The Maths of General Relativity (5/8) - Curvature
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- Опубликовано: 15 сен 2024
- In this series, we build together the theory of general relativity. This fifth video focuses on the notion of curvature, and the different tensors that are used to characterize it.
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This is such a great channel, I hope to see you guys grow
Thanks :)
They are already big but in french 😁
3 totally different concepts that make General Relativity a difficult subject:
1) We work in 4 dimensions.
2) The dimension of time has a totally different behavior than the spatial ones you learn geometry on.
3) This 4D space-time is not always flat but can be curved, and the curvature might be intrinsic (no need for a 5th dimension where the 4D space-time is embedded and curves).
On top of all of this, you have to choose coordinates systems that might be as weird all the above concepts.
These are all different ideas and all work together at the same time in General Relativity. Awesome work explaining some of this mind fuckery! You are an incredible educator.
An important point is that the formula at 7:55 is only valid for orthogonal coordinates(as stated), but the formula at 8:01 is true for any coordinate system and is the general definition.
👍👌
You are doing a great work. The intuition behind the mathematics is really important.
Thanks ! I think so too :)
Best series on GR ever. Logically sequemced and concise explaination of key princples.
I suddenly understood GR which I failed to do for decades. Many thx for that wonderful Christmas present!
*Sees Premire Time.* Noooooooooo! This is an active Series! My mind was being blown and absolved all at the same time.
I sincerely WISH this was available 15 years ago. It makes SOOOOOOO much sense.
I can't stand it I'm ready to jump out of my own skin so many conceptual walls were being knocked down!
I feel so tingly!
It took years to understand General Relativity. I wish this was available a decade ago. I wonder if this type of material would be available if there were no lockdowns.
Why wouldn't it be available without the lockdown?
These videos have already been published over a year ago on the main ScienceClic channel over a year ago - in french dub though. So the majority of the work has already been done I believe.
It's was available in french berofe the english chanel started. It was before lockdown
@@lounesz.5156 I just noticed there have been many General Relativity educational videos created in 2020. Veritasium, PBS Spacetime, and Minute Physics have all created new videos on General Relativity. Even at Science Asylum there was a new video on tensors, but it may have been produced before the lockdowns.
based covid
Absolutely brilliant! Richard Feynman himself would have applauded, he who was so apt at explaining complex concepts in a clear and engaging way. It really makes you want to go back and listen to it again to make sure you understand every bit of it - at least, every bit of what’s presented.
Thanks again for taking the trouble to make this effort - it sure isn’t wasted.
Btw, The University of Western Australia has a program to bring Einstein’s Relativity to school kids (in high school), which seems to be quite successful. I am sure they will find these videos very useful.
Bro u are the only teacher I found in my whole life who can teach relativity to even a 10 std student..ur my favourite and love you bro...👍
std?
@@llamatown8160 sus
I'm in 7th grade and I can understand and work on general relativity problems.... I have studied quantum mechanics too.
@@brett_webber233 nice to hear
@@brett_webber233 can you do calculus as well?
This gets Crazy complex but exponentially more interesting 🧐🤔
Thanks for simplifying and explaining to us so well such a fundamental topic.👏👏👌🤓😁
The humanity says thank you.
Get rid of the “the”.
Saying “the humanity” would be referring to being humane and not all people
@@biblebot3947 isnt it the opposite?
2:14 Note that the upper arrow moved to the right keeps pointing to the same "who knows what" direction, and the lower arrow moved to the right does not have the same behavior.
@silverrahul Nevertheless, as Collaz said, the upper and lower arrows move to the right in a different manner.
More specifically, the upper arrow moves "parallel" to itself, but the lower didn't. That makes the comparison not very convincing since it's normal the resulting orientation is different.
That said, perhaps the results would *also* have been differents if the arrows had moved in the same manner.
@silverrahul I stopped the video at this point since I’m siding with Jolulu on this one, perhaps there is an explanation further in in the video, but for now I can’t help myself from getting into the discussion. The lower arrow keeps the head pointed towards the “north pole” while the higher arrow keeps it angle intact relative to our perspective which makes it deviate from pointing towards the North Pole and instead pointing eastwards.
