An Introduction to Curvilinear Coordinates in Differential Geometry
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- Опубліковано 11 чер 2024
- The equations of General Relativity are written in the language of curvilinear coordinates, where mathematical objects like Basis Vectors, Metric Tensors, and Christoffel Symbols dominate the landscape. But where do these concepts come from and how can we connect them to our physical intuition? Join us to find out. Plus -- hotwheels! (Note: some familiarity with vector calculus is highly recommended.)
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Conceptualizing the Christoffel Symbols Video:
• Conceptualizing the Ch...
Metric Tensor Trilogy:
• The Metric Tensor
Contents:
00:00 - Intro
00:50 - What are Curvilinear Coordinates?
03:28 - Basis Vectors & Parametric Basis
08:00 - Coordinate Acceleration & Levi-Civita Condition
11:47 - The Christoffel Symbols
12:53 - Characterization of Arbitrary Coordinates
14:42 - Characterization of Polar Coordinates
19:12 - Geodesics
20:31 - Curved Surfaces - Наука та технологія
![ПОЛНАЯ ИСТОРИЯ ЭКЗОРЦИЗМА [Топ Сикрет]](http://i.ytimg.com/vi/qBySr6Bt0B8/mqdefault.jpg)
Babe wake up. New Dialect just dropped
I never slept.
At what speed? 9.8 m/s^2?
It didn't "drop". It merely followed a spacetime geodesic.
@@justanotherguy469 c 👀
@@rebase nice 👌
This channel is criminally underrated. 😫
Cause knowlegde is power. In our case, knowledge of tensor algebra, tensor calculus and differential geometry is absolute power 😎
I'm confused, shouldn’t it be relative power?
I'm confused, I was told that Differential Geometry was really hard, but this is unbelievably clear. Thank you.
😂
These videos are the new standard for differential/Riemannian geometry education (just as 3b1b's linear algebra videos were instantly the new standard for linear algebra education). I am grateful to live at during a time in which they exist.
Eigenchris also has a very good series of videos on differential geometry. (search for "Eigenchris")
Absolutely correct. 3b1b actually helped me to understand linear algebra. And these dialect videos are amazingly helpful for me to conceptualize on an intuitive level. I've got to get there first before I can go to the abstract
3b1b's series is the only reason I was able to pass linear algebra. I'm grateful as well.
I am a novice in multivariable calculus, but I find these animations beautiful and intuitive. You have your own distinctive style of animation and explication, masterfully juxtaposed, and I implore you to hold fast to it. Thank you and keep up the great work!
Thank you for watching and we appreciate your kind encouragement!
Oh this is exactly what I needed in my study of differential geometry!
Completely agreed! There were so many times where I just felt like I was doing the maths and didn't really get it. This made everything so much clearer to me.
Dialect consistently amazes me with their mind-blowing content. Animations are top-notch and even provide insightful revelations on how to approach thinking about these matters. I am eagerly anticipating their next videos. Thank you so much!
This is the best explanation of this I have ever seen. Great job.
New dialect video day is always a good day.
Thanks for a masterpiece of video graphics. The visualization and utilization of the driftwood flow gradient is very original and useful.
This material and presentation are simply ground-breaking, astonishing.
This is the single best video I’ve ever seen summarizing Reimannian , geometry, and general relativity. Absolutely brilliant. Bravo bravo!!
This is brilliantly illustrated and explained. I would recommend opening with a more directly stated context for how it relates to the arc of the long-term narrative, a simplified summary of the main conclusions, and a brief preview of the next step in our quest. It is very evident to some, while predictably opaque to others.
If this becomes a series by DIALECT. It will be my dream come true.
I've never seen such a clear explanation of this. This is so well put together and crystal clear. Can't express it enough. Job well done!!!
Thank you for watching!
Why is this so good? How long does it take to make a video with all these awesome animations? The last one wasn’t even that long ago!
Great for providing such a quality content video with this level of clarity and imagination building 🎩
A beautifully thought out and illustrated video. Great for beginners to differential geometry before they jump into relativity!
