Regarding the 2-sphere Christoffel calculations, there is a visual TYPO at 30:54: cotv is the value of the Gamma u-u-v christoffel symbol, as shown in the top half of the video, not Gamma v-u-v, as shown in the lower half, or as shown again during the ending title screen of the video. Indeed, the cotv and zero values should have their places swapped in the lower half of the screen at 30:54. Our apologies for this mistake, and thank you to our commenters for pointing it out. One of these days we'll make it through a Christoffel video without some sort of indices typo!
What a masterclass! You have changed the course of Riemannian geometry history for the better with this video. Such concepts have never before been so readily accessible
@@xypheli this video is a sequel to that one, I think. The whole series will stand as one of the canonical references of the subject for years to come.
I was drowning in my GR course because they don't give a visual explanation of what the Christoffel Symbol tells you. But now I am saved!! Thank you from the bottom of my heart
When I watch these science, engineering and math videos, the cleverness of these inventions and discoveries is amazing. The focus on the subtle nature must be so intense that only a few humans could achieve it.
@dialectphilosophy Your channel doesn't get enough praise for what you have done. You have made understanding tangent spaces, manifolds, the metric tensor and other concepts in GR so easy and approachable unlike anywhere else online. I spend hundreds of hours, literally, watching lectures online over the years on GR with the Standford Lectures by Leonard Susskind being one of the best. And you have managed to condense the rigour given by Prof Susskind in 30 mins without sacrificing any of the abstract concepts or important points. Your channel is godsent and I hope it grows and gets the appreciation it so deserves. This is probably my third or fourth watch of this video and its not because I don't understand but instead because it is so good at its explanation that I can rewatch it to reinforce the revelations that you have imparted. Keep up the great job that you and your team are doing and I look forward to when your channel reaches the level of appreciation to comparable channels like 3Blue1Brown and Veritasium (your channel is already on that caliber in terms of the quality of your content) which have also done a tremendous job at science communication. I absolutely love this channel, every single video that your team has published is a gem!!
Such a heavy subject handled with the OUTMOST of professionalism! You guys are first and foremost masters of pedagogy, and storytelling... I wish more and more people find and watch your videos, because you are a very special channel on the entirety of RUclips! As always, I can't wait for more - I like exploring relativity with you guys - keep it up!
Incredible. The connection between parametric space and the manifold is the question of what observers are. That's the crux of the quantum mechanics x general relativity puzzle. We are so close to seeing the full picture, and it is beautiful.
wow this is a FANTASTIC explanation. so intuitive! indexing the peaks and troughs of the manifold to a flat surface is such a genius simple way of defining such a surface!
Thank you so much for watching! Remannian Geometry is essentially just cartography, so credit goes to the first map-makers and circumnavigators of the world! ⛵ 🗺→🌍
Oh that's cool! I myself have discovered that GR has fractal spacetimes as solutions. My profile pic, a meromorphic function with fractal structure, is one of them as it's a fractal, and all 1D metric tensor fields are Ricci-flat, I.E. vacuum solutions.
It's refreshing to see that very advanced mathematics still often breaks down to little more than the basic fundamentals being done lots of times. Step 1: The triangle and its rules Step 2: ????? Step 3: Differential geometry
Question... The curvature of space is a mathematical tool that we can dispense with by bending the coordinates of the body towards the planet. The question is why do we need Riemannian geometry? Let's suppose that there is a body that approached a planet and its coordinates were bent... Why do we say that it is a curved space-time?@@dialectphilosophy
I'm a lot older than you but I will say the same thing If dialect ,eigenChris, Brian Greene, Lawrence Krause , and a special shout out to Leonard susskins theoretical minimum lectures at Stanford, had been around on RUclips when I was 15, I would definitely be doing the dream of my life theoretical physics
Currently finish up my Bachelor's in Applied Physics, and hopefully starting my Master's in Astrophysics next year. These videos are gonna be such a great help when I get General Relativity!
Amazing! You just made that differential equations class I took 16 years ago from a Russian that barely spoke English worth taking. And here I was wondering when that would come in handy. The applications to differential geometry are very interesting. I'm curious if you guys are going to take a look at the Ehrenfest Paradox.
It's always great when the outsider/skeptic has a far clearer exposition of the mathematics of the problem than anyone in the mainstream can come up with.
They usually look into it more critically, hoping to get a flaw, hence have a deeper insight than mainstream participants that just absorbed everything as it is.
@@chaoticmoh7091 Yes, it's really sad because mainstream science would certainly benefit from its practitioners having an adversarial understanding of their positiosn-- youtube has really driven that point home to me. Watching people like Susskind fumble in his GR lectures over how the Christoffel symbol relates to the convective/material derivative or even a great lecturer like Schuller twist into a pretzel over the twin paradox makes me think we are being held back by the fact that our contemporary academic scientists don't think as critically about what they are saying as they should, perhaps because they are afraid of being grouped as "cranks". I think it is holding us back.
Dialect is only really a skeptic about the whole one way vs two way speed of light thing in Special Relativity(SR). GR lets you do whatever coordinate transformations you want and also has things like ergospheres. As such, that SR debate is considered moot in GR. Dialect is only rebelling against the misconceptions in their head.
