Important Note, PLEASE Read: The formulas for the Christoffel Symbols provided at 14:38 are ONLY valid when the parametric bases are orthogonal, i.e. when the transformed e-u and e-v basis vectors are perpendicular to one another in the euclidean space. They will NOT work if the basis vectors are skewed with respect to one another in the euclidean space (such a skewed parametric basis is depicted in the accompanying visual at 14:38.) The most important step in finding the Christoffel symbols is decomposing the derivative of the basis vectors (the "acceleration vectors") into components of the parametric basis. To affect this decomposition in the video, we took the dot-product of the acceleration vectors with the basis vectors. However, using the dot-product to decompose a vector into its components only works if the component bases are perpendicular, not if they are skewed. This obvious oversight on our part was stimulated in part by the fact we have only ever worked with examples of bases that were perpendicular (e.g. polar coordinates) and therefore failed to notice that the formulas did not work for the general case. The formula for the general solution to the Christoffel symbols will be explored in the follow-up video to this one; keep an eye out for its arrival soon! Also a minor correction at 16:06 -- the dot product expression in parentheses for the sixth Christoffel symbol is not strictly correct; the basis derivative vector should read de-theta/dr, not de-r/d-theta. However, thanks to the Levi-Civita condition, these two derivative vectors are equivalent, and so can be swapped/substituted for one another.
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.
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
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!
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!
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.
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.
@@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 👍
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
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.
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.
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 🙏
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 :)
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.
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.
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.
Sort of but not entirely -- what you are likely thinking of is more along the lines of the Jacobian. The Jacobian is a coordinate transformation matrix which transforms the basis vectors of the parametric space at every point into vectors of the euclidean space. The metric tensor meanwhile could be viewed as a transformation of the coefficients of arc length from the parametric space at every point into coefficients of arc length in the euclidean space. The Jacobian and the Metric Tensor are however closely related. In short, if one is given a vector in the parametric space, one can use the Jacobian to transform that parametric space vector into a euclidean space vector. Meanwhile, if one is given an infinitesimal arc-length path at a point in the parametric space, one can use the metric tensor to convert that parametric arc-length into euclidean arc-length.
4:18, should't curvilinear coordinates instead change slower in larger areas? As it takes traversing more cartesian gridlines to traverse the same amount of curvilinear gridlines?
That would be the case if the hot-wheels were traveling at the same euclidean speed over the grid. But they are traveling at the same parametric speed, meaning each hotwheel has to reach the next gridline in the same amount of time as its neighbor. So a hotwheel traveling over a shorter gridline can take its time and travel more slowly, while a hotwheel traveling a longer gridline can't delay and has to travel more quickly. In more mathematical terms, we're looking at the change in euclidean distance per parametric unit. So if the change in euclidean distance is large for a single coordinate grid, then the coordinates there are changing "more quickly". There is some confusion that can enter here because we're equating changes in distances to a sense of changes in time -- most accurately we should say that a single parametric unit "represents" more euclidean distance if it spans a larger euclidean area, and then if we look over the course of several units, the representation of euclidean distance per parametric unit itself can be changing by a large or small amount.
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.
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.
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.
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
Strictly speaking you're correct -- the formula should read like that. However, due to the Levi-Civita condition, those two derivative vectors are the same, so the value calculated ultimately remains correct. Regardless, thank you for taking the time to point that out!
@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.
@@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 :)
7:37 Arrrrg. The law of cosines equations contains no cosine. I can accept that the metric tensor solves for the squared terms, and not the roots of those squared terms. Pythagoras, and all that.
great content- my only comment is that i wish you'd just say "if we take the limit" instead of talking about "infinitesimals," since the argument doesn't become any more complicated, only much more formally correct. your infinitesimal coordinates being linear is just a vector space spanned by the directional derivatives.
half convinced that the whole "special relativity doesnt yey have a physically meaningful interpretation" thing is just an excuse to teach general relativity
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
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?
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...
Important Note, PLEASE Read: The formulas for the Christoffel Symbols provided at 14:38 are ONLY valid when the parametric bases are orthogonal, i.e. when the transformed e-u and e-v basis vectors are perpendicular to one another in the euclidean space. They will NOT work if the basis vectors are skewed with respect to one another in the euclidean space (such a skewed parametric basis is depicted in the accompanying visual at 14:38.)
The most important step in finding the Christoffel symbols is decomposing the derivative of the basis vectors (the "acceleration vectors") into components of the parametric basis. To affect this decomposition in the video, we took the dot-product of the acceleration vectors with the basis vectors. However, using the dot-product to decompose a vector into its components only works if the component bases are perpendicular, not if they are skewed. This obvious oversight on our part was stimulated in part by the fact we have only ever worked with examples of bases that were perpendicular (e.g. polar coordinates) and therefore failed to notice that the formulas did not work for the general case.
The formula for the general solution to the Christoffel symbols will be explored in the follow-up video to this one; keep an eye out for its arrival soon!
Also a minor correction at 16:06 -- the dot product expression in parentheses for the sixth Christoffel symbol is not strictly correct; the basis derivative vector should read de-theta/dr, not de-r/d-theta. However, thanks to the Levi-Civita condition, these two derivative vectors are equivalent, and so can be swapped/substituted for one another.
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!
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.
If this becomes a series by DIALECT. It will be my dream come true.
