The Meaning of the Metric Tensor

[@MrAguc01]

This is a fantastic video, with well explained concepts and perfectly illustrated! I have a PhD in Physics and a master in Mathematics but despite all this, I have never seen the tensors concept so well explained. Other videos are always focused in the matematical operations and interpretations but here you can see the how it relates to real world. You make it so simple in this video that I will recommend this to my students! Your series are a very welcome addition to explain the beuty of Physics and Math! Please keep the excellent work and I will keep helping you as well.Thanks!

[@dialectphilosophy]

Thank you so much for your support! Feedback like yours definitely keeps us motivated, and your generosity is equally appreciated. Hopefully it will help your students wrap their head around the physics a little more easily. More is coming, so stay tuned!

[@stevecrothers6585]

:-: Crothers, S.J. Proof that Einstein’s field equations are invalid: Exposition of the unimodular defect, Physics Essays, Volume 34: Pages 420-428, 2021. Received: July 8, 2021; Accepted: August 14, 2021; Published Online: September 17, 2021: vixra.org/pdf/2206.0082v1.pdf Abstract: Albert Einstein first presented his gravitational field equations in unimodular coordinates. In these coordinates, the field equations can be written explicitly in terms of the Einstein pseudotensor for the energy-momentum of the gravitational field. Since this pseudotensor produces, by contraction, a first-order intrinsic differential invariant, it violates the rules of pure mathematics. This is sufficient to prove that Einstein's unimodular field equations are invalid. Since the unimodular form must hold in the General Theory of Relativity, it follows that the latter is also unsound, lacking a proper mathematical foundation. www.physicsessays.org/browse-journal-2/product/1896-2-stephen-j-crothers-proof-that-einstein-s-field-equations.html Stephen J. Crothers: The General Theory of Relativity: Its Faulty Mathematical Foundations kzbin.info/www/bejne/ZnS2fWpmZ6-Gmrs

[@alemorita92]

I did my undergraduate research and master’s on GR. Had I seen a video with this quality back then, I wonder how much faster I would have grasped concepts in GR. Fantastic video, explanatio, and animations!

[@HarpSeal]

Is this channel accurate on gr?

[@jaimemat3]

De eso se trata, de acelerar la curva de aprendizaje en modo autodidacta, buscando el mensaje que calce con nuestro mood del momento.

[@pushpendrayadav7069]

Where are you from?

[@stevecrothers6585]

:: Crothers, S.J. Proof that Einstein’s field equations are invalid: Exposition of the unimodular defect, Physics Essays, Volume 34: Pages 420-428, 2021. Received: July 8, 2021; Accepted: August 14, 2021; Published Online: September 17, 2021: vixra.org/pdf/2206.0082v1.pdf Abstract: Albert Einstein first presented his gravitational field equations in unimodular coordinates. In these coordinates, the field equations can be written explicitly in terms of the Einstein pseudotensor for the energy-momentum of the gravitational field. Since this pseudotensor produces, by contraction, a first-order intrinsic differential invariant, it violates the rules of pure mathematics. This is sufficient to prove that Einstein's unimodular field equations are invalid. Since the unimodular form must hold in the General Theory of Relativity, it follows that the latter is also unsound, lacking a proper mathematical foundation. www.physicsessays.org/browse-journal-2/product/1896-2-stephen-j-crothers-proof-that-einstein-s-field-equations.html Stephen J. Crothers: The General Theory of Relativity: Its Faulty Mathematical Foundations kzbin.info/www/bejne/ZnS2fWpmZ6-Gmrs

[@Blake_47]

Hey I'm an undergraduate physics student and I've some questions

[@eigenchris]

Another great video on the metric. I've been through pages Schutz and MTW, desperately trying to understand things. It really helps to have a video like this to consult first.