There's a lot going on in this video! Definitely recommend viewing the prior videos in our Differential Geometry Playlist (linked in the description) if you feel like you're struggling with this one.
@dialectphilosophy for the section of the Christoffel equation at 20:00 by "exterior" basis vectors you mean the vectors on the outside to the vector at the bottom left corner, correct? Likewise, are the three basis vectors actual stand in for neighbouring vectors? Meaning, since each vector in a vector space can be decomposed into components of the basis vectors, you are using the basis vectors to stand in for the actual vectors themselves. Likewise, what do you mean by the interior vectors? The reason i'm asking is that sometimes these terms have a strict mathematical definition or they can just be descriptive (simply the English language meaning) so I don't know which context is being used.
Thanks for asking. Yes, correct, the use of "interior" and "exterior" during that portion of the video is only meant to distinguish these vectors visually; it is not a reference to anything rigorous or mathematical. As to your second question, the three basis pair comprises all the relevant neighboring bases that are being compared to the original basis pair in order to take the derivatives of the basis vectors and get to the Christoffel symbols along the manifold there. Strictly speaking, they need to be compared in an infinitesimal neighborhood to be most accurate, whereas in the video they are compared approximately over a larger neighborhood (additionally the magnitudes of the vectors have been scaled down for visual clarity). Remember that what these basis vectors actually "are" are "bar scales", that is, they take us from magnitudes and directions in the parametric space (the "map" of the manifold) to actual the magnitudes and directions along the manifold.
Amazing video; this is super helpful for crystallizing this knowledge. Looking forward to next getting a better geometrical understanding of Lie brackets/torsion?
This is genuinely one of the most incredible maths videos I've seen - school is so finished with people out here teaching like this for free on the internet!
I absolutely agree...I attended a prestigious university and this video blows all that out of the water. What one can learn just in this one video covers what years at a University failed to teach.
Hey Dialect, can you do a video on the Michelson-Morley experiment? In your other videos you talk about the aether and how even Einstein later in life said he was open to it, but the experiment seems to me to show that the aether does not exist. I’ve tried to read some of the explanations online that try to disprove the null result of the experiment but I don’t quite get them.
Have been watching the video for 25 minutes now I'm starting to consider the possibility that I should know what a manifold is before watching this one...
Mathematicians will give you very abstract definitions of a manifold; however for the purposes here, they're nothing more than just a surface that can be curved or flat. If you find this video too difficult (and it certainly cannot evade that accusation) we recommend starting with the prior videos in our Differential Geometry playlist. Thanks!
@dialectphilosophy oh, thanks for the insight! I had thought this was an independent video when I started watching it, binge watching all your channel is definitely gonna make it easier for me to understand, these videos have an astonishing quality and I'm really glad I came across your channel!
You should look into Complex-Riemannian Manifolds. They're the same as Real-Riemannian Manifolds except that they allow complex coordinates and components and they must have complex structure. I made a post about them on my blog.
That sounds neat! You should hop on our discord and share your blog post there... there would definitely be others interested in learning about that as well.
All one would have to consider is instead of being limited to only one observer, rather having two observers should clearly indicate the second observer getting “shorter” as their feet start to vanish as they travel in any direction away from the initial observer. This would indicate motion in a 3rd dimension or from a strictly flat flat lander perspective the gradual vanishing of the second person from bottom up in relation to the distance traveled by the second person. Oh also a rotation in the direction traveled by the second person with 0° being the angle from the normal vector of the initial person and increasing positively in the direction of travel of the second person in relation to the intital person. An effect never observed on the ball earth.
Yes; it requires a bit more exposition than what was given here. However, we can tell you that the derivative vectors shown in this video that lay in the tangent plane on the sphere were all covariant derivatives. When dealing with an extrinsic curved 2D surface, a covariant derivative is simply a vector's extrinsic derivative along the manifold with the normal component thrown away or subtracted out, as depicted in the video at 10:10
I've always wondered how would a world with Lp norm/metric look like (distance(x;y) = (x^p + y^p)^(1/p)) ; is there a way to calculate Cristoffel symbols for that and see the curved space ?
You certainly could! However, you would need to define something analogous to an angle in this case. One route to doing such would be through the inner product. So if a dot a equals a^3, and b dot b equals b^3, what does a dot b equal? Once you've answered that question, you can relate your one-dimensional lengths to two-dimensional areas, and build the components of your metric tensor. Once you've built the components of the metric tensor, you can use the exact titular equation here for the Christoffel symbols to solve for how the basis vectors will change with changes in coordinates. From there you could in turn calculate curvature. However, "seeing" this curved space would be difficult, since what we can "see" or physically visualize is only euclidean metric spaces.
@@dialectphilosophy By "seeing" I mean see the curved manifold on which this metric is "applied" (like we can see the sphere which is the curved manifold on which spherical metric applies) ; I'm not sure what I'm saying makes sense by the way ^^' And thanks for the answer and the tremendous work you put into your videos, I have the feeling you're accomplishing something which can become really big !
yes, the metric tensor is a function of the where you are on the manifold. The Christoffel symbols involve derivatives of the basis vectors (i.e. how they change when taking an infinitesimal step)
@ Dawg, the segment on the recap of the first video, shows the stick person moving one basis vector in parametric space, and a single corresponding, transformed vector in Cartesian space. It did not intact, change continuously.
