Thank you, Dr. Caroll. As a matematician, it's perhaps the best explanation of a homotopy groups to a layman I've ever seen. And in the case if you're interested: - Bolyai was Hungarian and Lobachevsky (Лобачевский) was Russian. Actually we in Russia usually refer to hyperbolic geometry as "Lobachevsky geometry". - Yep, homeomorphisms are defined as continuous bijective maps, not necessarily smooth ones. - Technically speaking, the spaces you're working with when you speak of homotopy etc doesn't even need to be manifolds. But it's probably too much of a rigor :)
As someone with a bachelors, masters, phd and a postdoctoral in nuclear physics I can attest that these lectures are superb and the best thing on the internet right now covering the topic. Kudos Sean! Just bought the first book of the series and anxiously anticipating the other two.
"lighten up, experts" :D ...and that is precisely why it is so hard to find a good class - it is hardly ever fun, but this is. Sean, I LOVE this! I'm not too bad at geometry - but always felt too intimidated (mostly by the 'experts' in my class) to actually pursue a scientific career. Turns out I have been using parallel transport all along in gamedevelopment for steady camera motion along a path :)
Every class -- from math to software engineering. There's one guy in the front row who gets pedantic on every minuscule point. There's another guy in the front row who does not know what a pointer or vector is.
This lecture is one of the best explanations of particle physics and cosmic divergent galaxies in terms of the Theory of General Relativity. The Riemann curvature tensor is highly well defined here.
Sean is by far my favorite intellectual and specifically, theoretical physicist. We are super fortunate to have you, thanks for your wealth of generosity for bringing us this knowledge and humanity for making it accessible and understandable! I hope someday to catch a talk of yours in person, that would really be something!
Thanks for making more advanced videos! I was just listening to Eric Weinstein talk about how we need more advanced physics information out there for the general population vs the usual pop-sci physics stuff, and this series is definitely setting the bar high on advanced educational content!
Dr. Carroll - I've watched many of your videos and you have inspired me in many ways. That being said, that you referred to Gauss as a "dick" was the coolest. You are human after all. You rock, sir.
Thank you so much for these. As a hobbyist and someone who never retained any of my math education, attempting to find a clear definition of a Riemann Curvature Tensor or any similarly complex concept has proved very difficult. I'd be very interested if you made these lec..videos into a book. Kind of like a 'Road to Reality' except for people with smaller hat sizes.
The reason why hyperbolic geometry was the first non-Euclidean geometry discovered, is that it is easy to show that no parallel lines is inconsistent with the other axioms as they were then currently formulated of Euclidean geometry. For example there are an infinite number of different lines between the north and south pole of a sphere which contradicts the first postulate of Eucliden geometry - two distinct points determine a unique line. So this is why the focus was on many (infinite) number of parallel lines through a point not on the line and parallel to the given line.
All these videos are profoundly informative. You aren't going to get this level of knowledge from most other videos on UA-cam with maybe the exception of Science Asylum, Veritasium and Ask a Space Man. Great work Sean!
For some reason when he said "So, there's good news and bad news when it comes to topology" around 52:45 that struck me as hilarious for some reason - I love the off-the-cuff style of these lectures.
Yes. People, even a major at maths, should have to recognize pedagogic excellence, especially about vectors and high level maths, I saw young people fight and fail, fight...because math is not easy, even to those that understand the concepts but cannot do the calculations, nor those who don't see in three, four or more dimensions and suffer for that. In metric fields ...What is keeping a parallel postulate, Riemann Curvature tensor parallel transport.....the connection, the curvature...smoothly deformed spaces, topological invariants.
One hurdle I had with understanding non-Euclidean geometry was a notion of line. I was so used to “straight line” type of thinking that it was really hard to imagine that whole geometry would not blow up if lines were not straight. To my surprise I found that Euclid didn’t give definition of a line. He defined it algebraically by describing properties any line has to have such as “To draw a straight line from any point to any point.” (thanks Wiki). This made me appreciate how advanced ancient Greeks were because it looks very much like modern math. Once I realized that “line” is anything with requested properties it became easier to understand other geometries. It seems we can even have y=x^3-lines and Euclidean theorems would still hold. Amazing level of generality!
This series is astoundingly good. Thank you very much for your time,Dr. Could you show a bit of the math about parallel transport in the Q&A? For example, do parallel transported vectors change their length when changing direction? Maybe a radial velocity becomes tangential velocity in a curved spacetime?
Leibnitz tried to prove the parallel postulate by a proof of contradiction - by using a different postulate and looking for contradictions. But when he discovered that the resulting geometry was perfectly free from contradictions he was certain he had made a mistake and never published it - which says something about the respect people had for Euclid. We've found it in his personal papers.
@@SrValeriolete I think it's definitely a combination of the two. Leibnitz came from a period that would have would have placed the "classical genuises" on a very high pedestal.
Thanks, Sean! If you want a 'one word' for the videos - call each a 'presentation' on the Relevant Topic! Avoids 'lecture', which for some (not me!) has ominous memories of exams etc.
