General Relativity Lecture 2 

00:20 On notation 03:06 Extrinsic and Intrinsic curvature 07:54 Back to the original question of a fake or real gravitation field if the space is intrinsicaly flat or curved 10:21 Tensor fields (starting with scalar fields) 14:01 Vector Fields (covariant vs contravariant) 16:39 Vector basis (covariant and contravariant components of a vector) 19:46 Projection of a vector into an axis 23:11 Definition of the metric in terms of covariant coodinate basis 23:24 Lenght of a vector in terms of the metric (relating this notion of the metric in the end to our original notion of the metric as a multiplication of partial derivatives as seen in lecture 1) 25:12 Difference between covariant and contravariant indices (introduction to lowering the index) 29:26 Covariant and contravariant vectors (from lecture 1) 30:56 Back to tensor alaysis (already seen in lecture 1) 34:16 Notice the symmetry of the covariant and contravariant indexes with the partial derivatives 39:00 How a tensor transforms 43:47 The invariance of tensor-described laws (T=0 equation holds in any frame of reference -> T=W holds in any reference frame). This is WHY we specifically use tensors for GR. (Notice that this is invariance happens because we define the way a tensor transforms in a linear way (sumation of coeficients)) 49:15 Operations on tensors (that yield new tensors) 52:24 1) Addition of tensors: yields new tensor with same rank as the two added tensors 54:24 2) Multiplication of tensors (also called tensor product): yields new tensor with a higher rank than each of the tensors multiplied (if T is rank (a,b) and W is rank (c,d) then TW is rank (a+c,b+d)). This can be seen as a generalization of the outer product (takes 2 vectors and yields a matrix) 1:01:19 3) Contraction of tensors (+ a lemma): yields a new tensor with the elimination of two of its indices (one covariant and the other contravariant) (if it's the contraction between two vectors or the contraction of a 2-index tensor the it yields a scalar). This can be seen as a generalization of the trace of a matrix or the inner product between two vectors 4) Differentiation (covariant) of tensors (Seen in lecture 3) (Another opperation that can be applied to tensors is raising or lowering indices but for this we first need to understand how the metric works) 1:16:36 The metric tensor (how many independen components it has) 1:23:54 How the metric tensor transforms: the metric tensor is really a tensor 1:28:52 The metric has an inverse (no zero eigenvalues, symmetric)

These lectures are the best thing I have ever found on RU-vid. Nothing like becoming scientifically literate in my spare time. I still can't believe these are available for free online. I feel honored to be able to learn from the great mind of Leonard Susskind. I am in your debt.

How long until the professor asks for a like and subscribe?

I am a 13 years old who accidentally fall asleep with youtube on and somehow I've ended up here Not complaining though, I've been interested to learn general relativity for a long time now. I probably won't be able to understand, but I will try my best

Having struggled with primes in tensor calculus, finally you have kindly simplified it in this beautiful presentation. Thank you.

To supplement Susskind's explanation/motivation of contra vs covariant vectors: It's very easy to show that if you transform both vectors in a dot product using the same rule, you get a different number. Do this with a single vector (since length is calculated by taking dot product), and you find you've changed the length of a vector simply by making a coordinate transformation. That is clearly nonsense. Therefore, the two objects in the dot product cannot transform by the same rule, in fact, they must transform by factors which are inverse of each other, so they will cancel and give you the same number before and after the transformation. The fact that one 'copy' of the vector transforms under one rule while the other transforms under the inverse rule is what defines the one copy as contravariant and the other as covariant. In fact, you can probably get even simpler and just point out that the components of a contravariant vector must transform inversely (hence 'contra') from how the basis vectors transform in order for the object to be the same before and after the transformation. After all, this is what "covariant" means: the components "co-transform", i.e. transform by the same rule as, the basis vectors, i.e. inverse to how the components transform when referred to the basis vectors. In fact, you can really drive home the intuition by considering the simplest possible case: unit conversion. If you go from measuring a stick with certain units to measuring it with units half as long, the number you quote as the length of the stick must get twice as big in order for the stick to have the same length in both unit systems.

I woke up watching. Guess sleepy me is smart

Learning math from a physicist is WAY easier than learning it from a mathematician. This video was super helpful, though. I've been reading Einstein's The Foundation of the General Theory of Relativity and I've been getting stuck in the Mathematical Aids section (for I am a first-semester Sophomore. Tensors of rank 2, Christoffels, etc are still a bit outside my grasp). I became stuck on the expansion of equation 20a (Section B, Subsection 9), but this video (although not solving what I needed solved) gave me the knowledge I needed to do it on my own. My university's own physics and math professors either wouldn't or couldn't help me, so I lately thank this program that Stanford provided to help satisfy curious minds and the professor for practicing Einstein's principle of explaining things simply :)

lecture 1: 300k, lecture 2: 100k

What is that Leonard eats during lectures? It looks quite dense. Also, is that warm coffee? The lectures are quite long . . . . important questions.

