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 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 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.
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...
@@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 th-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!
@@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.
@@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'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!
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 suggest you study eigenchris' playlists on tensors for beginners, tensor calculus and relativity by eigenchris. He taught me general relativity extremely well.
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.
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.
@@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.
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 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!
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...
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.
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.
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. 💯✨
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.
Now that I've watched this video and perfectly understood what Christoffel symbols mean geometrically after literal months of know-how digging, I can confidently say that *nothing* is complicated, when it's explained in a proper matter. Thank you.
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
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!
Wishlist for a future video: Introduction to gauge theory! From the perspective of gauge theory (basically *all* of modern field theory), the coordinate system is a gauge. Christoffel symbols are used to make derivatives gauge-invariant. That is, covariant derivatives. So once gauge theory is understood, Christoffel symbols are easy. But how to introduce gauge theory? I recommend an analogy by finance. It is very amenable for visualization! See www.amazon.com/Physics-Finance-introduction-fundamental-interactions/dp/1795882417/ or arxiv.org/abs/1410.6753
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?
1) The Christoffel symbols can be used to express "distortion" relative to any coordinate system. The Christoffel symbols are in fact always defined with respect to some coordinate system. Here is how I would think about it: A manifold contains one or several coordinate systems that connect its points to open subsets in R^n. Given a point on your manifold (corresponding to an event in spacetime), you will also have coordinates for nearby spacetime points, meaning that there will be directions corresponding to changes in coordinates. For example, if you use polar coordinates at one part of your manifold, then there will be an "+r" direction, and a "+theta" direction, which will get you to nearby spacetime points. This might sound trivial, but it really isn't, as the "raw" spacetime points are only a mathematical set of points. The coordinate systems are basically what glues them together. The Christoffel symbols are basically derived from how the curvature should be expressed using these coordinate directions. So if you change from polar coordinates to some other coordinate system, the christoffel symbols themselves will also change (and sadly not according to the tensor transformation law, since you're BOTH changing the coordinate system AND the tensor itself at once, but I won't go into details on this). 2) I don't think it's interesting to think in terms of coordinate systems. They're not significant in any way. You can construct any coordinate grid, it's not an invariant property of spacetime. There are basically two viewpoints of GR - one in which tensors are defined through coordinate systems, but transformed in a way that their "meaning" remains unchanged through coordinate transformations, and another where tensors are coordinate independent objects, but can be represented in a coordinate basis when appropriate. The ONLY thing you should be concerned with are the tensors themselves, not the coordinate system, or grid lines. Rather, a better question to ask is "does the curvature tensor change for different observers?" as that would be a more physically meaningful question.
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
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
I was stuck at this part too. Basically, after rewriting everything out the long way, you can rearrange the e0 vector terms to one side and the e1 vector terms to the opposing side if the RHS of the equation. Then, no one can stop you from interpreting the expanded form as a summation across alpha for the basis vectors
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
@ 2:04 The velocity-components are derived from a position-vector: r = x1 . e1 + x2 . e2. There the e's are dependent on x1(t) and x2(t) too. Why don't we need the Christoffel Symbols in dr/dt like we need them in dv/dt?
That is a very good question and the answer is subtle but crucial : Beware, there is no such thing as a "position vector" in relativity. There are position coordinates, and the velocity vector's components are indeed the derivatives of these coordinates, but the coordinates are *not* the components of a geometrical vector. This is because your surface may be curved, in which case you can only define vectors locally, near a point, where the surface is almost flat.
@@ScienceClicEN Yes, of course. It's all about vectors intrinsic to the surface. I forgot all about that. Thank you for your fast and clear reply. Inspiring videos!!!
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
I am confused why the coordinate system is warped. Is that us trying to simulate the existence of masses warping it, or is that warping due to some other reason?
No for the moment it's simply because we want a model that works for any type of coordinates (this is what we call a "covariant" formalism, our equations won't depend on the grid we use)
In a way, they describe the derivatives of the basis vectors along the grid. We will see in the next episode that they can be expressed from partial derivatives of the metric
@the l33t hamm3rbro Not really no, because they are just artefacts coming from our choice of coordinates. So their units are related to the units we choose for our coordinates (which are completely arbitrary)
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.
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 ??
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.
1) How many eventually are the components described by the indexes? 16 or 64 ? 2) Since in GR the dimensions are equivalent the term dV/dt can be seen as ds/dt.dt (an acceleration , which means a force) or ds/dx.dy ( a space curvature) ? Or both ? And What about ds/ dt. dx ? Thank you.