The 5th postulate is a little more general: It says that if the angles in the interior add up to exactly 180, then they are parallel. But it should be noted that both angles don't have to be 90 degrees. For example look at this figure ( =/= ). If the diagonal line that cuts through the equal sign, forms two angles that add up to 180 (same side), then they are parallel. Just a fun fact! =D This episode was one of my favorites!
fab. it never occurred to me that you couldn't compare two vectors at different points in space without bringing them together . lol. i find that genuinely very very thought provoking :) all wrapped up with locality and what something being space really means
I don't understand the parallel transport example on the sphere. The orange vector starts out pointing straight up, but as we progress up the geodesic he has it lean over more and more, which isn't keeping it parallel. If you keep it parallel then it will still be pointing straight up when you get to the north pole. From this I conclude that parallel transport doesn't keep the vector parallel with its original value. So what does it keep it parallel with?
Of course they each are huge subjects which deserve videos of their own-they're not gonna get them; I tried to squeeze both of them into a single video. -Sean Carroll, Physicist
I love learning about the history of these ideas and the people that brought them to us. I love your explanation of the Reimann theory. I love it and I appreciate it. I want to know this. ❤️
Are the Gaussian circles rings or disks? I'm talking about the section where Gauss speaks of seeing geometry from the point of view of someone living on the circle.
Could you give a short motivation on (co)homology groups as well in the Q&A please? I struggle to get an intuitive approach there. Thank you for this series!
I found this good. It is not what you might think of as motivation, but it worked for me. Good presentation of *calculating* the groups. ua-cam.com/video/YNBi4Ix3cY0/v-deo.html
I found the way you and Riemann think about space utterly complicated. I usually think of space as a infinitesimal graph (points and links considered equal and random at the smallest scale).. so the shortest path is the shortest sequence of links (and the length is the number of links).. the straight line is the links that lead the furthest (using the shortest path definition) from some other point that defines your direction (here, a point alone doesn't carry direction).. etc.. there's can't be "unparallel transport of vector" without curvature.. there can't be "rotation" of a point (since a point has no dimension).. there can't be a vector or angle definition without multiple points (for the same reason)
So I clearly went off on a tangent, learning this via quantum mechanics* rather than my usual field (3D / physically-modelled computer graphics)... but honestly, this is the first explanation of non-Euclidian geometry I've ever understood. I've been using vectors in similar ways for so long now that - seeing parallel transport demonstrated like this - I can't believe this didn't dawn on me long ago (I was never good at 'math theory', but if I can visualise it in my head, I get it just fine). The topology stuff I could imagine including at some point in the near future as well; for example, mapping textures to arbitrary geometry, possibly using curvature tensors to project texels in 3-space based on surface normals. * Thanks Sean; I got here via some of your quantum mechanics talks, and I think I may be hooked.
Very informative and very well done, it shows that you sir are a teacher! :) There were a lot of things that I finally understood and others that I heard of for the first time. It is an art to put together basic and high knowledge and the mix to be understandable by any listener... I really wish that you sir never stop doing this series but I know that sometime in the future you really have to move on...:)
Enjoying these lectures very much! Actually feel I can begin to understand these topics better, and the Topology part seems as if it foretells the development of string theory(?)
Thank you for this wonderful series, Professor Carroll. There is one thing in this video that would like to comment on: at multiple places, 1:04:46 for instance, you referred to a member of the homotopy groups as "topologically equivalent maps", which I found a bit misleading. The members of the homotopy groups are "equivalent classes" of maps, instead of maps themselve. Any two "topologically equivalent" maps in fact represent the same member in the homotopy group. I think this should be pointed out as it is somewhat important for what follows, and it is not too hard for the non-professionals.
If there is no 'natural' or 'canonical' way of defining parallelism, i.e. of saying which vector at one point is 'in the same direction' as another vector at a nearby point, then what constrains our definitions of parallelism and therefore of curvature? Given any curve, can't I just define the velocity vectors to the curve at each point to all be parallel to each other, and thus the curve is trivially straight? But then I can make any curve at all 'straight' and the concept seems to lose meaning. If the metric determines the connection, which defines parallelism, which determines curvature, then what determines the metric? Doesn't the metric represent a coordinate system, and are we not free to choose any coordinates we like? But perhaps it is whether or not you can find a coordinate system in which the metric is the Minkowski metric that tells us about curvature or flatness.
If this goes on, Sean will end up with hair like the much-used photo of Albert Einstein :) Apart from that, I think Seans videos in general is amazing because I actually understand stuff, that I didn't expect myself to understand... (or rather.. I understand what leads to the theories, even when some of the theories are hard to wrap your mind around because they are counter-intuitive...)
Agreed! He's working on a QM textbook. Can't wait for that. Would also like a QFT book from him. He should do what Susskind did and work these videos up into a book series. He's so good at explaining without leaving the important stuff out.
So a Tensor is basicly a equation field (for some quantity) that you apply to every (or some) degree of freedom in a certain space and get an answer, wether its air pressure in the atmosphere, even a frequence in a song or curvature in some dimensional space, at a certain point (or any point in this certain space) ?
