The Maths of General Relativity (4/8) - Metric tensor

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.

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Ahah thanks I am glad you like it !

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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.

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I wish people taught not just general relativity, but also special relativity this way

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that might be the future

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special relativity is "relatively" simple. proper courses with deep explanation are better.

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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!

[@ScienceClicEN]

Exactly ! What we call "proper time" is actually simply the measure of distances along the worldlines of objects

[@ibmmbi2317]

It isnt Minkowsky Metric but Minskowsy MATRIX

[@AndrewStarostin]

@@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

[@ibmmbi2317]

@@AndrewStarostin Oh ok.

[@ibmmbi2317]

So within this Universe is made up of space(s). What is beyond the universe made of then?

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Fantastic! I never seen GR explained so succinctly

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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.

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[@hrithikravi8117]

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?

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@@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: kzbin.info/www/bejne/Z3TYhYGsmr19iMU 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: kzbin.info/www/bejne/iYKvqaWobrNrsK8 and this one on variational calculus: kzbin.info/www/bejne/iYKvqaWobrNrsK8

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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.

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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.

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Probably, the only video on KZbin 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. 👍

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Goosebumps

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Is this the same order in which mr. Einstein formulated the equations? what was going inside his mind?

[@ScienceClicEN]

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.

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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 😁.

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I thought the previous videos were amazing and the math will only get harder. But damn, this is amazing.

[@ozzymandius666]

Excellent. Now I want to know how to integrate the metric tensor over a trajectory on a sphere.

[@colejohnson2230]

He gives the formula/components in the vid, I'd think it would just be a very simple integration of those

[@underfilho]

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

[@breveennkukan3603]

Wow! I didn’t expect coming to realize why integrals are so important. Such a great video.

[@akhilanr1233]

A wonderfully simple explanation for a beautifully complex Theory

[@DavidBeaumont]

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.

[@stevewhitt9109]

Very BEST explanation of the Metric Tensor on You Tube.

[@stephanieparker1250]

Thank you for color coding the various symbols, it helped me keep track! 🙌

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Again, thank you so much for this series!

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Best advanced physics KZbin videos ever.

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Brilliant. What a great way to explain concepts.

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That's a brilliant way to illustrate metric tensor...👏

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Applaud this systematic approach! Sharing with my friend circle

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Great work! I'm excited to see the next video.

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As usual very well explained!

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Brilliant explanations and I now finally understand concepts I have tried to read elsewhere

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I'm so happy you are making those videos...

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This was amazing. I can't wait for more :)

[@joaopedroaguiarfmatos1473]

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

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What the hell .... this is so awesome .... really really simple while packed with all the details ... thank you very very much ....

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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

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yeah non euclidean geometry as well as riemannian geometry and conics are a bit hard to visualise on a regular cartesian plane

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Wow...!!!! Excellent....helped me understand the physical significance of metric tensor...thanks..highly appreciated..

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[@ScienceClicEN]

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.

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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.

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I watch this video through concentrated mode and diffused mode. It helps !-)

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Love these type of explaination ♥️

[@thevegg3275]

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?

[@user-xt9vo3fp2j]

🙂😊 Almost 15 minutes, but you gave us so much! Very well explained.

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Excellent explanation. Thank you.

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best GR video i have ever came across :') , thank you sir

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Thanks another very professional video!

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Best playlist of all time on GR😍

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Subscribed. Wish my physics lecturers in uni had explained things so well for then I would have understood it back then.

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Very good! Congratulations!

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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!

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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.

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Really love the videos! Ok if when you show the equations changing could you either keep up the old equation then introduce the new, or slow down the terms changing.

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Very good tutorial!! Thank u

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Great class, thank you.

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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

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I'm a physics undergrad student and... holy shit.

[@jack.d7873]

Brilliant description of the Metric Tensor. Please do videos on how a constant speed of light = a block timed Universe.

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Fantastic video again!

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Thanks !

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This is some great content

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This video gave me that lightbulb moment

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Adjust the contrast and sit further away. :)

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Thank you so much What a amazing video

[@romanbednarskiy4017]

Hi! I have no words to describe how cool your channel is. Marvellous, deeply thoughtful with clear explanations of extremely sophisticated abstractions. Would you excuse my poor knowledge of physics and explain, how GR metric tensor relates to a tensor in quantum physics / computing? Is it just the same concept of snapshotting the precise moment or there is something more? Thank you.

[@eammonful]

Gorgeous video