Relativity 107c: General Relativity Basics - Curvature, Riemann Tensor, Ricci Tensor, Ricci Scalar

You are free to continue watching to the next video, but if you feel you are getting confused, here are some other videos on curvature here:

0:00 Introduction
1:56 Riemann Curvature Tensor
8:12 Riemann Tensor Components + Symmetries
15:29 Riemann Tensor - Geodesic Deviation
19:04 Ricci Curvature Tensor
23:46 Ricci Curvature Scalar
25:46 Curvature of Rindler Metric
33:19 Summary

[eigenchris]

I made an unfortunate typo around 32:09. The final result of zero is correct, but the intermediate steps are not. A corrected version is below:

R^μ_vαβ = ∂_αΓ^μ_vβ - ∂_βΓ^μ_vα + Γ^σ_vβΓ^μ_σα - Γ^σ_vαΓ^μ_σβ
(Set μ=α=t, and v=β=x)
R^t_xtx = ∂_tΓ^t_xx - ∂_xΓ^t_xt + Γ^σ_xxΓ^t_σt - Γ^σ_xtΓ^t_σx
R^t_xtx = 0 - ∂_xΓ^t_xt + 0 - Γ^t_xtΓ^t_σt
R^t_xtx = - ∂_x(1/x) - (1/x)(1/x)
R^t_xtx = - (-1/x^2) - 1/x^2
R^t_xtx = 0

I apologize for the mix-up. Staring at indices for hours makes it easy to miss small errors.

[skyfall-tp]

Who would have expected 2 eigenchris videos a day? ;)

[CrowKunCGS]

This channel is a gold mine 📚💛 There is huge advanced knowledge, teached in a simple and pleasant form, for free. Thank you so much! 😁 I really appreciate your content and I think your work will be really important for many people over the years. Congrats! 👨‍🏫🏅

[prosimulate]

You are amazing this is superb, I’m a 51yr old Chemical Engineer, but I love this field of maths, but it is brick hard to understand in the books, I have tried, I get some way but your explanation and voice is just perfect! I salute you buddy!

[PM-]

I've probably watched your 5 minute topology video about 10 times by now.

[RichardLibby]

The condensed nature of this presentation is an excellent complement to the various books I've been reading, trying to understand General Relativity. Books of course are invaluable because they are meticulously complete, but the cost of reading them is in the sorting between the main ideas and the necessary but myriad details. Here, we have the main ideas and how they are connected, with occasional brief mentions of where important details are to be found.

[tinkeringengr]

Your instruction style is so powerful. Your impact is great!

[myaseena]

bruh you are so good at explaining stuff it is amazing and almost relaxing

[ГулзодаРахимова-бъ]

Amazing! Thanks a lot for these all videos about GR. It was really hard to me to understand the meaning of curved geometry and its quantities and couldnt find the appropriate book with simple explanation. By watching your videos finally I have some simple and understandable imagination of what Im calculating and doing now. Good luck for your career)))

[HighWycombe]

I love these videos, but confess I'm finding them more and more challenging. There's no shortcut to this. Before trying to follow these GR videos, I need to go back and study ALL the Tensors for Beginners videos, then ALL the Tensor Calculus videos. I've already watched the Special Relativity videos, but for anybody who hasn't I'd recommend watching those too. THEN you'll have a fighting chance to finish these last few GR videos!

[dorothyyang]

When I read the book about GR, I often feel confused. Now you make evething clear for me! Thank you!

[tomkerruish]

Thank you so much! I feel like I'm finally getting a handle on the Ricci curvature and scalar!
24:40 I'm surprised you didn't throw in a TARDIS reference.

[phillipafrayan]

Awesome video.
It is One of the best channel on youtube.

[emiliosani]

Excellent classes, marvelous professor. Just a minor biographical typing mistake in the beginning: Gauss died 1855, not 1885.

[maujo]

Dear eigenchris:

I've been struggling for years to find a GR course at this "entrance" level.

Which textbook[s] would you recommend that approach GR at the level you use in your videos?

Thanks for your great work! I will use your videos in my class!

[kanikapathak]

@eigenchris After seeing your video on "Finite Fields of prime order", I wonder, will you ever make a series on Abstract algebra ?

[justingerber]

On the Del_{[V, W]} T term in Riemann curvature:

Basically it is exactly as you say about needing to "cover the gap". You can find a clear motivation for defining the Riemann curvature tensor this way in Lee Riemannian Manifolds (I have 1st edition, don't know if there are more) Chapter 7. In flat space we have (for general vector fields X, Y, Z) that:

Del_X Del_Y Z - Del_Y Del_X Z = Del_{[X, Y]} Z

That is, If you parallel transport Z along X then Y and then again along Y and then X you will see a discrepancy exactly due to the gap [X, Y]. So for the definition to make sense in flat space you need to include that third term. We measure how much the result deviates from the flat space result to describe curved space.

[zzzoldik]

When gravity make thing shrink, it happen in one reference frame or for all reference frame? Could you answer this question please

[pong]

Excellent Video! By the way, what plans do you have for this channel after GR?

[gman]

Do I have this correct? A metric defined over a manifold uniquely determines its curvature… but curvature doesn’t uniquely determine a metric. That is, different metrics can yield the same curvature?

Where I’m getting confused is why two different metrics, like the metric belonging to a Rindler observer and the metric belonging to an inertial observer, can yield the same curvature. Does this mean that a metric tensor is coordinate-based? I assumed that, being a tensor, the metric shouldn’t be coordinate dependent. Or am I confusing something here?