Tensor Calculus 26 - Ricci Tensor/Scalar Properties 

I can’t wait for your general relativity series

We were all living in darkness, then one day God saw us, listened to our prayers and then He sent us eigenchris to shed some light and teach us what we could not see through the darkness.

What a delicious video series from our beloved Eigenchris! Twenty-sex in-depth video lectures on tensor calculus? On youtube? For free? What a delight. Thank you Eigenchris, you are an educational maharishi.

I just wanted to take a moment and express my gratitude on how concise, thorough and clear this whole series has been. I happen to be a math professor who is NOT an expert on the subject so this has been an excellent guide to the the main definitions and results of mathematical physics related to GR. I should also point out that your humility is a testament to your greatness. Keep up the good work and thank you for simplifying our lives!

Overall impressions of the whole series of videos. I finally reached the end of the series of videos. I can't believe I ate the whole thing ... like sitting alone in front of a large empty pizza box ... Yes, I even plowed through the "long boring proof" of the second Bianchi identity of lecture 26 from [10:23] to [16:38]. I found value in seeing a double Lie bracket and using the Lie bracket as one of the inputs of the Riemann tensor. (It remains beyond me how anyone could have managed to establish such a complicated proof.) A year ago, I was very excited with the first few videos on tensor algebra. So clearly and brilliantly explained. The occasional comment about "some of you might wonder why" came exactly when I was indeed wondering why - as if the teacher was witnessing incomprehension developing ... The explanation of the why was most welcome. A sense of being overwhelmed started to develop when I realized how much more material there was yet to view. But I persevered, taking notes. There was a promise of being led to General Relativity. There was a sort of epiphany when the unit basis vectors where shown to be the same as the directional derivatives. Wow ! And a clear explanation of the infamous λ parameter of parametrized curves with the use of speed and acceleration. Wow again ! I finally caught up on the backlog of videos. Next, a sense of a loss occurred when no new video was appearing for months. The level of mathematical complexity picked up but video 25 dealing with the derivative of the volume form really yanked me out of my comfort zone. It really was a nice touch advising that video 26 was the last one of the series. I had been sort of discouraged by the ugliest of all ugly formulas, regarding the volume form. That one truly is ugly as sin. The sneak preview of the GR field equations was very interesting. I am not too concerned with the delay to 2020 for the GR series. First of all, this will give me time to review at least the last 10 or so videos - glad I kept such meticulous notes. And 2020 is now really just around the corner. In the meantime, I'll also be busy boning up on the Arduino and Raspberry pi. What a wonderful experiment this all was. A stellar teacher who understands the difference between teaching and mere telling.

Just completed Both your playlists on Tensors and Tensor calculus. It was a very intuitive and logical dive into the world of Tensors - something that can really help many trying to understand General Relativity. A proper understanding of Tensors can help many get a better grasp of GR. I really appreciate all the effort that you have put in making this series .

This truly does feel like the end of a journey. I am happy that RU-vid brought me to your channel recently. I just watched a few of your videos, and got interested in the subject. Then, I decided I actually want to watch the whole series, and take notes. So I did. I religiously watched 1-3 videos per day, taking notes. During the proof of the second Bianchi identity, I started to question that decision. I almost gave in to the temptation of not writing an arrow over my vectors. But I did not. I am incredibly grateful that you made this video series. It inspired me, along with a few other factors, to change the direction of my undergraduate degree from Statistics to Geometry/Mathematical Physics.

Fifteen years ago, I was a physics major, but I couldn't wrap my head around General Relativity, so I switched to experimental physics. Unfortunately, I found experimental physics to be rather dull, and I eventually ended up as a software engineer. I attended a prestigious high school and then went on to the top engineering university in my country. My best friend from high school, the smartest student in our class, even won a silver medal at the World Physics Olympiad. But even he was baffled by General Relativity and gave up physics to become a dentist. Had eigenchris been around when I was a student, I'm convinced that we both would have stuck with physics. Less money but maybe more happy.

I was enthralled from beginning to end of the series. You are a master of exposition. Mega thank you.

My "Relaltivty" playlist (Special and General) is here: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-bEtBncTEc6k.html

Hey man you are a gem to the world !helping the self learners like me. I would be so grateful if you could provide the notes to these lectures as well like the zip file you provided for 'Tensors for beginners' series.

Been making notes on Tensor algebra and have just made it to the end of Tensor Calculus. I am about to go through the relativity playlist but before I do I just want to take some time to give my sincerest gratitude to Chris. I am about to embark on postgraduate studies in September and I will be studying this stuff more formally. This series has really helped me to better understand all the big ideas in an intuitive way. I feel more confident going into my postgrad studies now. Even Andrew Dotson has rated these videos (especially the Riemann curvature video which was just amazingly explained). You have done a great service to the physics/maths community in making these videos Chris. I give you my most sincere gratitude for going through the hard work of understanding this and arranging it in such a graphic and intuitive way.

