The Maths of General Relativity (4/8) - Metric tensor
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- Опубликовано: 6 сен 2024
- In this series, we build together the theory of general relativity. This fourth video focuses on the notion of metric tensor, its relations to the Christoffel symbols, and physical distances.
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Alessandro Roussel,
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How is it that you are able to explain this so clearly?
I am genuinely amazed. And somewhat smarter now.
Ahah thanks I am glad you like it !
I feel the same...I took a GR course last semester and I'm still struggling to come to terms with it entirely..watching this and revisiting it will bridge the barrier.
One of the MOST UNDER-RATED Science channel on RUclips.
This is why I love RUclips. Content creator like you makes this platform amazing.
That backround music keeps me charmed all along the story...
interstellar theme would have taken it to the whole new level.
Yeah bt 3blue1brown bg musics are better for this kind of things.
This is the song!
ruclips.net/video/3OCqSLc5J-c/видео.htmlsi=xUr2N2x6729qd77x
13:34... I lost it! I literally spilled my drink all over me, this was mindblowing!!
Yep that's when I went 100% crazy :D
I wish people taught not just general relativity, but also special relativity this way
that might be the future
We definitely see some higher quality educational videos since the Covid-19 outbreak.
special relativity is "relatively" simple. proper courses with deep explanation are better.
@Bhavesh sinha Personally I find minutephysics' videos too simplified. They use analogies that make it easier for the layman to understand, but aren't very accurate for people who know more math
You may search for sl in youtube.There are a lot of info about sl, esp in MIT .
Fantastic! I never seen GR explained so succinctly
Took me awhile to understand the meaning of distance by Minkowski metric at the end, but then I finally got that this distance actually represents the rate of proper time. And it is not the same in every direction of the space-time. I feel enlightened now. Your videos are amazing!
Exactly ! What we call "proper time" is actually simply the measure of distances along the worldlines of objects
It isnt Minkowsky Metric but Minskowsy MATRIX
@@ibmmbi2317 Minkowski metric is the function which determines the distance between two points in the Minkowski space-time which is defined/calculated using this matrix
@@AndrewStarostin Oh ok.
So within this Universe is made up of space(s). What is beyond the universe made of then?
I've been struggling for years to understand the math of GR. This video series is filling in most of the gaps and making it make sense for the first time in 25 years of trying. Thank you so much!!!
I wanted to know GTR but always hated and feared maths.
But you not only explained GTR perfectly but made me love maths as well.
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.
Problem with most lectures about ART (or other complex subjects) is, that they dive deep into the subject without simplifying it. When you read scripts from university professors about that, you get "beaten" by most abstract multi-dimensional tensor analysis (immediate brain death).
But in these videos you get a general glimpse step-by-step without overloading you.
I am looking forward for the rest of this series.
You make me feel stupid. I have no clue what he's been talking about. All videos 😢
In my opinion everybody should see this, truly, a remarkable video! The way it gradually connects all the dots and filling gaps between concepts, bringing them together, astonishing! Thank you very much for sharing this, sometimes I get lost in the formulas, this is eye-opening!
Probably, the only video on RUclips to have this clear and precise presentation of mathematical foundation of GR.
This is like a visual supplement to courses on Relativity from scientists like Suskind and Carol .
Awesome work. 👍
Just want to say that your last video which mentioned the geodesic equation sent me down a RUclips rabbithole which resulted in my learning a lot about variational calculus and Langrangian mechanics, which is was really interesting, so thank you for that. :)
Mind sharing your rabbit hole? I had some confusion surrounding the Christoffel symbols. Specifically, when we obtain the vector de1/dx0, and resolve them into their components, what components exactly are they?
@@hrithikravi8117 Sheesh, I don't remember the specific videos at this point and they're way too far back in my watch history to find them that way, but I just searched "Christoffel symbols" to see if any of the results showed up something I'd watched and I found this one: ruclips.net/video/1CuTNveXJRc/видео.html
I don't remember if it was any good, but it's worth a try.
Edit: Just realized there's a search tool got watch history. I think the above is the only video I watched on Christoffel symbols, but I remember this one on Langrangian mechanics being really good: ruclips.net/video/SQLxrr9N8zM/видео.html
and this one on variational calculus: ruclips.net/video/SQLxrr9N8zM/видео.html
I too went down a rabbit hole of trying to learn general relativity. Eigenchris has a good playlist on tensor calculus and general relativity now I understand it extremely well.