Really well done and thank you for your efforts 🙏
I love how in your videos you explain what the symbols used in GR mean in a way that I can understand. Your videos are like a Rosetta Stone for understanding the language of GR.
boy, oh boy; I see a new dialect video, I simply watch it right away ❤;
Thank you for breaking it down so clearly! And the animations are just 🔥
The present animation is very clear. I like it 😊
Amazing! Well explained!! Even after 8 mins into the video.
The QUALITY 🤩
Outstanding! ❤ Yet we still wait for the lesson how to deduce Riemann-Christoffel curvature tensor with metric tensor for an arbitrary manifold ✊
It's a few videos in the series down the line, but not too far off!
@@dialectphilosophy If you refer to the lesson when Christoffel symbols are evaluated I remember it very well, but in that lesson we never mentioned Riemann-Christoffel curvature tensor and its properties. You promised to talk about it much more then 👍
You use determinants. You can convert the paramentric vector space to the curvilinear one by using determinants as a linear transformation.
The parametric space can be represented as the column space of a Jacobian matrix (of coordinate derivatives). Each coordinate derivative is the eigenvalue of the Jacobian matrix.
Because of this, you can take the determinant to explain a chain of linear transformations to apply to the columns of the matrix, thus specifying how to transform into the curvilinear coordinates. This can be done in any number of dimensions, including greater than 3. 😜🔥
This is called a differential-form, and that's how you do it.
Essentially, the differential form becomes a change of basis matrix that converts back and forth between the parametric space and curvilinear space. An arbitrary manifold's surface will be specified by curvilinear coordinates. This allows you to convert back and forth between euclidean flat coordinates and the native local area of the manifold in question, in order to study an analyze the manifold.
Now, in terms of physics, I think the error screwing us up on a theory of everything all this time is a universal false assumption that space is continuous. Discretize space, aka finite difference methods, and that is a big step in the right direction in my opinion. But anyways. The real bizarre thing is that the human brain thinks in terms of euclidean spaces when there is no empirical basis for perpendicularity in the real world -- it is just a figment of our imagination.
I think Dialect means he’s planning on making the video in the future, as in a few videos down the line from now. Riemannian curvature requires a few more pieces of understanding but at this point you really aren’t far off
Very, very stoked to see where the next videos in this set go
Even though I don't have time to watch this video for a while, I felt like I had to put it on in the background for the algorithms sake. Your videos are a gift to mankind and you deserve a lot more views, thank you sir.
great channel damn very good animations and explanation that easily covers topics that rich kids pay $120k per year to achieve at 'college'.
Curvilinear Algebra, you've fixed everything.
A finally the next episode in this story. It feels like it has been an eternity.
This is just top-notch, amazing content!
great graphics and explanations, thanks!
You did a really good job on this topic.
Top notch animations!!
your voice is hypnotizing
Wow! What a fantastic visualization
When production is this good, math videos become the most interesting entertainment option in 2024.
brp why teachers dont make slides like this much easier to understand rather than just showing bunch of formulas
Fantastic as always.
Thanks fa Visuals visuals visualization !
Why the fuck didn't I get recommended this or notified? I've been so excited for a new video of yours. I'm subscribed and have watched everyone single one of your videos. Fuck youtube.
I have a degree that is analogous to European PhD in physics. Particularly in experimental physics. Yet this topic has always been over my head. I can grasp the basic concepts yet complete mastering is out of reach...
I always feel like a main character from "flowers for Algernon" novel. No matter how hard I try I just can't fully conceive such complicated things. My cudos to people who can though! They make our world a better place.
Absolute Banger!!!
Thank you!
Great video!
Just like every time, my day is made when I see a new Dialect upload!
I don't really have anything to note about this video because its already a masterpiece! I can't wait for more!
My only question is going to be about the nature of General Relativity... I am starting to get the feeling that there is a misconception about General Relativity and Newtonian Gravity... Many people seem to believe that Newton had no idea what Gravity is and where it comes from and that somehow Einstein with General Relativity gives us an intuitive explanation to not just how gravity behaves, but also what it is and where it is coming from...