Questions for the Q&A: Is the first homotopy group of a 1-Sphere mapped to Euclidean 3-space (a circle in Newtonian space) trivial? It seems like the winding number of a circle around a missing point is irrelevant, as it can just go 'above' or 'below' the missing point to avoid it as it smoothly transforms. (This would generalize to an n-Sphere mapped to an n+2 space, I assume?) Some versions of the story Physics tells of reality depict black holes as *actual* holes... is this equivalent to "missing points" in spacetime in any way? In other words, does the formation of a black hole fundamentally change the topology of the universe? Alternatively, is that what lead to the idea of black holes leading to other universes, analogously to the way a 1-Sphere can map to two different 1-Spheres? With respect to curvature... we often see mass depicted as a depression in a rubber sheet or 2D wireframe plane. In this analogy, black holes are depicted as depressions that go so deep that a hole is torn in the rubber sheet and/or fabric of the universe. But we also often hear the verbal description that the singularity is the point at which "curvature becomes infinite". But in that depiction, the curvature at the bottom isn't infinite; in fact it's very nearly zero, with all of the real curvature happening at the event horizon. What would it *really* mean for curvature to become infinite? Is there any way you can think of to visualize this more accurately? Could this have any interesting implications for the true nature of the singularity? This is, of course, assuming GR, not quantum gravity (in which, I presume, the singularity is not expected to persist and will turn out to have been an artifact of the math of GR being pushed past its domain). The disc with opposite points defined, and the way the even-numbered windings can contract to zero while odd-numbered windings can only contract to 1 reminds me of the way some of the curled up dimensions are depicted spontaneously unraveling into macroscopic dimensions in String Theory... is that a coincidence? Is that where that particular topology example is heading later on, or am I off base on that similarity?
BTW: If anyone was annoyed he left us hanging what the fundamental group of the plane minus two points is: it is the "free product" of two copies of the integers, which is indeed not abelian.
Every episode that I watch, I get a brain ache, but its always a good brain ache! This video has deformed my brain into both a coffee cup and a donut, ...hmmmm donut!
There must be something to how you say what you say, cause I had an idea about non Euclidean geometry imagining a version more like a sine wave than a diverging or converging parallel line. Then after thinking about it, that's basically GR which you almost immediately mentioned in the video as I had the thought.
also dont know if he talked about this but when Reinmann died his house keeper threw out a whole bunch of papers that he was working on. apparently Reinmann didnt publish unfinished work so we most likely lost some incredible discoveries 😞
11:28 The need for a more powerful way to describe curvature for GR than just hyperbolic or hypobolic, since different areas of spacetime are curved differently 13:25 we can describe curvature using the relation of circles with fixed radii to their circumference. This is a quantitative characterization of the curvature of a surface; C = 2(pi)(r), or C < 2(pi)(r), or C > 2(pi)(r), where you could imagine characterizing it exactly. And think about what it means to have a circumference warping up or down, that would *change* the radius based on how up or down it is, *unless the surface is curved* . That’s why this is such an adept description of curvature. After that, you can imagine defining this relation bit by bit, describing the curvature of a surface slowly, with infinitesimal increments, also being able to capture infinitesimal changes in the curvature, aka calculus, and so now you aren’t limited to describing surfaces with fixed curvature. 21:10 the metric. It is a generalization of Pythagoras’ theorem, where it tells us the the (physical) length of an infinitesimal curve C, from the coordinate related measures a and b. And it takes the form C = (alpha)a^2 + (ß)b^2 + (gamma)ab, since it is related to a quadratic equation [ which generally look like (a + b) (a + b) ]. 24:00 a Manifold is a space that is curved. 28:20 parallel transport. How parallel transporting a vector along 2 different paths will keep the vector pointing in the same direction on a flat surface, but not in a surface with curvature. This is another characterization of a way that curvature shows up in geometry. 33:30 this is a reflection of how separated vectors in a curved space don’t have a unique way of measuring if they are parallel (or the same, since vectors. This relates to cosmologists and galaxies (in our curved spacetime), but I don’t understand how velocities relate to the issue of parallel-ness of vectors in a way that would make the example make sense. 39:00 We’re getting some real payoffs here. Bearing in mind there is no unique way to measure if separated vectors are parallel in curved space, that means that the vector V1 will be pointing in a different direction than V5 where we just go in a loop. This is true for the triangle path on the sphere, and for a little parallelogram loop in a more generalized space with arbitrary curvature. So, with that in mind, we can use our infinitesimal calculus trick to define curvature using the difference between the V1 and V5 vector for our parallelogram, and this is called the Reimann Curvature Tensor. 47:37 The connection between Newtonian vs Least Action and Parallel Transport vs Shortest Distance and Differentiation vs Integration is crazy and I’m not fully grasping it. Could do with replaying this part
48:25 Topology. Phew. Gonna take a break before I start this section So first we were trying to find a way to talk about curvature in a manifold quantitatively. Now we don’t care about curvature, and are interested in trying to talk about some of the invariant properties of manifolds that are homeomorphic (the same, you can smoothly map them). 52:00 topology tells us that you need to have the same dimensionality in order to have the possibility of being homeomorphic 54:00 Homotopy. It’s one articulation of topological invariance, amongst others, that physicists prefer because of its physical relevance. And it’s about mapping arbitrary circles/spheres that we characterize as S ^n into |R ^n. 55:37 For some reason, for this type of representation, we care about having a fixed base point, and a directionality, like clockwise or counter-clockwise. Not sure what the resultant analogues are for higher dimensions. 57:40 in the circumstance where we remove a point from the |R ^2 plane to learn some concepts, the word “remove” is doing a lot heavy lifting, and I don’t have a real understanding of it. But in this circumstance, the homotopical topological ways of mapping the circle into the (Plane - a Point) can be characterized by the ‘winding number’, as loops with different winding numbers can’t be made to be homeomorphic. And the Group that fundamentally encapsulates the range of those winding numbers is 7/_ or *Z* . 1:03:28 So here we can see how just because we have different spaces that have homotopical mappings with the same Fundamental Groups doesn’t mean that the topological dimensionality of those spaces is the same. And to be crystal clear I’ll note that ‘different spaces that have homotopical mappings’ refers to [a n-sphere] being mapped to [some definable space, like IR ^2, or IR ^2 - Point, or even S ^1 ] where [some definable space] is what we’re are changing and comparing. Sean also talks about things like non-Abelian [], where u can’t change the order of going around A and B, disks with opposite sides indentified, Z + Z, S^1 x S^1, when 1+1=0 for disks, how that relates to the physical counterparts like domain walls, cosmic stings, dipoles, textures. But I don’t have any mental reference points for the relevance of these topics, so I should come back when I do
Heh, the line between theoretical physicists and mathematicians is somewhat blurry, and indeed, a fairly recent distinction. It wasn't that long ago that _most_ of the people who distinguished themselves in the advancement of science also distinguished themselves in the advancement of math.
