The Maths of General Relativity (8/8) - Summary and Applications

[@ScienceClicEN]

For the brave who try to solve the 2 exercises I proposed, here are the answers [SPOILERS]: Exercise 1 : (1) For a person on Earth, we can use the same formula as for the space station, but replacing the altitude r by the radius of the Earth (6370 km), because the person is on Earth, and the speed v by 0 because the person remains almost motionless (contrary to the station which is in orbit moving pretty fast). When we perform the calculation we get roughly 1.000 000 000 7 (that is 0,7 nanoseconds per second) (2) To obtain the ratio of time dilation between the astronaut in the station and the person on Earth, we divide one by the other (we want to consider how the former's time evolves compared to the latter's). We hence calculate 1.000 000 001 / 1.000 000 000 7 ~ 1.000 000 000 3. The astronaut in the space station undergoes a time dilation of roughly 0,3 nanoseconds per second with respect to the person on Earth. Even if the astronaut is further from the Earth, their time runs slower because of the speed of the station (special relativistic effect). Exercise 2 : [EDIT, the conservation of energy is actually not needed for this] We use the 2nd geodesic equation : (dvʳ/dτ) = -(GM/r²)(1-2GM/rc²)(vᵗ)² + (GM/r²c²)(vʳ)²/(1-2GM/rc²) And the norm of the velocity vector : ||v||² = (c²-2GM/r)(vᵗ)² - (vʳ)²/(1-2GM/rc²) We identify that (dvʳ/dτ) = -(GM/r²)||v||²/c². Since ||v||² = c², we obtain the desired result : (dvʳ/dτ) = -GM/r²

[@TiberiusMoon]

Knowing the answer wont tell you much, but understanding the question would tell you more than the answer will.

[@ΠΡΟΔΡΟΜΟΣΠΑΠΑΔΟΠΟΥΛΟΣ-ζ9ρ]

Congratulations. The Energy of the relation E = mc²vᵗ(1-2GM/rc²) is measured from a relatively motionless inertial observer A at infinite (as you said). Generally in a free fall from infinite distance will be valid the temporal relation [P(0) at infinite] = mc^2.(1)= const = [P(0) at every point of the orbit] , [ wiht "(0)" in P(0) we symbolize the covariant down index 0 of time t], because from known theorem we get that in your example the metric tensor is temporal static: d[g(μν)]/dt = 0. Therefore, at some K(t, r) point (or four-event) this gives E = U^(0).P^(0).g(00) = P(0)[g(00)]^(-1/2)= mc²(1-2GM/rc²)^(-1/2), because g(00) [U^(0)]^2 =1 and P^(0).g(00) = P(0) and also where we symbolize the contravariant component U^(0) where "^(0)" in U^(0) symbolize the contravariant up index 0 of time t and obviously the P^(0). And where g(μν) is the metric tensor, and t0. We point out that here we use the General System of coordinates where at point K(t,r) its observer is at rest relative to 3-space and so he is defined by a four-velocite that has only the time-component different from 0 and the fact that on the orbit the product of the observer's world velocity U with the body's world momentum P at some point K(t,r) will be invariant (as product of two world vectors) in all the systems of references (with curvilinear coordinates or not). A useful tool in energy calculations that may be correlated with the problem if the sum of positive and negative energy of the universe is zero. For example the quontity - c(PU) + mc^2 is the negative potential energy at the point K(t,r) of the orbit.

[@particleonazock2246]

Exercise 2!?! Yeah, I'm not a calc freak, so I'm not among the brave to have solved the exercises.

[@jacoblund8289]

Thanks so much for providing these exercises, they were really nice to do! Interestingly, for exercise 2 I didn't need to use conservation of energy; I just used the expression for the norm of the velocity vector to solve for vᵗ in terms of vʳ and then substituted that back into the 2nd geodesic equation. From there, most of the terms canceled out including all the vʳ terms, so I didn't need to relate it to energy or anything else) leaving behind the equation for Newtonian gravity. Not sure if there's something I'm missing here, but this worked perfectly fine for me and I was wondering if there's any particular reason that conservation of energy has to be used

[@krispoels2888]

You don't need the expression for the energy, just multiply the equation for the norm of v with GM/r²c², and add that equation then to the 2nd geodesic equation (lhs1 + lhs2 = rhs1 + rhs2), and you get immediately (dvʳ/dt) + GM/r² = 0 + 0 : )

[@maxwellsequation4887]

The series was one of the most beautiful series ever made and compares to 3b1b's calculus series. Brilliant work. The mathematics of quantum field theory now....