Congratulations!
The correlation (at the end of video) of proper time dτ = ds of movment obsarvator O' with the "external time" dt of a relatively motionless observer O by use only a single square root is astonishing. And the distinction between distances between imaginary, complex and real, in the end is wonderful, because it leads to conclusions of significant natural significance.
Bro I don't know how to thank you..I'm in lack of words to express to you my support..u are the best teacher I have seen so far
Oh man, I never thought I'd be able to discuss general relativity with people. This channel is crazy!
This channel is too underrated!
Thanks! This youtube channel is beyond this world!
Blown away by the quality of these videos. They raise the bar extremely high.
The less I understand, the more excited I get. Damn it! can't stop watching!!!
Initially I couldn't understand much of what was going on this series and I was honestly baffled most of the time. I gave up. But after watching the some introductory lectures and gaining somewhat of a theoretical understanding, I came back. Honestly, this series makes a world of difference when it comes to intuitively understanding the theoretical concepts. The animations and the examples you choose to explain new ideas are great. It is amazing I could actually understand after struggling for so long, trying to grasp what the matric tensor is. Thank you for your time and energy in making these videos. I really appreciate it.
These videos are excellent, thank you so much for them.
For someone first getting into GR, the math and jargon can seem so daunting that learning might seem so complicated that people don't bother. These videos give insight with things most people are familiar with and no doubt motivate people to learn this beautiful theory.
This is such an amazing video.. It deeply extends your understanding of the reality with such an ease.
Glad you liked it !
I'm from Houston tx and I started getting interested in science while listening science vids while driving as a way to escape politics and other useless trash.
It's been 5 years since then and I've learned deep concepts on relativety, quantum physics, electromagnetism and more.
I thank you from the bottom of my heart for the gift of knowledge and a continued love of learning.
I never thought that I would be able to learn this stuff.
You're going to make me have a good grade in GR this semester! Thank you so much
Hi friend how are you i hope you will get a better note in general relativity but can i ask you wich Studie in high school are studing i understand all about relativity
These are insanely complex topics from physics and mathematics perspective... and yet you managed to put them in amazingly simple terms that everyone is available to understand, so well done my friend well done 🤓😁👏👏👌
“This is more commonly called ... light.” :D
Exactly, it seems so weird to talk about something crossing no distance/time between two points, but that's exactly what light does all the "time".
*head explodes*
Goosebumps
I’m so lost, yet I keep coming back !!
I guess I enjoy knowing how did they reach their understanding. (In Theory, of course)
I am hooked to your videos beyond measure. Been watching them on repeat (and once again a new comes out) in the hopes that I catch something I had previously missed!
Please keep them coming!
This course is absolutely amazing, it describes the theory very intuitively and precise. Keeping moving forward.
The physics is amazing, but the way information is layed out here is even more amazing. Anyone who wants to teach anything to anyone should strive for this level of clarity.
This is amazing! I am reviewing all the videos of this series because I am taking GR this semester and I am understanding it and loving it :) Thank you so much for providing a means for a simple student like myself to have an easier first approach to these concepts and mathematical tools. Well done :D
Is this the same order in which mr. Einstein formulated the equations? what was going inside his mind?
This series is structured roughly in the same way we teach general relativity today, as a "finished" theory (no theory is ever "finished" though). But Einstein's journey in discovering all that was very much more cahotic and a lot less "linear". He was also greatly helped by many colleagues. Some of the things in this series were not discovered / studied by Einstein originally.
@@ScienceClicEN i have also thinked that
you are the best physics channel in RUclips. thank you for your videos. can you make a series about the math of string theory?
The satellite example at the end was THE BEST explanation I've ever heard of why nothing moves faster than light and why no time passes when you move at the speed of light. Thank you.
Excellent. Now I want to know how to integrate the metric tensor over a trajectory on a sphere.
He gives the formula/components in the vid, I'd think it would just be a very simple integration of those
if it's a metric that doesn't depends on the position, its like a ∫ds, like in a circle, ds with constant radius is ds=rdθ => ∫rdθ
wow!! Amazing work my friend! Never thought that the math of general relativity would be this easy to understand, and that's because it isn't, yet you make it easy somehow! Thank you for this series.