So I am starting to believe that that is sadly not the case. Sadly, Einstein still had no idea what Gravity is and where it is coming from either; After all, a curved spacetime isn't any less magical than a gravitational field... The big change from Newton to Einstein is that now we know that the Universe has a speed limit! The speed of light, therefore Gravity needs to respect that law. Thus we need to update Newtonian Gravity with General Relativity, to see how Gravity behaves in a Universe with an actual speed limit;
But neither theory actually makes anyone more intuitive to where Gravity is coming from... So for certain, General Relativity has been falsely "advertised" it seems by science communicators - as this theory that puts Newton into shame, explaining how Gravity isn't a force but the outcome of acceleration and all that jazz...
But in reality all we have, is just a mathematical model, where its manifolds can be subjects to higher dimensions and non Euclidean surfaces... Ok - its just mathematics, we can do whatever we want with them because they are in just in our brains... But does this General Relativity give us any real intuition about the nature of Gravity and its real origins?
Maybe not; Gravity is a fundamental force... We have no real explanation to how ANY other fundamental force works either... Electromagnetism? You mean this magical EM field that permeates all of space?
I hope that Im wrong. I hope that General Relativity really does give an intuitive explanation to Gravity, in a way that we can understand it in an amazing AHA moment, and we are just too uneducated to do that now; Hence we are watching videos like these... But I don't really believe that anymore... I think that its just all mathematics and just like mathematics, everything is permitted; But if we say that "spacetime is Mikowskian therefore this and that", that sentence is still no less magical than a "gravitational field"...
Anyway, I hope Im wrong! Ill be watching these videos to find out :)
Great video. Are you preparing a curvature tensor video??
Yes. It may still be some ways down the line however -- we apologize, as we wish we could work faster, but there's a lot on our plates!
Superb !
Totally excellent!
At 7:34 shouldn't the law of cosines be c.c = a.a + b.b - 2a.b? (as opposed to + 2a.b)
I thank UA-cam for recommending me this video
new video, yay!
Great explanation! The only thing I'm missing is an explanation in terms of geometric algebra and the external product. This is important because geomathematic algebra is widely used in computer graphics and there is a lack of materials translating (or including) relativism into the language of external algebra.
We don't know much about exterior algebra, unfortunately, though we understand it ties closely to this sort of subject.
Thanks!
Thank you so much for your support!
Just awesome!
if you are still also having trouble wrapping your mind around Curvilinear Coordinates look at a Topographic map and pretend their is no Z up and Z down, like maybe you are building a flat foundation on a steep curvy hill and you want to know exactly how to make the foundation 10 meters by 10 meters.
Beautiful ❤
I'm currently looking into evolving spacetime entirely with tetrads(basis vectors), so I can't use the metric tensor nor the Minkowski metric. I'd be interested in a way to get the Christoffel symbols entirely from the basis vectors without having to embed my manifold in a higher dimensional space. I'm currently looking into Teleparallel gravity, or the tetrad formalism of GR. I look forward to your future videos on this!
Sounds like you're a bit ahead of us! We know Einstein was looking into similar subjects in his latter years (though if we recall correctly he was also suggesting the use of antisymmetric metric tensors as well) so it'll definitely something we'll be looking into down the line.
@@dialectphilosophy I look forward to it! The reason I'm dropping metric tensors altogether is because I've deemed them unnecessary and problematic. In GR, the determinant of the fully covariant metric tensor pops up a lot. However, the determinant is only a defined operation for mixed-index rank-2 tensors. If you convert the metric tensor to one of those, the determinant is always 1. When we normally take the determinant of the metric tensor, we mistreat the fully covariant tensor as a mixed index tensor. This is a gibberish operation though! Not allowed! This is why I've deemed it somewhat problematic. This issue can be avoided though if you avoid ever taking the determinant of the metric tensor.
What the metric tensor is supposed to do is define distances in spacetime. Lets say you have a given vacuum solution for a topological soliton of spacetime. Lets say this topological soliton is a particle, say a hydrogen atom. Vacuum GR is a scale invariant theory though, so we can scale this solution up and down however much we like. This means we can have an asymptotically Minkowski space where copies of these particles of different sizes coexist with each other. In real life though, all particles of the same type are the same size. You could make two rulers with atoms of different sizes and measure the same length and get two different values. The metric tensor had one job, and it has failed. This is why I've fired the metric tensor. A consequence is that nonmetricity is an undefinable concept in this new formalism I'm working on.
