General Relativity Lecture 3

(October 8, 2012) Leonard Susskind continues his discussion of Riemannian geometry and uses it as a foundation for general relativity.

This series is the fourth installment of a six-quarter series that explore the foundations of modern physics. In this quarter, Susskind focuses on Einstein's General Theory of Relativity.

[lucasbuvinic]

0:08 "General relativity has a reputation for being very difficult..."
05:27 On current investigation concerning GR: the detection of gravitational waves
07:34 Importance of GR in other areas of physics
11:17 The problem set out to resolve
12:08 Search for a quantity that indicates us if the space is flat or not (the curvature tensor)
17:59 Quick recap on the metric tensor
23:23 Raising and lowering the indices
25:44 Introduction to tangen spaces (making the metric equal to the kronecker delta locally)
26:33 Theorem on Gaussian coordinates
31:30 Demonstration of the theorem
39:25 The problem with the partial derivative of a tensor field
43:17 The partial derivative doesn't yield a tensor
45:29 Constructing the notion of a tensor derivative
47:33 Definition of the covariant derivative
53:03 The Christoffel symbols
54:14 Symmetry of the Christoffel symbols (not demonstrated)
59:20 Covariant derivative of higher rank tensors
1:05:08 Covariant derivative of the metric tensor (vanishes)
1:06:40 Finding the form of the Christoffel symbols with covariant derivatives of the metric tensor (here we use the symmetry of the Christoffel symbols)
1:17:36 "No no no no no no no no"
1:18:37 "No!"
1:21:44 Parallel transport and how to tell when there's curvature
1:23:54 Scissors
1:29:13 Applying the parallel transport to the covariant derivative
1:38:01 The Riemann curvature tensor

[vkoptchev]

I am not sure how useful this would be as a first encounter with GR, but as someone who has gone through the rigorous algebraic derivations elsewhere, I really appreciate Susskind's concentration on the physics here. GR must be assimilated from different angles, and these lectures provide!

[bigtimernow]

"General Relativity has a reputation for being very difficult. I think the reason is because it's very difficult."

[Draginx]

The way he calmly yet sternly says “no, no, no, no, no…..no” when his students say something incorrect is great lol.

[ashishgupta]

It's interesting to see the declining number of views as the lecture series progresses, lol.

[EdSmiley]

So cool when he mentions gravitational radiation, and that some day we will detect it. (around 6:43!) Since we now have, of course.

[hugo-mpyp]

lockdown journal day 35 - just been hypnotised by Mr.Susskind, everything is fine

[randymartin]

Susskind did this whole course of lectures in Jan 2009, and were a lot easier to understand than this 2012 series. He used polar co-ordinates to get the quadratic feel for the displacement derivatives, and he never once mentioned Gaussian normal co-ordinates. This Guassian stuff is not intuitive at all, I recommend watching lecture 3 and 4 of his 2009 lectures. In the 2009 Lecture 5, he used + signs, not - signs in the Christoffel symbols equation. Another side note, toward the end of both series of these lectures where he explains the integral of the proper distance from a point in space to the Schwartzchild radius notably in the lecture 7 of this series is much more difficult than in 2009. He didn't bother doing the integral in this series, he said do it yourself. I was running into all sorts of weird values using u-substitutions and trig identities. In the 2009 solution to the integral you didn't need all that as I think he used different boundary and terms in the integrand.

[happyraccoon]

This guy is brilliant, he has the time to lecture, do hits for Gus Fring, interact with Walter White, where does he find the time??

[qbtc]

I think the covariant derivative part of this lecture is the most difficult. It is at the heart of general relativity and you really need to know it well. I recommend the youtube videos "tensor calculus 15", "tensor calculus 16" and "tensor calculus 19" by eigenchris to get a better understanding. Actually, it would be a good idea to watch the whole tensor calculus series.

[marko]

YT algo often leads me here when I leave it playing and go to sleep and honestly, if I had this professor (maybe) I'd like physics a bit more. Seems like a very nice and intelligent person :)

[blindConjecture]

When he says he's defining a metric "at a point", he really means that he's defining a metric on the tangent space at that point. And therefore this is a statement about how the metric changes as it moves from one tangent space to other neighboring ones.

[mrwibo]

As a Mechanical Engineering student trying out a uni course in GR I want to thank you for saving my semester :). Great explanations!

[nickcastillo]

Tensors do depend upon the coordinate system. However one important feature is that tensor equations hold true in every coordinate system. That's the key. Tensors enable one to write down physics equations that are independent of coordinates. Physics should not depend upon who's making the measurements. There are many ways one can define tensors. Because for one there are many different kinds of tensors.

[AldinCV]

I fell asleep watching car related videos and woke up to this. Now I’m intrigued. Thanks YouTube algorithm

[aqouby]

A lecture on Riemannian geometry by Leonard Susskind? Yes. Yes please.

[nickcastillo]

The tangent space also enables us to define a tensor field. Which is the same as a vector field except at every point in space time instead of assigning a vector you assign a tensor and that is what this metric tensor business is all about. The geometry of the manifold varies from place to place as well as teh function you use to compute "distances" this is where the metric tensor comes in.

[nickcastillo]

Now the number of vectors that this function (tensor) takes in as input- that is the rank. The more vectors it acts on the more indicies. One important thing I did not mention was that tensors are linear functions since our tangent spaces are linear spaces. The metric tensor for example acts on two small displacement vectors and is thus a tensor of rank 2.

[sriramvecham]

there really are some brilliant students who ask perfect doubts which sir misses to tell just for us.

[AkamiChannel]

Congrats everyone! We made it to lecture 3!