Thanks for a masterpiece of video graphics. The visualization and utilization of the driftwood flow gradient is very original and useful.
New dialect video day is always a good day.
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.
Thanks!
Thank you so much for your support!
This is the single best video I’ve ever seen summarizing Reimannian , geometry, and general relativity. Absolutely brilliant. Bravo bravo!!
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!
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
This is the best explanation of this I have ever seen. Great job.
A finally the next episode in this story. It feels like it has been an eternity.
Amazing! Well explained!! Even after 8 mins into the video.
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.
Dude, you are unbelievable. This is some whole different level stuff!
This material and presentation are simply ground-breaking, astonishing.
When production is this good, math videos become the most interesting entertainment option in 2024.
boy, oh boy; I see a new dialect video, I simply watch it right away ❤;
This is just top-notch, amazing content!
Top notch animations!!
Great for providing such a quality content video with this level of clarity and imagination building 🎩
Thank you for breaking it down so clearly! And the animations are just 🔥
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.
great channel damn very good animations and explanation that easily covers topics that rich kids pay $120k per year to achieve at 'college'.
Very, very stoked to see where the next videos in this set go
At 7:34 shouldn't the law of cosines be c.c = a.a + b.b - 2a.b? (as opposed to + 2a.b)
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 🙏
The QUALITY 🤩
The present animation is very clear. I like it 😊
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 :)
You did a really good job on this topic.
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!
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...
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!
My brain tells me I understand my lexicon invites disparaging criticism
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.
great graphics and explanations, thanks!
Thanks fa Visuals visuals visualization !
What do you use for animations? GREAT
I thank UA-cam for recommending me this video
your voice is hypnotizing
Curvilinear Algebra, you've fixed everything.
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.
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.
Wow! What a fantastic visualization
Would I be correct in saying that a metric tensor is a function that would output a transformation matrix for every point in parameter space?
Sort of but not entirely -- what you are likely thinking of is more along the lines of the Jacobian. The Jacobian is a coordinate transformation matrix which transforms the basis vectors of the parametric space at every point into vectors of the euclidean space. The metric tensor meanwhile could be viewed as a transformation of the coefficients of arc length from the parametric space at every point into coefficients of arc length in the euclidean space. The Jacobian and the Metric Tensor are however closely related.
In short, if one is given a vector in the parametric space, one can use the Jacobian to transform that parametric space vector into a euclidean space vector. Meanwhile, if one is given an infinitesimal arc-length path at a point in the parametric space, one can use the metric tensor to convert that parametric arc-length into euclidean arc-length.
Great video. The visuals are incredible. Can I ask what software do you use to generate, for example, the grids and the flow of the vector fields?
4:18, should't curvilinear coordinates instead change slower in larger areas? As it takes traversing more cartesian gridlines to traverse the same amount of curvilinear gridlines?
That would be the case if the hot-wheels were traveling at the same euclidean speed over the grid. But they are traveling at the same parametric speed, meaning each hotwheel has to reach the next gridline in the same amount of time as its neighbor. So a hotwheel traveling over a shorter gridline can take its time and travel more slowly, while a hotwheel traveling a longer gridline can't delay and has to travel more quickly.
In more mathematical terms, we're looking at the change in euclidean distance per parametric unit. So if the change in euclidean distance is large for a single coordinate grid, then the coordinates there are changing "more quickly". There is some confusion that can enter here because we're equating changes in distances to a sense of changes in time -- most accurately we should say that a single parametric unit "represents" more euclidean distance if it spans a larger euclidean area, and then if we look over the course of several units, the representation of euclidean distance per parametric unit itself can be changing by a large or small amount.
new video, yay!
Thank you!
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.
At 7:40, is the cosine rule not -2ab ?
Fantastic as always.
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?
Which Video Editor has been used to make these Video ? I like to try it
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.
Absolute Banger!!!
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
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!
16:03 Christoffel number six is wrong , it should be (partial of e sub theta over dr) dotted with ( e sub theta)
Strictly speaking you're correct -- the formula should read like that. However, due to the Levi-Civita condition, those two derivative vectors are the same, so the value calculated ultimately remains correct. Regardless, thank you for taking the time to point that out!
brp why teachers dont make slides like this much easier to understand rather than just showing bunch of formulas
In polar coordinates, what happens at the origin?
i have a hard time in understanding how did he transform the basis vector in parametric space into christoffel symbol, can anyone give me a help?
19:28 here's the part i am confused
@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.
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 :)
wow so much clear now
I'll take your word for it. 😅
sire, What software hath thy used to make Thee Animations?
Great video!
Superb !
best best best best best best best best best best best best best best
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.
Totally excellent!
Nice video... though I have to say "curvy-linear" catches me every time :)
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!
7:37 Arrrrg. The law of cosines equations contains no cosine. I can accept that the metric tensor solves for the squared terms, and not the roots of those squared terms. Pythagoras, and all that.
Just awesome!
Beautiful ❤
great content- my only comment is that i wish you'd just say "if we take the limit" instead of talking about "infinitesimals," since the argument doesn't become any more complicated, only much more formally correct. your infinitesimal coordinates being linear is just a vector space spanned by the directional derivatives.
That's how we used to
My brain is straight spagehetti now
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
Spacetime curvature is the crack of science.
I love love love your videos, but that purple is burning my eyes 😂
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
❤
Is it wrong to be addicted to dialect? 😁
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
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
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?
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...
Who is dialect?