OMG i'm using that skirt comment forever. instead of "thanks, it has pockets!" it'll be "thanks, it's hyperbolic!" (amazing they can fit all that extra area with negative curvature and still no room for pockets)
Great video! I liked the bit clarifying the Ricci tensor. I knew what the scalar curvature and Riemannian tensor meant, but I didn't know there was a meaning to the Ricci tensor. That was really nice to find out.
This is why, imho a cardioid best describes the overall curvature of the universe. There seem to be limits, deep voids and event horizons. The point tangent to circular orbit, and the node where the tangent is 90° to circular, radial to origin.
Given a 2D chart, wouldn't the first definition of curvature be where the derivative of a line isn't constant? Therefore where the second derivative isn't 0? It seems like that would be a concept which could be extended beyond two dimensions, although the object that would describe might actually require more dimensions to describe it. I feel like that's probably what Riemann Curvature is probably describing in a more formal way. It also seems like this only applies to Euclidean space. If you use a non-Euclidean Geometry, as I would argue might be a better model for the Universe, these curvatures might only be localized and there may be a hidden curvature which can't be directly observed. While I recognize that the inverse square law for gravity strongly suggests that there isn't an additional space dimension, I wonder if that is in part because our current definition of curvature is beholden to that property. I think things are more interesting when space is really a 4+ dimensional Klein Bottle of infinite size. Locally I think that would be indistinguishable from a 3 dimensional space. I thought the carrier for gravity might be in that higher dimension, and maybe it is, but that has other consequences I'm not yet ready to recognize.
Your intuition of using derivatives is actually nearly on the money. The problem is that fake curvature can be made if you use a non-rectilinear coordinate system, which I explore in my video on Special Relativity. For hidden curvature, you bring up cool points which Einstein himself actually addressed! So far, the definition of curvature is based on differential geometry which is local. In practice, it means the way 4D spacetime looks like in a 5D space, is completely unknown! However, as only local curvature affects things within the space, for the purpose of physics, we can ignore this greater question. Something for mathematicians to figure out for us ;)
Nice video! Thanks! @7:18 you said that Ricci curvature is basically scalar curvature but with a direction but the subtitle says, it's sectional curvature instead of scalar curvature. Which one is correct?
This was a superb presentation. I would like to know how to calculate the curvature of spacetime though. How does parallel transporting vectors on a curved triangle reveal the amount of curvature?
The mathematics is done by doing this vector dragging with derivatives. If you think about it, a derivative is about taking a tiny step in one direction, and observing the change that makes in another variable. You can set up vector (tensor) derivatives to derive against X, then Y, then backwards on X then backwards on Y to drag a vector around a parallelogram. In a flat surface/coordinate system, this gives zero. In a curved manifold, this won't be zero somewhere, and just like ordinary derivatives, the function can tell us where. More detail can be found on eigenchris' amazing TH-cam series on curvature, or from searching Riemann Curvature Tensor on Wikipedia.
5:49 this is breaks my flat and linear brain - how can you translate a vector in parallel way in a space where you have ether no parallel lines at all or infinite sets of them? The parallel translation of a vector should be undefined there, but you are trying to define something through it. How is that suppose to work?!
You are actually pretty sharp. Vectors are only be able to exist in a flat vector space, and surely there are many arbitrary ways to parallel transport a vector. The key is to both translate the vector, and the vector space underneath it. On a geometric level, a manifold (curved surface) can be approximated by an infinite number of flat surfaces. An example of a 1D manifold is a function on a graph, which we can display on flat 2D, one dimension higher. Imagine x^2, and a vector parallel to x=1. Say I wanted to parallel transport it to x=0. I know I need to rotate it by the derivative, though lengths won't be preserved very well. This approximation works better the smaller the gap between the X values. In fact, it produces a unique and linear method to parallel transport in all dimensions (embedding the space into a dimension one higher, taking the derivative in that space, and using it to move vectors). Eigenchris made a really good video series going into it with more maths, detail, and visuals.
@@mindmaster107, it's not sharpness, I just was wrecking my brain thinking about how to describe vector spaces on curved surfaces since 6 years ago. I think I understand calculus good enough but I never learned differential geometry in the university. I really haven't learned a lot through university courses so I'm trying to reteache myself all the math. Now I'm going through the "Elementary Calculus: Infinitesimal Approach" and I wanna teach myself Geometric Algebra in parallel to define multivariable calculus on bivectors and multivectors, so Physics would be really interesting after that. I think I have capacity to understand how the curvature works but I wanna define it through the stuff I know and I don't know differential equations and I expect I need them for this.
4:55 "The only place (...) is the North pole" Not really, there's one other place some 1.16km from the South pole, for example. It's my favorite part of the riddle.
