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.
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..
@user-ky5dy5hl4d U dont realize all this shit is easy asf for a mathematcian. I am also a mathematician and i invite you to define a tensor product of algebras. Lets see if u can understand the slightest😂
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.
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.
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.
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 !
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??
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!
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%+
As a non mathematician, on a lifelong journey to try to understand tensors, and tensor calculus...more than half a century, this seems very promising. In my old age I've had to pick my battles, and understanding tensors is one I haven't let go of just yet. Thank you for doing this!
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.
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
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!
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.
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 !!!
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.
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!
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.
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.
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.
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.
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 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.
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.
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 !
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.
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 video is absolutely perfect to have a first idea of what the hell is the metric, your content will reach a looot more people, the production is top, thx for sharing your knowledge
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!
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.
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).
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.
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.
This is an excellent video for those who want to understand the tensor elements of Einstein's general theory of relativity without getting lost in the derivation. Congratulations on making such a video that is intuitively simple to follow.
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!
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!
My physicist husband has been trying to explain this to me for years. This is a magnificently illustrated clear explanation of this difficult subject to understand. I really appreciate your incredible effort and wish you the very best for your channel. Now going to look up your other videos... Bravo!!!
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!
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!!!
Well done. As a non-mathematician, your explanation was easy to understand. As a Natural Philosopher, I can say that there is no force or field of gravity, that the measured value of gravity on earth known as the gravitational constant is a local value and is a product of the electromotive force that exists between the ionosphere and earth with a 300-400k electrostatic potential. It is this emf interacting proportionally with mass that causes objects to fall on earth at the same velocity. This simple classical model I present explains why GC fluctuates and is affected when storms are overhead. Furthermore, when Einstein derived GR, he did so for a space devoid of all electric and magnetic fields, a space where the planets were isolated islands in space. The last half a century of satellites coupled with abilities to perceive in other frequencies than visible light has revealed that this is not the case. We can measure the passage of time via a mechanized quantization, aka, clock, but we cannot detect time. We can measure the area or a volume of space with a mechanized quantization , aka, a ruler, but we cannot detect space. We can measure the effect of gravity on earth, by dropping an object in freefall, but we cannot detect gravity. Believing in GR is a faith based enterprise for it relies on a belief that these things that are undetectable are real. Religion is that which only requires belief, where-as, science is that which questions belief. Mathematical Constants of modern consensus science are the God of Modernity(Prof of physics Alexander Unzicker). Regardless of what the mathematical proofs may claim, if the math is not a true reflection of reality, if the foundational postulates are false assumptions, then the math is moot.
This is really helpful! I majored in physics many years ago, and am trying to learn a few new subjects like GR. This video was just right for where I’m at, thanks!
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
I have a phd in math. I found this video explaining the intuition behind the metric tensor really excellent! (Also, the quality of the animations is amazing!)
This is awesome how you make it understandable for the average joe that I am who never even thaugt of trying to understand what tensor is all about. A real big thank you.
@dialectphilosophy I'm rewatching your videos on the metric tensor and your new video on Christoffel symbols and I must say again that your explanations are enlightening and so intuitive to follow. I don't know if your team teaches or lectures but we need more instructors in the classroom who can communicate challenging concepts in such easy and clear manner. Whenever I want to review concepts in general relativity I draw for your videos and will continue to share them to anyone interested in the topic. Please keep up the good work and i'm awaiting the many revelations you've promised in special relativity and a few of your other videos. Great job to you and your team
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 .
I really appreciates your effort. This type of Explanation is not available in TH-cam. This is very useful and helpful for us. Please Make more videos. Keep it Up. 👍👍👍
"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 is AWESOME! Such a wonderful description of a very complex subject. I must commend you for taking the time to put this together so well. I tried to say thank you with my wallet. :)
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 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 !!!
At last - thank you so much! the light clock is a beautiful way to start understanding special relativity, its so simple, yet the implications are almost unbelievable. True though. So I have been searching for a similarly simple way to start along the path of understanding general relativity, and up until today, it was beyond me. Thank you SO MUCH, its actually as simple to grasp as the light clock, it just needed someone to avoid the freaking math! Woohoo, happy day!
