The Maths of General Relativity (3/8) - Geodesics
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- Опубліковано 7 гру 2020
- In this series, we build together the theory of general relativity. This third video focuses on the notions of geodesics, Christoffel symbols, and the geodesic equation.
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Alessandro Roussel,
For more info: www.alessandroroussel.com/en - Наука та технологія
This video is more difficult than the previous ones, don't hesitate to watch it several times, especially the example at the end. Next week we will introduce the metric tensor !
I can’t wait to see your next video❤️
Btw I love the BGM, what is the name of it?
I just want to say, I mighta stuck with calculus had I had these videos in my life when I was learning it. You're doing god's work here, and I hope you get all the clicks and ad revenue from millions of math and physics students trying to understand and visualize what they've been told to learn in abstract number form.
I have dreams about motion, pure motion. Am I a weirdo?
This video is awsome
I wish this video existed back when I was learning GR. I'm pretty sure this is the best introduction to Christoffel symbols currently out there. 3B1B tier.
agreed
agreed
agreed
agreed
agreed
This is the clearest explanation of complex geometries I've ever seen. Your videos are absolutely fantastic!
Man, I studied this stuff in a traditional way and I always had a severe struggle to deeply understand Christoffel Symbol and intrinsic curvature... but hey here it looks so natural with the help of these graphics!
Thanks a lot. I wonder if you also teach living in real life because you are so clear...
I'm having heavy 3b1b vibes with this series. I'm still with you, tho it took more time to comprehend this part. Thanks for awesome animation and explanation!
Ok, now probably my high school knowledge has found its limit here. I hope to be able to keep up with further explanations by the time I’m in college! hopefully next year...
Even though it is complicated, it doesn't hurt to go through all of the points he makes to see what you understand and what you don't. This is a good practice for the future as well if you ever want to study ahead for a class.
I'm a physics' student and I focus on GR. I need to tell you that it's hard to understand something like this.
@@sanath8483 I’ll try to do that! I still see someone who knows calculus like a superhuman or something. Just imagining that theoretically you could deduce the most important equations of all time by yourself is so impressive. I’m probably overestimating it, due to my anticipation to learn and habit to watch a lot of physics videos just for fun
Tbh only those with preliminary knowledge understand certain ascpects, as the vid will assume you know them. Othereise...it would have to teach several years of maths when explaining something lol
E.g. anyone that hasn't done derivatives/differentiation (dy/dx) won't know that it denotes the gradient or rate of change. When dealing with velocity..rate of change is obviously acceleration or deceleration. Anyone that hasn't studied maths and physics to a sufficient academic level simply wont know that.
...I got thrown off the boat at those 8 Schileoochoff values whatever they're called.
EDIT: he explains Chrystoffel symbols. So i got some understanding
It's great that you're using youtube/the web to learn about these interesting topics! I did the same in high school, fell in love with the creative thinking that goes into physics problems, and am now working on my physics phd. Don't worry about not understanding a topic or even about being wrong if you try to connect the dots with limited information, you are always laying the groundworks for a more complete, functional, and satisfying understanding.
hope the english channel grows, simply brillliant! and the animation of the equation being re-arranged is super helpful
Does he have a channel in another language?
@@joseville their primary channel, @ScienceClic, is in french. They also have a spanish channel.
This is by FAR the best explanation of Christoffel symbols I've EVER seen, holy moly
one of the greatest General Relativity series on whole UA-cam, simply and short! sincerely grateful!
One of the...? Pls tell me the other once! :)
@@leonackermann3098 there are also Russian ones, since I am Russian, but this will certainly not help you. Check also Alex Flournoy videos for more technical details, he downloads lectures on GR right now. Famous Susskinds´ are also good
@@stevenmarshall1119 Ah I know already Susskind and Flournoy ;) also post the russian just because of random interest if I will understand a single word ^^ Greetings from Germany
@@leonackermann3098 ua-cam.com/channels/yeF4TqSnmLDXWk-qRWTezQ.html, einfach LightCone. Ich bin ein Russe, aber seit langem ich lebe und arbeite in Deutschland. Schöne Grüße dir auch, lieber Unbekannte!
@@stevenmarshall1119 Okay dann check auch mal aus Physics Videos bei Eugene, für 3D Animationen.