Buuuuhhh!!! That’s a super tight video. I just wish you’d go over how the equations arise from their equivalents which you claim “we know that…equals…” but regardless this is truly a masterclass. The only other issue I have here is when assuming a “flat lander” on a curved surface is a violation of the flat landers 2D universe. I am under the notion that a straight line is not a straight line on a curved path because when viewed perpendicularly that straight line on a curved path is a curve and or a circle if the manifold is spherical. Just because the flat lander doesn’t turn left and right doesn’t mean he’s going in a straight line because on a spheres surface if there is another person at his original location, when person one walks in a “straight” line he’d notice he is somehow rotated and seems lower in relation to person two at the original point.
Thank you for watching! The concept of a flatlander is an interesting topic and requires a much greater dive into mathematics and philosophy than we had time for here. Our use of the term flatlander here means that the observer "lives" in two dimensions and is not aware of the existence of a third dimension. The video probably makes this confusing because our observer is three-dimensional and poking out of the manifold, when they should in reality be a flat two dimensional thing living in the manifold. It just didn't look as cool to have a 2d observer sliding around the manifold as it did to have a 3d one walking around atop it 🙃
@@dialectphilosophy I thank you for your response and I want to clarify, I did not want to come off like as if I am criticizing your video. THIS VIDEO IS AN ABSOLUTE MASTERPIECE 🤯 the ideas you present are better described here than I've ever seen anywhere else and all through University...what I meant with my comment was I just wanted to bring awareness to this notion of "curved" 2-Dimensional Planes...I am firm believer that curvature cannot exist for any plane unless a third dimension exists to curve it in... Two dimensional Planes can only have 1 dimensional lines possess curvature within the 2-Dimensional Plane. For the same reasoning as before, if you're only in a 1-dimensional line, the only way that line can have curvature is by introducing a second dimension otherwise it has nowhere to curve.
Dialect, you are mixing up two very important concepts here; the Levi-Civita condition and the Schwarz-Young-Clairaut theorem. The first one states for two vector fields X and Y that nabla_X(Y)-nabla_Y(X)=[X,Y], where nabla_X(Y) is the covariant derivative of Y along X, and [X,Y] is the Lie bracket/commutator. The 2nd one states that basis vectors/partial derivatives commute; [partialx;partialy]=0. These things look similar, but are completely different concepts. In more advanced theories, like Einstein-Cartan, the Levi-Civita condition no longer holds, but coordinate lines still have to close on the manifold, fx.. (I can elaborate on this and the importance of it in advanced theories, if someone is interested.)
Moreover, the Levi-Civita condition is more than just this statement; it also states on top of the zero torsion condition, that the connection 1-form is metric compatible, nabla(g)=0, where g is the metric tensor.
Thank you for the clarification. We do understand there is a lot of subtley surrounding this concept, so we tried to mitigate it in the video by stating that the Levi-Civita condition requires the coordinates close "in this context" i.e. within the context of Riemannian geometry. We would however definitely love to hear more about the difference between these concepts as they pertain to more abstract geometries, as it's hard to understand how the coordinate lines could close without basis vectors commuting; so please feel free to reach out to us on our discord and elaborate more about it there!
@@dialectphilosophy No, you don't understand my comment. The mistake is at 17:48. The Lie commutator of the basis vectors have to commute for coordinate lines to close, but this has nothing to do with the Levi-Civita condition. Again: Levi-Civita condition means that the connection 1-form we pick is metric compatible and torsion free. We do this ad hoc for simplicity in GR. Young's theorem states that partials commute. These are different concepts.
@@milihun7619 It’s not a mistake. In Riemannian geometry, the condition of metric compatibility/being torsion free is equivalent to having the coordinates close, as Dialect wrote above. This follows straightforwardly from the fact that the Christoffel symbols with symmetric indices are equivalent. Stop being a nitpick
Eigenchris was the most helpful RUclips channel for relativity we came across; though it takes some patience as there are no real self-contained videos and it descends steeper and steeper into abstraction the further along it goes. Other than that our physics education has come from the traditional places; i.e. schooling, textbooks, science communication works, and a mashup of popular RUclips channels.
@@dialectphilosophy My question was-give me please another sources on your taste because i like your videos but they are not daily or something, so i need more to feed my interest. Ok, one is nice also, will watch, much appreciated.
We can’t disregard the notion that any point on a sphere from the perspective of the person at that point, is technically the highest point and all adjacent points are at a declination…hence why the exterior of a sphere cannot hold stable a fluid liquid like water, which will always tend towards the lowest point.
@@alexjohnward I don't think anyone can...liquids with the viscosity similar to that of liquid water and lower viscosity (thinner), can only be contained in concave surfaces not convex surfaces.
These are called Riemann-Christoffel Symbols because the Master himself found them about 10-15 years before Christoffel. Though, Riemann did not make it public, because he regarded them as trivialities, not worth a publication.