Sir, you have done a very noble service for mankind. thank you.

As a senior high-school student who's watched your entire Tensor Algebra, Tensor Calculus and around 50% of your Relativity series and taken the proper time to learn them, I must thank you for this awesome journey which you've bestowed upon this platform! Nowadays, knowledge is everywhere - yet, people take neither the time nor the effort to properly explain things in such beautiful detail! I have never come across such intuitive explanations of the three aforementioned subjects, and never will again! Again, thank you, Chris!

You should write a textbook on tensors! In all the years since dropping out of my Ph.D. program in chemistry, I had struggled with understanding tensors. Your video series cured that. I can’t wait to see you general relativity series, and I may just go crack open my old copy of Misner, Thorne, and Wheeler!

Can’t wait for the GR series! Great work!

Hi Chris. I think that your general proof of the second Bianchi Identity is totally Heroic! 😅 Thankyou. Dr-J

i would be awesome seeing almighty eigenchris teaching us mathematics of QM..........

I've come this far from Functions in 8 months. I don't know if it's a big job or a small job, but I thank everyone who helped. Exist. Now we are one step closer to our goal. There is no stopping until we reach our goal. Shame on those who do not stop. IN DURDURANA

Just coming to the end of this series, thank you so much for taking us on this journey! The clarity of your explanations is really exemplary, your course is a gem! And with respect to the somewhat more complex proofs towards the end of the series - with a physics background myself I can live live with proofs that work “in almost all situations”, especially anything that is encountered in the real world, whilst I trust that mathematicians will have put the rare exceptions and extreme cases on a solid foundation. I can only encourage you to keep going where your curiosity takes you, and continue sharing your learnings! The following comments may perhaps be useful for some viewers - at some points of the series I used additional resources as a change of perspective helps my learning. There are 2 recent books that I found particularly helpful to complement the series: The first book would be “A visual introduction to differential forms and calculus on manifolds” by Jon Pierre Fortney with excellent illustrations and step-by-step derivations of the math foundations for differential geometry. The second book is “Visual Differential Geometry and Forms” by Tristan Needham. This book uses a somewhat different mathematical notation that is geometrically inspired and goes back to Newton’s Principia, so this takes some getting used to. However, the practical examples of constructing geodesics or doing parallel transport on the peel of various fruits are really fun and helped me visualise some concepts better. The history excursions are fun, too.

Wow! This was tough staff, but you made it comprehensible and enjoyable (without compromising rigour). Quite a feat. Looking fwd to watching the GR videos!

Finished the series, Eigenchris. Many thanks on your fine work.

I am a computer science student and I found this series so useful and I am already thinking how can I solve certain problems with this new toolbox, thank you so much!

I watched this video series over and over again till I have some kind of intuition on abstract concepts of tensor calculus. Without this series, I have no idea how to find a route to climb up the Mt. Everest of General Relativity theory. On these days, I am also watching your relativity series. It Is A Great Pleasure To Meet Your New Uploaded Video. But I recommend you not to hurry - never hurry. I believe your job is already outstanding. You don't need to hurry up or make yourself overloaded to prove that your job is outstanding. Your job is worth to wait. If somebody complaints that Christopher Nolan or James Cameron are not not so diligent to make a movie every month, It's not their problem. So, please take your time. from South Korea

Man these videos must take a lot of effort to make. Thanks for the great job Eigenchris.

Just finished your video series and I want to say thank you very much! Great explanations.

I started seing you series on linear tensors in the first half of 2020, then i wanted to begun this but i didnt haved enought knowledge and needed to focus on my tests to get into university, after the first classes in university i begun reading throw the books and at some point i remenbered this series and realized i already haved the knowledge to see it. Now after a few months of using this as study material i have finaly ended, i cant say enought thanks to you for how much you chaneel helped me learn. I an not kiding when i say that i learned more from you than from any of my uni teatchers until now. The proff is from my notebooks from college the only one that has a similar amount of pages to yours is the Linear algebra and analytic geometry one, and that one i studied almost solo due to corona.

I am deeply impressed by this lecture series. I will return to these lectures over the next couple of months, not only to refresh some of the concepts presented here, but also to take note of how the course is structured and in what ways it outshines most university lectures. Apart from being helpful to students, this is an absolute masterclass in how to construct a lecture series that is both engaging and informative! There is so much more I'd like to write about the ways in which I appreciated this series, but that would be better expressed in an essay rather than in the youtube comment section.