[@maxbaharev2712]

That would be great! That series would have to be bigger though.

[@infinummjb]

QFT without using the word "weird" once

[@hafizajiaziz8773]

For QFT series I recommend viascience

[@sciencewithsigns3215]

I also want

[@iliketrains0pwned]

@@infinummjb Use "spooky" instead

[@inverse_of_zero]

Thank you for producing an incredibly intuitive and neat package of KZbin videos that summarise General Relativity in an easily digestible format. This is an incredibly valuable educational resource for the Internet!

[@emiilio_s]

These video series was such a masterpiece!

[@hoodio]

fr, how does he have so few subs

[@jx5967]

@@hoodio it's difficult to understand math🌝

[@hoodio]

@@jx5967 yeah ik but look at 3blue1brown or pbs spacetime for example, they have huge followings

[@raider.3060]

It was not. It is.

[@achyuththouta6957]

@@hoodio It has few subs because hardly anyone is interested in science and whenever I say something about black holes to start a conversation I'm labelled as " nerd" and what not

[@soumavakundu5850]

Being a physics undergraduate this is one of the best things i saw and it gives me the much needed intution before taking a rigorous university level course in general relativity.Hope to see more advanced physics topic like maths of qft while will provide much needed intution and help people know beforehand a overview what to encounter in a college level course.Waiting for the next masterpiece by ScienceClic.

[@khoalb]

Well I still don't understand General Relativity, but this is the closest I've ever come. Thanks for the videos!

[@Xcalator35]

Lol! I'm with you Albert...still don't get it...but closer nevertheless

[@bobnob4393]

It doesn't seem that complicated really. The mathematics involved in calculating it are hard but it's beautiful how simple the physics is I feel

[@mohdazminishak6387]

If Albert still can not understand it, what more us, the non Albert

[@cosmiccomedy7394]

I've gotten it and ungotten it hundreds of times now haha

[@leon_noel1687]

The theory isnt that complicated philosophicaly compared to the math (in contrast with the rest of fundamental physics) so probably you understand pretty much.

[@OFF0Dansk]

I was lost in the mathematics but I cannot stop myself from being infatuated with theories from big to small. This was amazing.

[@EmilianoMarino]

Best series of videos about General Relativity.... Congratulations.

[@marcelobrinholli8201]

Alessandro has a deep understanding of General Relativity so as to explain it in such a clear way to the non-initiated ones. He has a unique schematic view of complicated topics 99,9% of the professors should be deeply envious about. He creates the videos and the stunning illustrations. He composes the background music. Plus, he is young and handsome. What's left for the mortals?

[@charlesfountain3946]

Thanks for all your efforts to make this topic so accessible and entertaining for those of us who enjoy tensors and relativity as such beautiful mathematics. Anyone can see you have really gone the extra mile to break things down better than some of the other mainstream university courses. Hope there is more to come than can motivate the viewers to study this topic for fun and or profit. .

[@web3risk]

This is by far the most educationally packed series I've ever seen on KZbin. No hype, no regurgitation of the same material. Great visuals, clear application of formulae, precise demonstrations, and a gradual building on prior topics. Outstanding!

[@Incalor]

As a person with no background in theoretical physics with a great interest in learning the same, I think the greatness of this video is best exemplified by the fact that even I could understand atleast 40%-50% of the concepts. Thanks, great work.

[@mirkothe12]

Wow, wow, wow!!!! This is a masterpiece of understanding most difficult concepts and making them accessible to others! Big thanks!

[@УэстернСпай]

I can't believe you compressed an entire graduate GR course into like 2 hours.. and very elegantly too!