I don't think you can have a winding number greater than 1 on an *invertible* map of the kinds you're showing. For a winding number of 2 on the plane-minus-a-point, the line must cross itself somewhere. Same for a map to a circle, no?
Using small differentaion neighbourhoods seems to run into trouble when expressed as fields. If you lived on donut and figured out a metric field could one figure out that for ex -100 x and +100 x refer to the same point? If a worm hole formed the jump from flat to coffee mug topology seems hard to represent. With the special relativity sphere, de-sitter space I have wondered whether it is two disconnected pieces or one connected one. Define the shape as events 1 second from the central event. If you draw it it seems to be a future bowl and a past bowl. One could try to draw a great circle on it by picking a tangent and parallel transporting in the direction pointed. Because the bowl has a lightlike asymptote far away from the center it seems it could be possible that the proper lenght over all of the coordinate space could stay finite. If that compares to 2pi would it be fair to characterise it as flat, positively or negatively curved? It also seems that as one goes into far west future and far east past the distance between them approaches zero. Would this be sufficient to conclude/guess that the branches actually connect that way? If you have an asteroids screen the argument that the flat rendering places some locations far away is poor argument that the points are not closeby on a torus. Is there a way to make such connectivity judgements for arbitrary potentially weird spaces?
Say we have a gravitational sphere like earth orbit, is the “shortest distance” based on a 2-D manifold using the least amount of necessary force to go from point A to point B or would force be arbitrary when calculating the distance, where we should rather think in a 3D curveless manifold? Does gravity warp a flat spacetime into a curved one where the “shortest distance” is based on the amount of energy/force needed to alter trajectory rather than viewing it from a 3D Euclidean geometry where spacetime is flat? Does this mean we really live in a 2D world that is complied in our heads as a flat 3D world
It is the greatest gift that some people could spend time to teach, to interact and respond.
Thank you, Dr. Caroll. As a matematician, it's perhaps the best explanation of a homotopy groups to a layman I've ever seen.
And in the case if you're interested:
- Bolyai was Hungarian and Lobachevsky (Лобачевский) was Russian. Actually we in Russia usually refer to hyperbolic geometry as "Lobachevsky geometry".
- Yep, homeomorphisms are defined as continuous bijective maps, not necessarily smooth ones.
- Technically speaking, the spaces you're working with when you speak of homotopy etc doesn't even need to be manifolds. But it's probably too much of a rigor :)
As someone with a bachelors, masters, phd and a postdoctoral in nuclear physics I can attest that these lectures are superb and the best thing on the internet right now covering the topic. Kudos Sean! Just bought the first book of the series and anxiously anticipating the other two.
You need one to understand this stuff! 😵💫
That was the best description of the Riemann curvature tensor I've seen, these videos are much appreciated
This series is the absolute best thing in the world right now.
Keep up the great work Dr. Carroll !!!
Yes, u should also check "#your daily equation" with brian greene
'best thing in the world right now. ' - and probably for oodles of light years around - in any direction!
@@Amir-vw6rk Thank you for the hint. Another one perhaps are lectures on viruses by Vincent Racaniello -- ua-cam.com/video/Pfs6SChEXmc/v-deo.html
Yes,
I just assist courses sincec 2012...Dark Matter ...
.
This series is the best thing that happened to UA-cam since Leonard Susskind's "Theoretical Minimum"
"lighten up, experts" :D ...and that is precisely why it is so hard to find a good class - it is hardly ever fun, but this is. Sean, I LOVE this! I'm not too bad at geometry - but always felt too intimidated (mostly by the 'experts' in my class) to actually pursue a scientific career. Turns out I have been using parallel transport all along in gamedevelopment for steady camera motion along a path :)
Every class -- from math to software engineering. There's one guy in the front row who gets pedantic on every minuscule point. There's another guy in the front row who does not know what a pointer or vector is.
Hello everyone 👋 welcome to the biggest ideas in the universe. Im your host sean carrol... Always glad to hear this! You are super charismatic!
This lecture is one of the best explanations of particle physics and cosmic divergent galaxies in terms of the Theory of General Relativity. The Riemann curvature tensor is highly well defined here.
A new video from Dr. Sean! Stopping everything and starting to watch! =)
really appreciate this professor！ you are doing something grate！
Sean is by far my favorite intellectual and specifically, theoretical physicist. We are super fortunate to have you, thanks for your wealth of generosity for bringing us this knowledge and humanity for making it accessible and understandable! I hope someday to catch a talk of yours in person, that would really be something!
