Tensor Calculus 25 - Geometric Meaning Ricci Tensor/Scalar (Volume Form) 

What a great lecture. You are the first person to explain the Ricci tensor and scalar and its interpretations in a simple way.

Every time I see the word "Eigenchris" in the YT notification is Christmas again!

Best explaination of the Ricci Tensor contraction I have ever come across. I was always looking for a physical explanation for the contraction .Most if not all text books gloss over Riemann tensor contraction without a good physcial reason. Thank you for this awesome video and your great work .

I'm currently in the middle of my final year physics project on the failure of Smarr's Law to describe the energy of extremal black holes. It requires a base knowledge of tensors and I can fairly safely say that this video series has pushed my progress in that regard well beyond what it otherwise would have been. Thanks very much and I look forward to the next episode!

Thanks you , the series is so clear and helpful to build up intuition when I learning GR. That’s why I come back to the series again and again.

Dude, you are literally saving my graduation, I love you

User Zxymr pointed out I had made a typo where I was writing the determinant as "deg" instead of "det" pretty consistently throughout the video, so I decided to re-upload with this fixed.

The best we can ever find on YT!! The way you extended Line element geodesic deviation to a volume element approach using metric and levi-civita determinant..... I'm totally impressed... Thank you!!

Thank you for speaking so clearly and plainly in this video.

Thanks for these videos. I am self studying general relativity and I wasn’t satisfied with the common definition of contracting a tensor by summing on an upper and lower index. I searched all over to find some kind of derivation of the Ricci tensor and why the Riemann tensor is contracted using a geometrical argument. Of course, many books just say the right side of the Einstein equations is a rank two tensor so the left side has to be, and, therefore, we contract the Riemann tensor. Geodesic deviation along the geodesics that particles travel in space leads to the contraction. This is exactly what I was looking for. Good job.

An enormous thanks for the series. I was really looking for an introduction to the geometrical meaning of Riemann en Ricci tensors as well as the scalar, and this surely helps. I got the second paper which was an "interesting" read, and I'm sure by now you got the proof, you seem to do a great job of puzzling out the math (I think the proof of the multi sphere was pretty straightforward in that paper, but if you want a self head butt moment I'm happy to discuss, this of course was years ago, and your experience will have grown significantly) Also it doesn't seem it's doing a lot of geometrical interpretation as the title seems to promise.

Finally I have been able to understand what is the geometric significance of Ricci Tensor and Ricci Scalar. Great video. Thanks for your effort.

People have pointed out the error around 19:09. I think the correct proof that sets y = mu_j is as follows. Assume y = mu_k and k =/= j, we can reorganize into -R v^z v^x s^{mu_k}\prod_{i=1, i e j}^{D} s^{mu_i}_{i} \sqrt .... Notice that the thing to the right of "-R v^z v^x" is the volume created by s_{i} (i = 1...D except i=/= j) and s_{k} (k =/= j). Note the last vector s_{k} coincides with one of the s_{i} (i =/= j) vectors, so the volume is 0. That means the right side will be 0 in this case.

Thank you for this wonderful continuation to this unique series. I was wondering if when you talked about the volume subtended by a set of vectors it would have paid to include the idea of signed volume.

From pretty reliable sources I heard that this year the Noble Prize for videos on Physics goes to ... eigenchris!!! I'll congratulate you when the official announcements are made ;-)

A great job you have done , that contents were very well explained and described .Thank you

Amazing video, thanks! It's really great to get a clear and fast intuition on the matter. Regarding the volume comparison quotient, have you tried Petersen's book Riemannian Geometry (2016)? I know there's an entire section (I believe in chapter 7) about Ricci curvature and volume comparison, where it also talks about the Bishop-Gromov comparison theorem. I don't remember if this formula comes up though, but maybe it helps. Thanks again!

U r my legend bro , gr8 lectures bro. U are even better than my professor.

Be nice to yourself! You do exceptional work.

