Tensor Calculus 20: The Abstract Covariant Derivative (Levi-Civita Connection)
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- Опубліковано 24 лип 2024
- Previous Covariant Derivative Videos:
17 - Flat Space: • Tensor Calculus 17: Th...
18 - Curved Surfaces: • Tensor Calculus 18: Co...
19 - Intrinsic Definition: • Tensor Calculus 19: Co...
Error at 23:00. You can't do "division" like when using Einstein notation to cancel terms on both sides of an equation. The conclusion is correct though. Basically since the expression must be true for all vectors, you can put in vectors of all zeros and a single 1 to check that the tensors are equal component-by-component.
Came back here to refresh your memory, eh?
Thanks to Euclid for inventing geometry, thanks to Isaac Newton for inventing calculus, thanks to harmann grassmann for inventing linear algebra ,thanks to Riemann for taking them together, thanks to Einstein for getting rid of the summation and thanks to eigenchris for explaining them.
and thanks to you for learning them :)
its actually einsteins wife though…
@@chenlecong9938 rip mileva maric
@@chenlecong9938is IT? Wow..
@@Noah-jz3gtyeah,wish I have one
U have no idea how you have helped us from tensor torturing, wasting countless hours and saved money.
Please keep going with general relativity, fantastic presentation. You have made tensors calculus, algebra
as delightful as it can be, and this way opened the doors of GRelativity. I never write comments, this is my first in ten years
So you can imagine how respect gratitude you command on us by your video.
Every week we were waiting for your videos. Thanks chris.
same here!
Same here!
same here ! Maximum respect
Same here :0
Me is the opposite I always find ways to criticize the content I see on youtube as many usually upload unusefull content principally in spanish I dont know why. but this man is really on another level. makes think so easy to understan yet in a formal mathematical construction that books fails to give in an autocontained manner. he is engenier and physist but this man i also a mathematican gives presice definitions
Nothing in this life matters but your videos. Not love, not family, and not Forknife. Just your videos about tensor calculus.
I looked up "Forknife" because I was full of doubts about whether something going by this name deserves to be mentioned in one sentence with these tensor calc vids. Well, my doubts were justified. But then again, de gustibus non est disputandum. :))
Having arrived in my sixties I couldn't agree more, hahaha!
that escalated quickly.. but yeah, tensor calculus moves are fancy :)
And the legend of eigenchris continues
This is one of the most lucid presentations of differential geometry I’ve ever seen. Seriously good, man.
I think by now there's a growing community of people who wakes up every morning and the first thing it does is checking YT for a new video on a certain series on not (for instance) WWE wrestling or new videogame tips but on TENSOR CALCULUS. :)
Why not all three? There's enough hours in the day. :)
@@williamwesner4268 first thing tho! :D
This video progression was really necessary to understand the concept of covariant derivative and parallel transport.
Finally, the abstract description encompasses in a powerful way all the concepts involved. I recommend watching the videos in the proposed order. Great job !
Thanks. I remember finding the covariant derivative insanely confusing when I first learned it. At this point I don't think it's that big a deal. Glad the videos helped.
I just finished this video series. Thanks a lot for the effort you put into these videos Chris.
The best 10 minutes in this entire series in the first 10 of this video. My recommendation is watch this before watching the rest of the series, it helps motivate all the rest and gets you out of thinking in limiting ways about vectors--they are NOT arrows in 2 or 3-d! At least not only that..
This series of lectures explains tensor calculus in a very clear manner. This is a very difficult subject and his use of examples to explain the concepts makes understanding easier. Highly recommended.
You’re the best. Thank you so much for this series, it’s been truly invaluable.
You are a life saver with these videos. Never been more interested in learning Tensor Calculus than I have been watching your videos. Please keep up the good work! I’m so looking forward to the rest of this series and the future teased series (i.e. General Relativity)
A Thank you for all the offers you offered your classes in a distinctive and excellent
Your videos are leigendary! Literally, this series is beyond helpful, it's a lifesaver.