No bears on Antarctica. 🤷 If we really want to be pedantic, Polar bears have clear fur which only looks white. So the better question isn't what color was the bear, but what color is its fur. You're bound to make twice as many friends at the party.
Not just one - there's an infinite sequence of circles close to the south pole which work! Provided you can somehow find a bear, that is. You just need to make sure that after walking 1 km south, you end up in a place where the “circumference” of the Earth is 1 km divided by an integer. That way, you end up walking around the south pole an integer number of times before walking back north to where you started.
Perhaps the claim that relativity would have never been discovered if not for its prerequisites is a bit much? Or is this just semantic/philosophical, whether you count is as discovered or invented... I guess its also how you read into that, whether you read "if not prerequisites by this point => relativity never" or probably, closer to the intended "prerequisite before relativity, but could happen later"... maybe its just unfortunate wording, anyway, cool vid
General relativity almost certainly wouldn’t have come about without differential (Riemannian) geometry, as Einstein himself commented on during interviews. On the topic of discovered/invented, the discovery would be observing things like black holes. Invented would be the maths used to theoretically predict them. Physics being halfway between discovery and invention makes it quite fun to study, and something I plan to make a video on as well.
@@mindmaster107 Wouldn't have come about in his time? What I meant was, surely, even without Riemann and Einstein, given enough time and vaguely monotonic progress, eventually some other smarty-pants is bound to come up with a Riemann-like theory after which someone else can come up with an Einstein-like theory, maybe its not described quite the same way, but if there is a truth to be found and we're capable of finding it, we eventually find it?
I guess as a physicist myself, that statement feels a little bit entitled, given how hard research actually is to find these theories. The universe gives no hints, and thats why research is so hard… and so satisfying. Imagine challenging the universe, and winning. That and trying to get funding. Physics research, both the how and why, is something I’m going to make a video on, so look forwards to that!
![ILLSLICK - KILLSHOT REMIX [FULL VERSION]](http://i.ytimg.com/vi/RIK8RrqIAzE/mqdefault.jpg)
Wow! This video has helped me massively. Your presentation style is so engaging. Thank you so much!!
OMG i'm using that skirt comment forever. instead of "thanks, it has pockets!" it'll be "thanks, it's hyperbolic!" (amazing they can fit all that extra area with negative curvature and still no room for pockets)
Great video! I liked the bit clarifying the Ricci tensor. I knew what the scalar curvature and Riemannian tensor meant, but I didn't know there was a meaning to the Ricci tensor. That was really nice to find out.
It's quite amazing all of these abstract things have some physical analogue!
More things to love about diff geometry
ScienceClic's videos on the mathematics of General Relativity might help you some more: th-cam.com/play/PLu7cY2CPiRjVY-VaUZ69bXHZr5QslKbzo.html
This is why, imho a cardioid best describes the overall curvature of the universe.
There seem to be limits, deep voids and event horizons.
The point tangent to circular orbit, and the node where the tangent is 90° to circular, radial to origin.
Criminally underrated!
This video is great! It expanded my understanding of GR
Given a 2D chart, wouldn't the first definition of curvature be where the derivative of a line isn't constant? Therefore where the second derivative isn't 0? It seems like that would be a concept which could be extended beyond two dimensions, although the object that would describe might actually require more dimensions to describe it. I feel like that's probably what Riemann Curvature is probably describing in a more formal way. It also seems like this only applies to Euclidean space. If you use a non-Euclidean Geometry, as I would argue might be a better model for the Universe, these curvatures might only be localized and there may be a hidden curvature which can't be directly observed. While I recognize that the inverse square law for gravity strongly suggests that there isn't an additional space dimension, I wonder if that is in part because our current definition of curvature is beholden to that property. I think things are more interesting when space is really a 4+ dimensional Klein Bottle of infinite size. Locally I think that would be indistinguishable from a 3 dimensional space. I thought the carrier for gravity might be in that higher dimension, and maybe it is, but that has other consequences I'm not yet ready to recognize.
Your intuition of using derivatives is actually nearly on the money. The problem is that fake curvature can be made if you use a non-rectilinear coordinate system, which I explore in my video on Special Relativity.
For hidden curvature, you bring up cool points which Einstein himself actually addressed! So far, the definition of curvature is based on differential geometry which is local. In practice, it means the way 4D spacetime looks like in a 5D space, is completely unknown! However, as only local curvature affects things within the space, for the purpose of physics, we can ignore this greater question.
Something for mathematicians to figure out for us ;)
Awesome! That's a really nice explanation of the various kinds of curvature. I'll be sharing this for sure 🙂
I can't believe this only has 182 likes.
Nice video! Thanks! @7:18 you said that Ricci curvature is basically scalar curvature but with a direction but the subtitle says, it's sectional curvature instead of scalar curvature. Which one is correct?