I thought that this series would end couple of episodes ago, but the big ideas keep coming!! Awesome!
Thanks for making something that is waaay over my head a lot easier. Teachers and professor like you should get paid like professional athletes.
Thanks for making more advanced videos! I was just listening to Eric Weinstein talk about how we need more advanced physics information out there for the general population vs the usual pop-sci physics stuff, and this series is definitely setting the bar high on advanced educational content!
Thank you for going into the “Mathyness” in this pop physics video. So grateful. Thanks!
Dr. Carroll - I've watched many of your videos and you have inspired me in many ways. That being said, that you referred to Gauss as a "dick" was the coolest. You are human after all. You rock, sir.
These lectures are in a class of their own!
I don't want the lockdown to ever end if means Sean will go back to his daytime job. Please keep it up, Dr. Carroll. This is super helpful!
Sean! Thank goodness for you, my man! You keeping me (kinda) sane during the lock down. Thanks so much!
Thank you so much for these. As a hobbyist and someone who never retained any of my math education, attempting to find a clear definition of a Riemann Curvature Tensor or any similarly complex concept has proved very difficult. I'd be very interested if you made these lec..videos into a book. Kind of like a 'Road to Reality' except for people with smaller hat sizes.
DrCar I’m already a huge fan and watched hundreds of your talks. I ran into this randomly searching for answers “geometry of the universe “
The reason why hyperbolic geometry was the first non-Euclidean geometry discovered, is that it is easy to show that no parallel lines is inconsistent with the other axioms as they were then currently formulated of Euclidean geometry. For example there are an infinite number of different lines between the north and south pole of a sphere which contradicts the first postulate of Eucliden geometry - two distinct points determine a unique line. So this is why the focus was on many (infinite) number of parallel lines through a point not on the line and parallel to the given line.
My favorite series. I appreciate the detail you get into, compared to most. Excellent vids 👌🏻
All these videos are profoundly informative. You aren't going to get this level of knowledge from most other videos on UA-cam with maybe the exception of Science Asylum, Veritasium and Ask a Space Man. Great work Sean!
Marvin Ash has some good ones on QM, too. IMHO
For some reason when he said "So, there's good news and bad news when it comes to topology" around 52:45 that struck me as hilarious for some reason - I love the off-the-cuff style of these lectures.
Great explanation of Riemann curvature tensor
Yes. People, even a major at maths, should have to recognize pedagogic excellence, especially about vectors and high level maths, I saw young people fight and fail, fight...because math is not easy, even to those that understand the concepts but cannot do the calculations, nor those who don't see in three, four or more dimensions and suffer for that. In metric fields ...What is keeping a parallel postulate, Riemann Curvature tensor parallel transport.....the connection, the curvature...smoothly deformed spaces, topological invariants.
Officer, I was not driving. I was parallel transporting my velocity vector. I don't need a license for that.
20:58
Metric: infinitesimal length.
24:25
38:31
43:00
46:21
1:00:32
1:08:28
One hurdle I had with understanding non-Euclidean geometry was a notion of line. I was so used to “straight line” type of thinking that it was really hard to imagine that whole geometry would not blow up if lines were not straight. To my surprise I found that Euclid didn’t give definition of a line. He defined it algebraically by describing properties any line has to have such as “To draw a straight line from any point to any point.” (thanks Wiki). This made me appreciate how advanced ancient Greeks were because it looks very much like modern math.
Once I realized that “line” is anything with requested properties it became easier to understand other geometries. It seems we can even have y=x^3-lines and Euclidean theorems would still hold. Amazing level of generality!
I'm ready to buy the book(s) from this series of lect- ups, of videos. Absolutely stunning material, thanks prof. Carroll
This series is astoundingly good.
Thank you very much for your time,Dr.
Could you show a bit of the math about parallel transport in the Q&A?
For example, do parallel transported vectors change their length when changing direction?
Maybe a radial velocity becomes tangential velocity in a curved spacetime?
If geometry becomes Euclidean in small scales then how does the parallel transport of the vector change it ??? please anyone answer
Great explanation of a tensor - thank you so much for these lectures - videos.
Leibnitz tried to prove the parallel postulate by a proof of contradiction - by using a different postulate and looking for contradictions. But when he discovered that the resulting geometry was perfectly free from contradictions he was certain he had made a mistake and never published it - which says something about the respect people had for Euclid. We've found it in his personal papers.
I don't think it's just respect, geometry on non-flat surfaces seems wrong because we don't experience it most of the times.
@@SrValeriolete I think it's definitely a combination of the two. Leibnitz came from a period that would have would have placed the "classical genuises" on a very high pedestal.
Thanks, Sean! If you want a 'one word' for the videos - call each a 'presentation' on the Relevant Topic! Avoids 'lecture', which for some (not me!) has ominous memories of exams etc.
These videos mean so much to me! Thank you, Sean!
Thank you so much Prof.Carroll for great series.
I'm no mathematician, but has someone explored the potential idea of parallel lines that converge asymptotically?
Really like the communication in this one
Incredible as always and great timing! I’ve been teaching myself differential geometry in an attempt to ready myself for Riemann Geo and GR : )
Superb lecture. Thank you.