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. :)
Woah i was struggling with the basic of tensors and this video explains so elegantly..Thank you. I'll take this as my first stepping stone in studying tensors and further to General relativity
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..
@user-ky5dy5hl4d Ask a flat earther they will give an accurate answer
@user-ky5dy5hl4d U dont realize all this shit is easy asf for a mathematcian. I am also a mathematician and i invite you to define a tensor product of algebras. Lets see if u can understand the slightest😂
@@mrmaestrouk "I don't understand it" therefore it's wrong, wtf is wrong with you lmao
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.
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
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?
Commenting to help the algorithm, this guy totally needs more subscribers
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.
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??
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.
Two years later, and this remains the most intuitive and well communicated depictions of the Metric Tensor on the internet.
Simply brilliant.
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.
As a non mathematician, on a lifelong journey to try to understand tensors, and tensor calculus...more than half a century, this seems very promising. In my old age I've had to pick my battles, and understanding tensors is one I haven't let go of just yet. Thank you for doing this!
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.
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
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!
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.
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 !!!
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.
Man, if these videos were around in the 90s when I studied physics, I might have actually understood what I was studying. Excellent video!
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!
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.
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!
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.
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!
Wild top quality stuff for a person who just fell down the hole of relativity
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 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!
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.
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 !
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.
This is may be the BEST video i´ve ever seen about physics !
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 video is absolutely perfect to have a first idea of what the hell is the metric, your content will reach a looot more people, the production is top, thx for sharing your knowledge
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!
This has been the clearest and most relatable introduction to tensors I have ever come across. Thank you for describing it so well.
An 18-year-old me would have hugged you and jumped up and down watching this video! Thank you!
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.
Impressed by the simplicity and clarity of this presentation
Brilliance is on another level. These are the teachers are needed at universities...
One of the best explanations I have ever seen. This channel is so underestimated. Keep on doing great works like this.
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
Brilliant. Utterly brilliant.
TY for your time and willingness to share and allow this availability to others. This is my second video. Keep them coming.
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.
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 can't imagine how much time this must have taken! Great job, extremely informative!
More people need to watch your stuff 👍
Absolutely brilliant!! The quality of the animations and voice over at every step is just phenomenal. Bravo and kudos to all concerned.
This is an excellent video for those who want to understand the tensor elements of Einstein's general theory of relativity without getting lost in the derivation. Congratulations on making such a video that is intuitively simple to follow.
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!
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!
My physicist husband has been trying to explain this to me for years. This is a magnificently illustrated clear explanation of this difficult subject to understand. I really appreciate your incredible effort and wish you the very best for your channel. Now going to look up your other videos... Bravo!!!
Thank you! 😂 😂 😂
Excellent. Without this one can't appreciate General Relativity
I had actually stopped learning General Relativity when I got to this tensor part. But wow you explained it so well!!!!
this complex you are making much simple,Great work
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!
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!
Truly intuitive. It’s that rare moment when you say - yeah, I really think I get this at last! Thank you.
this might have been the clearest explanation i've ever seen! instantly subscribed
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!!!
Well done. As a non-mathematician, your explanation was easy to understand. As a Natural Philosopher, I can say that there is no force or field of gravity, that the measured value of gravity on earth known as the gravitational constant is a local value and is a product of the electromotive force that exists between the ionosphere and earth with a 300-400k electrostatic potential. It is this emf interacting proportionally with mass that causes objects to fall on earth at the same velocity. This simple classical model I present explains why GC fluctuates and is affected when storms are overhead. Furthermore, when Einstein derived GR, he did so for a space devoid of all electric and magnetic fields, a space where the planets were isolated islands in space. The last half a century of satellites coupled with abilities to perceive in other frequencies than visible light has revealed that this is not the case.