I don't remember the last time I've binge watched a science channel woahh
I thought the previous videos were amazing and the math will only get harder. But damn, this is amazing.
Bro, I legit have no idea how you only have 646K subs. Your work is very good, you deserve way more.
13:55 "Nothing moves faster than light."
We hear this everywhere, but hearing it after being shown the reason has a different feeling to it, a stronger impact 😁.
I've actually watched the whole series in French many times and I'm in school to learn about GR and QM/QFT so I was already familiar with that concept, but that line still gave me goosebumps 😁.
I’m watching this all the way from Afghanistan 🇦🇫 and I’m loving it.. don’t stop making these and also when could we expect the next video in this series?
This is the GREATEST video on RUclips hands down.
I never thought I'd understand GR, but you sir have done it. Also, the background music fits the mood absolutely perfectly. Keep at it mate.
I'm waiting for all the 8 videos to come out so that I can share them indefinitely with all the souls who've expressed interest in understanding GR to me
This concept of tensors and how they need to be integrated along the path is phenomenal. I was concerned as to how to compute things on a non Euclidean space with our math and coordinate systems that are flat. But thinking of the tensors as a "field", in which each point is the error between the actual coordinates and our (which is actually the one who's curved in comparison to the real world) is brilliant.
I am currently following this series by programming on an implementation of non euclidean spaces. Right now I have a "black hole" that you can orbit!
I may show you in twitter oncr I'm finished playing
yeah non euclidean geometry as well as riemannian geometry and conics are a bit hard to visualise on a regular cartesian plane
Really nice presentation. The only thing that's a little strange to me is showing time horizontally and space vertically. Having worked with space-time diagrams before, that was a bit unexpected.
now it makes sense, tahnk you, I had modern physics in college and my professor wasnt even close of explaining the tensors to us, I was so confused because I really didnt know why nothing could be faster than light, thank you
I need to pause this right now cause my mind is absolutely blown to bits, I don't usually watch youtube but for the past 3 or 4 days I've religously watched your channel, I have no idea how I found it. But these explanations are the exact thing I've been looking for to quench my thirst for knowledge in this feild of physics.
Very BEST explanation of the Metric Tensor on You Tube.
Wow! I didn’t expect coming to realize why integrals are so important. Such a great video.
Best advanced physics RUclips videos ever.
This series is awesome, however there is one deficit in the expanation. The actual values in an example and what these values mean on inspection.
Ex. Minute 8:28
I did the calculations and came up with the following matrix for the metric tensor wrt this example.
I got
--------------------------
phi = -8 deg (west of prime meridian)
theta =8 deg (north of equator)
--------------------------
top row left
d theta
R^2 =~16,000,000 mi = (4,000 mi)^2
top row right
d phi
0
bottom row left
d theta
0
top row right
d phi
R^2 cos^2 (20 deg)=~15,000,000 mi = (4,000 mi)^2 * cos^2 20 deg)
plugging in to distance equation
ds^2=R^2 d(theta)^2 + R^2 cos^2 (20 deg) d(|phi|)^2
ds^2=R^2 d(8)^2 + R^2 cos^2 (20 deg) d(8)^2 (-1)
ds^2=16,000,000 d(8)^2 + 16,000,000 cos^2 (20 deg) d(8)^2 (-1)
ds^2=16,000,000 64 + 16,000,000 (.94) 64 (-1)
ds^2=1,024,000,000 - 61,603,840,000=60,579,840,000
ds=~250,000 mi (about 10 times around the earth)
Where did I go wrong here, please?
Even though I will have to watch this multiple times to get an actual grasp, I am a huge fan of these videos. It is brilliantly made. Thanks!
You had my mind blown away at 13:35. I love you man, please keep your channel going.
This is just beautiful and brilliant. You are doing an amazing job!
I wish you had shown the derivations from minutes 2:00 and 5:50
For 2:00 check out quadratic forms. Basically the idea is that we can choose a grid such that it *is* the Pythagorean theorem. And therefore with another grid it must be the Pythagorean theorem but with a linear change of coordinates (so for example instead of x²+y², a change of grid gives you (ax+by)²+(cx+dy)², which is a sum of terms containing x², y², and xy)
For 5:50... the proof is left to the viewer :p Otherwise check out covariant derivatives. The idea is that the metric tensor should have a 0 covariant derivative (roughly speaking geometry must stay coherent along the surface), which forces the Christoffel symbols to take this form. The calculation is a bit painful but it's quite straight forward and its a good exercise to do. Try it !