@@PerpetualScienceThat sounds very interesting. Again you’re talking a bit over our heads here, but definitely feel free to contact us via our email and send any of your work you’ve done on the subject.
@@dialectphilosophy Well it's still very much a work in progress, so nothing's published yet. I'll probably just keep on polishing it. It might end up as part of my PhD dissertation.
Take a peek at the Tetradic Palatini Action and the Spin Connection...
Hey man, amazing video as always! May i ask, how do you do the animation? Do you use 3D software such as blender or unreal?
It's a combination of programs, but this one was mostly made with blender. Just don't give our secret away 👀 🙃
@@dialectphilosophy Ah thanks dude! Dont worry, our secret is safe! Just a quick question: in blender do you animate everything manually or do you use scripting to get the shapes right? I want to try doing some similar animations myself in the future :)
My brain tells me I understand my lexicon invites disparaging criticism
Will you do a video on covariant and contravarient components?
Yes. We'll probably be doing a series on tensors down the line, and that will be a crucial component.
Hey friend! Could you share which technology (software) you use to create those math-related graphics like formulas, surfaces, etc.?
We'll be putting those up on our Patreon soon!
sire, What software hath thy used to make Thee Animations?
If electromagnetism was actually warping of spacetime in this wave like curvilinear space, would we be able to actually detect this as our matter would be warped with the waves? Basically asking if there was a way to detect if EM radiation could even be experimentally shown to be a high frequency gravitational wave. Due to it warping matter we likely would only see it acting on one axis at a time.
So basically the levi civita connection is the derivative of the wedge product?
Hi,
I hope you are well these days.
I have been thinking for a while how the world could look like in a spacetime with positive curvature.
It's clear that the space have a finite size (without any borders or special points in it), but should it be true for time, as well?
How could work causality in this world?
Should the wavelength of a particle in harmony with the finite amount of space/time?
Thanks in advance!
Robert Fuszenecker
In polar coordinates, what happens at the origin?
I think the description you gave for how something needs to accelerate in the parametric space in order to travel in a straight line in the Cartesian space might be incomplete. I tried simulating motion in polar coordinates and checking whether it would correspond to a straight line in Cartesian coordinates and found that if I simply took the opposite of the sum of the christoffel symbols with the upper index that refers to the same coordinate I’m using the object still accelerates in Cartesian coordinates. If I first multiply each christoffel symbol by the components of the velocity that correspond to its lower indices and then add up their opposites I do get a straight line in Cartesian coordinates.
Hmm... in the first example, where the Christoffel components are simply summed, the parametric velocity components are both equal to one. Before the ending however, we mention that for parametric velocities not equal to one, you have to scale the Christoffel components by the respective velocity component. Did you perhaps miss that part of the video, or are you saying that you did something differently than us?
@@dialectphilosophy Well the example given makes it look like you first sum the opposites of the Christoffel symbols and then multiply them by the velocity component corresponding to their upper index. I found that I seem to need to instead multiply the opposites Christoffel symbols by the velocity components corresponding to their lower indices first and then sum them together.
Which software use for this video graphics?
i would assume mostly programming geogebra
There is no way that is geogebra. Looks like Blender to me.
The Discord Invite link doesn't work :(
Thanks for letting us know! We're not sure why it stopped working, but we've since updated it so maybe try again!
@@dialectphilosophy Thank you!
best best best best best best best best best best best best best best
Spacetime curvature is the crack of science.
❤
👍
Maybe you should make also videos of how you make these videos :-)
We wish we had the time too 😂 We are going to try to start uploading some behind-the-scenes stuff on our Patreon soon
@10:20 I don't see the intuition behind why this should be true. I have two cars moving in the "horizontal" green directions, and I take the difference in their velocities to get a vector (V1 let's call it since I don't want to type out the full name). I do the same thing for 2 cars moving the "vertical" purple directions, to get a difference of velocity vector V2. I don't have any "feel" for why these vectors V1,V2 should be exactly equal.
It's a tricky point. The first thing that's necessary to clarify is that the condition is only infinitesimal; so if you are looking at too large a region it will not hold. But in the infinitesimal space, essentially, if one coordinate basis vector expands or contracts along its own direction, then the other coordinate basis vector must rotate and pick up a component in the direction of that expansion/contraction in order to compensate for this.