I think my confusion may be embedded in a higher dimension of confusion.

If you don't understand how matrices multiply you probably shouldn't be in this class lol

I don't understand any of this but I'm interested.... I guess that's a start.

Anyone who has wrapped a brithday or Christmas gift should understand what intrinsic flatness means. Try wrapping a book, as compared to a bastket ball or a saddle. Try then wrapping a cylinder or a cone.

oh wow, I've read several tensor analysis and tensor calculus books via self studying and i've taken enough prerequisite materials like vector calculus formally at my school but this by far is the best explanation of all the maths you need for GR

I still cannot get myself out of the stick and hole metaphor. Great lectures.

He reminds me of the best professors we have in my faculty of Physics. They prepare their lessons, they lay it out step by step, calmly, solving questions and always open to discussion and delivering every bit of information in an ordered manner. This is a really hard part of Physics so having a person like this that can explain it so brilliantly is a miracle.

I bought a complete tensor calculus book and 100% self-learning for one year , and there are some minor doubt? After looking at the General Relativity for one video lesson , all my previous doubt fully clears ! Thanks for the ENLIGHTEN VIDEO knowledge sharing .

If you get confused about the covariant and contravariant vectors, I suggest watching these videos: ru-vid.com/group/PLdgVBOaXkb9D6zw47gsrtE5XqLeRPh27_ They made it much more clear for me.

+atrumluminarium focus on learning the notation first, it makes everything easier. The notation will be overwhelming at first, but you will soon find that it will all be very simple; learning GR is next to impossible without the notation, and this notation is important to generalize laws and such across coordinate systems (the importance of this is obvious; space is not exclusively euclidean, and if one were to say try to define laws in a place where space is curved due to gravity, one would find these laws would differ when compared to the laws for a flat surface, but by a predictable amount). Think of vectors and scalars as extensions of tensors: A scalar is a rank 0 tensor; there is no index, and therefore a scalar will be the same in any coordinate system. To think of this, think about running 6mph on a track vs running 6mph on a straightaway; the concept of speed is scalar. Even though one will be moving in a different direction on the track when compared to the straightaway (due to turns and whatnot, this results in the velocity vector being different on the track when compared to the straightaway), one will still be running the same 6mph, the scalar part of velocity. A vector is a rank 1 tensor. A contravariant vector is one that is contracted with a basis vector (do you remember learning about i hat, j hat, and k hat? These are basis vectors) to form a vector. A contravariant vector has an upper index, and a basis vector has a lower index. These two indices are "contracted," which gives the vector (So the contravariant vector V^i, when contracted with ei, gives V^iei=V). A covariant vector is a vector dotted with a basis vector. This is when V is dotted with e, or V*ei gives Vi. A way to think of this is the covariant vector is when the vector adds the "extra" lower index from the basis vector; the vector itself is unchanged since the basis vector has a value of one. I hope this helps; rewatch these lectures if you need to, and try to look for other sources on vector analysis besides these lectures. Once you understand this notation and vector calculus, GR is a breeze.

Does anyone know what the name of the cake that he's eating is? I'm studying in the library and I'm kinda hungry and the way he eats them makes me want some :)

I love the example he used with the triangles to explain the distinction between intrinsic and extrinsic curvature.

absolutely... youtube university.... so many great lectures though few hold a candle too this man.

GR is not easy but some of the students here make it more difficult then it really is…lol.

Man...I was riding high after Lecture 1 on GR. 'Einstein Smeinstein...this isn't that hard' I told myself. Halfway through this lecture I feel like a 3rd grader that wandered into a multi variable calc lecture.

"It's pretty clear which end the stick has to go into. It has to go into the thing with the hole. You can't try putting a hole into a hole or a stick into a stick. You can only put the stick into the hole, and another stick can slide into another hole and some more holes to put more sticks into... And general relativity is a lot like that..." Leonard Susskind, 2012

For those who did not understand tensors, these are nice introduction videos: ru-vid.com

The key to making through these videos is to set playback to between 1.5 - 2 x. These are really all one hour lectures.

