, it does not matter what direction you are moving, the distance element remains the same. This holds true for any metric on a manifold technically, including the spacetime manifold. The components of your tensor more describe the changing relationships between old and new axes/bases.
In that altitude scene we technically cheated by using an extrinsic geometric component (since the vertical component does not lie in the manifold but in the exterior space) -- something which we felt guilty about doing. However, we felt it was a more intuitive, easily visualized approach to introducing the idea of components, and technically speaking, the correct metric components could be recovered from that information. We'll probably try to clarify that issue in our follow-up video -- we hope you'll stay tuned!
@@dialectphilosophy Glad you like our videos! I understand. It was just nitpicking, it doesn't affect the quality of the video at all. I've definitely subscribed, cannot wait to see the next videos!
@@dialectphilosophy
I was a bit perturbed by that extrinsic axis as well, but only a bit. Some amount of hand waving is to be expected, and as long as the general idea comes across, we're good. We're already going along with the grid analogy, after all.
happy to c u sir
Today I learned the politically correct terms to call out errors. "Precision" and "Imprecision". How's that for pedantry?
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.
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!
Thank you for watching! We'd be happy to know you are sharing this in the classroom or in texts, and no need to credit us, we're just doing this for the betterment of the TH-cam learning community as well as our own understanding.
What textbook did you write? It sounds intriguing. Indeed, our goal with this video - and the ones that will follow it - was to try to explain an advanced mathematical/physics concept to a lay person. Hopefully we can keep it up!
@@dialectphilosophy our textbook is "The Beautility of Math." It's a portmanteau of the words beauty and utility to emphasize that we want students not only enjoy mathematics for its own sake, but appreciate how we use it in the real world.
You're right, i just finished my school and when i had read the title, i had not understood anything and thought it would go above my head.. and now i have got what metric tensor is 90%+
@@everythingisalllies2141 and your take is BS.
"You have become the very thing you swore to destroy."
@@jmcsquared18 I love the term "Beautility" as a description for mathematics, really hits the nail on the head.
Commenting to help the algorithm, this guy totally needs more subscribers
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!
We can promise you no butter bar LTs were involved in the making of this video
If you can quantify a crap ton, we can add it to the tensor! Thank you for serving, soldier.
I have seen hundreds of videos about general relativity for 15 years but this is so far the best explanation. Best wishes for your channel
I see in the previous comments that viewers of "Demystifying The Metric Tensor in General Relativity" feel they have clarity of what you are presenting that exceeds any treatment of this subject they have experienced for decades. The very same is true for me. You have done a remarkable job with this topic, "What is the Metric Tensor" and "How does it work". I could not help smiling throughout your video as it all became clear. Thank you indeed !
Appreciate the kind words, thank you. Glad to hear it helped your understanding, and if you liked it, make sure to check out the sequels!
After watching the video I understand what the metric tensor is how it works. What I still don‘t understand: What is the function of the metric tensor??
What does it have to do with the map conversions??
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
We're glad it could help! We didn't give a general definition of a tensor in this video since it is a much more abstract, broad mathematical tool, but will touch on it in later videos.
Dang, _yes!_ It's freaking ingenious.
Wow wow WOW! The title and thumbnail got me, then I looked at view count and thought, "it must not be super great." I watched not even the 1st minute and thought, "oh this is GOOD". I watched he rest and then thought, "This IS the BEST!" You should be so proud of the effort you put into this. It is a MASTERPIECE!
I'm an engineer, and have been studying General Relativity. I just finished a class in differential geometry; your explanation has helped me better understand some of the concepts.
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.
Within a couple minutes, I was already like "who is this guy, and does he have any more videos???" Fantastic video, thank you!
Thank you for watching! We're planning a series of highly-visual GR videos (on the metric tensor, connection coefficients, curvature tensor, etc.) for this spring & summer, so stay tuned!
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.
This is an awesome video. Clear, focused script, excellent visuals, engaging presentations. I am disappointed to see that it has only garnered less than 40 thousand views in nearly half a year. It should be 4 million at least.
Gosh thank you! We're just glad it's been helping as many people as it has so far!