Such a clarity of idea... You are superman💗 Thankyou so much for these videos

that general formula can be derived from the general formula of calculating determinant (that permutation formula)

I'm loving this set of videos (thanks!) and so far it's been very understandable, but there's something here that I am missing in this installment: At 14:42 you have shown that the first derivative of the volume form is zero. But surely that means that the second derivative of the volume form is also zero? I'm guessing there is something about the vectors a, b, and c in the first part that makes them different from the u, w, and t in the second part, but I cannot see why you cannot simply apply the reasoning in the first part to u, w, t too, and show that the first derivative is zero, and that therefore the second derivative is also zero. What am I missing please?

I'm hung up on one thing. When taking the derivative of V with respect to lamba, I assumed that you we have to use the product rule. Generally speaking g depends on the spatial variables which, along the geodesic, depend on lamba.

any mistake in this derivation? if we take the con variant derivative of a Vector S,we should obtain something different from the volume derivation in 18:10. Nabla v S = d/lamba (S^[u^j] _j e_j )= [d(S^[uj] _j)/dlamba] (e_j) + (S^[uj] _j ) d(e_j)/dlamba where we can write d(e_j)/dlamba as connection factor. if we take the derivative again,we will obtain Nabla v Nablav S = -R(S,V)V and the component i : [Nabla v Nablav S]^i = -R^uj j s^y _j v^z v^x and R is a factor that consists with lot of connection factors and partial derivative of connection factor,in this case,we TAKE the vector S and take the second derivative of it and obtaining it component. As you assume the volume is a scalar (therefore it is ordinary derivative),that mean you have [d(S^[uj] _j)/dlamba] this part only,there is no "connection part" as this is "ordinary derivative" as you said.and you take the derivative twice and it will remain the logic in same form. in this case,we TAKE the COMPONENT of vector S and take second derivative of it. it seems the left side and right side are not equal because you take the derivative in different ways? d^2(S^[uj] _j)/dlamba seems not equal to [Nabla v Nablav S]^i as the former one only involved ordinary derivative but the latter one involved the whole convariant derivative.

Hello ! Thanks for your video. But i have a question about the part at 19:00. Why did you replace index y with the Uj index while y index doesnt depend on Levi-Chavita symbol ? So i think we need to take all values of y instead of just one value like in the video.

29:20 The ratio of curved to flat surfce when d=2 in fact is dimensionless and u recover the formula u derivated when plugging d=2. for higher dimentions that ratio isnt dimentionless because the surface of an hypersphere grows with radius to the power of n-1, so the ratio cannot be dimensionless. In the other hand ricci scalar for higher dimentions on spheres is = n(n-1)/r2 , but i cant even start thinking with that dimensionless 1 in front of the formula...

On 10'32'' I believe it should be sqrt(det(g~)) and not sqrt(det(g)).

great presentation of concepts. thank you.

Hi Chris, I don't understand the slide starting @ 13:54. I get that the volume is made up of sums of products of w, a, b, and c components. I also see that the w tensor is a multilinear map. But I am not seeing how this translates to the product rule that you apply at 14:00. Care to elaborate/explain? Thanks!!

In 13:49 you said covariant derivative doesnt change volume so equal to zero, but in the last that part you give equation volume change under the parameter lamda -R (V) + .... . And you said again R negative mean volume shrink. Which one truth, when object move along geodesic or curve path, the volume change or not?

Great program! Want to let you know that the notes in github for this section is missing the 2nd derivative volume discussion and the Ricci Scalar sections

Hi Chris, at 18:20 double derivative of s component equals the Riemann contracted term.. I did not get how so.. Any links for its derivation please?

Please can you do a video on the Electromagnetic Field Tensor?

Errr, what is a "separation vector" for geodesics? Any vector between geodesics? Pick any 3 for a 3-d space?

At 26:47 how positive 1/6! becomes -ve 1/6! and also the sign after that changed for higher terms?? Please explain

well explained sir,thank you very much

Very nice content as always and nice try to connect Ricci tensor to the volume form but it seems unfortunately that relation between components of Riemann tensor and the second covariant derivative of the separation vector shown @ 18:10 does just take into account regular part of the derivative and when you include all the other terms of the covariant derivative of separation vector the relation is not true anymore

Hello, I could not understand the correction (1st minute). If our space is 2D then how the surface moving along v direction (v vector does not lie on the space spanned by {e1 e2} base vectors). In the Ricci tensor definition we summed sectional curvatures for all base vector directions. However, this correction make the situation seem like we have 3D space and Ricci tensor measures the change of area in the surface that is perpendicular to the motion vector (v). Therefore, we should sum sectional curvatures for base vectors of the surface to which v vector is normal. I am really confused.