Indeed we'd recommend watching our "conceptualizing the christoffel symbols" video; around the half-way mark we do an infinitesimal walk-through of the Levi-Civita condition and it becomes a lot clearer there. We didn't want to go into the infinitesimal picture here in this video, but without presenting it the Levi-Civita condition is almost impossible to grasp intuitively.
⤴️
Good morning
Is it wrong to be addicted to dialect? 😁
holy shit.... :O
all these squares make a circle
all these squares make a circle
all these squares make a circle
all these squares make a circle
all these squares make a circle
Who is dialect?
curvy linear
The very question is: How does this reconcile with the "Matrix Theory"?
It was actually the study of General Relativity, with its emphasis on coordinative maps and metrics, that first led us to the idea of Matrix Theory. So yes, they will be reconciled soon...
Personally, I don't believe in the existence of perpendicularity.
half convinced that the whole "special relativity doesnt yey have a physically meaningful interpretation" thing is just an excuse to teach general relativity
My cat’s breath smells like cat food.
As does…
I love that smell. Oh, his ;little breath. 🥰
Fun fact: Einstein invented this type of geometry because one day he was so drunk he couldn't draw a simple square but wouldn't admit it
Much further back in history, Gauss
Sure beats blackboard and chalk
Good video, but I really think this is the wrong geometry for relativity or space-time. I think wrong geometry may be the flaw in SR.
We don't actually disagree with you there; what we've learned is that these "alternate" geometries are actually just descriptive conventions; two geometries can in fact describe the same theory. We tackle the geometric conventionalism of SR in our video "The Loophole"; we have yet to seriously treat on it in GR, but we are headed definitely headed there.
@@dialectphilosophy Thanks for the reply. Much appreciated, and I am looking forward to what perspectives you present (for write or wrong).
"There is nothing either good or bad but thinking makes it so" - Shakespeare
>
I have been working over a number of fundamental geometries that Einstein could have had in his mind, before it was possibly re-interpreted or misinterpreted by his mathematician colleagues.
Each geometry "appears" to go some way toward describing the concept of "combined" space-time in SR, but fail at the extremes or have quirks.
In any functional geometry (in my mind visuals) that describes relativity with combined space-time (distance-time m/s) I cannot find any natural x,y,z(3D) x,y,z+t(4D) and even a xt,yt,zt appear to break spacetime into space AND time. I can create the 2 spherical histograms of any point observer for past photons in m/s along the radius lines in my mind which is more complex than it first appears, but in both cases I think anything other than radial coordinates (ones using 2D polar surfaces) seams to break the mathematical description back into space and time separately.
>
So, we have 2 possibly functional distance-time m/s radial (spherical like) geometries that the mind can grasp visually and conceptually (when it learns how to) and any attempt to describe it in our 2D, 3D paper breaks it. And the math based geometry appears to recreate space-time in mans image and breaks it too lol
It really is difficult to translate into human language outside of the mind :)
>
Good luck with your quest and take care. If I see you down the rabbit hole I'll give you a hello :) Thank you for your videos and perspectives.
>
P.S. Not an indentured physicist. I lean heavily upon philosophy and cognitive psychology in an attempt to describe the human reality and then step beyond to the external reality (what physics looks for).
Kind of lift the veil of the human condition to see outside in a more real raw way.
Polar Bear lost his job to Hot Wheels!
He required a lot of computational time and effort 😂 However he will be back... at some point!
in-fin-i-TES-uh-muhl
I want to hear a debate between the creator of this channel and a mainstream physisist. The subject: Does the ether exist?
There's nothing to debate, there are several experimental measurements that when combined together will give you the Lorentz transformation, and _only_ the Lorentz transformation. That's a requirement for any luminiferous aether theory... and for special relativity... and any theory using the Lorentz transformation is going to make the exact same predictions.
Aether is the source field where 1) Force and Motion 2) Inertia and Acceleration 3) Capacitance (Permeability) and Resistance (Permittivity) >>>> into >>>> Magnetism, Electricity, Di-electricity and Gravity >>>> manifesting into >>> Mass, Matter and Energy.