I'm a late comer to your series but have been mesmerized since stumbling onto it. Your presentation is truly without equal.
This whole series for tensor calculus is so amazingly helpful! Thanks a lot.
Your videos are true gems. These explanations are the best I've ever seen so thank you very much for everything
Most helpful! Best post I've ever seen. Thank you for making this difficult subject accessible. Text are hard to follow and put me to sleep, in total contrast to your great presentations here! Your a top professor for sure.
Thanks. I remember the covariant derivative took me forever to understand. I'm glad if these videos made it more accessible.
Incredible videos. These are helping so much with my differential geometry class. Thanks for making this stuff so accessible and easy to learn!
Thank you Chris. You probably have no idea how many people you have helped with this series.
As of today, exactly 50,000 views. Many thanks for your efforts!
Oh my!!! I had struggled so much with these ideas trying to study Riemannian Geometry using De Carmo's textbook. Thanks for providing these intuitions!
This video serie is fantastic! looking forward for more~
Just a masterpiece of pedagogy... Thx a bunch !
This is the masterpiece of tensor calculus , one of the hard subject in mathematics.
eigenchris, the best teacher! That's all I can say, thank you.
Many thanks for your excellent video courses. Things I particularly appreciate: The content, of course. Also, clear explanations with examples and motivation of concepts, clarification of names of math structures. Your speaking speed is just right for me -- If I miss something I can pause and go back a bit, but am not waiting for next sentence while watching eye candy. No overhead of intro video clip - just right into the good stuff of each topic. Lots of work you have done / are doing. Thanks again and best wishes.
Thanks. Glad you like them.
Need More videos on this topic eigenchris.You are a wonderful teacher. Please open a patreon account, so we can donate and contribute in your efforts.
Great lecture in a great series. Much appreciated.
Thank you so much! It is so easy to understand with your explanations!
My mind is expanding just like the universe.
L MFOA
You are an absolute gem!
Completed the entire series just in time to wish you a happy New year. Thank you so much for this wonderful gift or your time and talent.
My wish for the New year : your laser-sharp insights on divergence, curl, Green and Stokes theorems. But I know I am asking for a lot, considering all the time you have already put in.
I don't plan on making videos on those in the near future. You could check out Khan Academy's video on those topics.
Sorry but I feel there are enough videos on those topics already.
@@eigenchris I understand, You already gave so much and I was embarrassed asking for more. Happy holiday.
Hi Chris, You already know that your videos are amazing and I can imagine how much of your time and energy must go into this. So thank you so much. I would love an opportunity to help you out either by means of donation or any grunt work you need help with to do to get these videos out even faster.
I think in my next video I'll post a link to a tip jar. Thanks!
I did find these video helpful. Thanks a lot for your work, you're great!
Lucid Explanation, thanks for helping me self-study tensor calculus as an undergraduate physicist. I am now on my way to tackle Gen Relativity.
Thank you, thank you! I love you! Your work has been soo soooooo enlightning!
I'm very glad to hear it.
Thanks a lot Chris.... I understand it now.... Have a nice weekend..... Again... 👍👍
@eigenchris is God's gift to us mortals. Thank you for creating this wonderful series. These lectures are highly recommended for self learners, students of math, physics, engineering of all ages. The creator has mastered the art of online teaching using visuals, text and explaining complicated concepts in easily understood layman's terms instead of high falutin gibberish, a rare gift. Would place him in the pantheon of my most respected teachers along with Sal Khan and Andrew Ng.
Love your videos man!
Thank you for this excellent summary. Very helpful.
this video series on covariant derivative is a must !!!
Thank you so much for your priceless videos and for making those things accessibles for guys like me, your contribution to true knowledge is incredible. Concerning the sequence of this video : metric compatibility then covariant derivative of covectors then tensors maybe you could begin first with covariant derivative of a covector (covariant derivative of a scalar which is covector of a vector and then leibniz rule and second order symmetry), define covariant derivative of a tensor, apply this to the metric tensor (leibniz rule), impose that covariant derivative of the metric tensor is null (metric doesnt change so lenghts and angles are preserved) and then the metric compatibiliy appears by magic.