We haven't looked into the historical aspect of it, but it's often the case that certain features of a mathematical theory are often attributed to individuals who popularized them as opposed to first discovering them.
Most science people on RUclips are confused what gravity is! They keep. talking about Newtonian concepts, That is we are on the earth due to the pull of gravity ie the apple falls due to gravity. This is very confusing. Not many are using Eisenstein relativity and the earth is accelerating towards you.
@@PerpetualScience She claimed curvature can be measured internally in one of her latest videos. But she can't explain why time still ticks at the same rate and angles of triangle are still 180° in the local reference frame of the observer who is falling into black hole.
Hey dialect I haven't caught up on your videos for a while sorry. I love the way you think though and I love the pic of your brain. See I want to take music and use it to teach humans how to think. And I'm trying to get 30% of the population at least up to 200 IQ. You know the whole experiment with the rats were one group of rats after a certain way and another group across the country acted a the same way. I think if I get 30% of the population to 200 IQ the rest of follow. And I'd love to talk to you about calculating the speed of gravity. Because I haven't found anybody trying to calculate it yet and that's the only way you humans are going to get off this planet. Holy Spirit of Humanity and I do recommend you to other people that are geniuses. And I do think you're a genius or I wouldn't be bothering you.
SR wrong due to reference frame mixing and bad math. GR follows as incorrect. “The Final Theory: Rethinking Our Scientific Legacy “, Mark McCutcheon for proper physics including the CAUSE of gravity, electricity, magnetism, light and well..... everything.
We would agree with you, with the caveat that it's the interpretations that are incorrect, not the mathematical formalisms. See our other videos, for instance our recent release, "Einstein Was Wrong". Relativity, both SR and GR, are mathematical formalisms without any real physical interpretation, which is why understanding where the math comes from is so important to us.
@ “We admittedly had to introduce an extension to the field equations that is not justified by our actual knowledge of gravitation”, Einstein.So Einstein ‘knew’ what ‘gravitation’ was (no) and blundered anyway. “One must not conceal any part of what one has recognized to be true”, Einstein. Gravity is simple Galilean relative motion. The earth is approaching- expanding at 16 feet per second per second constant acceleration- the released object (apple). “The Final Theory: Rethinking Our Scientific Legacy “, Mark McCutcheon for proper physics including the CAUSE of gravity, electricity, magnetism, light and well..... everything.
VERY ORIGINAL!!! (but not really) THIS IS EPISODE # 385.402 OF THE SAME SUBJECT. IN OTHER WORDS, THE "WHEEL" IS RE-INVENTED 385.401x WITHOUT IMPROVEMENT.
Regarding the 2-sphere Christoffel calculations, there is a visual TYPO at 30:54: cotv is the value of the Gamma u-u-v christoffel symbol, as shown in the top half of the video, not Gamma v-u-v, as shown in the lower half, or as shown again during the ending title screen of the video. Indeed, the cotv and zero values should have their places swapped in the lower half of the screen at 30:54. Our apologies for this mistake, and thank you to our commenters for pointing it out. One of these days we'll make it through a Christoffel video without some sort of indices typo!
What a masterclass! You have changed the course of Riemannian geometry history for the better with this video. Such concepts have never before been so readily accessible
Right, it clearly gathers both concept and math
Donate money to his channel
There‘s an older video of dialect about this topic from about a year ago, which is also brilliant!
@@xypheli this video is a sequel to that one, I think. The whole series will stand as one of the canonical references of the subject for years to come.
@ that could be true; haven’t watched the first one in ages, but I hope u‘re right🙏
I was drowning in my GR course because they don't give a visual explanation of what the Christoffel Symbol tells you. But now I am saved!! Thank you from the bottom of my heart
When I watch these science, engineering and math videos, the cleverness of these inventions and discoveries is amazing. The focus on the subtle nature must be so intense that only a few humans could achieve it.
@dialectphilosophy Your channel doesn't get enough praise for what you have done. You have made understanding tangent spaces, manifolds, the metric tensor and other concepts in GR so easy and approachable unlike anywhere else online. I spend hundreds of hours, literally, watching lectures online over the years on GR with the Standford Lectures by Leonard Susskind being one of the best. And you have managed to condense the rigour given by Prof Susskind in 30 mins without sacrificing any of the abstract concepts or important points. Your channel is godsent and I hope it grows and gets the appreciation it so deserves. This is probably my third or fourth watch of this video and its not because I don't understand but instead because it is so good at its explanation that I can rewatch it to reinforce the revelations that you have imparted. Keep up the great job that you and your team are doing and I look forward to when your channel reaches the level of appreciation to comparable channels like 3Blue1Brown and Veritasium (your channel is already on that caliber in terms of the quality of your content) which have also done a tremendous job at science communication. I absolutely love this channel, every single video that your team has published is a gem!!
Thank you! Appreciation and feedback from individuals like yourself keeps the channel going and let's us know we're doing something right 🫠
efficient, rational, complete, without resorting to metaphor. Bravo!!
Such a heavy subject handled with the OUTMOST of professionalism!
You guys are first and foremost masters of pedagogy, and storytelling... I wish more and more people find and watch your videos, because you are a very special channel on the entirety of RUclips! As always, I can't wait for more - I like exploring relativity with you guys - keep it up!