The 5th postulate is a little more general:
It says that if the angles in the interior add up to exactly 180, then they are parallel. But it should be noted that both angles don't have to be 90 degrees. For example look at this figure ( =/= ). If the diagonal line that cuts through the equal sign, forms two angles that add up to 180 (same side), then they are parallel. Just a fun fact! =D
This episode was one of my favorites!
Really enjoying Sean Carroll”s videos!!!
Still the best videos anywhere on the internet
Thank you so much for making these!
21.23 just lighten up guys. Awesome!
fab. it never occurred to me that you couldn't compare two vectors at different points in space without bringing them together . lol. i find that genuinely very very thought provoking :) all wrapped up with locality and what something being space really means
This was great, love the longer vids, and the new pop-ups edits above your head is appreciated.
I don't understand the parallel transport example on the sphere. The orange vector starts out pointing straight up, but as we progress up the geodesic he has it lean over more and more, which isn't keeping it parallel. If you keep it parallel then it will still be pointing straight up when you get to the north pole. From this I conclude that parallel transport doesn't keep the vector parallel with its original value. So what does it keep it parallel with?
Of course they each are huge subjects which deserve videos of their own-they're not gonna get them; I tried to squeeze both of them into a single video.
-Sean Carroll, Physicist
I love learning about the history of these ideas and the people that brought them to us. I love your explanation of the Reimann theory. I love it and I appreciate it. I want to know this. ❤️
Are the Gaussian circles rings or disks? I'm talking about the section where Gauss speaks of seeing geometry from the point of view of someone living on the circle.
"These are hard concepts": such an understatement after having just summarized an entire semester of differential geometry in 45 minutes.
I am starting to get it. Thanks Professor
Could you give a short motivation on (co)homology groups as well in the Q&A please? I struggle to get an intuitive approach there. Thank you for this series!
I found this good. It is not what you might think of as motivation, but it worked for me. Good presentation of *calculating* the groups.
ua-cam.com/video/YNBi4Ix3cY0/v-deo.html
🤯 Thanks Doc!😷 Phenomenal gift to us all, and a delightfully casual presentation that keeps me coming back for more.
Thanks for helping me out with continuing my education.
+
39:49 Hi Dr. Carroll. Thank you for the great lectures, it would be very helpful if you can make a separate lecture on tensor calculus.
It's nice to know there a positively curved universe where the circumference of a circle is exactly 2r
Great! I've heard of this tensor - it's so nice to see it! Thank you!
This is one that I've been waiting for.. looking forward to watching this later!
I found the way you and Riemann think about space utterly complicated. I usually think of space as a infinitesimal graph (points and links considered equal and random at the smallest scale).. so the shortest path is the shortest sequence of links (and the length is the number of links).. the straight line is the links that lead the furthest (using the shortest path definition) from some other point that defines your direction (here, a point alone doesn't carry direction).. etc.. there's can't be "unparallel transport of vector" without curvature.. there can't be "rotation" of a point (since a point has no dimension).. there can't be a vector or angle definition without multiple points (for the same reason)
So I clearly went off on a tangent, learning this via quantum mechanics* rather than my usual field (3D / physically-modelled computer graphics)... but honestly, this is the first explanation of non-Euclidian geometry I've ever understood. I've been using vectors in similar ways for so long now that - seeing parallel transport demonstrated like this - I can't believe this didn't dawn on me long ago (I was never good at 'math theory', but if I can visualise it in my head, I get it just fine). The topology stuff I could imagine including at some point in the near future as well; for example, mapping textures to arbitrary geometry, possibly using curvature tensors to project texels in 3-space based on surface normals.
* Thanks Sean; I got here via some of your quantum mechanics talks, and I think I may be hooked.
Very informative and very well done, it shows that you sir are a teacher! :) There were a lot of things that I finally understood and others that I heard of for the first time. It is an art to put together basic and high knowledge and the mix to be understandable by any listener... I really wish that you sir never stop doing this series but I know that sometime in the future you really have to move on...:)
Oct onions by Sean will be a real treat.
Enjoying these lectures very much! Actually feel I can begin to understand these topics better, and the Topology part seems as if it foretells the development of string theory(?)
Now we hitting the good stuff
Do graph theory I hate math but love when I can shortcut the work and just see the concepts.
You really explain things amazingly well........................
These are amazing.
Thank you for this wonderful series, Professor Carroll. There is one thing in this video that would like to comment on: at multiple places, 1:04:46 for instance, you referred to a member of the homotopy groups as "topologically equivalent maps", which I found a bit misleading. The members of the homotopy groups are "equivalent classes" of maps, instead of maps themselve. Any two "topologically equivalent" maps in fact represent the same member in the homotopy group. I think this should be pointed out as it is somewhat important for what follows, and it is not too hard for the non-professionals.
Love the blackboard!
@47:05 I'm "incredibly complicated abstract stone(d)" by these lectures.
Fantastic Lecture!
If there is no 'natural' or 'canonical' way of defining parallelism, i.e. of saying which vector at one point is 'in the same direction' as another vector at a nearby point, then what constrains our definitions of parallelism and therefore of curvature? Given any curve, can't I just define the velocity vectors to the curve at each point to all be parallel to each other, and thus the curve is trivially straight? But then I can make any curve at all 'straight' and the concept seems to lose meaning. If the metric determines the connection, which defines parallelism, which determines curvature, then what determines the metric? Doesn't the metric represent a coordinate system, and are we not free to choose any coordinates we like? But perhaps it is whether or not you can find a coordinate system in which the metric is the Minkowski metric that tells us about curvature or flatness.