Oh my goodness my subtitles are wrong thank you so much for noticing
This was a superb presentation. I would like to know how to calculate the curvature of spacetime though. How does parallel transporting vectors on a curved triangle reveal the amount of curvature?
The mathematics is done by doing this vector dragging with derivatives.
If you think about it, a derivative is about taking a tiny step in one direction, and observing the change that makes in another variable.
You can set up vector (tensor) derivatives to derive against X, then Y, then backwards on X then backwards on Y to drag a vector around a parallelogram. In a flat surface/coordinate system, this gives zero.
In a curved manifold, this won't be zero somewhere, and just like ordinary derivatives, the function can tell us where.
More detail can be found on eigenchris' amazing TH-cam series on curvature, or from searching Riemann Curvature Tensor on Wikipedia.
@mindmaster107 derivatives, of course! Thank you so much
Thank you, mate, really enjoying your videos!
5:49 this is breaks my flat and linear brain - how can you translate a vector in parallel way in a space where you have ether no parallel lines at all or infinite sets of them? The parallel translation of a vector should be undefined there, but you are trying to define something through it. How is that suppose to work?!
You are actually pretty sharp. Vectors are only be able to exist in a flat vector space, and surely there are many arbitrary ways to parallel transport a vector.
The key is to both translate the vector, and the vector space underneath it.
On a geometric level, a manifold (curved surface) can be approximated by an infinite number of flat surfaces.
An example of a 1D manifold is a function on a graph, which we can display on flat 2D, one dimension higher.
Imagine x^2, and a vector parallel to x=1.
Say I wanted to parallel transport it to x=0. I know I need to rotate it by the derivative, though lengths won't be preserved very well.
This approximation works better the smaller the gap between the X values.
In fact, it produces a unique and linear method to parallel transport in all dimensions (embedding the space into a dimension one higher, taking the derivative in that space, and using it to move vectors).
Eigenchris made a really good video series going into it with more maths, detail, and visuals.
@@mindmaster107, it's not sharpness, I just was wrecking my brain thinking about how to describe vector spaces on curved surfaces since 6 years ago. I think I understand calculus good enough but I never learned differential geometry in the university. I really haven't learned a lot through university courses so I'm trying to reteache myself all the math. Now I'm going through the "Elementary Calculus: Infinitesimal Approach" and I wanna teach myself Geometric Algebra in parallel to define multivariable calculus on bivectors and multivectors, so Physics would be really interesting after that. I think I have capacity to understand how the curvature works but I wanna define it through the stuff I know and I don't know differential equations and I expect I need them for this.
4:55 "The only place (...) is the North pole" Not really, there's one other place some 1.16km from the South pole, for example. It's my favorite part of the riddle.
No bears on Antarctica. 🤷 If we really want to be pedantic, Polar bears have clear fur which only looks white. So the better question isn't what color was the bear, but what color is its fur. You're bound to make twice as many friends at the party.
@@R.B. Unless you brought the bear with you. And painted its fur purple first.
Not just one - there's an infinite sequence of circles close to the south pole which work! Provided you can somehow find a bear, that is. You just need to make sure that after walking 1 km south, you end up in a place where the “circumference” of the Earth is 1 km divided by an integer. That way, you end up walking around the south pole an integer number of times before walking back north to where you started.
Perhaps the claim that relativity would have never been discovered if not for its prerequisites is a bit much? Or is this just semantic/philosophical, whether you count is as discovered or invented... I guess its also how you read into that, whether you read "if not prerequisites by this point => relativity never" or probably, closer to the intended "prerequisite before relativity, but could happen later"... maybe its just unfortunate wording, anyway, cool vid
General relativity almost certainly wouldn’t have come about without differential (Riemannian) geometry, as Einstein himself commented on during interviews.
On the topic of discovered/invented, the discovery would be observing things like black holes. Invented would be the maths used to theoretically predict them. Physics being halfway between discovery and invention makes it quite fun to study, and something I plan to make a video on as well.
@@mindmaster107 Wouldn't have come about in his time? What I meant was, surely, even without Riemann and Einstein, given enough time and vaguely monotonic progress, eventually some other smarty-pants is bound to come up with a Riemann-like theory after which someone else can come up with an Einstein-like theory, maybe its not described quite the same way, but if there is a truth to be found and we're capable of finding it, we eventually find it?
I guess as a physicist myself, that statement feels a little bit entitled, given how hard research actually is to find these theories. The universe gives no hints, and thats why research is so hard… and so satisfying. Imagine challenging the universe, and winning.
That and trying to get funding.
Physics research, both the how and why, is something I’m going to make a video on, so look forwards to that!
@@mindmaster107 Carry on the noble pursuit my friend, the fun is in the chase anyway
Food HMS