16 years old here studying quantum mechanics, GR, SG, and differential geometry. Spent one month trying to study tensor and got really confused until I found your videos. Absolute beautiful and a great way of helping me to understand these concepts. Thank you Chris!
Hi CHRIS
We haven't forgotten you. Please don't forget us. We would like to see more videos maybe till you can make us the favor to explain Einstein equation. MAybe we can help some way. You are one of the best showings where ideas are coming from.
I will be making more videos. I'm just taking a break now. I've been making videos continuously for about a year and I'm a bit burned out.
I do plan on starting a new series that explains General Relativity from the basics in 2019.
@@eigenchris Sounds good Chris! Do you accept donations?
@@SpecialKtoday I'll probably start a PayPal donation box in 2019 and announce it in my videos. Thanks.
it is a very useful series i have ever watch thank yoy very much , please give use more series ...
Great video. In mathematical terms, the covariant derivative generalizes the directional derivative of a vector field with respect to an other vector field. Since for a submanifold of R^n there is the ambient space, R^n, we just have to project the directional derivative to the tangent space, to get at least the tangential component of the directional derivative, namley the amount of the derivative that corresponds to the manifold (its tangent space). The abstract definition cannot take into account an „ambient space“, so what we do is very typical for objects in math:
Look at the analogous object in R^n, take its properties as the defining properties for the abstract object and define the new object between spaces related to the manifold that carry enough structure (here: the tangent bundle/ tangent space). And here you go, that’s the covariant derivative. What often confuses people is that in the definition there is no formula. Here is why: Since you already gave the defining objects, just look what these properties do to basis vectors. The expression you end up with is your formula.
Hi, just wanted to say that your videos are simply brilliant and please keep going.
I also wanted to understand general relativity so I looked up tensors and tensor calculus, and, who would have thought, it's quite complicated. I'm making progress though and your videos are helping a lot.
Thanks. I plan to add a couple more videos in this series. After that I will do a short series on general relativity.
Great series! Thank you
Damn, this was really nice. I think all physics students who are taught GR should first be taught these things rather then just making them learn index gymnastics of the tensor. This was really insightfull and i probably would come back to these lectures again and again (since i binged watched it from the start without carefully following the calculations). Thank you so much for taking your time to do this.
I am following lectures on GR by Susskind and I couldn't digest covariant derivative. Someone in the comments suggested your playlist, and i am glad to have followed it. I wont continue from here on, as i only needed to understand covariant derivatives. But if i ever require the concepts from later lectures, surely i would continue.
Edit: After typing this comment, i checked the topics of other lectures, and now i really want to watch them all (I really dont have time, as i have planned to finish a lot lectures before my holiday ends.)
Yeah, the Christoffel symbols and covariant derivative took me way too long to understand. I ran into the same problem of learning "index gymnastics", but not really understanding what's going on. They main trick I've learned in this playlist and my relativity playlist is that tensors are much easier when you write out both the components and the basis. Writing transformation rules using only the components is possible, but not very enlightening most of the time.
Brilliant explanation. Thank you.
Thank you very much. one of the best lectures.
This just keeps getting better! Thanks again :D
I'm glad you like them. This is more or less the point I wanted to stop at when I started the series. I think I'll end up doing a video on the Riemann and Ricci Tensors as well, but this series is basically done other than that.
@@eigenchris Awesome! I think you've done a great service to us all! Thank you so much :)
Incredible! It is amazing how easy to understand and interesting becomes Tensor algebra and calculus with your videos. Congratulations for your work and your clear mind. I will stay alert just in case you start another series of whatever the subject
eigenchris easier to read Gravitation now that you’ve done these videos?
This is incredible.
These videos have been so helpful
Glad to hear it.