All i have to say is this is one hell of a beautiful explanation tbh
Dude, with this level of clarity in understanding (and hence explaining), you will probably solve the Theory of Everything. I feel enlightened.
4:16 will be my go to visual when remembering Christoffel symbols.
Fantastic visuals!
Thank you so much for this series!
Nice! Can you explain what's going on?
@@joseville Each basis vector e_i can vary in all coordinates, generally. Hence, we consider all possible deviations from the initial state of the basis vectors in all coordinates, i.e. the derivatives of all basis vectors with respect to all coordinates, x_j. The derivative d(e_i) /dx_j is our desired change.
Now, the change of this basis vector e_i is more in the direction of the coordinate x_j if the velocity component in the direction x_j is larger. Hence, we have to add a velocity component v_j to our derivative here. That makes our term as
( d(e_i) / dx_j ) * (v_j). This right here is our 'component' for the deviation of our basis vectors.
To get the direction part of the term, we multiply a basis vector to this term, which would indicate the directions of change of basis vectors, in terms of basis vectors (if you get what I mean). This, coupled with the initial derivative change consists the Christoffel Symbols (_gamma^i_kj v_j e_i).
Hope you get what it means. I'm sorry, the math notation is a lot weird here. Feel free to ask if you have any problem.
@@aaryanchokshi3862Is the spacetime being used here Euclidean?
@@tahamuhammad1814 No, the case discussed in this video is for any irregular grid.
@@aaryanchokshi3862can you help me, I'm so freackin confused. It just doesn't make sense for the basis vectors to change along the grid, that's against definition. If our vector space is 2-dimensional (not necessarily Euclidean), there may exist many bases (plural of basis) but each basis is a unique set of two vectors. Now, when we give coordinates to that space (which is the same thing as making a grid on it) we choose a two vectors that can form a basis (let's call those vectors a and b). We will choose these vectors such that there is a unique way to write every point in that vector space as xa+yb where x and y are scalars.
I can't wait for the 4th part. I am still waiting for the equations of frame dragging, gravitational lensing and gravitational waves.
Finally, a great visualisation helps me to comprehend "Christoffel Symbols ".
I've attempted to teach myself general relativity several times at this point, and the Christoffel symbols was always a stumbling block for me. But not anymore, this brought it home finally. The other aspect I find tricky for me is keeping the upper and lower indices straight in my head. Excellent excellent animations. That's what does it for me. Thank you very much.
Hello I am a bachelor and this was the first time I formally study general relativity. I can say that your work helped me a lot! It was brilliant! I believe this is the best material on the internet to explore the concepts behind this subject.
I found your channel three days ago and I have watched every video atleast twice. this is by far the most comprehensive and visually effective way I have seen these concepts be presented. I am honestly blown away. I'm currently studying in my undergraduate for astronomy/astrophysics, and while I still have two years until my relativity courses, this is some killer brain food. physics is all about those connections, and you have strengthened these connections for me before I would need them. Thanks a million, as you should have a million subscribers for the work you do here.
I've been a casual student of GR for a long time, and this is the most intuitive and clear explanation I've encountered to date. Kudos, sir!
I suggest you study eigenchris' playlists on tensors for beginners, tensor calculus and relativity by eigenchris. He taught me general relativity extremely well.
The greatest introduction of Christoffel symbols I've ever seen. Well done, loving the presentation.
The example at the end really helped with my intuitive understanding of the maths.
I love watching these videos, even though I understand very little, and I know nothing of this math, the philosophy is fascinating
My god, I have studied Physics ... and this, is by a fair stretch better than any book or lecturer that I have encountered ! Thx!
This is the absolute best explanation of Christoffel-symbols I have ever seen!! Very, very well done!
I totally agree with the comments already made: these are the clearest explanations I've seen to date on these subtle GR concepts! And the graphical animations are to become a legend. There are other excellent ones out there (e.g. by Eugene Khutoryansky), but they tend to assume a bit more background from viewers. So these here complement the others rather than substitute for them (horses for courses) - but even in and of themselves, they are a model to emulate!
This is just the video I've been looking for. Incredibly clear and accurate; I only blame youtube for not introducing me ScienceClic so far
I love how giddy I get over the release of a new general relativity video
terrific; i wish these videos were longer; i'm no night owl but i could spend the whole evening watching this
Clarity!! You, sir, are a gift to humanity 🎁
Best explanation of GR in history!