[@stevenschilizzi4104]

This series of eight videos must certainly be the best ever introduction to GR that I’ve ever come across. Thank you so much for this brilliant effort. It should be a “must see” series for all budding physicists or for anyone who would like to get a serious insight into what GR is really about. Surely a great teaser for year 12 high school students who have had enough math to at least guess how some of the concepts work, and if not kindle the curiosity to go find out. It would be absolutely fantastic if something similar could be done for quantum field theory, for which there exists no real user-friendly introductions. Of course, in that case, viewers will need to have a bit more background, e.g. be already familiar with basic non-relativistic quantum mechanics and the Schrödinger equation, as well as with Hilbert spaces and the like. For this, I can recommend Alexandre Zagoskin’s “Quantum Mechanics: A Complete Introduction”, 2015, also as an e-book, which I have found, as a non-specialist, very understandable and very helpful, even though it is a bit hard-going at times. I’d recommend a first reading without getting hung up on the details, just to get a feel where it’s all going (as later examples or explanations illuminate earlier text), but in making sure you understand they key ideas and concepts, and only then, in a second reading, linger on the details (which are the bare minimum) to really understand what’s going on, though the very notion of “to understand” takes on a whole new meaning in quantum physics!! (Everyone must work that one out for themselves)... A series on QFT would fill in a gaping void that just begs to be filled in! But you have first deserved a bit of a break after such an effort! Thanks again for a brilliant piece of work.

[@pietroghsvf]

This is one the best videos series I've ever seen on youtube. Aspiring to be a Professor but still only a graduate student, I'm in awe at what is to me the epitome of didactics, explaining complex topics thoroughly, beautifully and clearly. Congratulations!

[@andrewevans5761]

This whole series has been absolutely amazing. I've been trying to understand these math concepts for a long time and they have suddenly become clear. Thanks very much.

[@couposanto]

It is one of the best explanation about general relativity I have ever seen. Congratulations !!!

[@Albator102004]

I've been binging this whole series before sleep and I'm just totally fvckflabbergeisted... These were the most well crafted videos I have ever seen on any educational subject. Everything is on point and especially the visual production which is perfect. The 6ish episode journey to the "4D representation" of the time component of the tensor around the black hole was a spiritual revelation ! Of course I'll have to rewatch the whole thing..

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[@lilyh4467]

Thank you so much for making these vids! They have been a perfect intro to me for getting into GR!

[@tobiashaist4852]

Perfect ... just perfect. Thanks a lot for this course. When I went (long time ago) through my one year relativity course, I think I learned not much more than what managed to teach in 1...2 hours. Should be a must for every physics student.

[@mr.comic_art3516]

This playlist was the best! I’m 15 and I could grasp the concepts easily, very well explained, clear and concise. Great videos!

[@RemoCODclips]

this series was aboslutely amazing. Im very bad at math and my major is nothing alike, but i have a deep passion for reality, the universe and everything in between. Watching these serieus and actually understanding the math behind it was such an eye opener for me

[@esgotometer]

Thank you so much. I have always been a Physics enthusiast but nothing like this video series has ever awaken such a passion I didn’t know existed within me. I thank you from the bottom of my heart.

[@nezalco7725]

I understood everything from beginning to end, but that music though... It's perfect.

[@Diego0wnz]

Thanks so much, amazing series, and you’re JUST 23!!! I’m hoping to follow you on this amazing science journey for a lifetime

[@MartinLeggewie]

I just completed all 8 videos. Wow. Thank you. I will not claim that I understood everything, but as others have also commented before, I do have the impression that I at least came a little closer to the concepts of general relativity. Again: Thanks a lot for taking the time and effort and energy to create these videos - must have been a lot of work.

[@ScienceClicEN]

Thanks for the support! Very glad you liked the series

[@imagine.o.universo]

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.

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[@brilliantbutlazy3.14]

Thank you for this series... I know a lot of people would appreciate this very much. Sharing this with all my science friends. 👍

[@straussi4]

very nice video serious. I've watched the serious the second time and I still don't understand many of the things. But I understood the fundamental concept of special realtivity, and I am addicted to these mindblows you get, when you suddenly understand the thing, that is always talked about. Thanks for breaking down so complex mathematics to the simplest model, that remains intuitive and complex at the same time. I'm watching this in 11th grade nether bavarian "Gymnasium" (which is like high school) and I more and more get the feeling to understand the very fundamental topic "Special Relativity".