1) The Christoffel symbols exist in N³ versions, where N is the dimension of the space / spacetime (so 4³=64 for our 4D spacetime). However the two downstairs indices are symmetric so in reality it's only N²(N+1)/2 components (40 in 4D) 2) I did not really catch this question, what are you calling exactly s ? The idea is indeed that dV/dtau is the acceleration of the object (where the derivative is the covariant derivative of the vector, not the derivatives of the components)
@@ScienceClicEN I mean that "s" can be seen as the segment containing the space -time dimensions (the same that you have in special relativity : ds^2 = dx^2 + dy^2 +dz^2 - c.dt^2). So you can do the second derivative both respect to time (t) or space (x,y,z). If you derive respect (t)) can be an acceleration(force) . If you derive respect space (x,y or x,z or z,y) it's a space curvature. I want to go deep in this because i'm tired to hear people saying "Gravity is not a force, but just a curvature of space-time". For me it all depends from the point of view. Thanks again.
Oh I see, so if I get it right technically what you called "s" is rather the spacetime coordinates of the objects. But then, these coordinates are parametrized by only one variable : the proper time tau, so the only meaningful derivative is that with respect to tau. This corresponds to taking the (covariant) derivative of the velocity vector along itself. And this gives always 0 for a geodesic, so there is no acceleration. But check out my video "Are forces illusions ?", I explain why indeed gravity *is* a force, but it's an inertial force, a sort of "illusion" which is not really a physical force.
@@ScienceClicEN Hello, ScienceClic English ! I'm surprized and pleased how quick are your answers. Thank you. Of course there is no acceleration for an observer traveling along a geodesic.(an astronaute in a shuttle). However if you look at the final equation of the geodesic, you may notice that you need to add a "correction term" (Christoffel term also called "affinity connection") in order to annull dv/d tau, so that the total sum is = 0 . For an external observer on the earth this "correction" is equivalent to a work made by the gravitation field to bend the space time, and it has opposite sign of dv/tau. After all, also the bending of the space time has a cost ! To this work the shuttle would reply with the acceleration dv/ d tau but the result is null. It seems to me that you already start from the position that a geodesic has not acceleration just because it's a geodesic. I will check with pleasure your video " Are forces illusions?". Regards.
You are right that Christoffel symbols can be interpreted as a force : an inertial force. Actually, all inertial forces (like gravity, but also the centrifugal force or the Coriolis force) are merely due to Christoffel symbols, that appear when we use non-inertial coordinates. But that's the crucial point : Christoffel symbols (inertial forces) depend on our coordinate system. In particular, we can always find a coordinate system in which all the Christoffel are 0 for a given object. So they do not tell us the curvature of spacetime, but only the curvature of our grid / coordinates. This "correction term" is not a correction to the curvature of spacetime, but a correction to the curvature of our coordinates.
Yes, infact you are differentiating with respect to the components of the vector a.k.a. the direction the object is moving along that component. I think that this means you are looking at how much the basis vector is deviating with respect to the "straight" basis vector which is following the path component. Like, lets suppose we are on a geodesic, i go straight but the metric is weird and my grid changes as I move, in that case the Chrystoffel symbols would tell you how much the basis vectors are changing from what they would be if they simply stayed the same following the straight line if they were on a homogeneus grid. They tell you how the grid changes from the "flat" case (a.k.a basis vectors being constant) to the "not flat" case (a.k.a grid being different and so basis vectors not being constant anymore).
You can still have curved paths in GR, but just like in Newtonian mechanics it requires a force to be acting on the object. If there's no force, the object moves along a geodesic.
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?
At minute 4:17 you show the eight Christoffel symbols, but not the values. It seems you can use the dot product to get the values or you can measure directly on the grid of the tangent vector space. I tried this and I got numbers like .3ex and .85ey by reading off the screen. But almost every example I see these values look like 1/r or 0. I never see .65r for instance. What am I doing wrong?
At minute 4:07 you show a graph of 110 Christoffel symbol on thecurvilinear csys. Using that information only, how do I measure the length of the white vector. Unlike the Cartesian coordinate there are no numbers on the curvy linear grid.
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?
Q: WHY is the object changing it's direction? A: BECAUSE spacetime isn't flat everywhere! Q: WHY isn't spacetime flat everywhere? A: BECAUSE mass and energy bend spacetime!
General relativity explains the direction of an object dropped from a height, but how does it explain its impact on the ground with increasing momentum?
Note that objects cant EXACTLY move in a straight line as theres never vaccum, and due to gravity and particles colliding with the object it will always be moving at a small angle
i'm still trying to understand, the christoffel symbols are defined on wikipedia as the derivative in the local basis, dotted by the local basis. is that the general formula? or does this work only when the local basis is orthogonal?