Finally a good animation with *deep in content and maths* about general relativity.
BTW, I genuinely recommend the video series of Stanford university about general relativity for those of you who look for diving into the maths of the subject...
This video is fking awesome. The best video i have ever seen about Christoffel symbols. Finally after months i can understand the meaning of christoffel symbols and how operate with them.
Really really thanks for your time and dedication. Take all my fking money and please be happy, you deserve it.
Thank you so much, these videos have been tremendously helpful in giving a visual intuition on how Chrisoffel symbols work with the vector components and the basis components. Hours with Leonard Susskind's lectures and still could not picture where the heck these symbols fit in. They are connections but he never explained how they are connected.
Brilliant, like the way you explain it in both detailed and complicated but also simplified terms.
The airplane example was beatiful, great explanation, hope this chanels skyrockets some day!
Thanks :)
“Christoffel symbols measure the extent to which our coordinates deviate from straight lines along the grid”. Great freaking summary!
This has been the most insightful video on GR that I have ever encountered. Thank you so much. This has helped me over a huge hurtle in my understanding of the topic.
These series of videos are the best on the internet
This is a beautiful and clear explanation! Thank you!
Oh my god! The best series on GR I have ever saw!
Just a physics enthusiast who has always been keen to learn relativity and man I deeply admire how awesome this content is. Great effort and skill. Kudos to you and thanks for making things so intuitive. 💯✨
Amazing. Please don't stop doing these videos!
Thank you very much for such clarity and crispness of the presentation.
The best explanation i have ever really found
I look forward to the next episode with great interest
Great explanation. Thank you! I spent some time to understand all the derivatives, but in the end it was clear!
I'm currently learning Tensor Calculus and this video is just perfect
Excellent explanation. Great series so far!!
3Blue1Brown of science ❤️❤️ Thank you guys
Outstanding series.
VERY BEST vid on Geodesics with Christoffel Symbols and intuition! And I finally understand it :)
Excellent visuals. I love your explanations. Keep them coming.
Thanks !
Waiting for the next part. Well explained, luckily I still remember differential calculus and I can still follow.
Thank you for this video!
When you finish the series, will you do a video where you use GR to calculate the motion of Mercury around the sun? It will be pretty awesome!
Eso estaría de lujo!!
I didn't know that I needed every second of this video.
If you guys want to see the entire series, go on the normal ScienceClic channel, where the series is in full but in french. I found that using autotranslated captions works fine tho
Excellent video. Can't wait for more :)
Thank you so much for making such videos!
Thank you very much! This video gave me a geometrical understanding of Christopher's connection.😊
Sir this is best explanation of gtr. Helps a lot to a beginner in gtr. Thank you for all the effort
love this series on relativity
I absolutely love how you are explaining things, but I have a question at 4:47 or so. After substituting in the derivative of the alpha'th basis with respect to proper time(which itself took me fifteen rewinds to understand), e_gamma and e_alpha seem to cancel each other out. How is this possible? Isn't gamma a variable used for Einstein notation with the one in the Christoffel symbol? A little bit of extra explanation will be incredibly helpful. Again, I love your explaining methods.
I struggled a bit there too. But I think it makes sense. There’s no reason we can’t relabel gamma, alpha, and beta to alpha, mu, and, nu respectively since we’re expanding them out anyway (it’s not like there’s any significance to alpha appearing in both the left and right side). If you accept that, then it’s a matter of accepting that if two vectors are equal, then their components are equal too.
Helps to recognize that the basis vectors are linearly independent since they form a basis
Thank you for the great effort
Two questions:
1) Is it correct to say that the Christoffel symbols describe how the grid is distorted relative to a flat Cartesian grid of the same dimension?
2) Does the distorted grid look identical to all observers or do different observers use different grids?
quality content! Amazing series keep it up man!
Thanks !
This is just amazing! Thank you so much.
Amazing video , thank you
Great graphics!
This is a great video for someone who knows GR and seeks a refresher. But obviously it will confuse people new to the field as it glosses over so much intuition
hey man, your videos let people like me a 13 year old understand, thanks for that
Brilliant explanation.