I believe what is lacking is proper usage or visualization of geodesics vs curved lines. In the video, both arrows move upwards in a straight line (geodesic) while the sideways motion is not (can’t remember if the lower lateral movement is on the equator) or at least not the upper lateral movement. Point is, if you have two vectors on spherical coordinate system, moving the vectors in a straight lines only (geodesics) but in different orders like in the video should render the effect the video is supposed to demonstrate. Instead he is actually n o t moving the vectors in straight lines with the same angle in reversed orders, by not following a geodesic the lateral movement is actually curved. Meaning the lower vector is moving laterally in a straight line if on the equator or with a flatter curve compared to upper vector’s lateral movement if not on the equator.
@@johanpersson8156 Hello, this point gave me some concern, but the video is correct. The vector moves along the coordinate so as to keep the same angle to the tangent along that coordinate. The lower arrow starts at 90 degrees and remains at 90 degrees to the tangent along that line of latitude. The higher arrow starts at 60 degrees (or thereabouts) and keeps that angle to that line of latitude.
No body:
Physicists:
Let's make learning GR harder by naming variables in such a way that they're indistinguishable when lecturing!
It's impossible to denote Tensors any other way. If you are a physics student, just know that there are much harder things in the world.
@@aniksamiurrahman6365
I think they were referring to "mu" and "nu". They could have easily named them something else.
@@APaleDot No matter what system u adapt, I believe they'll end up just as complex.
@@aniksamiurrahman6365
It has nothing to do with the complexity of the system. It's specifically about how the names of variables sound when spoken aloud.
@@APaleDot They're greek letters, and they are not pronounced like in the video. µ is pronounced "mi", and ν is pronounced "ni". Not really that important to distinguish them either, just need to know that they are indices. Anybody can mix them up or switch them out. Not like it's suddenly much harder just because the letters are similar. They don't matter that much.
The quality of the Video is simply gorgeous. You've made such a good work
Thank you :)
Oh wow. I've never really understood curvature until now. Thank you!
Whoa! Calculus anyone?
@@aniksamiurrahman6365 Vector calculus
I love how you include the equations.
I consider these videos a great gift and I would like to express my gratitude.
THANK YOU!
Those equations simply explode, it must be tedious to calculate by hand!
Yeah it's the kind of thing you only do once. Thankfully wolfram alpha lists all of these values related to several coordinate systems
It is! But it is worth and a good practice to calculate all those objects (metric tensor, Christoffel symbols, Riemann curvature tensor, Ricci tensor and Ricci scalar) by hand. It takes a while but this will teach your brain :)
You really need to understand Vector Calculus to get a conceptual grasp of Tensors because Tensors are extensions of vectors.
Even though it's so hard to grasp and understand... I didn't skip evn one second throughout this series.... Just because of your way of teaching!!!!
I LOVE this video series so much! Can wait for the next installment. The visuals with your fantastic explanations help to demystify one of the most intimidating topics in all of physics.
So clearly explained, it feels like I’m cheating somehow. I’m just starting to learn GR and I seem to have landed on the big ladder square of Chutes and Ladders. Thank you for making these videos!
If anyone wants the curvature tensor deep dive, eigenchris does a great job also.
just wanted to say great job with this series! i'm in high school and find your explanations amazingly clear and cohesive. keep doing what you're doing, we really appreciate it!
Thank you very much, it's great that in highschool you're already interested in such topics !
No words to describe it. Simply amazing your work.
As a high school student watching this, I don't necessarily understand the math. I understand the math and notation in small pieces of equations but when woven together, it is beyond my current understanding due to my limited knowledge in math. However, though I may not see the fine details in the mathematics, I understand how each piece of the equations (like the metric tensor and the Ricci tensor) plays a role in the motion of objects and the general ideas being set forwards. It is very hard to craft lessons and explanations in that way, where both experts and novices get something out of it, but you have done it perfectly. You are incredible and I can't wait to see more! Have a great day and keep up the fantastic work!
Coming from a programer's background and very interested in the sciences, I would love to see someone (or myself if I find the time) create a program that you could manipulate the fabric of spacetime and see how that your changes in the inputs would affect an object in the output. Something like KSP I suppose... but you can change the fabric of spacetime.
A small suggestion, since we can safely assume, the audience of this content are going to be familiar with fundamentals of calculus, and you people are know how to lucidly show concepts, it would be great if you could include a video explaining the math behind the formula derivations!