[@stevecrothers6585]

Crothers, S.J. Proof that Einstein’s field equations are invalid: Exposition of the unimodular defect, Physics Essays, Volume 34: Pages 420-428, 2021. Received: July 8, 2021; Accepted: August 14, 2021; Published Online: September 17, 2021: vixra.org/pdf/2206.0082v1.pdf Abstract: Albert Einstein first presented his gravitational field equations in unimodular coordinates. In these coordinates, the field equations can be written explicitly in terms of the Einstein pseudotensor for the energy-momentum of the gravitational field. Since this pseudotensor produces, by contraction, a first-order intrinsic differential invariant, it violates the rules of pure mathematics. This is sufficient to prove that Einstein's unimodular field equations are invalid. Since the unimodular form must hold in the General Theory of Relativity, it follows that the latter is also unsound, lacking a proper mathematical foundation. www.physicsessays.org/browse-journal-2/product/1896-2-stephen-j-crothers-proof-that-einstein-s-field-equations.html Stephen J. Crothers: The General Theory of Relativity: Its Faulty Mathematical Foundations kzbin.info/www/bejne/ZnS2fWpmZ6-Gmrs

[@jrv634]

Wow! I rarely write any comments in KZbin but I must say the level of detail and simplicity with which you describe these concepts is astounding. Hope you get millions of subscribers soon, not only you deserve it but they would hugely benefit from this content.

[@TheMemesofDestruction]

^ This :-)

[@ihatethesensors]

I completely agree and I can't wait for the next one. The graphics were amazing! A difficult subject described simply - it's beautiful.

[@-danR]

Dialect has suddenly become not only the most illuminating school of introductory SR and GR, but a model _master-class_ on _how to explain SR and GR_ . And I speak as one who has watched dozens of youTube channels on the topic and 100's of hours, and many years. It is in danger of rendering channels like eigenchris, The Science Asylum, SciencClic, Eugene Khutoryansky practically marginalized, if not irrelevant. They had better stand up and take notice: This is how you actually explain things. If I have any objection at all, it's that the (excellent) ambient music could be reduced to, say, 2/3's volume.

[@TheMemesofDestruction]

@@-danR We’re all entangled. But yes this Channel’s communicated knowledge per minute value is extremely high! ^.^

[@squarerootof2]

Maybe if he introduced some cute cats mixed with some really stupid pranks and some tik-tok style hoes, he would reach a million subs...

[@nemvus_]

The amount of this video that is 3D animated is impressive. I imagine it takes a lot of clever work to get the 3D shapes to behave that way in the video. Excellent quality content. Needs more views.

[@chyldstudios]

I'm astonished at the level of detail in this video. Well done!

[@worldwithinadream]

I have bachelor degree in physics and I find this metric intuition as clear as it could be. Better than in any book I read.

[@stevecrothers6585]

>>>

[@stevewhitt9109]

This was amazing. After years I finally get the intuition.

[@HassaanFareed]

How can someone be so perfect in explaining such a difficult concept. Your order of explaining things is perfect for the proper understanding. And I can understand how much difficult it is to produce animations of this quality. Keep making such an amazing content for us ❤️

[@mikegale9757]

Beyond brilliant. You really got it down to the brass tacks. Testify!

[@jmchez]

When Professor Goodstein of Caltech got a grant to film his lectures and use the latest 3d animation back in 1983, he created one of the greatest teaching tools in physics still fantastically useful to this day, "The mechanical World and Beyond". I've been wondering for a while how much better the animations could be done with today's technology. Frankly, not that much better. but your videos do fill in the gaps in some areas and are a worthy successor to Goodstein.