@@ScienceClicEN alright; thanks for the video and the explanations
This series has been a great introduction to GR for me. I haven't studies it properly before this, and now Im learning it on my Christmas break. Thank you to the people at @@UCWvq4kcdNI1r1jZKFw9TiUA.
I'm a physics undergrad student and... holy shit.
I majored in Physics and it's embarrassing I finally understand it. If it existed 20 years ago, I wouldn't give up Physics. (I manage to live by ML)
Me too.
1)Suppose a rocket accelerates in an empty universe, undergoing an inertial force: which would be the grid of coordinates around it ?
2) Suppose a light beam, made of plane waves, is bent by a gravitational field : is the light velocity the same in all points of the plane? (In periphery as near the rotation centre)? Thank you for your "proper" time !
ScienceClic - "...The satelite will never be able to reach such points. _Nothing_ moves faster than light."
Miguel Alcubierre - "Rev up the warp drive! I'm gonna bend spacetime and *_harness the power nothing_* to move faster than light!!"
But yet, light inside the warp drive would still move faster than you ;)
I watch this video through concentrated mode and diffused mode. It helps !-)
this is the first time I get an intuitive understanding of the Lorentz factor. It's the inverse of an object's "speed" through time
at 7:30... after all that theory about the metric tensor... i really thought: "shit, now I really would like to have an example", sad, but maybe it is just too hard to give one...
"as usual... let's summarize all these concepts with an example" :D
NICE
Ahah I couldn't stop without a concrete example ;)
0:11 Oh the jouissance of mathematical name dropping!!! I think physicists have it more than mathematicians... But i guess it is because mathematicians don't allow themselves to express feelings that easily..... Anyway this is a great series good job!
Brilliant explanations and I now finally understand concepts I have tried to read elsewhere
A wonderfully simple explanation for a beautifully complex Theory
This video gave me that lightbulb moment
Adjust the contrast and sit further away. :)
That's a brilliant way to illustrate metric tensor...👏
The fact that distance between two point gets smaller when they are when you take it away blows my mind
When you simplify the metric tensor and you are left with just 16 differential equation for Christoffel symbols instead of 64. Math so great
I don't actually get everything explained, mathematical part especially, but it is so damn cool I love it
Wow, your videos really are something else. This made the intimidating equations and concepts really intuitive, thank you
Thanks ! Glad you liked it ! For the moment it's not planned but maybe in the future. Quantum physics is a really broad subject compared to General relativity, so I might do videos in this format but about specific subjects.
These videos aren't a substitute for practice. but holy hell are they great for enabling visual learners like me to conceptualize and understand the core concepts much better than stuffy and dry textbooks. if a picture says a thousand words, then at 24 pictures per second x 855 second long video, you're saying 2 Million words with visuals alone.
Brilliant description of the Metric Tensor. Please do videos on how a constant speed of light = a block timed Universe.
Originally, I only just wand to learn something about Reimann Manifold, but this fantastic explain of GR addicted me instantaneously when I started the video, after which let me had to firmly believe that we are living in a virtual word.
This is genuinely an amazing series. Thankyou so much for making it. I'm excited to see the rest of it :)
Hi, many tanks! Albert Einstein did the heavy lifting. We only have to understand. And that is my goal. I’m super curious. Best wishes, Ralf
Thank you! I, with an avid layman interest in GL formulation, gained a lot of insight into it. There is one vagueness that is still persisting in my learning. Every now and then, you refer to "proper time", that I am unable to fix within the relatively concept.
Glad you like the series ! If you want to rewatch it in episode 1 I explain the notion : basically proper time is simply a graduation of equally spaced intervals along the worldline
@@ScienceClicEN thx! I recall the diagram now.
Then, the duration as such (the equally spaced interval) of that proper time is contextual, right? It could be seconds, or days, or light years.
Yes ! But if we want it to correspond to the "real" time that we experience the spacing of the intervals must be so that the spacetime-velocity vector has length "c"
@@ScienceClicEN ok! I am 70 🙂. I won't tease myself with maths intrigues! Thx for your responses.