Fantastic series! Looking forward to the new series, your method of explaining these concepts is unparalleled!

Great playlist, I think I understand tensors better now, thank you!

I'm really grateful to you for this series. I've always been fascinated by General Relativity, unfortunately many people always that one needs a really deep mathematical background, which is not my strong point, to understand this theory. But after watching your videos, I realized that while it does require some mathematical knowledge, it's not as complicated as people say (at least the basic concept). And this series will definitely help me in my exam next month. For this I would like to thank you once again

Getting productive! Thanks for the new vid, now I am falling behind.

This video series was too good! Thanks for making such detailed visualisations with examples, it really really helped a lot! THANK YOU!!

This series has been wonderful! It really helped me to put pictures and meaning to all the math. I look forward to your general relativity videos! ❤

This is really the great works of Tensor calculus and the foundation to GR.

Great series man.. I was struggling in understanding tensor calculus for two years.. Now it becomes clear to me. Thanks for this. Waiting for General relativity series..

You have already so many gratifying comments, but I can't help adding mine: for both algebra and calculus you made a VERY VERY WELL DONE JOB! I never had such a clear explanation of vector spaces and all the attached stuff. Thanks a lot. Looking forward your GR video series...

Thank You so much Chris!! You have explained these concepts in an amazing way. I have learnt a lot from your video series. Keep up the good work.

Again, very nice video series. You are the best!

Wonderfull work! This is the best tensor lessons to understand what it really is and to use it. I tried by reading some books some years ago but I've never get the essence before seing your channel. Thanks!

lots to consider here.. but glad i completed the series! gonna do some light courses on spinors… maybe one day soon i’ll be able to implement something cool with all this knowledge of tensor calculus!

Enjoy your videos so much! Also happy with myself to make to your last video in tensor calculus🙂

Absolutely well structured and best step by step approach to understanding tensor calculus

Finally completed the series. Great stuff.

Thanks a lot for your fantastic explanation!!

Suggestion for your upcoming series on GR : an explanation of the Mars perihelion and how GR solves it. And no, you definitely do not have to be concerned with how much time you need to create the new series. You are the master of your own time and we'll get what you give when it is good and ready, period.

Thank you so much for this series Chris, this was very useful. Can't wait for your GR series.

Awesome series.

Finally come to the end thank you for your videos !!!

first thanks for all your effort and hard work you have done at this course i want to know from you the name of a best textbook that you see in your opinion is good and explain easy as your style of explaning at tensor calculs i trust your opinion very much thank you sir very much ❤

it was really helpful , its worth watching this lecture then going to university every day and wasting time there.

YES

Thank you for making this series.. 👌👍

Thanks for your lecture, enjoy the ko-fi.

i feel like sometimes that field equations are dimensionally inconsistent...because on left we have curvture and on the other side we have energy....dont both have different dimensions? i know i am wrong ...correct me(by the way....very nice vedio)

Thank you eigenchris!

Bravo! Great series!

Hah :) Some time ago I wasn't sure if I would go through this playlist... thought the tensor algebra would be enough. But here I am. And I'm certainly moving to the general relativity playlist. Woo-hoo!

can't wait for the GR series!

How do I linearize the Ricci tensor according to the metric? When it comes to gravitational waves in flat space, Einstein's field equation is rearranged according to the Ricci tensor. Then you assume that you are in a vacuum and therefore the energy-momentum tensor = 0. So you get Ricci-Tensor=0. Now we linearize this at the point of the Minkowski metric and obtain D`Alembert operator of the metric tensor =0 and thus a linear wave equation. I have not found anywhere how to do this. I would like to know how to linearize the Ricci tensor in other metrics. For example, when gravitational waves propagate in the Schwarzschild metric.

A question about the proof of the second bianchi identity. You proved it at first in local inertial frame and after in a general case. But since these are tensorial relations, proving them in one coordinate system shouldn’t be enough to be true in any coordinate system, since tensorial relationship shouldn’t change changing coordinate system?

Hi eigenchris, I really enjoyed watching your tensor calculus videos and learned a lot and new concepts on how to look on already familiar things. May I propose an Idea to you: In the end and sometimes inmidst of your videos you have these nice summary slides. Could you make them available to your audience (not just for me of course) in PDF format or something similar? Now that I am learning GRT in master courses I wish I had your formulas summed up. Since this is a lot to ask for and will cost you extra working time, I'd definitely drop you another cups of Coffee for that. Even if you dont like that proposal or dont even have time for that, still thanks a lot for these high quality lectures. :)

as far i am concerned, u dont deserve a coffee...u deserve all the cafè!