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.
Think of it this way : You choose a basis vector, then you transport it along a coordinate, and then you decompose its variation into its components. This gives you three indices : which basis vector do you choose ? along which coordinate do you transport it ? and which component of the variation are you interested in ? These are respectively the two subscripts and the superscript of the Christoffel symbols. So the Christoffel symbol with indices ^a_b_c is the a-th component of the variation of the b-th basis vector along the c-th coordinate (sometimes it's the opposite convention for the order of the two downstairs indices, but it doesn't really matter because they are symmetric)
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
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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...
one of the greatest General Relativity series on whole TH-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 th-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.
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.
This is by FAR the best explanation of Christoffel symbols I've EVER seen, holy moly
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!
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.
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.
Dude, with this level of clarity in understanding (and hence explaining), you will probably solve the Theory of Everything. I feel enlightened.
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.
The greatest introduction of Christoffel symbols I've ever seen. Well done, loving the presentation.
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
This is the absolute best explanation of Christoffel-symbols I have ever seen!! Very, very well done!
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.
My god, I have studied Physics ... and this, is by a fair stretch better than any book or lecturer that I have encountered ! Thx!
I can't wait for the 4th part. I am still waiting for the equations of frame dragging, gravitational lensing and gravitational waves.
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.
All i have to say is this is one hell of a beautiful explanation tbh
Clarity!! You, sir, are a gift to humanity 🎁
I love watching these videos, even though I understand very little, and I know nothing of this math, the philosophy is fascinating
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!
Finally, a great visualisation helps me to comprehend "Christoffel Symbols ".
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...
The airplane example was beatiful, great explanation, hope this chanels skyrockets some day!
Thanks :)
Brilliant, like the way you explain it in both detailed and complicated but also simplified terms.
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.
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
Oh my god! The best series on GR I have ever saw!
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.
The example at the end really helped with my intuitive understanding of the maths.
These series of videos are the best on the internet
Best explanation of GR in history!
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. 💯✨
“Christoffel symbols measure the extent to which our coordinates deviate from straight lines along the grid”. Great freaking summary!
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.
Now that I've watched this video and perfectly understood what Christoffel symbols mean geometrically after literal months of know-how digging, I can confidently say that *nothing* is complicated, when it's explained in a proper matter.
Thank you.
3Blue1Brown of science ❤️❤️ Thank you guys
I'm currently learning Tensor Calculus and this video is just perfect
Amazing. Please don't stop doing these videos!
The best explanation i have ever really found
VERY BEST vid on Geodesics with Christoffel Symbols and intuition! And I finally understand it :)
Thank you very much for such clarity and crispness of the presentation.
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 visuals. I love your explanations. Keep them coming.
Thanks !
Great explanation. Thank you! I spent some time to understand all the derivatives, but in the end it was clear!
This is a beautiful and clear explanation! Thank you!
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!!
Wishlist for a future video: Introduction to gauge theory!
From the perspective of gauge theory (basically *all* of modern field theory), the coordinate system is a gauge. Christoffel symbols are used to make derivatives gauge-invariant. That is, covariant derivatives. So once gauge theory is understood, Christoffel symbols are easy.
But how to introduce gauge theory? I recommend an analogy by finance. It is very amenable for visualization!
See
www.amazon.com/Physics-Finance-introduction-fundamental-interactions/dp/1795882417/
or
arxiv.org/abs/1410.6753
I look forward to the next episode with great interest
I didn't know that I needed every second of this video.
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
Outstanding series.
Excellent explanation. Great series so far!!
Waiting for the next part. Well explained, luckily I still remember differential calculus and I can still follow.
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?
1) The Christoffel symbols can be used to express "distortion" relative to any coordinate system. The Christoffel symbols are in fact always defined with respect to some coordinate system. Here is how I would think about it: A manifold contains one or several coordinate systems that connect its points to open subsets in R^n. Given a point on your manifold (corresponding to an event in spacetime), you will also have coordinates for nearby spacetime points, meaning that there will be directions corresponding to changes in coordinates. For example, if you use polar coordinates at one part of your manifold, then there will be an "+r" direction, and a "+theta" direction, which will get you to nearby spacetime points. This might sound trivial, but it really isn't, as the "raw" spacetime points are only a mathematical set of points. The coordinate systems are basically what glues them together.
The Christoffel symbols are basically derived from how the curvature should be expressed using these coordinate directions. So if you change from polar coordinates to some other coordinate system, the christoffel symbols themselves will also change (and sadly not according to the tensor transformation law, since you're BOTH changing the coordinate system AND the tensor itself at once, but I won't go into details on this).