Please please please keep making more videos! Differential geometry and then General Relativity!
This helped with my DG course so much
I'm giving this incredible guy more money.
Excellent lecture. Thank you.
U are brilliant! Thanks a lot!!🥳🥳
I seldom leave a comment, but this video series is soooooooo great!!!!!
Just thought I would check in with you to see if you were working on some more videos--they are great!
I plan on making more, but I've been busy lately Thanks for the support though!.
thank you for the video
After 28:49: "I hope you find these videos helpfull". From my perspective, that's The Understatement of The Year 2018". Thanks, Chris.
I'm glad you like them. :)
@@eigenchris You better you bet, Chris!
Woooooow! Fantastic! This straightforward tutorial series help me understand concepts that I can never understand by merely reading textbooks, which always tends to build purely abstract terminologies to show off their intellectuality. I also have trouble in truly understanding Lie derivative, Lie groups, Lie algebra, ... anything associated with Lie... Pls make tutorials on those topics with basic examples. Thank you so much. -- By a student in computer graphics.
I have also been having trouble understanding Lie groups and Lie algebras. I don't think I will be able to make videos on these for a while. There are 3 videos on Particle Physics by Alex Flourney (videos 6,7,8) which have helped me understand Lie groups and Lie algebras somewhat... at the very least I understand that a Lie algebra is a vector space of tangent vectors at the identity element of a Lie Group, and the exponential map helps you go between Lie group and Lie algebra. I don't really understand more than that at this point.
@@eigenchris
==============
thank for your help,i can provide some useful link for you to learn more stuff and make videos
i dont know manifold and killing vector/lie derivative either.
but i think this might be useful to you(similar teaching style?)
lie derivative
ua-cam.com/video/HG3TTsx8PR0/v-deo.html
Killing vector
ua-cam.com/video/ZXrwhhQAEss/v-deo.html
i also think this might be useful(general relativity with no gap ) if you make your video..
because the lecturer said he will not skip any detail when he teaches the course...
ua-cam.com/video/JzCX3FqDIOc/v-deo.html&lc=z23xyvdgdmithhhvhacdp43bic1l3qbjetiioa1qpu1w03c010c.1575076378780604
You are the best
great videos!
Like a drink of water in the desert! One thing I'll note is a little extra info for people like myself who don't work with Einstein notation in everyday life: The renaming of summed indices may be jarring and may raise questions when there are 2 or more terms. If you work it out you'll see that the only times you cannot rename a summed index are: (1) there are other terms that use the new letter but it is not summed, and (2) the new letter is summed in all terms, but the range of summation would differ in each term (e.g. k = 1..2 in one term but k =1..3 in another term). Except for that, renaming always works.
awesome!
We all badly need you to keep on sorting out & explain all this tensor thing. Hope will you come to curvature tensor ... & Einsein field equation. I thing mosto of us are willing to reward you for you time (as I do with wiki) but dont see any "donate" key. Greate !!!!
Hello, How are you doing?
When are your GR series videos coming?
In the metric compatibility expansion in terms of the basis vectors, you already used the torsion-free property by swapping the lower indices comparing to the definition in the upper-right corner.
Also, when you take the covariant derivative of the dot product, you wrote the answer as a zero vector. Shouldn't that be a scalar?We can see in the summary that the covariant derivative is expressed in the same space than the tensor field given as the input.
This channel is awesome! Please keep going as we are many to enjoy your videos (which have, clearly, no equivalent on UA-cam)!
Thanks
BTW - 28 minutes. Thanks a lot Chris
You always have to use the proper definition of the torsion. You are right that the lie bracket of your basis vector FIELDS is 0, but if you construct two general vector fields from those (you have a position dependent linear combination) you will not get 0 because of the product rule. So either write as a definition that nabla_e_j(e_i) equals the other way around, or with general vector fields it equals with the lie bracket of the two, since it's always the case for general smooth vector fields.
Excellent.