Does anyone have any tips on how to study this stuff? I can sort of get the logic but there is a lot to keep track of and be comfortable with in Tensor Analysis (I'm not even sure I understand fully the difference between covariant and contravariant tbh) :/

I fell asleep to wake up with this on, I am confused

The first time after years I understood that stuff. Most books are going to some level of unnecessary details, thereby hiding the basic principles. REALLY good lecture - RESPECT! Symbolically I would like to invite Professor Susskind to a cup of coffee :-)

I suggest excellent tensor calculus lessons by prof. Grinfeld; can be found on youtube according to the spirit of prof. Susskind of the essentials

Hello, from mathematical point of view, can you please let me know if my understandin is correct? We have a vector space - let's say 2-dimensional - with two bases: base B_1={e_1,e_2} that is a standard canonical base on cartesian plane i.e. e_1=(1,0), e_2=(0,1) and B_2={f_1,f_2} that is a linear transformation of B_1. We have a vector V that can be represented as V=a_1e_1+a_2e_2 and V=b_1f_1+b_2f_2. So here a_1 and a_2 are so called "covariant components of vector V" as they are the dot products of e_1 and e_2 (of length 1 each) with V (so projections on x and y axis). And b_1 and b_2 are called "contravariant components of vector V" as they are just representation of V in base B_2. Now we apply a transformation to our vector space (not necessairly linear, hence the outpus is a space) given by the functions y_1=y_1(x_1,x_2) and y_2=y_2(x_1,x_2) that are continuous and smooth that gives us a space (primed space). And we want to have a primed components of transformed vector V and than we can use the formulas visible on 37:57 to calculate a_1', a_2', b_1' and b_2'?

Countless thanks to Prof. Susskinds and Stanford University for making this highly informative materials accessible to the world with ease.

You too can be this good at Physics - but only if you consume 6 cookies and 2l of coffee every hour.

Dog, this is cool, really cool ...

Awesome lesson!!! , great teacher. The technical stuff are in tons of books but what he does here, its not

Wow, he is seriously patient!

possibly the only lecturer who can be allowed to take a big mouthful of cake immediately prior to speaking!

If any of you are having trouble with this strange definition of a tensor I would recommend reading chapters 2 and 3 in Robert Wald's "General Relativity" textbook. He gives a very good definition on what tensor is and shows how to do algebra and calculus with tensors.

anybody link to that paper in the board>??

Nono no Albert is always right imagination always bring "Notation" to whatever is in thought. You lose what you have with what fits.

sticks and holes... the history of the world

Sir I watched it carefully, I got small idea about calculus of relativity. I wish to get more. My question is what was you were eating? Cake, bread or chese?

from 37:24 one will have that the derivative are equal to the partial derivative, isn't it strange??? dy^{m} / dx^{n} = partialdy^{m} / partialdx^{n}

I am enjoying listening to this lecture series As a mathematics major I find his treatment of mathematical things almost blasphemous, in a good way. I think he thought to himself that General Relativity is so obvious that he could teach this to anybody and you don’t even have to understand higher mathematics. Its just about manipulating notation …. you can almost do this without thinking … I’m not so sure. He reminds me of my physics friends from university. As a math major working rigorously defining and proving things ,,,,, and then hearing my physics friends working with these things, integration, differential equations vector fields, gradients, curls etc …. like they were hammers ….. it always made me laugh and Im still laughing …. It works and physics is being done this way. I prefer listening to Roger Penrose for my physics. That said I am enjoying greatly these lectures. Thanks

What a brilliant man and a patient teacher. Every time a student asks a misconceived question he gets straight to the heart of the misconception without ever getting irritated. Despite the fact that I'm only passively watching, I can feel the structural logic of tensors solidifying in my mind. The whole upper/lower indices thing is about the deltas versus their reciprocals.

At 23:30 Prof skips from geometry to metric tensor, for covariant components. See the geometric meaning by drawing the dot product of V dot e1, on e1. It is a length along e1 of magnitude the lengths of e1 times the length of V time cosine of the angle between them. That is the projection of V on e1 if e1 has length 1. As e1 increased in magnitude, the dot product also increased in magnitude. So the covariant component is projection times. Magnitude of e1. Right?

Professor Suskind is an awesome teacher. Never get tired of listening to him.

Smith Shirley Williams Patricia Rodriguez Barbara

Its energy of existence is time there in there is no space. Because you can change one position or point in existence forever without space time.

john malkovich

thanks old man !

Jackson Shirley Harris Shirley Young Eric

Is it only me who sees the list, Lecture Collection | General Relativity, in reverse order. That is, in my browser, the list of videos places the first lecture at the bottom. It would, of course, be neat to have the first lecture at the beginning of the list.

When man be calling a whiteboard a blackboard, you knows hes had dealings with the Einsteins of old.