[@nosnibor800]

Thanks that was well described - my God the amount of work you put into the graphics. The 4th dimension adds time. Then we need to understand how mass causes tensor to change. Thanks

[@JoEbY-X]

I have a degree in mathematics and studied some physics but never formally studied relativity, but decided to learn it myself. After finally thinking I had a grasp of special relativity, I tried to tackle general relativity but never quite got a grasp of what a metric tensor is the few times I tried. Your video is the most amazingly intuitive treatment I could possible imagine. I get it now! Thank you! Also, I conceptualize special relativity just as you did at 6:16 with a light clock measuring time perpendicular to the velocity. But I never quite understood why the spacetime interval involves subtraction instead of addition. I can easily perceive time dilation and length contraction with a Minkowski diagram, but the image at 6:16 and your description of the spacetime interval as simply being the static proper time FINALLY made it click!

[@dialectphilosophy]

Thanks for watching! We put that visual of the light clock in exactly for that reason, so we're glad to hear it clicked for you!

[@abnereliberganzahernandez6337]

Man so you are mathmatican AND didnt had a course con diferencial geometry? Imean I am computer scientist AND love math. But It Is interesting to see that no body fully understands everything

[@stevecrothers6585]

>>

[@agrajyadav2951]

bhai kya gajab video banaya maja aa gayi, pasta khate khate dekha, metric tensor bhi samajh gaye aur swad bhi samjho double ho gaya, mauj kara di matlab. Adbhut bro keep it up!

[@casaroli]

Thank you for finally making this intuitive. No textbook I’ve ever read has been capable remotely achieving this.

[@manymany1191]

I really like the pauses in narration at the right spots to give me a moment to think through what's just being said.

[@jeanklein2198]

May I share my praise to whoever scripted and executed this wonderful video. Excellent quality content

[@jack.d7873]

Newly discovered your channel Dialect. And as someone who's familiar with Special Relativity wanting to understand General Relativity, your visual presentation along side clear, slowed foundational GR language is absolutely brilliant. I feel it in your video's, your goal and target audience is too teach this subject to people who have zero knowledge of GR. And as someone who feels like your target audience, I must say again, the ways you teach are mindbogglingly sensational! I'll give you a donation for your service to humanity. You've earned it.

[@dialectphilosophy]

Thanks for watching, we appreciate your support!

[@nooneinparticular3370]

I watched the video and I honestly don't understand how such a small channel can manage to have such amazing animations, great job! Anyways, as a mathematician with physicist friends, in my experience what happens is that they don't have the mathematics physically and historically motivated, and so they're just left confused by it all, and it's only in hindsight that they realize what all those maths were saying physically, so this channel is certainly plugging quite a glaring hole in that regard! As it regards to tensors in particular, the issue I see is that the average physicist will ask you "How should I interpret _a_ tensor?", but the problem is that there is, to my knowledge, no good explanation or physical motivator for _all_ tensors that would allow physicist to go "Ooh, okay!"; you essentially need to go type by type, and then explain what the whole dropping and raising indices magic is. So far as I know (Gravitation is on the mail and I'll start getting serious with GR once it arrives), the only books that give a decent explanation and motivation for such geometrical concepts are (an honourable mention might be Jürgen Josts edition of Riemann's "On the Hypothesis Which Lie at the Bases of Geometry", which is a short and sweet book) Lee's "Introduction to Smooth Manifolds" and "Introduction to Riemannian Manifolds"; the latter even has the occasional detail about pseudo-Riemannian/Lorentzian manifolds. Which segways into my last point: I know it's hard, but I really hope you guys make justice to the distinction between regular old Riemannian manifolds (which are very intuitive) and the Lorentzian spacetimes one deals with in GR (which are completely alien to the uninitiated), and in particular explain at length how the mathematicallty seemingly innocuous condition of changing the signature of the metric tensor so as to ensure everyone measures _c_ to be the same everywhere has in fact huge mathematical (and therefore physical) consequences.

[@dialectphilosophy]

Thanks for watching! The interpretation of what a tensor "is" was something that frustrated us for the longest time, and we couldn't get good answers on it. The discussion of what defines a tensor in the abstract, general sense is something we dropped from these videos altogether, because we finally realized it just wasn't necessary. What a tensor is or isn't in terms of physical meaningfulness varies depending on context. And yeah, the Lorentzian vs. Riemannian signature was a little too much to squeeze into this video. More forthcoming!