A++
Hi man, in first place, I hope you don’t care about my English, cause it isn’t my nature language. I’m from Brazil, and I really enjoy your videos, I’m following you since the “tensor for beginners” playlist, where, in one of these episodes, you showed us your educational plan, which include, after these tensor calculus season, a differential geometry series, and after that one, general relativity videos. I saw that you aren’t more continuing these plan, and this really worried me, cause I really, as I already said, REALLY enjoy your videos (they help me a lot), and, therefore, I don’t want them to stop. I hope you read this post, and do a forward transformation with your old plans (this would be awesome xD), so, thanks for your attention and for the knowledge you’ve been sharing with us, and, simply by. 😁
Hi. I'm glad you like my videos. My tensor calculus series basically "became" a differential geometry series starting at video 15 (geodesics). I plan to make 3 more videos in this series on curvature and torsion. After this I will start work on a short series on General Relativity.
Thanks for your answer 😁. I’m glad too you’re gonna continue the videos. But, something seems really strange for me; in the most part of the books that I’ve looked about differential geometry, a prerequisite was topology, how did we learn it without this topic, and do you have some thoughts about making a playlist about it ? =)
There are basically two "versions" of differential geometry. There's the "classical" version that Gauss did and the "modern" version. The classical/Gauss version is all about 2D surfaces that live in 3D space (sphere, cylinder, torus, etc.). The modern version is more abstract and is about "Riemannian manifolds", which are abstract curved spaces of any dimension. I feel most of the important ideas can be understood using the classical/Gauss approach. The modern approafh requires you understand the definition of a manifold, which requires topology. I find the definition of manifolds is somewhat overly complicated and not needed if you just want the basics, so I don't talk about it.
Give this guy some money ! I have twice!
YOU HAD MENTIONED SEVERAL MONTHS AGO YOU WERE PLANNING ON DOING MORE VIDEOS ON THE 1) Torsion Tensor, 2) Riemann Curvature Tensor, 3) Ricci Tensor--FOLLOWING UP ON YOUR TENSOR CALC VIDEOS. . DO YOU STILL PLAN ON DOING SO? REALLY LIKE YOUR EXPLANATIONS. THANKS.
thank you!!
20:42 would you mind explaining where the expression on the top-right came from?or you derive that in the other video in the tensor calculus playlist?
Good Day Chris, which one of your tensor calculus Video did explained about tensor density.... Have a nice weekend... 👍👍🙏
I don't think I talk about tensor densities... I briefly touch on the volume form in Tensor Calculus 25, which behaves like a density.
Hey Chris, please don't keep us waiting. You started this, and now not posting a video is just plain cruel. We need this, we really do. We already acknowledged that you have great powers of explaining this difficult concept. I understand that you are busy and making these videos probably take up huge time, but great powers come with great responsibility.
26:09 and 27:36 is it not easy to see that covariant derivative(DC) of the metric MUST be 0, since g_ij is just a dot product of e_i and e_j, and the DC of a dot product is zero.
So one can "see" it immidiately without lengthy calculations (right? )
how do i support your channel.
wow, before watching this series, just purely looking at all these mathematical symbols feels like looking at egyptian hieroglyph. Now everything makes total sense. This whole mathematical thing is just a way to measure and calculating changes in an always-changing space, aka a manifold
A question:
Does a Connection differ from Levi-Civita, does it not preserve lengths and angles?
Hi Chris, can you clarify that parallel transport on a sphere is possible through geodesic paath only ? Because by other path rate of change of the vector is not zero .
@20:30 why the i,j,k arrangement in the 𝝠 (covector connection) is not the style of 𝚪(vector connection)?
Because alpha is a covector, is (j) components go on the bottom. This means that the (j) components of the 𝝠 in order to do a summation.
Very good explanations. I don´t think you are mentioning that you are multiplying both sides by g^{im} in the end at 15:44, where g^{im} is the i:th row, m:th element of the inverse matrix to the matrix-representation of the metric g.