One of the best videos I've ever watched - genuinely. Thank you for ths!
This is pure magic. Wonderful, comprehensive presentation and careful explanation. Thank you for this. If only all teaching was this good.
Wow! Talk about a riveting Saturday night show! Thank you for the outstanding quality of this video. Can't wait for the next episode.
Incredible. The connection between parametric space and the manifold is the question of what observers are. That's the crux of the quantum mechanics x general relativity puzzle. We are so close to seeing the full picture, and it is beautiful.
Here's hoping your channel blows up with success!!!
Amazing, this video is a gold mine, thank you so much for bringing this material down to common folks like me.
This is truly an incredible visualization and top notch explanation!
Great vid, love the animations (and the sound effects haha)
wow this is a FANTASTIC explanation. so intuitive! indexing the peaks and troughs of the manifold to a flat surface is such a genius simple way of defining such a surface!
Thank you so much for watching! Remannian Geometry is essentially just cartography, so credit goes to the first map-makers and circumnavigators of the world! ⛵ 🗺→🌍
Fantastic~!!
Thanks a million! Your generous support helps keep this channel going 🥰
this guy is truly great for teaching these deep and with great visualisation
Great storytelling mate. 👍
Incredible work! Thanks
Excellent explanation! Thank you!
Another Great one. Congrats on breaking 100K subs. 🤜🤛🏼
Thanks!
Thanks a million for your support ☺
I love this channel for explanation graphic of high mathematics. Thank you 😊
Thank you for watching!
Very impressive! Thank you so much!
Dialect is pure infotainment ♥️🧠
I have an interesting paper I wrote a bit ago on the symbols as derived in a fractal space manifold using finsler directional geometry.
Oh that's cool! I myself have discovered that GR has fractal spacetimes as solutions. My profile pic, a meromorphic function with fractal structure, is one of them as it's a fractal, and all 1D metric tensor fields are Ricci-flat, I.E. vacuum solutions.
Amazing! Thanks for your work!
NICE Visuals VISUALS visuals animation and explanation! Enough said !
So well and precisely explained, thanks
It's refreshing to see that very advanced mathematics still often breaks down to little more than the basic fundamentals being done lots of times.
Step 1: The triangle and its rules
Step 2: ?????
Step 3: Differential geometry
This poignant comment is both humorous and true 😂
Question... The curvature of space is a mathematical tool that we can dispense with by bending the coordinates of the body towards the planet. The question is why do we need Riemannian geometry? Let's suppose that there is a body that approached a planet and its coordinates were bent... Why do we say that it is a curved space-time?@@dialectphilosophy
Great summary
Great ending question
Thank you a lot. Kindly upload classes on parallel transport and curvature soon.
Amazing!!!
BEAUTIFUL
Outstanding! 💖
This helps so muchhhhhhhhh!!!!!!!!!!!!!!!!!
If διαλεκτ and EigenChris had been on YT around 2009, I might had graduated in GR 🤣
I'm a lot older than you but I will say the same thing If dialect ,eigenChris, Brian Greene, Lawrence Krause , and a special shout out to Leonard susskins theoretical minimum lectures at Stanford, had been around on RUclips when I was 15, I would definitely be doing the dream of my life theoretical physics
Really amazing series. Keep going☺
Currently finish up my Bachelor's in Applied Physics, and hopefully starting my Master's in Astrophysics next year. These videos are gonna be such a great help when I get General Relativity!
All the best! I have done GR just last year, so feel free to comment here for any doubts.
@ey3796 thank you! Which of the subtopics were the hardest for you if I may ask, and what was so challenging about them?
Amazing! You just made that differential equations class I took 16 years ago from a Russian that barely spoke English worth taking. And here I was wondering when that would come in handy. The applications to differential geometry are very interesting. I'm curious if you guys are going to take a look at the Ehrenfest Paradox.
Babe wake up. The new Dan Brown novel just dropped
Thanks for helping that poor little guy trying to navigate a manifold generate his christoffel symbols so that he doesn't get lost!
It's always great when the outsider/skeptic has a far clearer exposition of the mathematics of the problem than anyone in the mainstream can come up with.
They usually look into it more critically, hoping to get a flaw, hence have a deeper insight than mainstream participants that just absorbed everything as it is.
@@chaoticmoh7091 Yes, it's really sad because mainstream science would certainly benefit from its practitioners having an adversarial understanding of their positiosn-- youtube has really driven that point home to me. Watching people like Susskind fumble in his GR lectures over how the Christoffel symbol relates to the convective/material derivative or even a great lecturer like Schuller twist into a pretzel over the twin paradox makes me think we are being held back by the fact that our contemporary academic scientists don't think as critically about what they are saying as they should, perhaps because they are afraid of being grouped as "cranks". I think it is holding us back.
Dialect is only really a skeptic about the whole one way vs two way speed of light thing in Special Relativity(SR). GR lets you do whatever coordinate transformations you want and also has things like ergospheres. As such, that SR debate is considered moot in GR. Dialect is only rebelling against the misconceptions in their head.
amazing video, thank you
What a masterpiece !