[@HesderOleh]

This comment got me thinking that some of the math youtube channels should organize to make explainer videos for math wikipedia pages. Wikipedia pages are really great in most of the sciences, having both high level explanations along with some easier to understand explanations, but most of the math wikipedia pages just assume you know everything about the concept and are just wanting to look up some small detail of definition.

[@nooneinparticular3370]

@@HesderOleh I know exactly what you mean, as I've found myself with the same problem too many times. I think the worst ones were either the page for "Spinors" or the one about the "Clebsch-Gordan coefficients"; the ones about the mathematics behind physics are the worst offenders in my experience.

[@bofink5377]

Thank You for the very clearly and straight forward explained Metrics and mapping. This clears many problems for me, trying to really understand the General Relativity and Gravitation in a more visual way, than just the plain raw mathematics.

[@elindauer]

The visualizations in these videos are amazing. The whole series is beautifully instructive. Great work.

[@flmbray]

What an amazingly simple and straightforward explanation. Although I not a physicist, I do watch a lot of these types of videos, and I've seen explanations of the metric tensor before. But this is the first (and only) one that really made it clear. The analogy to old maps and how they depict a curved space in a stretched fashion for our benefit was absolutely perfect. It all feels so natural now that I'm sure I will retain this new knowledge. Thanks! You earned a subscribe from me!

[@tanyachou4474]

One of the most underrated channel on KZbin! Thank you !

[@yashasvikatara1244]

I cannot put it in words how amazed I am that you made such an intricate concept so easy to understand. I'm really greatful, thanks a lot ❤

[@mickwilson99]

Bloody good start. One of my favourite analogues for tensors is playing games like Railway Life. The further you in the cab are from "events" (e.g. your destination at your scheduled arrival time) then shorter are your "kilometers" and the faster your "hours" pass. A great way for short-cutting the boredom of being stuck in proper-time-and-distance for a 500km run across Germany to France

[@nixledude4586]

Thanks for taking the time to demystify things for us! Understanding GR has been an objective of mine for a while now and video series like yours bring that goal one step closer. Thanks again.

[@dialectphilosophy]

Your goal and ours are one and the same! Thank you so much for your generous show of support, and make sure to stay tuned for our upcoming videos on the Christoffel symbols, geodesics, and more!

[@brianyeh2695]

The effort required for this video is beyond words... Keep going man. I truly admire your great explanation. Other videos just focus on the math operations rather than giving intuitions for what they are doing. For example, you explain the shrinking of areas near the north and south pole by playing around the transformation matrix and squeezing the element in the animation at the same time! That's extremely helpful to understand!

[@Maderum]

Wow! The animations and simplicity of explanations on this video blew my mind! Well done sir, well done.

[@matthewbusche4547]

I've been looking for a simple explanation of GR tensors for years. This was fantastic. I too hardly ever write comments, and only hit the like button once or twice a year. Here I've done both. Thanks so much.

[@cesararf]

Amazing explanation. Until now I didn't understand anything on metric tensors although I had heard of them. You deserve a massive thumbs up. 🎉

[@sinekonata]

Nah, thank you. So much. I can't believe we now get this much better of an idea just because someone on the internet decided to put big effort into visualising, explaining and then writing in video what he understands. We're pretty lucky. This kind of shit used to be only accessible if you had direct written correspondence with a physics master, waiting forever for his letter to come back or travelling to his university.

[@DelhiUniversity-j9s]

Just awesome. We lost the essence of physics behind those equations and maths. No one describes it visually like you did. Thanks a lot.