Thank you! I just saw the last two videos and learned a lot!! Great work! I was wondering, do you know about connections between curvature and weighted networks (where we can determine length and metric by the weights of the links between the nodes)? Nir

Another masterpiece.

Another exceptionally well-made video on a very complex topic! Chris, if you are so kind, can you please explain (around the 19:00 min mark) how duplication of indices inside the parenthesis affects the otherwise "fixed" Levi-Civita symbol outside (which is the only thing that can be zero)? In other words, why is it not possible to have mu_1=1, mu_2=2 and mu_3=3 (and thus epsilon_{123} = 1) while y = mu_1 = 1 and j=3 which yields the (possibly non zero) product of components s_1^1*s_2^2*s_3^1?

Hi Chris, I think at 10:34 , square root of det g~ should be used instead of square root of det g and also in rest of video.

Excellent, Excellent, Excellent !!!!!!!!

10:44 Shouldn't it be u~^i and w~^j instead of u^i and w^j? 29:44

How is it possible that the ricci tensor in the sphere is always positive. I mean: if the geodesics are converging or spreading depend on if we are going towards the poles or towards the equator, so how can it be always positive.

At 27:59 the higher order terms of the Taylor series are expressed with a O(r) exp 4 notation - what is this ?

Wait. How can we subtend lesser area compared to flat circle? I can only imagine area getting bigger due to a bulge. How come in negative curvature, the area is decreasing?

So could we define "parallel" geodesics as curves for which the flat space term of the second order volume derivative is 0 since then the volume change is strictly due to the curvature of space and not the geodesics being "at angles" with each other?

I'm finding these lectures immensely helpful thankyou, but also a lot of information to remember, so if I'm in error please forgive... at 22:09 you write the inverse metric tensor as a 'normal' g, with superscript ij. Am I incorrect to think that previously the inverse metric tensor, g^ij , was written as g^ij in a different font? in order to distinguish the inverse metric from the 'normal' metric g with subscript ij

Question! I can't seem to find the answer anywhere... I see that Ricci(v,v) corresponds to volume change along a geodesic. But what about Ricci(u,v), where u and v are different vectors? Would this correspond to a mixed derivative of volume?

Thank you for the quick reply but my previous question was not about zero r3, it was about letter "O" r4 - what does the "O" notation mean ?

19:28 I don't understand why "y = mu_j" Can someone explain this? I understand why all of the mu_i's need to be different, but the mu_j is appearing in the upper index of R so it shouldn't affect what value of "y" is allowed.

At 13:00 you stated along geodesic, volume can change, then at 14:40, you stated volume is preserved by the covariant derivative along a geodesic. Why are these not contradictions. Thank you.

Thanks a million ... I am just confused about manifolds of negative value R ... why for the same circumference they will actually have less Area ... After all, what's a good example for a negative Ricci scalar / tensor shape ? ( a sphere viewed from within ??? ) Thanks again for the brilliant effort : )

Do you think Einstein used pictures or just math?

@20:32, it looks like you only tell us the geometric meaning of Ricci curvature Ric(v,v), not the full Ricci tensor Ric(v,w). Is there no good geometric meaning for the whole tensor, only the curvature part?

Question. What does the superscription on u and w mean? Is that power?

At the minute 28:45 you mention area, its mean volume or not?

At 20:20, is the vector *v* just d/dλ?

Grazie mille!

It's too late for me to explain of Ricci scaler.

gr8 work!!!!!!!

27:22 metric tensor?

after all that nasty stuff from 16:48 onward till 20:26 ,firstly,I actually understood the entire workings….but,in the end,we have a Second order Derivative of Volume.How does that relate to volume itself?? At 20:26,you said Ricci Curvature is the proportionality of volume.well,I can fathom that if you have V instead of second derivative of V on the Left Hand Side of the equation.but not the otherwise-it makes no sense! Unless you’re saying the second derivative of V is V itself…..

i think you uploaded this twice

Please translate This video into arabic

similairly lol

LOL