It took me a long time to understand this video. An excellent video but I think that I might need to see it twice or more.😅
There's a lot going on in this video! Definitely recommend viewing the prior videos in our Differential Geometry Playlist (linked in the description) if you feel like you're struggling with this one.
@dialectphilosophy for the section of the Christoffel equation at 20:00 by "exterior" basis vectors you mean the vectors on the outside to the vector at the bottom left corner, correct? Likewise, are the three basis vectors actual stand in for neighbouring vectors? Meaning, since each vector in a vector space can be decomposed into components of the basis vectors, you are using the basis vectors to stand in for the actual vectors themselves. Likewise, what do you mean by the interior vectors? The reason i'm asking is that sometimes these terms have a strict mathematical definition or they can just be descriptive (simply the English language meaning) so I don't know which context is being used.
Thanks for asking. Yes, correct, the use of "interior" and "exterior" during that portion of the video is only meant to distinguish these vectors visually; it is not a reference to anything rigorous or mathematical.
As to your second question, the three basis pair comprises all the relevant neighboring bases that are being compared to the original basis pair in order to take the derivatives of the basis vectors and get to the Christoffel symbols along the manifold there. Strictly speaking, they need to be compared in an infinitesimal neighborhood to be most accurate, whereas in the video they are compared approximately over a larger neighborhood (additionally the magnitudes of the vectors have been scaled down for visual clarity).
Remember that what these basis vectors actually "are" are "bar scales", that is, they take us from magnitudes and directions in the parametric space (the "map" of the manifold) to actual the magnitudes and directions along the manifold.
@dialectphilosophy Thanks for the clarification
was waiting ... masterclass...
Damn this video is amazing I'm blown away
Amazing video; this is super helpful for crystallizing this knowledge. Looking forward to next getting a better geometrical understanding of Lie brackets/torsion?
Yes indeed, that'll fall under the territory of the concept of connections. Thank you for watching!
Thank you!
This was awesome 😎
I love this
right on time parallel to my GR lecture currently lol
This is genuinely one of the most incredible maths videos I've seen - school is so finished with people out here teaching like this for free on the internet!
I absolutely agree...I attended a prestigious university and this video blows all that out of the water. What one can learn just in this one video covers what years at a University failed to teach.
Very nice! There is just a little mistake when you grouped the results at time 30:54, where you swapped the values of Tvuv and Tuuv.
Oh gosh crap 😤 Thank you for pointing that out.
Sehr gut
Hey Dialect, can you do a video on the Michelson-Morley experiment? In your other videos you talk about the aether and how even Einstein later in life said he was open to it, but the experiment seems to me to show that the aether does not exist. I’ve tried to read some of the explanations online that try to disprove the null result of the experiment but I don’t quite get them.
Where was this video during my Diff Geo class?
The math is cool but I was hoping for a big picture explanation, like how do the symbols help?
Have been watching the video for 25 minutes now
I'm starting to consider the possibility that I should know what a manifold is before watching this one...
Mathematicians will give you very abstract definitions of a manifold; however for the purposes here, they're nothing more than just a surface that can be curved or flat. If you find this video too difficult (and it certainly cannot evade that accusation) we recommend starting with the prior videos in our Differential Geometry playlist. Thanks!
@dialectphilosophy oh, thanks for the insight!
I had thought this was an independent video when I started watching it, binge watching all your channel is definitely gonna make it easier for me to understand, these videos have an astonishing quality and I'm really glad I came across your channel!
You should look into Complex-Riemannian Manifolds. They're the same as Real-Riemannian Manifolds except that they allow complex coordinates and components and they must have complex structure. I made a post about them on my blog.
That sounds neat! You should hop on our discord and share your blog post there... there would definitely be others interested in learning about that as well.
@@dialectphilosophy Ok, will do!
Glad to see that your trashing-fellow-youtubers is something of the past. Excellent class.
All one would have to consider is instead of being limited to only one observer, rather having two observers should clearly indicate the second observer getting “shorter” as their feet start to vanish as they travel in any direction away from the initial observer. This would indicate motion in a 3rd dimension or from a strictly flat flat lander perspective the gradual vanishing of the second person from bottom up in relation to the distance traveled by the second person. Oh also a rotation in the direction traveled by the second person with 0° being the angle from the normal vector of the initial person and increasing positively in the direction of travel of the second person in relation to the intital person. An effect never observed on the ball earth.
Will you do a video on the covariant derivative?
Yes; it requires a bit more exposition than what was given here. However, we can tell you that the derivative vectors shown in this video that lay in the tangent plane on the sphere were all covariant derivatives. When dealing with an extrinsic curved 2D surface, a covariant derivative is simply a vector's extrinsic derivative along the manifold with the normal component thrown away or subtracted out, as depicted in the video at 10:10
I've always wondered how would a world with Lp norm/metric look like (distance(x;y) = (x^p + y^p)^(1/p)) ; is there a way to calculate Cristoffel symbols for that and see the curved space ?
You certainly could! However, you would need to define something analogous to an angle in this case. One route to doing such would be through the inner product. So if a dot a equals a^3, and b dot b equals b^3, what does a dot b equal? Once you've answered that question, you can relate your one-dimensional lengths to two-dimensional areas, and build the components of your metric tensor.