2) I don't think it's interesting to think in terms of coordinate systems. They're not significant in any way. You can construct any coordinate grid, it's not an invariant property of spacetime. There are basically two viewpoints of GR - one in which tensors are defined through coordinate systems, but transformed in a way that their "meaning" remains unchanged through coordinate transformations, and another where tensors are coordinate independent objects, but can be represented in a coordinate basis when appropriate. The ONLY thing you should be concerned with are the tensors themselves, not the coordinate system, or grid lines. Rather, a better question to ask is "does the curvature tensor change for different observers?" as that would be a more physically meaningful question.
quality content! Amazing series keep it up man!
Thanks !
Excellent video. Can't wait for more :)
love this series on relativity
Thank you so much for making such videos!
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
The best explanation of what the Christ awful symbols Are!
Thank you for the great effort
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
I was stuck at this part too. Basically, after rewriting everything out the long way, you can rearrange the e0 vector terms to one side and the e1 vector terms to the opposing side if the RHS of the equation. Then, no one can stop you from interpreting the expanded form as a summation across alpha for the basis vectors
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 feel like the general relativity is really convenient after this nice interpretation and explanations
Is there anywhere we can see the intermediate steps that got skipped when dealing with the equations?
If you play the video in 0.25x speed you should be able to see the steps, I don't think any steps were skipped
@ 2:04 The velocity-components are derived from a position-vector: r = x1 . e1 + x2 . e2. There the e's are dependent on x1(t) and x2(t) too. Why don't we need the Christoffel Symbols in dr/dt like we need them in dv/dt?
That is a very good question and the answer is subtle but crucial : Beware, there is no such thing as a "position vector" in relativity. There are position coordinates, and the velocity vector's components are indeed the derivatives of these coordinates, but the coordinates are *not* the components of a geometrical vector. This is because your surface may be curved, in which case you can only define vectors locally, near a point, where the surface is almost flat.
@@ScienceClicEN Yes, of course. It's all about vectors intrinsic to the surface. I forgot all about that. Thank you for your fast and clear reply. Inspiring videos!!!
Thanks !
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
just blown my mind away
I am confused why the coordinate system is warped. Is that us trying to simulate the existence of masses warping it, or is that warping due to some other reason?
No for the moment it's simply because we want a model that works for any type of coordinates (this is what we call a "covariant" formalism, our equations won't depend on the grid we use)
Are Christoffel symbols like a partial derivative?
In a way, they describe the derivatives of the basis vectors along the grid. We will see in the next episode that they can be expressed from partial derivatives of the metric
@the l33t hamm3rbro Not really no, because they are just artefacts coming from our choice of coordinates. So their units are related to the units we choose for our coordinates (which are completely arbitrary)
3:17 which coordinates??
what do X0 and X1 mean? you suddenly introduce them from no where
It's from in the first episode, we first have to choose a coordinate system before doing any calculation
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
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 ??
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.
Very instructive, some of the maths are unexplained such as the swapping of indices.
1) How many eventually are the components described by the indexes? 16 or 64 ? 2) Since in GR the dimensions are equivalent the term dV/dt can be seen as ds/dt.dt (an acceleration , which means a force) or ds/dx.dy ( a space curvature) ? Or both ? And What about ds/ dt. dx ? Thank you.
1) The Christoffel symbols exist in N³ versions, where N is the dimension of the space / spacetime (so 4³=64 for our 4D spacetime). However the two downstairs indices are symmetric so in reality it's only N²(N+1)/2 components (40 in 4D)
2) I did not really catch this question, what are you calling exactly s ? The idea is indeed that dV/dtau is the acceleration of the object (where the derivative is the covariant derivative of the vector, not the derivatives of the components)
@@ScienceClicEN I mean that "s" can be seen as the segment containing the space -time dimensions (the same that you have in special relativity : ds^2 = dx^2 + dy^2 +dz^2 - c.dt^2). So you can do the second derivative both respect to time (t) or space (x,y,z). If you derive respect (t)) can be an acceleration(force) . If you derive respect space (x,y or x,z or z,y) it's a space curvature. I want to go deep in this because i'm tired to hear people saying "Gravity is not a force, but just a curvature of space-time". For me it all depends from the point of view. Thanks again.
Oh I see, so if I get it right technically what you called "s" is rather the spacetime coordinates of the objects. But then, these coordinates are parametrized by only one variable : the proper time tau, so the only meaningful derivative is that with respect to tau. This corresponds to taking the (covariant) derivative of the velocity vector along itself. And this gives always 0 for a geodesic, so there is no acceleration.