Seshnag R follow prof Leonard susskind if you want to learn the mathematics behind these nice explanations
I would absolutely love if you’d do a similar episode on Weyl curvature! I understand Ricci curvature but I’m struggling with intuitively understanding Weyl curvature.
Absolutely a fantastic series about all the ins and outs of General Relativity. Probably the best I have seen sofar. Modern computer graphics make it easier to understand. Well done!!!!!
Thank you ! Glad you like it :)
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.
Outstanding! I read so much and watched so much videos about R, but you are the first to simplify that concept to make it understandable for me :)
I've been waiting patiently all week for this, and I'm not used to that delayed gratification but damn is it good
About ready to dive into my old copy of Thorne and Wheeler Gravitation for bedtime stories! This series is a fun reference of visualization.
Thank you for making me able to follow more complex ideas with visual presentations
THANK YOU. I LOVE EVERYTHING OF YOUR VIDEO CONTENT AND YOUR VOICE. LOTS OF BEST QUALITY MATERIAL SUPPORTED BY CLEAR EXPLANATION, IT IS SUCH A PLEASURE, THANKS FOR SHARING YOUR KNEWLEDGE AND PASSION OF PHYSICS WITH US!!!
This is best science channel in YT... you deserve a special plate.... every vid is a masterpiece!!!!
Great material. The visual explanation of the Christoff symbols and Curvature Tensors are stunning, I already studied the math but this helped a lot into getting a "physical" grasp of the subject. Thanks so much for all the time and effort you put into this series of video .
Really short and well visualized explanation
Great series! One suggestion I’d make is to include the interesting history behind the discovery of each of the concepts. And maybe a few links for “further reading” too
Super excited to get to the EFE's. Keep up the great work! This channel will hit viewer critical mass soon enough
Please, make more videos! They are indeed absolutely eye-opening and expand my horizons of knowledge immeasurably!
So fun. During this video, when he explained curvature and the "R", I suddenly understood what they mean when they say "is the universe flat or spherical". Keep up the good work.
I was today years old when I finally found a video series that explains the math AND application of GR!
Really cool, such clear explanations and the animations help with understanding concepts more intuitively
My head hurts 😂
Same. But thinking of R tensor as a tool and not something physics helps a bit
@@abhijithcpreej go for weyl tensor
That's a necessary condition before understanding GR
Take a Ricci aspirin.
I love this channel so much dude!
WOW.
Thanks for making such a great explanation with awesome animation.
I have read a few concepts from the book but it's becoming more clear watching this!
Thanks a lot. I can't wait for another video.
these visualizations and explanations are really great!
Thanks !
Oh I wish you could be more formal with the math, I mean on another series. But for easy digesting, this is already great.
Sir I tried learning gtr now for atleast 2 months and were not getting anywhere, these vids are mind blowing.
I could completely follow the concepts now, thank you a lot.
This is one of the best way to teach
Hope if u can even cover black hole curvature and other possible curvatures
Simply wonderful, ScienceClic. Thanks. Stay safe of Covid.
First video I've seen on this channel - fantastic stuff!
Once you're finished this outstanding series, can you make some really in-depth videos about black holes? From a general relativity perspective?
Simply fantastic, can't wait for the following videos
Why in god's good name do you always have to blow my mind at the end of the video???
Wow, it requires someone brilliant to make something complex seem so obvious. Awesome stuff.
Fantastic accent, fantastic visuals, fantastic explanations. Fantastic channel.
But in curved space, the christoffels, riemann tensor, ricci tensor, ... variate by position. And the postion itself is a function of pathlenght (or proper-time).
So if you really want to calculate lightpaths, you have to completely write out and solve the geodetic Differential Equation:
d² x(τ)^i/dτ² = - Γ(x(τ))^i_uv dx(τ)^u/dτ dx(τ)^v/dτ --> And solve for the functions x(τ)^j that give you the position-coordinates at every given time (lightpaths).
So, here's my question: Abusing Tensorflow2.x with multiple nvidia-gpu support, is it possible to make relativistic raytracing in realtime?
A gameengine that could calculate bended lightpaths around massive objects and disturbed spacetime. Looking around corners, looking to the past, simulate Warpfields, and so on ...