A small doubt. Isn't the metric compatibility a result of basic mathematical facts so is compulsory to be satisfied by any connection? If so how can there be other covariant derivatives(connections) not satisfying it??
Hi eigenchris, hope all is well. Have a quick question regarding the idea of metric compatibility, please: the formula at 13:48 says that when we take two vectors and parallel transport them together, the dot product is a constant. However, the righthand side of the formula seems to say that we transport one, then dot product and then add the inverse...would this not imply that we are taking the dot product of vectors that now live in different vector spaces (as one has been parallel transported and the other has not)? Thanks in advance :).
The covariant derivative outputs a vector that lives in the same tangent plane. You can think of taking an "initial" vector at the start of a path, and a "final" vector at the end of a path, then slowly shortening the path until it becomes a point, (similar to how you take ordinary functions by taking looking at a line connecting two nearby points and then taking the limit until the line becomes tangent to the curve). "Parallel transport" is just saying that the covariant derivative is zero, similar to saying a function is constant when its derivative is zero. Does that answer your question?
@@eigenchris I understand that to do anything useful with the two vectors, we'd need them to be in the same vector space, which we can achieve by 'connecting' two vector spaces. But, my lack of understanding here comes in the idea of dot producting a vector that has been parallel transported to some vector space with one from the original vector space (the righthand side of the aforementioned equation). If the parallel transported vector is actually 'moved' into the vector space of the vector it is being dot producted with, then surely the lefthand side of the equation doesn't necessarily make sense as it requires us to dot product v and u, living in different vector spaces. In terms of v and u: if we do derivative(d) of (v.u) = d(v).u + v.d(u), then if v and u are in the same tangent space, surely dv is no longer as it has been 'connected' to some other tangent space, and so d(v).u isn't a meaningful expression (and ditto for the v.d(u) term).
My issue really is just with the spaces that each of these vectors belong to. Thanks a load!
In definition, is the formula for covariant derivative (which includes Christoffel symbols) essential? or other formulas that obey 4 properties are also defined as covariant derivative?
I think the definition with the Christoffel Symbols is called a "linear connection" or "affine connection". This is pretty much the only one we care about in General Relativity. The Covariant Derivative can get pretty abstract and appears in other places too. For example, I think Quantum Field Theory has something called a "Gauge Covariant Derivative" and that doesn't use Christoffel Symbols. Instead it uses "Gauge Fields" or something. I'm not super familiar with it.
@@eigenchris thank you for your response.
I think it become even better if you use greek letters like μ, ν, γ for summation indices.
This practice helps us to make things clear so that we see which index is nominative use and which index is used just for summation in the equations.
As a matter of fact, many GR books incorporate this practice.
I was mostly under the impression that in GR textbooks, greek letters meant "4D spacetime" and latin letters just meant "3D space". Is there another reason to use one vs the other?
@@eigenchris
There is a very good reason.
In this point , let me say the fact.
Each time I see the tensor equation in your video, the first thing I do is
rewriting the equation on a sheet of paper so that roman letter indices used for summation be replaced by greek letter representation and indices used for nominative use stay the same.
This way I can separate main indices from dummy use ones so that I can focus on the main meaning of the equation.
@@g3452sgp What exactly do you mean when you say "nominative"? Is this for naming things, like the basis vectors? Or is it just any index not used for summations?
@@eigenchris
I mean index is nominative when it always shows up both side of the equation . Nominative index specifies the specific direction. Most of the time i,j index is used as nominative because they specify the specific direction rather than all directions as summation indices or dummy indices do.
On the other hand , summation indices or dummy indices like l,m in your video do not suggest the specific direction of the interest . They simply suggest all directions for repetition. They are local and they will show up only one side of the equation, right?
Therefore separating Nominative indices from dummy indices is important.
I think this is why many GR textbooks make use of greek letters to mark the indices are dummy.