Thanks sir 🙏

These lectures are old ... But they will never really become useless

The last student's question on the identity matrix - wtf. How low is the standard to get into Stanford?

Subtitles have not been included in the first five lessons of this video course. Why? Can you insert them? Thank you

diablo ii is like quantum physics, get impossibily hard fast

What are the prerequisites for a course like this?

By the way, his hands have been contaminated by the debris of white board pens. He should not touch his food. It is not good for his health, I am worried.

i love that he's always eating something...and talking with his mouth full :)

BringerOfBloood They are tensors, because even though they behave differently compared to scalars and vectors, they still have a well defined rule of transformation.

Why do I keep hearing the voice from Family Guy dad

can they start in class, covid is kinda over, it's contained at least

WATCHED ALL THE LECTURES, PREATY GOOD TEACHER...

0:50 IM DYING!!!! I think I found that way more amusing then I should have....

My favourite lecturer and lectures , I can totally understand these calculations. Brilliant

Did the front of Stanford get changed since 1990?

1:34 he's definitely dealing with a mathematician lol

see see i knew physics was homophobic! "holes are only for sticks" lol. kidding.

Lol I love when people ask questions and he gets frustrated. "Nononono... no no.. no..."

How clear the gentleman makes it, thank you Dr.

damn, how is this guy using videos that's 11 years old, is this guy still alive

great lecture, even though, what he said around the 1:00 Mark, could be misinterpreted very badly.

I think his notation is correct. Remember to sum over an index (say p), it has to appear twice; once as a superscript and once as a subscript (or "upstairs" and "downstairs" as he calls them)

Why isn't he screaming in his students faces?

Im only in 8th grade but i watch this lecture and imagining me standing there 20 years later. By the way i love this calm and consistent voice!

Contravarient is North, it has an n in it "Con" Cov"

I cant believe I understand all this.

>×24:47 V*V Is this simply a scalar?

Great day and enjoy each lecture

It imbedded in temperature

To distract at abstracttion

Why didn't my professor of differential geometry explain things like this? It's really all so clear. I love his face. The prominent nose, huge ears, and the eyes. A caricaturist's dream.

OK, search for "diffgeom1" on youtube: these lectures are by njwildberger.

Whats my man susskind eating ?

you should check out ocw.mit.edu

at 29:00 if e1 which is unit vector along x1 but not its not unity in magnitude but changing, that means e1 is also changing along x1 ,,,, so how can we take V =v1e1+v2e2+v3e3

7:21 Its white board not black board! Only error I could point out :p

Thanks Mohamed, but I could not open the link you sent.

at 29:00 if e1 which is unit vector along x1 but not its not unity in magnitude but changing, that means e1 is also changing along x1 ,,,, so how can we take V =v1e1+v2e2+v3e3

Horrible notation creating all the confusion. Most people don't understand relativity not because of the underlying logic but because of the horrible notation! If any linear algebra textbook uses Einstein's horrible notation, no one will be able to pass the final exam! LOL

The student at the end was freaking me out. Professor Susskind was consistently expressing genuine interest in his students' comprehension, and the student was edging on becoming a nuisance.

At first it was not clear to me, at 1:29:05, how we know that there are no zero eigenvalues of g. But now I understand: At a given point, P, in the manifold, the square of the length of a vector, v, in the tangent space of P, is given by v(g(P))v. If g(P) has a zero eigenvalue, then by definition there exists a non-zero vector, u, in the tangent space at P with (g(P))u = 0, implying the contradiction that u has length zero. Of course this argument will not work in a Lorentzian manifold where it is possible to have a non-zero vector with zero length.

great lecture! I do not follow the part where he derives that the metric tensor really does transform like a covariant tensor. A covariant tensor requires that all components transform as per equation shown at 1.27. However he derives this starting at 1.25 about a statement about the distance which is a single number (i.e. g-mn * dx-m * dx-n summed over m and n). So the actual equation derived in 1.27 is not a statement about each component of g, but rather that the equation holds if we sum over all m,n,p & q. Given that in general g is different at each point, and this equation must hold for all points - I can kind of grasp that the sums on each side can only be equal if every component m,n is equal - but it does not seem to be what he derived. Am I missing something?

Holes and sticks. I totally get this now lol. I have an LD in math lol. But I'm pretty good at anything else...b

Why does he say covariant components come from dot products with basis vectors? I thought covariant components are the vector components in the reciprocal basis, that is, the basis with contravariant basis vectors with upper indices. See Wikipedia article on covariance and contravariance of vectors. I don't see how you could construct the vector using his definition of covariant components and basis vectors.