If this goes on, Sean will end up with hair like the much-used photo of Albert Einstein :) Apart from that, I think Seans videos in general is amazing because I actually understand stuff, that I didn't expect myself to understand... (or rather.. I understand what leads to the theories, even when some of the theories are hard to wrap your mind around because they are counter-intuitive...)
Sean Carroll should write a textbook about everything :)
Agreed! He's working on a QM textbook. Can't wait for that. Would also like a QFT book from him. He should do what Susskind did and work these videos up into a book series. He's so good at explaining without leaving the important stuff out.
For a novice -- why exactly is the formula for infinitesimal length = Aa2 + Bab +Cb2 ?? --- if c2=a2 + b2
So a Tensor is basicly a equation field (for some quantity) that you apply to every (or some) degree of freedom in a certain space and get an answer, wether its air pressure in the atmosphere, even a frequence in a song or curvature in some dimensional space, at a certain point (or any point in this certain space) ?
Questions for the Q&A:
Is the first homotopy group of a 1-Sphere mapped to Euclidean 3-space (a circle in Newtonian space) trivial? It seems like the winding number of a circle around a missing point is irrelevant, as it can just go 'above' or 'below' the missing point to avoid it as it smoothly transforms. (This would generalize to an n-Sphere mapped to an n+2 space, I assume?)
Some versions of the story Physics tells of reality depict black holes as *actual* holes... is this equivalent to "missing points" in spacetime in any way? In other words, does the formation of a black hole fundamentally change the topology of the universe? Alternatively, is that what lead to the idea of black holes leading to other universes, analogously to the way a 1-Sphere can map to two different 1-Spheres?
With respect to curvature... we often see mass depicted as a depression in a rubber sheet or 2D wireframe plane. In this analogy, black holes are depicted as depressions that go so deep that a hole is torn in the rubber sheet and/or fabric of the universe. But we also often hear the verbal description that the singularity is the point at which "curvature becomes infinite". But in that depiction, the curvature at the bottom isn't infinite; in fact it's very nearly zero, with all of the real curvature happening at the event horizon. What would it *really* mean for curvature to become infinite? Is there any way you can think of to visualize this more accurately? Could this have any interesting implications for the true nature of the singularity? This is, of course, assuming GR, not quantum gravity (in which, I presume, the singularity is not expected to persist and will turn out to have been an artifact of the math of GR being pushed past its domain).
The disc with opposite points defined, and the way the even-numbered windings can contract to zero while odd-numbered windings can only contract to 1 reminds me of the way some of the curled up dimensions are depicted spontaneously unraveling into macroscopic dimensions in String Theory... is that a coincidence? Is that where that particular topology example is heading later on, or am I off base on that similarity?
Is the surface of a taurus an example of hyperbolic geometry, at least on the "inner" surface (the face that you can see the center from).
BTW: If anyone was annoyed he left us hanging what the fundamental group of the plane minus two points is: it is the "free product" of two copies of the integers, which is indeed not abelian.
Please share resources for reference, they would be of great help.
Every episode that I watch, I get a brain ache, but its always a good brain ache! This video has deformed my brain into both a coffee cup and a donut, ...hmmmm donut!
I understand QM, entanglement, special relativity, QFT, geometry etc, but I found the topology stuff really hard to follow.
There must be something to how you say what you say, cause I had an idea about non Euclidean geometry imagining a version more like a sine wave than a diverging or converging parallel line. Then after thinking about it, that's basically GR which you almost immediately mentioned in the video as I had the thought.
Thank you for putting out this content, this is very useful!
Awesome lecture, thanks so much!
also dont know if he talked about this but when Reinmann died his house keeper threw out a whole bunch of papers that he was working on. apparently Reinmann didnt publish unfinished work so we most likely lost some incredible discoveries 😞
11:28 The need for a more powerful way to describe curvature for GR than just hyperbolic or hypobolic, since different areas of spacetime are curved differently
13:25 we can describe curvature using the relation of circles with fixed radii to their circumference. This is a quantitative characterization of the curvature of a surface; C = 2(pi)(r), or C < 2(pi)(r), or C > 2(pi)(r), where you could imagine characterizing it exactly. And think about what it means to have a circumference warping up or down, that would *change* the radius based on how up or down it is, *unless the surface is curved* . That’s why this is such an adept description of curvature.
After that, you can imagine defining this relation bit by bit, describing the curvature of a surface slowly, with infinitesimal increments, also being able to capture infinitesimal changes in the curvature, aka calculus, and so now you aren’t limited to describing surfaces with fixed curvature.
21:10 the metric. It is a generalization of Pythagoras’ theorem, where it tells us the the (physical) length of an infinitesimal curve C, from the coordinate related measures a and b. And it takes the form C = (alpha)a^2 + (ß)b^2 + (gamma)ab, since it is related to a quadratic equation [ which generally look like (a + b) (a + b) ].
24:00 a Manifold is a space that is curved.
28:20 parallel transport. How parallel transporting a vector along 2 different paths will keep the vector pointing in the same direction on a flat surface, but not in a surface with curvature. This is another characterization of a way that curvature shows up in geometry.
