Cheating by having two ideas this week. But they are big ones. We talk about how “parallel transport” of vectors allows us to define curvature intrinsically, and leads us to the Riemann curvature tensor. Then we move to topology, in particular homotopy groups, which show up in physics all the time.
The Biggest Ideas in the Universe | 13. Geometry and Topology
And here is the associated Q&A video, where we talk a bit about embeddings, tensor components, topological defects, and more.
The Biggest Ideas in the Universe | Q&A 13 - Geometry and Topology
Any thought about the possible finding of axions:
https://www.scientificamerican.com/article/physicists-announce-potential-dark-matter-breakthrough/
Thank you for this series. What are the upcoming topics?
I like the pace, density of information, and humour. The Q and A videos are thoughtful and equally enjoyable.
I’ve struggled with the concept of tensors for a long time. Most references seems to go strait into their utility or decoding them. Those 20 seconds of saying “its a just map for coordinate transformations” was that obvious and unspoken bit I was missing. The way it was described leading up to it allowed me to visualise them. It’s communicating these central concepts in practical terms that provides the handle needed to grasp the rest.
Your explanation of Riemannian Geometry didn’t quite mesh with my intuitions about how calculus works, but I’m not familiar with subject so I’m not sure whether you’re glossing over something here or if I’m misunderstanding it.
So you have v1 and v2, defining the shape of an infinitesimally small parallelogram you will transport v3 around to get v4. You implied that distances were irrelevant in this formulation, so presumably only the direction of the vectors matter not their magnitude. But it seems to me that the ratio of the lengths of v1 and v2 should affect the result. Is that right? Is there a rule that you only use unit vectors so you don’t worry about this?
My second question was how you get v4. You implied that transporting v3 around an infinitesimally small parallelogram resulted in a different vector, but it seems to me that in most spaces (e.g. surface of a sphere) it should only be infinitesimally different. So if you want to get a different vector out of this you should have to do something like subtract v3, scale it up by what you scaled v1 and v2 down by, then add this to v3. Is that how this works?
Can you please also explain the Ricci tensor? How is it used to measure the curvature of spacetime?
On the development of non-euclidean geometry it’s just baffles me that elliptic geometry wasn’t recognised immediately as soon as Mercator, the great cartographer, created his world map around 1565. The mercator projection maps the globe to the 2D plane and thus deforms converging longitudes meridians on the sphere to parallel lines on the plane. This was done almost three centuries before Lobachevsky and Bolyai developed hyperbolic geometry.
The mercator map made rhumb line navigation easy. The what? The rhumb or loxodrome is an arc crossing all meridians of longitude at the same angle, that is, a path with constant bearing as measured relative to true or magnetic north. Doesn’t this sound familiar? The compass pointing north is the independent vector a navigator parallel transports. So it was all there for nearly three centuries and nobody saw it? Or is this just literally my 2020 hindsight?
Will you share your thoughts on the mystery of why zeros of Riemann’s zeta function and energy levels of atomic nucleus appear to have the same statistical distribution?
“A possible connection of Hilbert–Pólya operator with quantum mechanics was given by Pólya. The Hilbert–Pólya conjecture operator is of the form 1/2 + i H where H is the Hamiltonian of a particle of mass m that is moving under the influence of a potential V ( x ). The Riemann conjecture is equivalent to the assertion that the Hamiltonian is Hermitian, or equivalently that V is real.”
https://en.wikipedia.org/wiki/Hilbert–Pólya_conjecture
Great series!
Thank you very much for the perspective you give as a working world-class scientist and communicator!
In your discussion of homotopy groups, most of the maps cross each other. (That is, the maps aren’t one-to-one, I think?) Like R2 minus a point, anything with more than one loop has to cross itself. Should that not worry me? It worries me a bit. I guess the study of one-to-one maps isn’t so interesting? The crossings themselves don’t warrant mention?
Thanks!
Thinking a bit further along the lines of navigation. Can the mariner’s chronometer be thought of as an independent vector set at GMT and parallel transported to determine longitude making it the velocity vector for East-West movement around the shere?
Although the chronometer doesn’t point to any point in space it only refers to the set time at a specific point in longitude, Greenwich UK. The connection is the difference in time between 12:00 noon GMT and the noon sun at your location. Once the independent vector, the chronometer, is transported around the sphere and returned to its origin, Greenwich UK, the vector is off by a day. For parallel transportation eastward backward a day and conversely for westward forward a day.
How do you determine the metric without embedding the manifold in a higher dimensional Euclidean space? Also, don’t you need to kinda convert(embed?) the manifold to Euclidean space in order the use calculus?
Great series, thanks for your time.
I just want to add my thank you for you doing this series for all of us out here. You have a way of helping me understand things that always puzzled me, and I so like your sense of humor as well You make it more like we’re sitting together in a room having a discussion, not like a classroom lecture.
I also want to say it helps me to also see your hair style changing as you move along. I don’t feel so bad having taken to cutting my own hair now.
Thank you professor Carroll for this intuitive treatment of the beginning basics of GR math. Although I haven’t used linear algebra or calculus since the late 1970’s , you have inspired me to take another run at it in this context.
I assume there is a relatively simple, more generalized version of the Pythagorean theorem that allows computation of vectors (and therefore tensors) in Riemannian spaces. Could you share that with us?
You have made this difficult time worthwhile. Peace.
what is the relationship between the projective disk example and a Spinor. they seem related. sorry that question comes after Q&A published. maybe you covered
Does topology have anything to do with multiple universes or many worlds?
I’ve read how the branching of the wave function can spawn a new universe that would intersect with the one we are in. Doesn’t it make more sense that another spacetime (4d manifold) would harbor another branch of the wave function, than some place far away from our co-ordinates in this spacetime?