@@juliavixen176 I mean, the author of this channel tries very hard to re-introduce the idea of an ether... and most physisist believes it doesnt exist and has been disproven. So I think there is indeed something to debate lol.
Physicist here, although not exactly mainstream. Dialect's matrix theory is just GR but with a definite correct coordinate system which an observer can never determine. This naturally gives the exact same predictions as GR, and isn't really even a different theory. We have coordinate systems for black holes(Gullstrand-Painlevé raindrop coordinates) where the speed of light in the radial direction varies depending on whether it's ingoing or outgoing. This can be interpreted as spacetime flowing into the black hole, reaching light speed at the event horizon. Dialect hasn't done anything particularly innovative or controversial here, although it does go against the spirit of GR in some admittedly unmeaningful respects. Any debate on the matter would be purely on the philosophy of science, as the theory has no meaningful differences.
@@PerpetualScience ok ok. So you are saying that his interpretation doesn't give rise to new predictions or anything like that? It's just a different interpretation of the framework that produces exactly the same results as GR? What's the goal then?
"Curvy-linear" lol. Great video though!
Hi
Thank you .I need to repeat the videos because it contains very important ideas related to GR theory.
@@anglaislangletaire2098 OP just said hi..
Kriegeskorte 303 @37ans
Fam, you need to spice up your titles and thumbnails. 😭
Unbelievable! Shocking breakthrough proves spacetime can be curvier than Stacy's Mom! You won't believe what happened next..
Hahaha DENSITY! Lmao
you don't need ... wtf are you doing?
This man is undoubtedly great. However, he believes every single aspect of physics and its evidence for, other than Special Relativity for some reason?
Though of course it's not for some reason. It's for the same reason why primitive humanity believed the Earth to be flat or the center of the Universe. One day Dialect will include SR in his repertoire of physics knowledge. The theory of everything is dependent on it.
So funny story: I recently got around to reading up on Lorentz-Larmor (Poincaré) aether theory... and... it's litterally what every pop-sci entertainment thing describes as "Special Relativity", except with a completely undetectable dielectric material filling all of space which does _nothing_ . It makes the exact same predictions as Einstein's much simpler theory of Special Relativity, and is much much more obtuse and complicated than Special Relativity.
Aether theories are a complicated mess, only Lorentz's theory (with Larmor and Poincaré) actually match experimental measurements... but it doesn't work at all with gravity. So everyone uses Einstein's much easier and logical theory of Special Relativity which also does work nicely with gravity.
Undetectable aether makes everything worse. There are some other huge problems with aether theories that nobody needs to solve if there is no aether. Like, the aether needs to be a solid because light is a transverse wave... which can't propagate through liquids. It would need to have millions of times more tension than steel for a wave to propagate at the speed of light... but all solid matter and the planets can pass through it without slowing down (planetary orbital have been stable for billions of years). etc.
dogma really compared this guy to the flat-earth mindop
Are you implying that Dialect doesn't know SR? That's kind of peculiar. Regardless, it's clear that GR is more fundamental than SR, and it's clear that Dialect knows GR. Their matrix theory is basically just GR but with a definite coordinate system. Not meaningfully different.
@@juliavixen176 I once actually tried to generalize Lorentz Ether Theory(LET) and found that in order to reproduce GR, you'd need an electric permittivity tensor, a magnetic permeability constant, a velocity field for the ether, and a scalar field to dictate the size of particles. Altogether, it was equivalent to the metric tensor, but described in a very inconvenient and cumbersome manner. You could still use it, but it wouldn't be fun. Alternatively, you could describe everything with a metric tensor and pretend that it's the electric permittivity tensor, magnetic permeability constant, velocity field, and scalar field which are actually fundamental, but that you use the metric tensor because it's easier. At that point though, you might as well just use GR. Dialect's matrix theory is basically that approach, but not even bothering with anything other than the metric tensor. Is this even still an ether theory?
@PerpetualScience To be clear, I'm not saying he doesn't understand Special Relativity. But he certainly avoids it (revealed in previous videos attempting to explain SR in a new manner).
People try to prove SR wrong because it says our lives are designed. That we have no freewill. We're embedded in a block-timed universe.
GR is simply about geometry. SR has major implications on personal world view and leads to an overarching purpose for life.