33:30 this is a reflection of how separated vectors in a curved space don’t have a unique way of measuring if they are parallel (or the same, since vectors. This relates to cosmologists and galaxies (in our curved spacetime), but I don’t understand how velocities relate to the issue of parallel-ness of vectors in a way that would make the example make sense.
39:00 We’re getting some real payoffs here. Bearing in mind there is no unique way to measure if separated vectors are parallel in curved space, that means that the vector V1 will be pointing in a different direction than V5 where we just go in a loop. This is true for the triangle path on the sphere, and for a little parallelogram loop in a more generalized space with arbitrary curvature. So, with that in mind, we can use our infinitesimal calculus trick to define curvature using the difference between the V1 and V5 vector for our parallelogram, and this is called the Reimann Curvature Tensor.
47:37 The connection between Newtonian vs Least Action and Parallel Transport vs Shortest Distance and Differentiation vs Integration is crazy and I’m not fully grasping it. Could do with replaying this part
48:25 Topology. Phew. Gonna take a break before I start this section
So first we were trying to find a way to talk about curvature in a manifold quantitatively. Now we don’t care about curvature, and are interested in trying to talk about some of the invariant properties of manifolds that are homeomorphic (the same, you can smoothly map them).
52:00 topology tells us that you need to have the same dimensionality in order to have the possibility of being homeomorphic
54:00 Homotopy. It’s one articulation of topological invariance, amongst others, that physicists prefer because of its physical relevance. And it’s about mapping arbitrary circles/spheres that we characterize as S ^n into |R ^n.
55:37 For some reason, for this type of representation, we care about having a fixed base point, and a directionality, like clockwise or counter-clockwise. Not sure what the resultant analogues are for higher dimensions.
57:40 in the circumstance where we remove a point from the |R ^2 plane to learn some concepts, the word “remove” is doing a lot heavy lifting, and I don’t have a real understanding of it. But in this circumstance, the homotopical topological ways of mapping the circle into the (Plane - a Point) can be characterized by the ‘winding number’, as loops with different winding numbers can’t be made to be homeomorphic. And the Group that fundamentally encapsulates the range of those winding numbers is 7/_ or *Z* .
1:03:28 So here we can see how just because we have different spaces that have homotopical mappings with the same Fundamental Groups doesn’t mean that the topological dimensionality of those spaces is the same. And to be crystal clear I’ll note that ‘different spaces that have homotopical mappings’ refers to [a n-sphere] being mapped to [some definable space, like IR ^2, or IR ^2 - Point, or even S ^1 ] where [some definable space] is what we’re are changing and comparing.
Sean also talks about things like non-Abelian [], where u can’t change the order of going around A and B, disks with opposite sides indentified, Z + Z, S^1 x S^1, when 1+1=0 for disks, how that relates to the physical counterparts like domain walls, cosmic stings, dipoles, textures. But I don’t have any mental reference points for the relevance of these topics, so I should come back when I do
Any other non scientists here who just enjoy listening to Sean talk about cool shit? Half of the fun is just trying to keep up lol
Sean sure qualifies as a mathematician, better than a physicist. Very good presentation.Also interesting.
Heh, the line between theoretical physicists and mathematicians is somewhat blurry, and indeed, a fairly recent distinction. It wasn't that long ago that _most_ of the people who distinguished themselves in the advancement of science also distinguished themselves in the advancement of math.
I don't think you can have a winding number greater than 1 on an *invertible* map of the kinds you're showing. For a winding number of 2 on the plane-minus-a-point, the line must cross itself somewhere. Same for a map to a circle, no?
Major bummer! I already thought about tea, chocolate cookies and the video!
It is working after all!
Using small differentaion neighbourhoods seems to run into trouble when expressed as fields. If you lived on donut and figured out a metric field could one figure out that for ex -100 x and +100 x refer to the same point? If a worm hole formed the jump from flat to coffee mug topology seems hard to represent.
With the special relativity sphere, de-sitter space I have wondered whether it is two disconnected pieces or one connected one. Define the shape as events 1 second from the central event. If you draw it it seems to be a future bowl and a past bowl. One could try to draw a great circle on it by picking a tangent and parallel transporting in the direction pointed. Because the bowl has a lightlike asymptote far away from the center it seems it could be possible that the proper lenght over all of the coordinate space could stay finite. If that compares to 2pi would it be fair to characterise it as flat, positively or negatively curved?
It also seems that as one goes into far west future and far east past the distance between them approaches zero. Would this be sufficient to conclude/guess that the branches actually connect that way? If you have an asteroids screen the argument that the flat rendering places some locations far away is poor argument that the points are not closeby on a torus. Is there a way to make such connectivity judgements for arbitrary potentially weird spaces?
Say we have a gravitational sphere like earth orbit, is the “shortest distance” based on a 2-D manifold using the least amount of necessary force to go from point A to point B or would force be arbitrary when calculating the distance, where we should rather think in a 3D curveless manifold? Does gravity warp a flat spacetime into a curved one where the “shortest distance” is based on the amount of energy/force needed to alter trajectory rather than viewing it from a 3D Euclidean geometry where spacetime is flat? Does this mean we really live in a 2D world that is complied in our heads as a flat 3D world
Thank you so much for all that you do. And it ain't just coming out my black hole. I really mean it. Thank you!
Could you expound a bit more on the concept of "embedding" in spaces in the Q&A episode?
This one was a brain melter.
Good stuff.