Once you've built the components of the metric tensor, you can use the exact titular equation here for the Christoffel symbols to solve for how the basis vectors will change with changes in coordinates. From there you could in turn calculate curvature. However, "seeing" this curved space would be difficult, since what we can "see" or physically visualize is only euclidean metric spaces.
@@dialectphilosophy By "seeing" I mean see the curved manifold on which this metric is "applied" (like we can see the sphere which is the curved manifold on which spherical metric applies) ; I'm not sure what I'm saying makes sense by the way ^^'
And thanks for the answer and the tremendous work you put into your videos, I have the feeling you're accomplishing something which can become really big !
Hold up, im lost, aren't the e_v and e_u basis vectors changing continuously?
yes, the metric tensor is a function of the where you are on the manifold. The Christoffel symbols involve derivatives of the basis vectors (i.e. how they change when taking an infinitesimal step)
@ Dawg, the segment on the recap of the first video, shows the stick person moving one basis vector in parametric space, and a single corresponding, transformed vector in Cartesian space. It did not intact, change continuously.
@@DR_Sam_RUclips
Continuous change is approximated by tiny discrete steps. Calculus.
@ That gives a bad intuition of what’s actually happening then, not something I would appreciate trying to learn something like differential geometry.
@@DR_Sam_RUclips
The picture here is much more intuitive than any other explanation I've seen
Idk why but your video felt like an quantum insight too, is it by any means related to lie alzebras?
Is this a mathematical interpretation of how terrain following radar in aircraft works?
My understanding is that this is differential geometry 101 and that this is in all introductory textbooks on general relativity.
Buuuuhhh!!! That’s a super tight video. I just wish you’d go over how the equations arise from their equivalents which you claim “we know that…equals…” but regardless this is truly a masterclass. The only other issue I have here is when assuming a “flat lander” on a curved surface is a violation of the flat landers 2D universe. I am under the notion that a straight line is not a straight line on a curved path because when viewed perpendicularly that straight line on a curved path is a curve and or a circle if the manifold is spherical. Just because the flat lander doesn’t turn left and right doesn’t mean he’s going in a straight line because on a spheres surface if there is another person at his original location, when person one walks in a “straight” line he’d notice he is somehow rotated and seems lower in relation to person two at the original point.
Thank you for watching! The concept of a flatlander is an interesting topic and requires a much greater dive into mathematics and philosophy than we had time for here. Our use of the term flatlander here means that the observer "lives" in two dimensions and is not aware of the existence of a third dimension.
The video probably makes this confusing because our observer is three-dimensional and poking out of the manifold, when they should in reality be a flat two dimensional thing living in the manifold. It just didn't look as cool to have a 2d observer sliding around the manifold as it did to have a 3d one walking around atop it 🙃
@@dialectphilosophy I thank you for your response and I want to clarify, I did not want to come off like as if I am criticizing your video. THIS VIDEO IS AN ABSOLUTE MASTERPIECE 🤯 the ideas you present are better described here than I've ever seen anywhere else and all through University...what I meant with my comment was I just wanted to bring awareness to this notion of "curved" 2-Dimensional Planes...I am firm believer that curvature cannot exist for any plane unless a third dimension exists to curve it in... Two dimensional Planes can only have 1 dimensional lines possess curvature within the 2-Dimensional Plane. For the same reasoning as before, if you're only in a 1-dimensional line, the only way that line can have curvature is by introducing a second dimension otherwise it has nowhere to curve.
Can this stuff be applied to game development
Dialect, you are mixing up two very important concepts here; the Levi-Civita condition and the Schwarz-Young-Clairaut theorem. The first one states for two vector fields X and Y that nabla_X(Y)-nabla_Y(X)=[X,Y], where nabla_X(Y) is the covariant derivative of Y along X, and [X,Y] is the Lie bracket/commutator. The 2nd one states that basis vectors/partial derivatives commute; [partialx;partialy]=0. These things look similar, but are completely different concepts. In more advanced theories, like Einstein-Cartan, the Levi-Civita condition no longer holds, but coordinate lines still have to close on the manifold, fx.. (I can elaborate on this and the importance of it in advanced theories, if someone is interested.)
Moreover, the Levi-Civita condition is more than just this statement; it also states on top of the zero torsion condition, that the connection 1-form is metric compatible, nabla(g)=0, where g is the metric tensor.
Thank you for the clarification. We do understand there is a lot of subtley surrounding this concept, so we tried to mitigate it in the video by stating that the Levi-Civita condition requires the coordinates close "in this context" i.e. within the context of Riemannian geometry. We would however definitely love to hear more about the difference between these concepts as they pertain to more abstract geometries, as it's hard to understand how the coordinate lines could close without basis vectors commuting; so please feel free to reach out to us on our discord and elaborate more about it there!
@@dialectphilosophy No, you don't understand my comment. The mistake is at 17:48. The Lie commutator of the basis vectors have to commute for coordinate lines to close, but this has nothing to do with the Levi-Civita condition. Again: Levi-Civita condition means that the connection 1-form we pick is metric compatible and torsion free. We do this ad hoc for simplicity in GR. Young's theorem states that partials commute. These are different concepts.