Very intuitive, thank you!
just blown my mind away
I feel like the general relativity is really convenient after this nice interpretation and explanations
brilliant Alessandro.
Brilliant. Thank you.
Well done! ♡
The best explanation of what the Christ awful symbols Are!
Very instructive, some of the maths are unexplained such as the swapping of indices.
Well done!
Can you please wait Ill have to watch every other part quick
My favorite physics channel along with PBS Spacetime.
MARAVILLOSO GRACIAS POR EL APORTE
Absolutely beautiful
It should be noted that the "derivative" here is the covariant derivative, which is a more complicated mathematical object than the normal "instantaneous rate of change of a real numbered function" derivative. Its definition is related to parallel transport and computing the Christoffel symbols involves a whole bunch of work.
At this point in the videos we don't have curvature yet (at least in the first part of the video) so it could be interpreted as a "standard" derivative in R², but yes when it comes to curved surfaces and manifolds more generally this is the covariant derivative. I don't want to go too much in the details of technical notations with this series (you might have noticed that some derivatives should rather be partial derivatives too) but it's always interesting to add precisions in the comments, thanks ! For the calculations of the Christoffel symbols this will come in the next video ;)
Oh I see, the videos are also aiming to familiarize with the concept of using arbitrary coordinate systems, and don't assume curvature yet.
Viewers would definitely have to consider "weird" / "curvy" coordinate systems here, since "normal" / "straight" coordinate systems would have zeroes for all Christoffel symbols in flat space
@@JubilantJerry Yes that confused me also. This is a flat space with curved coordinates
Aand now it's getting tricky ^^
Can you please explain what happened at 4:44? Why did you change the variable names, and why/how did the basis vectors disappear? Especially, how did the upper index of the Christoffel symbol change from gamma to alpha?
We rename the indices on the right such that the basis vectors match on both sides (with index alpha), and then we project the equation onto the basis vector (we consider only the alpha component of the vectors on both sides)
These videos are great! Impressive animations.
1) Is there a coordinate free analogue of christoffel symbols, or is the whole purpose of christoffel symbols to talk about coordinates?
2) Are these related to connections? I am reading Penrose's book Road to Reality too so trying to tie the ideas together!
Thanks ! These are really good questions !
1) No, Christoffel symbols are (as you guessed) defined in terms of your coordinates, by definition. In particular they are not a "tensor" (in other words they have no "physical" meaning, they are just a set of arbitrary numbers, in particularly at a given point you can always arrange to choose a coordinate system such that all the Christoffel symbols vanish)
2) Yes ! A "connection" is basically a set of "Christoffel symbols" that describe how to transport vectors along the coordinates (technically speaking, the Christoffel symbols usually refer to a specific type of connection : the Levi-Civita connection, which is constructed using the metric as we'll see in the next video)
@@ScienceClicEN Great, thanks!
1) Would this special coordinate system (with vanishing christoffel symbols) have something to do with parallel transport of basis vectors? Or is it simpler and we just take the standard basis of the tangent space: the partial derivatives.
2) Cool! I look forward to the next video.
It's simpler than that : it's just comes from the fact that near a point, locally, you can always construct a coordinate system that is "cartesian", in the sense that the lines of the grid are "straight" near the point. These are called normal coordinates.
Great video! I have a question though: aren't the Christoffel symbols defined by the covarient derivative in a general curved space? I.e. ∇_a e_b = Γ^c_ab e_c instead of de_b/dx_a = Γ^c_ab e_c?
Yes I've been a bit sloppy on notations here to avoid having to introduce new symbols but it is technically a covariant derivative
@@ScienceClicEN thanks for the clarification! Just wanted to double check since I wanted to use some of your explication in my masters thesis haha
So, in this video you showed there are a total of 8 (2^3) Christoffel symbols for a 2-coordinate system. Am I right if I assume there are 3^3 (27) CSs for a 3-coordinate system or 4^3 for a 4-coordinate system?
3 coordinates, times 3 coordinates to measure each coordinate change, times 3 coordinates for the resultant vector.
Yes ! But the two down indices are symmetric so among the 4³ possibilities some are repeating
Great work! Didn't quite understand the last sentence at 6:59. Did you measure the thickness of the line?