But check out my video "Are forces illusions ?", I explain why indeed gravity *is* a force, but it's an inertial force, a sort of "illusion" which is not really a physical force.
@@ScienceClicEN Hello, ScienceClic English ! I'm surprized and pleased how quick are your answers. Thank you. Of course there is no acceleration for an observer traveling along a geodesic.(an astronaute in a shuttle). However if you look at the final equation of the geodesic, you may notice that you need to add a "correction term" (Christoffel term also called "affinity connection") in order to annull dv/d tau, so that the total sum is = 0 . For an external observer on the earth this "correction" is equivalent to a work made by the gravitation field to bend the space time, and it has opposite sign of dv/tau. After all, also the bending of the space time has a cost ! To this work the shuttle would reply with the acceleration dv/ d tau but the result is null. It seems to me that you already start from the position that a geodesic has not acceleration just because it's a geodesic. I will check with pleasure your video " Are forces illusions?". Regards.
You are right that Christoffel symbols can be interpreted as a force : an inertial force. Actually, all inertial forces (like gravity, but also the centrifugal force or the Coriolis force) are merely due to Christoffel symbols, that appear when we use non-inertial coordinates.
But that's the crucial point : Christoffel symbols (inertial forces) depend on our coordinate system. In particular, we can always find a coordinate system in which all the Christoffel are 0 for a given object. So they do not tell us the curvature of spacetime, but only the curvature of our grid / coordinates. This "correction term" is not a correction to the curvature of spacetime, but a correction to the curvature of our coordinates.
My favorite physics channel along with PBS Spacetime.
the basis vectors change as we move, and we can differentiate with respect to directions but aren't directions defined by basis vectors?
Yes, infact you are differentiating with respect to the components of the vector a.k.a. the direction the object is moving along that component. I think that this means you are looking at how much the basis vector is deviating with respect to the "straight" basis vector which is following the path component. Like, lets suppose we are on a geodesic, i go straight but the metric is weird and my grid changes as I move, in that case the Chrystoffel symbols would tell you how much the basis vectors are changing from what they would be if they simply stayed the same following the straight line if they were on a homogeneus grid. They tell you how the grid changes from the "flat" case (a.k.a basis vectors being constant) to the "not flat" case (a.k.a grid being different and so basis vectors not being constant anymore).
hey man, your videos let people like me a 13 year old understand, thanks for that
So, In classical physics we have fixed grids for flexible paths. In GR we have fixed geodesics with flexible grids. Is that It?
You can still have curved paths in GR, but just like in Newtonian mechanics it requires a force to be acting on the object. If there's no force, the object moves along a geodesic.
That's very simple way to explain the most complex piece
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?
At minute 4:17 you show the eight Christoffel symbols, but not the values. It seems you can use the dot product to get the values or you can measure directly on the grid of the tangent vector space. I tried this and I got numbers like .3ex and .85ey by reading off the screen. But almost every example I see these values look like 1/r or 0. I never see .65r for instance. What am I doing wrong?
At minute 4:07 you show a graph of 110 Christoffel symbol on thecurvilinear csys. Using that information only, how do I measure the length of the white vector. Unlike the Cartesian coordinate there are no numbers on the curvy linear grid.
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
Q: WHY is the object changing it's direction?
A: BECAUSE spacetime isn't flat everywhere!
Q: WHY isn't spacetime flat everywhere?
A: BECAUSE mass and energy bend spacetime!
General relativity explains the direction of an object dropped from a height, but how does it explain its impact on the ground with increasing momentum?
Absolutely beautiful
Note that objects cant EXACTLY move in a straight line as theres never vaccum, and due to gravity and particles colliding with the object it will always be moving at a small angle
i'm still trying to understand, the christoffel symbols are defined on wikipedia as the derivative in the local basis, dotted by the local basis. is that the general formula? or does this work only when the local basis is orthogonal?
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.
I wrote my article on Christoffel symbols and it got cited on Wikipedia
Aand now it's getting tricky ^^
I did not understand the Christoffel Subscripts and superscripts. :(
What do they mean?
Think of it this way : You choose a basis vector, then you transport it along a coordinate, and then you decompose its variation into its components. This gives you three indices : which basis vector do you choose ? along which coordinate do you transport it ? and which component of the variation are you interested in ? These are respectively the two subscripts and the superscript of the Christoffel symbols.
So the Christoffel symbol with indices ^a_b_c is the a-th component of the variation of the b-th basis vector along the c-th coordinate (sometimes it's the opposite convention for the order of the two downstairs indices, but it doesn't really matter because they are symmetric)
@@ScienceClicEN Amazing explanation!:)