@@ScienceClicEN The world line has usually time and space components, and so its intervals ; so the concept seems to be a mathematical artifice.
this is what world needs right now.
By seeing this series I will become the next Einstein!
But I am not able to understand I think just because I am in class 8.
It took me more than an hour to understand the previous part aka Geodesics. :(
Relativity.
I get older, they stay the same age.
-Albert Einstein
Space travel - The women get older I stay the same age (cries in Mathew McConaughey)
@ 2:09 Of course, each coëfficiënt is the dot-product of two normalized basisvectors (at each point of the grid).
This was amazing. I can't wait for more :)
Subscribed. Wish my physics lecturers in uni had explained things so well for then I would have understood it back then.
Best playlist of all time on GR😍
Applaud this systematic approach! Sharing with my friend circle
🙂😊 Almost 15 minutes, but you gave us so much! Very well explained.
Thank you for color coding the various symbols, it helped me keep track! 🙌
Thank you so much! Quick question about you as a person :-) : where/what have you studied?
Glad you liked it ! I have studied Theoretical physics and focused mainly on General Relativity. I did my undergrad in France at Sorbonne University, and then went to the UK for my Master's at Cambridge
@@ScienceClicEN thanks for your reply! Again, amazing videos... Thanks a lot for these. I really like how you visually show the equations and their calculation process. I also found your videos on SR and GR extremely well done and easy to understand!! Always looking forward to your next videos :-)
Wow. Now i am waiting for the maths of frame dragging and gravitational waves.
... the -1 made most of what I thought I understood completely wrong...
- It (initially) finally made perfect sense: If I move faster in space, I have to 'sacrifice' speed in time.... but it's the complete opposite...
-- It also made sense that I cannot move faster then c through space: I've just redirected all of the time-velocity into spatial, and have no more to use. (it had no problem with redirecting the vector further, and going backwards, though - which, still, in some sense agrees with my lay understating of general relativity)
... and oddly, it still makes sense to me... This concept just seems to be ... differently formalized? - Or maybe it will break soon... or I just don't see it...
also, moving all my velocity to be spatial, I had none left to advance in time. It all made perfect sense.
- hence light not moving in time, only in space.... except... it's the opposite; the internal clock of the light is stopped, and from outside it's definitely moving...
Yes actually this difference is because of the definition of what I call "speed" in the series (more formally this is known as "four-velocity").
In my video "We all move at the speed of light", which corresponds to your intuition about distributing speed between time and space, I talk about the components of the observed velocity of an object. The observed speeds are measured with respect to the observer's time. In particular, it's the observer's which can't be faster than the speed of light. And the observed speed through time is indeed 0 for light, as it is the rate of proper time divided by the rate of the observer's time.
However in this series (and in relativity in general) we don't use observed velocities, because even though if they are what we usually call "speed", they are arbitrary as they depend on the chosen observer. We rather use "four-velocity", defined with respect to the object's proper time. With this definition, there's no such thing as "the speed of light". Here, the spatial component of the velocity is not bounded, it can very well be larger than c. And the temporal component of velocity is actually the inverse of what we called "temporal speed" before : it's the rate of the observer's time divided by the object's proper time.
Now this second vision, with four-velocity, is much less intuitive because it seems to make no sense anymore (when spatial speed increases, temporal speed also increases which seems weird). But it actually does makes sense : because of the -1 in the metric, for a vector to conserve the same norm, if one of its components through space grows larger, its component through time must also grow larger, to compensate. So the idea that both time and space components must balance each other is still valid, but because of the -1 this seems a bit less intuitive. It's what we call hyperbolic geometry (instead of the usual euclidean geometry)
As usual very well explained!
I'm so happy you are making those videos...
Hi. I have a question about the Minkowski metric. If a photon travels in one direction in space then hit a mirror and return to the place he started in space he traveled two zero length paths but did real movement in spacetime (moved in time and not in space) how does it possible?
What the hell .... this is so awesome .... really really simple while packed with all the details ... thank you very very much ....
im studying string theory and when i have to create lagraungians i use tensors. The geodesic equations are really helpful as well as the christoffel symbol and the kronecker delta, but i mostly use the stress energy momentum tensor since it uses different vectors which areb easy to dot and i usually need basic quantites like them
8:00 i love the use of θ and φ, cause they're both cricles with perpendicular lines