[@stevecrothers6585]

;;;;Crothers, S.J. Proof that Einstein’s field equations are invalid: Exposition of the unimodular defect, Physics Essays, Volume 34: Pages 420-428, 2021. Received: July 8, 2021; Accepted: August 14, 2021; Published Online: September 17, 2021: vixra.org/pdf/2206.0082v1.pdf Abstract: Albert Einstein first presented his gravitational field equations in unimodular coordinates. In these coordinates, the field equations can be written explicitly in terms of the Einstein pseudotensor for the energy-momentum of the gravitational field. Since this pseudotensor produces, by contraction, a first-order intrinsic differential invariant, it violates the rules of pure mathematics. This is sufficient to prove that Einstein's unimodular field equations are invalid. Since the unimodular form must hold in the General Theory of Relativity, it follows that the latter is also unsound, lacking a proper mathematical foundation. www.physicsessays.org/browse-journal-2/product/1896-2-stephen-j-crothers-proof-that-einstein-s-field-equations.html Stephen J. Crothers: The General Theory of Relativity: Its Faulty Mathematical Foundations kzbin.info/www/bejne/ZnS2fWpmZ6-Gmrs

[@erikquintanilla7705]

Amazing content. Love the clarity

[@beln33]

excellent job, Dialect. I highly recommend watching this video after the one titled "Demystifying the metric tensor in GR". After watching both (and taking some notes), the importance of the metric tensor should be obvious to anyone interested in GR and cosmology. I'm looking forward to watch "part 3" with will go about the 10 component tensor for curved spacetime.

[@1three7]

I love your videos so much. I'm not someone with enough math background to keep up with most physics stuff. But I also hate how unhelpful it is to just get over simplified analogies. It's hard to strike a balance of being clear to lay people but also being accurate and describing it with the underlying math in an intuitive way. Your channel is one that excels at that in my opinion. The animations really help keep things clear. Plus I love the music too! Thanks for putting the effort in on these. It's greatly appreciated!

[@thorntontarr2894]

Unlike you, I find the music annoying and distracting.

[@DennisMHenderson]

@@thorntontarr2894 I foun/d it, ‘Hard; 2bete [0ff] 2it-2nite

[@cykkm]

Dialect, you're a genius of a teacher!!! Holy Great Guacamole with vuvuzela on a Pogo stick, this is the most visual explanation of the metric tensor that I have ever seen! And I am not only a physicist by education; I had a full GR course with applied diffgeo in my grad school (following the MTW book) 25 years ago. The _abstraction_ of a tensor in my head has always been a row of columns of rows of columns of etc. (rows are covariant, living in a dual space, columns are your normal contravariant vectors); the metric you can slice either way. This is how it's easy to connect the tensor product with the normal linear algebra dot products, but that's only another, if a more familiar abstraction. Can you maybe eventually develop similarly simple visuals for tensor calculus? I'm not even working in physics any more, but I love GR so much for its concise elegance, but it's still a mathematical, not a visual elegance. But I'm ultimately a visual thinker, and I never looked at the spacetime like that!

[@dialectphilosophy]

Thanks for watching and we appreciate the compliments! In terms of how to define a tensor in the abstract and general sense, there seem to be a variety of ways to go about it and mostly they all confuse us 😂. When (and if) we understand the mathematics more fully, we may very well consider a series simply about tensor calculus in-itself. Of course, we tend to be more interested in the philosophy of mathematics, namely when its justified to apply certain calculuses, so you're more likely to see a series skewed towards that.

[@-danR]

This _alone_ : 0:35 - 0:38 is one of the most expressive and illuminating 3 seconds in the entire history of GR pedagogy. And I speak as one who has watched 100's of hours of youTube explanatory videos on the subject. Now granted one would want some background material to understand what that elegant bit of choreography is describing. But one could dispense with those hundreds of other hours (scienceAsylum et al.) and replace them with the rest of Dialect's classes in Relativity, which are also master-classes in _how to explain Relativity_ .

[@SeaPpl]

U shud deserve a noble prize for this explanation.

[@SuperMenders]

Wow the best Introduction to Differential Geometry, i have ever seen!