Great lecture. Well done.

9:16,how do you know which connection coefficient relates to point P? Similarly,at 9:36,how come some of the connection coefficients reduced to zero and others didn't? How do you distinguish?????

Just to whet our appetite : can you publish a rough outline of the topics your upcoming GR series will cover ? How many videos ? Regards

when will you upload lectures on general relativity??? will you talk about the energy momentum tensor in your upcoming videos??

FUCKING UPLOAD ALREADY IM TOO EXCITED

@eigenchris Hello sir , Plz tell me - at 11:55 what is difference between Nebla z [R ( u,v) w] and ( Nebla z R )( u,v) w notations

For those who are getting a little impatient, look here for more on General Relativity: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-7G4SqIboeig.html

Great video as always!

Hey Eigenchris, Thanks again for uploading such a clear and well-presented course! I was just wondering - what sources did you find most useful in learning about Tensors and Tensor calculus? I have been looking for a good resource for intuition on the push back/pull forward and the wedge and double wedge product and was wondering if you had come across any whilst making your videos!

When contracting the Ricci Tensor, this is a 2 times covariant tensor, right? Thus. before contraction we need to use the metric tensor to transfrom it to a 1,1 Tensor. What is the correct formula for the resulting scalar (Ricci scalar) ? For 2 dimensions: R11*g11 + R21*g21 + R12*g12 + R22*g22 ? We need to multiply the "row of rows" (Ricci Tensor) with a "column of columns" (metric tensor), right? It should be the trace of "the matrix", but I'm too dumb to find the correct answer. Could you please enlight me? THANK YOU!!

Thanks for all! =)

Hi Chris. At 20:00, why can you raise & lower indices on the right side of the semi colon? I see why you can do it on indices to the left of the semicolon as you showed earlier in the video. Thanks in advance!! EDIT: Sorry, not raise & lower, but just change letters via kronecker delta. Question still remains - why can you apply kronecker delta to the basis vector direction on which the covariant derivative is acting on R?

Dear Eigenchris Will you speak about coordinate basis and non-coordinate basis and tetrad formalism on the next series?

Could you do a video on the Weyl tensor? Awesome videos! Keep up the good work!

I've been kind of misled by the assumptions that we can equalize G_mn and T_mn based on the fact that theyr's both derivatives are 0. Following this logic we actually can equalize any other things that has 0 derivative. Why can't we equalize g to T_mn?

Hello, Your series is very helpfull! I would like to ask if you can recommand me a book about GR to understand it in a deeper level. Thank you for your time!

Hi Chris. What books do you recommend on tensor calculus and differential geometry for self-study?

Many little gems of insight here! Thank you. Do you suppose that the conservation laws for T was a key element in Einstein's search for the modification of R that would also have zero covariant derivative?

Wow what a great series. Just one question. I found the choice of indices somewhat confusing at parts. I watched the whole series and the tensor algebra but I'm still confused about how to choose the indices of the terms confusing especially when multiplying by new terms. would you mind referring to a text or source for clarification?

thinks for this courase; its very great

Thank you so much! Can you add a video on frame formalism along with Cartan-Riemannian geometry?

First of, great series! Truly outstanding. Secondly, would it be possible for you to maybe upload these slides, so we can download them. I know there are some of them already on you Github page, but I am talking about the rest. Thank you!

So all these videos are enough to start studying general relativity ??

you keep calling R superscript mn the Ricci Tensor.But i thought Ricci Tensor should be subscripeted?The superscripted version should be the Ricci Tensor that’s raised to superscript by virtue of the Inverse Metric Tensor. So is it just colloquialism?

Hey - I was wondering if you had any books with good exercise sets for this and your other series? Thanks!

Hello Chris, I plan on writing some tensor calculus notes and would like to use your videos as a source, giving full credit, both in the document and in the page linking to them. Is that okay? I could not find any other way to contact you. Thanks for the great videos!

How to derive that "ugly formula" at 1:22 sir?

Dark Energy and dark matter are very controversial in theoretical physics as to date there has been no empirical detection of the same. The recent Null-experiments to empirically detect SUSY WIMPS and Axions has sent theoretical physics back to the drawing board regarding the physical explanation of dark matter. Obviously, there exists a principle which theoretical physics has not yet imagined nor discovered.

On other sources the indices on all sort of tensors in Field eqn. Are covarient .why?..

Could you explain Navier-Stokes equations ?

When can we aspect the new series ?... sir.....

You should write a Book! (by the way, if you already have one then I want to buy it)

16:33 w is there...i guess that's not to be, according to 6:43

I made it!

Awesome

Can you please explain lie deivatives

👍👍👍