@@milihun7619 It’s not a mistake. In Riemannian geometry, the condition of metric compatibility/being torsion free is equivalent to having the coordinates close, as Dialect wrote above. This follows straightforwardly from the fact that the Christoffel symbols with symmetric indices are equivalent. Stop being a nitpick
Hey, Dialect, do you have any physics channels you sub for? Could you tell us a couple?
For going indepth into mathematical physics, @eigenchris is a good one.
Eigenchris was the most helpful RUclips channel for relativity we came across; though it takes some patience as there are no real self-contained videos and it descends steeper and steeper into abstraction the further along it goes. Other than that our physics education has come from the traditional places; i.e. schooling, textbooks, science communication works, and a mashup of popular RUclips channels.
@@dialectphilosophy My question was-give me please another sources on your taste because i like your videos but they are not daily or something, so i need more to feed my interest. Ok, one is nice also, will watch, much appreciated.
How in the world A person discovers steps like this. How does a person come up with this idea?
We can’t disregard the notion that any point on a sphere from the perspective of the person at that point, is technically the highest point and all adjacent points are at a declination…hence why the exterior of a sphere cannot hold stable a fluid liquid like water, which will always tend towards the lowest point.
you can't.
@@alexjohnward I don't think anyone can...liquids with the viscosity similar to that of liquid water and lower viscosity (thinner), can only be contained in concave surfaces not convex surfaces.
👏👏👏👏
❤
Heavy Duty Niftiness
These are called Riemann-Christoffel Symbols because the Master himself found them about 10-15 years before Christoffel. Though, Riemann did not make it public, because he regarded them as trivialities, not worth a publication.
We haven't looked into the historical aspect of it, but it's often the case that certain features of a mathematical theory are often attributed to individuals who popularized them as opposed to first discovering them.
Math major!
Lol
Most science people on RUclips are confused what gravity is! They keep. talking about Newtonian concepts, That is we are on the earth due to the pull of gravity ie the apple falls due to gravity. This is very confusing. Not many are using Eisenstein relativity and the earth is accelerating towards you.
honey who here in 2024 dialect just skibidi
get ready for the einstein rizzler no cap kai cenat gonna steal my christoffels
omg, we so cooked with this pookie 🗣️ 🔥 🔥 🔥
what
💀?
Woahbb
Isn't the point of GR geometry. So why introduce such non geometric objects.
Sabine Hossenfelder dislikes this video
The more you know the more obvious it becomes that people like her are really projecting their own insecurities
Why would she?
😂
@@PerpetualScience She claimed curvature can be measured internally in one of her latest videos. But she can't explain why time still ticks at the same rate and angles of triangle are still 180° in the local reference frame of the observer who is falling into black hole.
Id quess she might like this.
TFTC
Hey dialect I haven't caught up on your videos for a while sorry. I love the way you think though and I love the pic of your brain. See I want to take music and use it to teach humans how to think. And I'm trying to get 30% of the population at least up to 200 IQ. You know the whole experiment with the rats were one group of rats after a certain way and another group across the country acted a the same way. I think if I get 30% of the population to 200 IQ the rest of follow. And I'd love to talk to you about calculating the speed of gravity. Because I haven't found anybody trying to calculate it yet and that's the only way you humans are going to get off this planet. Holy Spirit of Humanity and I do recommend you to other people that are geniuses. And I do think you're a genius or I wouldn't be bothering you.
Flatlanders lack acces to their reference frames 😂
Skibidi geometry
I fucking hate math ❤
You have to calculate 192 derivatives. This shit is stupid. "If it aint broken dont fix it"
Shimmisupremacy
christ these are aweful symbols!
SR wrong due to reference frame mixing and bad math. GR follows as incorrect. “The Final Theory: Rethinking Our Scientific Legacy “, Mark McCutcheon for proper physics including the CAUSE of gravity, electricity, magnetism, light and well..... everything.
We would agree with you, with the caveat that it's the interpretations that are incorrect, not the mathematical formalisms. See our other videos, for instance our recent release, "Einstein Was Wrong". Relativity, both SR and GR, are mathematical formalisms without any real physical interpretation, which is why understanding where the math comes from is so important to us.
@ “We admittedly had to introduce an extension to the field equations that is not justified by our actual knowledge of gravitation”, Einstein.So Einstein ‘knew’ what ‘gravitation’ was (no) and blundered anyway. “One must not conceal any part of what one has recognized to be true”, Einstein. Gravity is simple Galilean relative motion. The earth is approaching- expanding at 16 feet per second per second constant acceleration- the released object (apple). “The Final Theory: Rethinking Our Scientific Legacy “, Mark McCutcheon for proper physics including the CAUSE of gravity, electricity, magnetism, light and well..... everything.
VERY ORIGINAL!!! (but not really) THIS IS EPISODE # 385.402 OF THE SAME SUBJECT.
IN OTHER WORDS, THE "WHEEL" IS RE-INVENTED 385.401x WITHOUT IMPROVEMENT.
Thanks!
Thank you so much for your incredible generosity and support! You keep us motivated 😁
Thanks!