We can measure the passage of time via a mechanized quantization, aka, clock, but we cannot detect time. We can measure the area or a volume of space with a mechanized quantization , aka, a ruler, but we cannot detect space. We can measure the effect of gravity on earth, by dropping an object in freefall, but we cannot detect gravity. Believing in GR is a faith based enterprise for it relies on a belief that these things that are undetectable are real. Religion is that which only requires belief, where-as, science is that which questions belief.
Mathematical Constants of modern consensus science are the God of Modernity(Prof of physics Alexander Unzicker). Regardless of what the mathematical proofs may claim, if the math is not a true reflection of reality, if the foundational postulates are false assumptions, then the math is moot.
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.
I can literally feel my brain growing while I watch this, thank you.
Using the earth maps gives me a better appreciation for General Relativity. Well done!
This is really helpful! I majored in physics many years ago, and am trying to learn a few new subjects like GR. This video was just right for where I’m at, thanks!
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! This is the most clear, concise, easy to understand explanation I've come across for metric tensors! Thank you so much!
I have a phd in math. I found this video explaining the intuition behind the metric tensor really excellent!
(Also, the quality of the animations is amazing!)
Thank you!
super underrated video. it showed how simple the metric tensor actually is. you made it so clear in here!
Excellent explanation...never heard it described this well before! thanks!!
Congrats on hitting the algorithm jackpot. For once well deserved.
I recently took up general relativity and your explanation or just seeing "-2abcosθ" cleared up the kroneker delta for me.
Thank you.😁
Excellent, at the end, the best I have never seen about this matter!!! Congrats from Spain!!
You literally outdone it. I have been trying to understand the Mercator projection! But never got it until now.
The best spent 15 minutes of this day for sure. Thank you!
This is awesome how you make it understandable for the average joe that I am who never even thaugt of trying to understand what tensor is all about.
A real big thank you.
@dialectphilosophy I'm rewatching your videos on the metric tensor and your new video on Christoffel symbols and I must say again that your explanations are enlightening and so intuitive to follow. I don't know if your team teaches or lectures but we need more instructors in the classroom who can communicate challenging concepts in such easy and clear manner. Whenever I want to review concepts in general relativity I draw for your videos and will continue to share them to anyone interested in the topic. Please keep up the good work and i'm awaiting the many revelations you've promised in special relativity and a few of your other videos. Great job to you and your team
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 .
I really appreciates your effort. This type of Explanation is not available in TH-cam. This is very useful and helpful for us. Please Make more videos. Keep it Up. 👍👍👍
videos like this are so insanely valuable. very intuitive
"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 video will push civilisation forward. It is so important. Thank you!
Definitely one of the best educational videos... please make more!
This is AWESOME! Such a wonderful description of a very complex subject. I must commend you for taking the time to put this together so well. I tried to say thank you with my wallet. :)
Best teacher from the far side of the universe......knowing this stuff about tensors enabled you to navigate your way here...🤪🙂
So clearly written, drawn, presented. Thank you
Another fantastic explanation. This brings to life all the numbers that float on pages of text and mean very little. Thank you.
Thank you for watching! Glad it helped!
Amazing video quality and graphics. Very intuitive presentation.
This is the moment where Neo unplugges his cable and tells Morpheus: I know metric tensors... Glad I found your channel!
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 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 !!!
What a wonderful video! Thank you for spreading and explaining such profound and abstract concepts.
At last - thank you so much! the light clock is a beautiful way to start understanding special relativity, its so simple, yet the implications are almost unbelievable. True though. So I have been searching for a similarly simple way to start along the path of understanding general relativity, and up until today, it was beyond me. Thank you SO MUCH, its actually as simple to grasp as the light clock, it just needed someone to avoid the freaking math! Woohoo, happy day!
Best explanation of general relativity I've ever seen!!!
Brooooooo this explanation is what i was searching for♥️♥️
Good job, it`s clear the author understood the meaning of metrical tensor. Further, I think it will help persons to find out what`s space-time.
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 :-)
Woah i was struggling with the basic of tensors and this video explains so elegantly..Thank you. I'll take this as my first stepping stone in studying tensors and further to General relativity