Tensor Calculus 23: Riemann Curvature Tensor Components and Symmetries

[eigenchris]

Error at 20:35 : I mixed up the arrows pointing at r and 1/r

Fixed an error with one of the symmetries pointed out by AverageYoutuber17. R_abcd = R_cdab. In the original video I put a negative sign in front by accident.

[morganhopkins]

I can't thank you enough for all the hard work you've put into all your videos! These videos have been monumental in helping me achieve my goal of understanding general relativity. These series really mean a lot to me

[NoName-sylm]

Absolutely love the way you talk through these concepts in an intuitive way, it makes it a lot easier to visualize what's happening.

[igenggy]

Can't wait till next video. Big shout out from Poland man!

[WilEngl]

After a few attempts, I finally understood the value of tensors thanks to your series of video.
Thank you very much.

[orktv]

You've been doing an excellent job at making things intuitive, and I think it would have been a fun exercise to qualitatively check the Riemann curvature tensor (and its symmetries). For the example treated one could look at the u2 basis vector on the equator. Parallel transporting along u1 in this case preserves the angle (you're actually going straight), and transporting along u2 always preserves the angle. Moving first along u2 then u1 therefore yields a completely unchanged vector, but if going along u1 first makes you leave the great circle, and now the path veers to the left. This means that the subsequent transport along u2 introduces a slight change in angle, which results in a difference vector pointing purely in the u1 direction. Close to the poles, the spherical coordinate system effectively becomes the polar one, so the curvature should vanish, in accord with the sin^2(u1) formula.

[Why_Alex_Beats_Bobbie]

Another incredible video! This is the best intro to differential geometry I have every seen!

Around the 12:30 mark, I think we need to use the torsion-free property to conclude that the LHS=0. The dot product is indeed a scalar, but after the first partial derivative we are now differentiating wrt a (0, 1)-tensor/covector, which means we need to resort to Christoffel symbols this time. Luckily, they are symmetric on the lower indices (tensor-free) so we end up with zero anyway.

[imd]

I'm so thankful to universe that this man exists ❤️:-):-)

[abdullahmudasar]

When will you upload Ricci tensor...?Btw enjoyed the series...

[jnk]

so according to 22:55
if I travel from moscow to baghdad then from baghdad to cairo then from cairo to berlin then from berlin to moscow 'i will have a square area less than the square of sine angle between between moscow -baghdad-cairo-moscow
right, am I right Dr Chris?

[doctormeister]

Would it be possible that for some sort of coordinate system, the lie brackets would not go to zero? If they always go to zero, the 3rd term of the Riemann tensor seems nonsensical (Except for linearity but that would just be a slight of hand then)?

[djordjianfairpaltrusio]

THIS SERIES SO GOOD

UPLOAD NEXT VIDEO ALREADY

[fredrickvanriler]

I immensely enjoy your videos! My question is: in the future, will you be doing any tutorials on Quantum Mechanics, specifically, of the operations on the Schrödinger equation?

[yangyang]

Great Lecture! I have a question at 13:17, the fouth line to fifth line of the equation, why e_i can be factored out in this way? Would the order matter?

[aliesmaeil]

thank you, we are waitting for next video

[ilredeldeserto]

how do we calculate how many the (236) dependent components for space time are?
Is there a fast way?

[terribiliniandrea]

Question about formula at 10:26 R(ea, eb)(r*s). The Riem tensor has 3 slots accepting 3 vectors. Here you are feeding a function (dot product r*s) into the 3. slot. Is this correct ? What is this meaning ?

[larrydurante]

Hi Chris in a17.5 you cleared up a ton for me about the way engineers and mathematicians look at things. Anyway you could please explain the second covariant derivative of a tensor in components like you did for the first in 17.5 please. Theses videos are brilliantly done. I’ll support you with many coffees 😉

[Null_Simplex]

I'm a bit confused. The explanation about parallel transporting a vector w along a parallelgram represented by the vectors u and w is 3 variables, but the Reimann curvature tensor seems to require 4 variables. What does the fourth variable represent geometrically?

[tursinbayoteev]

Hi Chris, shouldn't it be Rd_bac the whole expression equal to? 4:03