[@stevenschilizzi4104]

Fantastic presentation. I’m eager to see the sequel. Thanks for putting all this together for us.

[@omsingharjit]

This channel is going to hit Millions sooner

[@hexeract9541]

i literally have a project due on this in two days, thank you so much

[@inverse_of_zero]

wow, this is an incredible video, with the concepts so well explained and illustrated! i have three mathematics degrees, with majors in physics and mathematics, and i have studied general relativity and differential geometry. but despite all this, i never fully grasped the concept of the metric tensor until you explained it so simply in this video. thank you! i will recommend this to my students; if only i had seen this when *I* was a student myself!

[@dialectphilosophy]

Thank you so much for your support! We're really glad the video helped and hopefully if you check back in a few months we'll be on to the connection coefficients and curvature tensor!

[@Ri-bc5ff]

Great job! Kindly make a video explaining concept of induced metric ,intrinsic and extrinsic metric

[@robertmoran]

The Art of seeing is key. The math explains what we are seeing. Stellar to the max.

[@deltalima6703]

This better not be clickbait. Could not resist the title

[@alamagordoingordo3047]

Great clarity. I can't wait for the next episode. Thank you!

[@png7]

These video came out just before I needed them, They are very detailed and explain a lot with out be confusing.

[@nicolaecoman7188]

Incredible animations and explanations. keep up the good work!

[@massimoacerbis8138]

One of the best video on general relativity ever I have watched that many on the topics Highly recommended

[@jewulo]

Hello Dialect. Awesome work. Simply brilliant. Thank you very much for this.

[@georgesos]

I don't know why your channel hasn't a million subscribers already. You rock!

[@MictheEagle]

Pure Gold! This is the school we all needed. Bravo! Keep it coming, and Thank you.

[@jimlbeaver]

Outstanding…thanks for the patient and diligent explanation

[@mathophile716]

came here to understand the polar bear one pre-lecture lmao. loved it

[@starfishsystems]

Good work here. I think that anyone with a basic undergrad understanding of calculus and linear algebra would have no problem with it at all. That's me, and I followed the narrative easily. Oh, some basic familiarity with spherical geometry helps too. Then the motivation for tensors becomes obvious, and the video brings these intuitions to life in a very satisfying and confirmatory way. There are appeals to intuition being made here, but they work very well. The reference to irregular map projections is brilliant, because it motivates the use of tensors as a pointwise expression of the general case.

[@jzblue345]

I'm so glad I found this channel.

[@cesarjom]

Very well explained and demonstrated with visualizations!

[@AurelienCarnoy]

Each year is incomparable. Enjoy the present 🎁

[@xgozulx]

"you are literally a space time cartographer" that is so true and such an epic way to put it :D

[@GizmoMaltese]

Best explanation of the metric tensor ever

[@Redant1Redant]

amazingly good explanation. I wish I had seen this 10 years ago. It would have clarified a lot. Thank you for this video.

[@abulkhondker3556]

This is a superb video. The 3D animation is out of the world (pun intended). I cannot wait to watch more videos like this.

[@BernhardWeber-l5b]

Very impressive and helpful video! Just a minor remark: on 02:06 I never heard anybody read "i.e." as "eye ee", it's much rather read as "that is".

[@EricKolotyluk]

Wow, that was beautifully explained... thanks. Not that I grok it now, but I can certainly appreciate it better than before.

[@RT-tn3pu]

Dude, you got down explaining this concept.

[@twisted4872]

Holy Higher dimensions . . .that was simply exquisite.

[@Alpasonic]

Thank you. It was extremely satisfying

[@jialixx]

This is amazing! Can't wait for your next video. Thank you for sharing.

[@JustSayNotoGravel]

This is excellent content: clear, concise script delivered at a good pace: very helpful visuals; great production values. I have only one request: please loose the music track. Your content doesnt need this crutch and the audience will find it much easier to follow the ideas if they are not having to unscramble speech and music. That caveat aside, thanks for posting!