You are a superb teacher, able to make a most difficult subject comprehensible to anyone willing to learn. I wish you would write a book on tensors, giving full instruction with the same clarity you have provided in this video. I'd buy it in a heartbeat and, as result, I would at last understand tensors!
Man, if these videos were around in the 90s when I studied physics, I might have actually understood what I was studying. Excellent video!
An 18-year-old me would have hugged you and jumped up and down watching this video! Thank you!
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.
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.
I can't imagine how much time this must have taken! Great job, extremely informative!
Excellent explanation...never heard it described this well before! thanks!!
Amazing video quality and graphics. Very intuitive presentation.
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!
More people need to watch your stuff 👍
I've been out out of physics for a long time, but 40 years ago I could actually do the math, at least in the sense of being able to manipulate the symbols. But as far as gaining an intuitive sense of what the metric meant I was on my own. I eventually did manage to come up with something like what's in the video, but it would have been very helpful to see it explained so clearly!
What an intuitive explanation! At 4:40 he actually broke the slope that instantly reminded me of almost anything we can express as basis vectors (back in college).
PS: back in college we didn’t know why the heck we’re deriving the field equations. Lol
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.
Thank you! Riemann tensor is planned as the next one in this series, though we might have to do an intermediate one with the connection coefficients first.
As far as I know, learning tensor idea from this video is far more better and meaningful than attending so many expansive college level courses Thank you!
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.
The way you peel the layers of the metric tensor concept and lay it bare, coupled with animation of every word uttered, makes the concept sink in easily and permanently. Brilliant !!!
Absolutely brilliant!! The quality of the animations and voice over at every step is just phenomenal. Bravo and kudos to all concerned.
Ex throretical physicist taught GR by one of the most famous theoretical physicists of the latter half of the 20th century. I must congratulate you on a brilliant exposition.
I recently took up general relativity and your explanation or just seeing "-2abcosθ" cleared up the kroneker delta for me.
Thank you.😁
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!
Wow, what a terrific exposition. Your communication, clarity, and visuals are outstanding. This is only the first video of yours I happened to stumble upon. Immediate subscriber.
I was studying general relativity by myself thanks to great lessons on internet and that was very interesting, I loved it. I understood what these diagonal metric tensor coefficients were (just distance, easy), but I didn't know what were these other components. A video of Scienceclic about general relativity told that was zero when we choose an orthogonal coordinate system.
I can't believe this was just the description of angles by just using the Al-Kashi theorem, I understand now ! All explains !
Instant subscribe. I graduated with a physics minor a couple years ago, always wondering what might have been, had I pursued a Major in physics. I watch many videos and struggle to understand concepts. On rare occasion, in a sea of explanations that I can't quite crack, I'll find a pearl. This video is a pearl. You've brought me insight more than the last 30 videos that I watched combined. We the people, are so lucky to have teachers like you making content. This video has encouraged me; If I keep searching for understanding, it will shine on me.
Sincerely,
Riley.
Thank you! We're glad it helped!
I am teaching a course on Cosmology to my astronomy club and this is the best video I have seen on the metric tensor. It will become part of the course and curriculum.
This has been the clearest and most relatable introduction to tensors I have ever come across. Thank you for describing it so well.
This is phenomenal. Thank you for the quality content. I'm discovering this channel and I can't wait to see other videos like this one. Bravo again.
So clearly written, drawn, presented. Thank you
This one of best explaination 25 years back my friend gave me book on tesonr i have been graduated i didnt much understand theory of relativity and concept of tesor i have see two three times more , Universities should be placed this video as teaching tools teahers should learn from this video hats off thank youu very much
i cannot express enough gratitude to you for making this video. i tried learning general relativity long time back, and struggled to find well explained content. this is by far the best. thank you
I had actually stopped learning General Relativity when I got to this tensor part. But wow you explained it so well!!!!
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 .
TY for your time and willingness to share and allow this availability to others. This is my second video. Keep them coming.
Brilliantly conceived & produced. And the attitude with which you answered ‘jmcsquared’ in the comments below earned both a subscribe and the first donation I have ever made to any TH-cam video.