Raytracing in realtime is impossible at the moment, but it can be done with some approximations in certain specific situations. Check out my personal channel "Alessandro Roussel", I am developing an algorithm to do some realtime relativistic "raytracing" around black holes (it's not really raytracing as my algorithm gets rid of integrals, but the maths that are involved are doing the raytracing in a way)
Can't thank you enough. I quit PhD some years back. Trying to get back to Physics. Needless to say that these are invaluable. Does anybody know similar series for Quantum Field Theory?
Fascinating. Makes perfect sense. But it is clearly a century old. There is about to be a new description of physical reality that will take the next leap into re-framing into an even more elegant model. This is well done, but it is not the finish line.
You are doing a great job with this series
Never seen such a Great explaination in RUclips you are the one , thank you very much
at 9:10 , it's important to comment that two particles will not come towards each other because of the test particle limit. People might get confused and ask.. "what about their mutual gravitation?"
if there is gravitation then it's not flat Minkowski space any more
Perfectly and succinctly explained, thank you.
Bro I study math and your channel is insaneeeeee keep up the work, you helped me out alot
This is dope! Looking forward to the rest!
Never thought I would be able to get mathematics behind General Relativity :) 👍👍
To say you do an excellent job would be a grosse understatement !!!
Great Video! Thank you. At last, time drawn on the horizontal axis. Way more convenient than the standard representation :)
I can't believe I am starting to understand GR even slightly, thanks a lot!
This one particularly was awesome🎉
Keep up the good work👍
Infinitely better than anything I've seen in a graduate level textbook or in Wikipedia.
I can't wait for the next few videos in this series.
Commuting diagrams! Category theory! I love it :)
Beautiful video
0 dislikes so far
Nice!!!
My first time watching a premier. .. I love this channel. :) See you in the comment section.
Good work guys❤️❤️ Thanks for such a great video ❤️
Awesom channel and very good video. But I would wish an extra Video about co- and contravariant representation of vectors and 5he coordinate transformation associated with it because it seems that students in physics sometimes swap its mathematical meaning. Good work.
Great video series! However, it does leave me with a couple of questions.
It would be nice to know how and why we choose the indices for the Ricci tensor from the Riemann tensor for higher dimensions. In 1+1 dimensions, it is somewhat easy to see why we choose the ones you stated. If you simply see which ones are 0, and which ones are the opposites of each other, it becomes obvious. However, the particular choice from a +/- pair is still a mystery to me.
Great job guys, again ;)
Ah--this makes sense. Thank you for explaining this clearly.
Perfect videos. My congratulations!
Become a big Fan of your content - Alessandro Roussel, a name i will remember :)
I wish you can make it more clear what are "defined/found out to be" and what "can be mathematically derived".
Great Channel, keep up the Great work
Thanks !
10:11 from here this is a great short flat earth debunk clip
Thank Q so much . A thousands thanks to you
Very great and simple, really thank you.
Brilliant expression.
...and now we know why GR is no non-intuitive. 256 component curvature tensor. This is why when calculating scenarios where the curvature is high, and rapidly changing, we need computers, it is impossible to do the calculations by hand in a reasonable time.
god these videos are so so good
i haven't seen anyone else try to get right into the nitty-gritty of tensor maths etc, and to do so in a way that is accessible for someone like me, who has a pretty basic grasp of linear algebra and that's about it
Your explanations are beautiful...thanks!
Great video! I have to say that mu and nu can be confusing to hear.
@@user-sq7um9kh2o The Greeks!
Thanks Sir for your wonderful videos, Wow!
So in curve space although path is differentiable doesn't hold for vectors the analogous of Schwarz theorem for scalar fields:
therefore to take into account variation of the bases vectors, that depends on path chosen, we need to compute Christoffel's symbols?
The Rieman tensor can be seen as a sum of the second derivative of the g-values (g'') along a circuitation : if the g-values are constant R=0. You showed that this is the case of the Minkowski flat space-time.
However we can have positive g'' and negative g'' balancing each other in some points of the circuitation. Eventually is always R =0. Which would be the geometry of space-time in this case ? Going down to a 3D surface is this the case of a conical surface?In this case two geodetics will meet towards the vertex ? Thank you.
Beautiful!!!!!!!!
"The vector R is the difference between two derivatives of the basis vector in opposite orders."
In the first episode you said that everything would remain understandable and intuitive.
The fact that you used this statement determined that was a lie.
I don't know what it is but this sure helps me understand MTW a lot more.
So grateful for this explanation.🙏🏼
u did a great job, congrats from brazil
great content!