[@gabor6259]

3:21 Szekeres is pronounced sekeresh. Edit: excellent explanation!

[@snakecsmaker]

I gave up science 8 years back. But this video made some dying neurons fired again. It feels always great! Thank you!

[@AlokKumar-ym8bl]

🎉 excellent information..thank you sir for making this highly informative article about space and time.

[@zerozzable]

Your videos are beyond brilliant. Keep it up.

[@a.andacaydn9736]

Another amazing video! I cannot thank you enough

[@roberthayter157]

Absolutely stunning graphics. Brilliant explanations.

[@oblivitus.]

You deserve a ton of veiws! Brilliant quality!

[@eitasaulo]

this channel is so underrated

[@陳佳頎-j9c]

三D動畫真的很不錯 也相當易懂的內容 持續更新肯定可以得到更多的定閱 而且也很少人會做出如此精細的科普視頻。

[@Minstorm34]

Insane quality from a

[@PamSesheta]

Beautiful! I’ve always wondered about that 4D mapping bit, was hoping for like, iunno a guess even. I could watch a video of a shadow of the 4D mapping with some “guesses” at what that mapping might be. Thanks for reminding me about that strange old diff geometry book I was working through for fun.

[@brianhay4024]

The clearest explanation I've seen.

[@umeshchandramakwana806]

Awesome ! Really a great clarity in explaining the complex concepts! 👍👍👍

[@colafish2152]

Amazing video, it give a vivid visual explanation to some very abstract mathematics-physics concept. Actually, I spend lot's of time to study these concept by myself, but I still can't fully understand these concept, what I learn from this video it worth then these book i read.

[@MrKreinen]

I've gotta tell you while I love the voxel space-time concept, when I went to write up and build a lil virtual solar-system w a warping space-time g/ tensor + Hub Const I was utterly unable to conceive on it in such simple clear straightforward terms (I'm not strong on my applied mathematics, nor am I great at coding), but OMG this videos methodic teaching method was just so illustrative. Thanks :)

[@dialectphilosophy]

“Voxel” spacetime is a great name for it! Going into 3+1 dimensions unfortunately complicates the geometry, as you add in a hyperbolic component. But we’re tackling it in our next video!

[@MrKreinen]

@@dialectphilosophy looking forward to it

[@mery.m]

wow one of the best videos i watched on youtube

[@ioanapopa3473]

Astonishing explanation of something that is so overcomplicated all the times. The information just flows in your had with pure clarity!!! Thank you for your effort! Please keep up doing such beautiful and content precious videos!!!

[@eduardogonzcerv4636]

Man, your animation is awesome!

[@TranquilSeaOfMath]

Great presentation! Well animated and explained.

[@labhsingh1320]

Very useful video for the beginners like me to have at least some intuition about the metric tensor

[@ΚωνσταντίνοςΛαζαρίδης-ξ9ι]

Really good explanation. Thank you!

[@Abon963]

This video is a Gem!

[@sumanthakur1519]

Really such a good explanation hats off sir keep it up

[@sonysantos]

Your videos are very good, both in content and visual effects. You deserve more subscribers!

[@coolwinder]

First time i dvel into relativity, but just amazing!! You are amazing, thank you for this pace of work!

[@MrKaar9012]

this is amazing. great job. very impressed by the 3D work, must have been a lot of creative coding lol

[@anandbavkar8572]

Excellent Work!! I wouldn’t be surprised if this leads to true understanding. Thanks a lot for making these videos.

[@sebastienouler2529]

Is it possible to see a thousand more videos like this one ? :) congratulations !!!!!!!!!!!!!!!!!!!!!!!!

[@Markoul11]

This video is a masterpiece! Bravo!!

[@lucasf.v.n.4197]

Best explanation and animation

[@victorvila1056]

I'm gonna have to rewatch this in a couple years if I get into physics. Not mad tho since these are great videos.