9:57 does this Christoffel expansion work in terms of derivatives though? What I mean is can you expand a second order derivative in terms of the first order derivatives using the connection coefficients? I feel like it wouldn’t be the case but if so, it wouldn’t make much sense to identify vectors and derivatives like this
I believe you should be able to expand any derivative using some combination of Christoffel symbols. The covariant derivative of a vector field gives another vector field. You can continue taking covariant derivatives to get more vector fields as much as you like, provided the field is continuous. You just need to be sure to apply product rule correctly and take the derivatives of both the vector components and the basis vectors.
HI Chris. I am taking notes, so Levi Civita connection is the covariant derivative together with the Christoffel symbols expressed in terms of the metric? And that is what you called the fundamental theorem?
For the purposes of this video, "connection" and "covariant derivative" mean the same thing (some mathematicians might argue, but this is the terminiology I use).
There are many possible covariant derivatives/connections, given by the choice of Christoffel Symbols. The Levi-Civita connection is one choice of covariant derivative, and its Christoffel symbols are defined using the formula in this video.
The Levi-Civita connection is special because it is the only connection that is torsion-free and compatible with the metric. This fact is the fundamental theorem of Riemannian geometry.
Chris,is there a book (or you lecture/video notes ?) that you would recommend that roughly follows the track of your Tensor and Tensor calculus videos ?I only stumbled into your excellent videos when reading Peter Collier's "A Most Incomprehensible Thing: Notes Towards a Very Gentle Introduction to the Mathematics of Relativity", and got stuck/lost on the section where Tensors are introduced and used.Thank you for spending the time producing these videos, which I have watch twice now, and am beginning to understand things.
My knowledge is mostly from reading tons of online pdfs and notes, and just picking out the parts I liked. I can't recommend a book.
Chris, thanks for the honest & quick reply.The two books I have bought, other people might be interested in as well are;"Tensors made easy" by Giancarlo Bernacchi (also available as a free PDF download) "Tensor Calculus Made Simple" by Taha Sochi. One thing about your videos that you do, that makes understanding easier is:- Fully expand out when multiplying out brackets and substitutions. Makes it so much easier to see where specific terms come from.- Use of colours for "matching" terms/tensors that are related.- Manually indicating when you are changing indexes, a lot of books just silently change tensor indexes and collapse results, making the derivation of some equations seem like "magic".
According to the fundamental theorem of Riemannian geometry , Levi-chivita connection is the only co-variant derivative which preserves both of torsion free property and metric compatibility property.
Is this understanding correct?
That's correct.
@@eigenchris Thank you.
Really like your explanations and clear graphics. Feel like I am beginning to understand most of this now but would like to know how all this fits into Einstein's field equation. Would the Ricci tensor require much more explanation beyond the Levi-Civita Connection? Can one use the intrinsic calculation? Is that what Einstein used? And what about some of the other tensors--could you briefly explain these..or is it possible to briefly explain them? Thanks for your excellent explanations.
My next 3 videos will be on: 1) Torsion Tensor, 2) Riemann Curvature Tensor, 3) Ricci Tensor. I hope to be done these before Christmas. They all make sense for both intrinsic and extrinsic geometry (general relativity uses intrinsic geometry, however).
There are probably lots of other tensors I have not heard of but I probably won't talk about them.
Thanks very much. When you cover the tensors you mentioned above, can you please give a short explanation of how these tensors apply to Einstein's field equation? Thanks
Only the Ricci Tensor really comes into play. I'll give a brief description of the role it plays in General Relativity.
Thank you, your videos are wonderful. I have a remark: In 28:51 you cancel out different terms to get Gamma = - Lambda. Since these different terms are part of a sum, I'm wondering if you did as a "trick" because the alpha's and v's should be put in evidence...
Yeah, that was a blunder on my part. You can't directly "cancel" them as if they were terms that can be divided. However, the formula should work for ANY choice of alpha and v, so you can conveniently choose alpha and v to each contain all 0s with a single 1, and prove that the elements of Lambda are equal to the elements of Gamma, entry-by-entry. I apologize for not explaining that better.