Thank You! You will go far with this quality of concept & narration.
It's so difficult, but you're absolutely correct - the greatest way to understand the geometry of a manifold is to not attach yourself to your choice of coordinates. It's an abstract idea and even the examples in this video are still very coordinate focused since they have to be to get general understanding.
Outstanding explanation! Happy to have found your channel with under 4k subscribers. It will grow rapidly.
You literally outdone it. I have been trying to understand the Mercator projection! But never got it until now.
I don’t know anything thing about mathematics. You made this subject so easy to understand. Great job. Your channel is a new discovery for me and I am telling everyone about it.
Thank you very much, we're excited to hear it!
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.
videos like this are so insanely valuable. very intuitive
Brilliance is on another level. These are the teachers are needed at universities...
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
I was impressed by the ruler animation at 2:00 in, then my man comes in 5 minutes later and cuts up the entire globe into a Robinson projection. Really beautiful explanation
Wow that was so clear and insightful and well produced. Subbed!!
Definitely one of the best educational videos... please make more!
Thank you for the very precisely explained video. I'm taught!
And... I'm fascinated by your content, and spent an entire sleepless night contemplating space endlessly falling into earth, and think I had some thoughts on the subject! And all of them made sense. Which is truly baffling. :)
We're glad you found the channel :-)
This is may be the BEST video i´ve ever seen about physics !
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
this might have been the clearest explanation i've ever seen! instantly subscribed
Amazing quality. Beautifully presented. Wow!
Absolutely genius how you explained such a complicated concept so clearly. Thank you!
Wow! Outstanding description of the Metric Tensor. I think the best I've ever come across. Great job
Wow! This is the most clear, concise, easy to understand explanation I've come across for metric tensors! Thank you so much!
Excelent video man!!! Love it!!! Good animations and an amazing script!!
Impressed by the simplicity and clarity of this presentation
Beautiful! Excellent presentation. Nice work.
Really great video. Very well explained and the graphics made it even better. Good job.
Amazingly clear! Great job with this video.
Using the earth maps gives me a better appreciation for General Relativity. Well done!
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
This is gold! Thanks for the insights into projections too
Truly intuitive. It’s that rare moment when you say - yeah, I really think I get this at last! Thank you.
super underrated video. it showed how simple the metric tensor actually is. you made it so clear in here!
Great video! Amazing animations! well done.
Wow, superb! 👏👏👏👏👏
This is the kind of videos that in a few minutes expands my understanding, my mind, in a way that, before the existence of TH-cam, was only possible after an arduos search and study of rare books.
THANK YOU!!!
Watched a ton of tensor videos today. This is the best one.
The best spent 15 minutes of this day for sure. Thank you!
I can literally feel my brain growing while I watch this, thank you.
Fantastic please continue this series. I cant wait to watch the next Videos.
Awesome. If I'd had this video before retiring I would've had an easier time teaching this concept to my physics students. What you've done here is basically the same approach I took, but your graphics are so explanatory and intuitive - I just love this. Excellent work!
Holy crap, this video was absolutely amazing and intuitive. Subscribed for sure!!!
Wow! A truly incredible explanation.
Banger musics in this one. Really good video aswell though!
Why thank you! Our music is composed in-house, so we appreciated that compliment :-)
"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 :)
this complex you are making much simple,Great work
This was a very good video, thank you.
I was critical of one of the other videos on your channel, but this one is excellent.
Good job.
This video was refreshingly good and well explained!
The animations in this video are top notch. Kudos.
Thanks! This is of great help in order to understand what is a metric tensor!
I remember my first two day hike in Kentucky's Red River Gorge, which was a strenuous introduction to some of the themes elucidated here so clearly.
Soooo.... I'm not a mathematician but was an art history major / writer, and I'm here to learn about tensor metrics because I'm reading a book on Einstein's general relativity. This was an excellent introduction, so thank you!
Best explanation of tensors I’ve found. Thank you.
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 !!!
Thank you for the clarity.
Excellent. Without this one can't appreciate General Relativity