Thanks! The "vertical" component of the vector (which here is almost zero which is why it seems we measure the thickness)
One thing I don't understand is why the basis vectors seem to be of arbitrary length? Shouldn't they just be the full length of the (coordinate component of the) velocity vector, like the component is? Like what determines, for example, that v1 = e1, but v0 is 1/2 of e0?
3:00 What do you mean that the vector stays the same even as the components vary? Heck, you can see in the diagram that the arrow is moving around, both the angle and magnitude changing. If it has a different angle and/or magnitude, it's not the same vector.
I'm talking about the gray vector. The red and blue vectors are the basis vectors that come with the coordinates, they can change throughout the grid, but the gray vector (velocity) does not change.
U born to teach 👏🏻👏🏻👏🏻👏🏻
That's very simple way to explain the most complex piece
Awesome videos, which tool do you use to make them?
Thanks ! I animate the videos on After Effects
4:44 please help in understanding what the new symbols "Mu" and "v" which replaced the "alpha" and "beta" at 4:44 please...
EDIT:
as per your suggestion i am seeing this video many times to understand more clearly, but this doubt is stopping me...
This is just a change in the name of the variables. These are so-called "dummy variables", their name is not important so we can change it to whatever we prefer
I still have one question, how are the length of the basis vectors changed? For example, in the earth lines example shown at the end, the length of the basis vector increased as basis vector was translated along the latitude. I expected it to remain the same.
Hi, first I would like to say that I think your videos are very good! Congratulations!
My small critique is relative to the passage in 4:42 where you write the formula for the derivative of the basis vectors and the components of the velocity appear in it. This is not reasonable, since the mentioned derivative does not depend on the components of velocity of a particular geodesics. This confusion arises because the starting point should not be a "total derivative" relative to the proper time, but rather a "directional derivative" in the direction of the curve. I hope you get my point, it is very hard to write these without equations.
Glad you liked the video ! Actually we usually denote the directional derivative along the curve as the total derivative with respect to proper time. Proper time after all is the unit graduation along the worldline, so the derivative with respect to proper time is precisely the directional derivative along the curve.
To Science Clic ---> Thank you for the video.
To everyone --> A small and hopefully easily understood discussion point. WHY ARE WORLDLINES STRAIGHT?
.
There's quite a lot of good discussions about the over-arching ideas already going on. So here's a small thing I noticed that's specific to just one point in the video.
At about 0:47 they (Science clic) present a partial explanation, a plausibility argument, that explains why worldlines tend to be straight lines when there are no forces acting. The argument is something like this " There is symmetry. If you could argue a reason to make the worldline turn left then the same things should apply to an argument that it could turn right. So the only solution is that it doesn't turn".
I like this reasoning a lot and it is certainly plausible but falls short of a conclusive proof. For any direction in space, it all seems reasonable, any turn in a particular direction can easily be balanced against an opposite. More formally, space is isotropic and behaves exactly as we expect. However, as they used to say in Sesame Street, one dimension on those diagrams is not like the others. Suppose there is a tendency for the 4-velocity of an object to become entirely in the time direction.
Let's give a concrete example: In our ordinary experience we do notice that (due to friction etc.) objects tend to lose all their spatial velocity and so their 4-velocity is then entirely pointing in the time direction. So, it's perfectly sensible to think - maybe this actually does happen all the time, everywhere. Even without friction, if you keep watching the 4-velocity of an object for long enough then it could start to turn and point entirely along the time direction. (Hopefully you can all see where I'm going with this).
The symmetry argument is harder to apply to the time dimension, it is our experience that objects move forward in time, not backwards. They have 4-velocities with a positive time component and never a negative time component. (We expect the objects proper time to increase when the co-ordinate time increases, maybe at a different rate but we certainly don't expect its proper time to decrease as co-ordinate time increases). So there is a reason to suggest that a turn toward the positive time direction is NOT going to be balanced by a similar argument for a turn to the negative time direction. You may also know that time is not so symmetric (isotropic and homogenous) as space,, in particular things were not the same at the time of a bing bang as they are now. This can be used as another reason to suggest that the symmetry argument wouldn't apply to turning the 4-velocity in the time direction.
Open to any thoughts and opinions from anyone.... Why are worldlines straight, really ??
Sir.... very eager to know..... How do u get so deep mathematically..... It's appreciable.....