[@alexeygunya6519]

Hi, dear colleague! Good work with the video about GR! Simple, understandable, pleasant)

[@bmx666bmx666]

Alessandro you made the best and the greatest series about gravity and general relativity. Bravo!

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[@MattiaConti]

I lose myself somewhere between the fourth and the fifth episode, but the series is very cool. I hope to understand better next time I will resee the episodes.

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I really hope this channel blows up because its content stands among the gretest physics and math channels on youtube

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I get put into a sort of trance when watching these videos. My mind wanders as I try to visualize the concepts that ScienceClic English has graciously bestowed upon me. Then I jump back into reality and realize how small we really are!

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This series has been so good! You are the 3Blue1Brown of physics.

[@-_Nuke_-]

MY MIND = BLOWN I can't help you enough... THIS is a testament to how insanely genius Einstein was... INSANE!

[@j1sh109]

This series is legendary in realm of physics. Feynman would be proud

[@adamraiyan]

I am very thankful to have these videos are out for free to the public. Thank you for solving my curiosity.

[@objectSIX617]

I've tried to understand relativistic physics for years in a qualitative way not having the background or patience to understand the math. This series expertly explains the math visually and I finally grasp the basic idea of the equations and how each tensor is used/ what it means.

[@ladislavhusty9466]

This was just a freaking insane series. This is just a whole another level of videos. I understand such videos take time to create. Please keep up this great work! Oh and thank you.

[@Darthvanger]

A masterpiece! The first time I got a sense of how it is really done. The calculations of the curvature and how it affects the movement.

[@maxqutekerman907]

This is the best explanation of GR I've seen. Extremely helpful. You have officially surpassed 3b1b in my personal list of best educational channels, claiming top spot.

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What an excellent articulation for such a complex topic. Undoubtedly the best content on this topic across whole web. This channel is way too underrated!!! Thanks for your efforts to educate us!!!

[@AndrewKishman-wx4hn]

This series was incredible. I hope you're authentically very proud of yourself. Hard to think of something more beautiful than to so generously explain and illustrate the math of General Relativity. This is by a mile the best introduction I've ever encountered. You're an exceptionally skilled writer and artist in addition to having a remarkably lucid understanding of General Relativity. And the fact that you offer it to the public for free officially makes you a badass. Thanks!

[@KH-xl9em]

luminous well done!

[@Walter-uy4or]

The best basic look at general relativity that includes some of the math that I have come across. Even if you want more detail, I would recommend watching these videos first. Really outstanding.

[@ハェフィシェフ]

I'm baffled by the quality of your vids, you're a brilliant explainer

[@ianweller4492]

Spectacular stuff, extremely well done series!

[@lost2weeks245]

This channel is a hidden gem

[@canyadigit6274]

I have learned the math behind general relativity, but have not seen it applied in general relativity for things like time dilation. Well done. This was such a great series.

[@m67esteban]

Thanks!

[@globaldigitaldirectsubsidi4493]

Possibly the best educational series of videos I have ever seen.

[@raavi.mohindar]

It just blew my mind and you brought an absolute clarity GR and I am hooked to learn more rather start to work on the math. Great content indeed!!!

[@PMLighthouse]

Amazing! your video series helped me make good progress thru GR and also make sense of Sir Roger Penrose "Road to Reality". Bought this book more than a decade ago along with other books, made multiple attempts to understand but progress was zero until I found your videos. I don't want to become an expert, it is very satisfying to realize that now I know what those field equations are.... Thank you very much ScienceClic Team....

[@navneetmishra3208]

THAT WAS A SUCCESSFUL GR SERIES!! Thank you so much! You guys are legends. Hoping for more such GR examples in future videos. Can't wait for the next series with such awesome videos!!

[@kris030]

This series was incredible! I've always wanted to understand how the universe works even tho I won't study physics. This is great for us "casuals" too! Keep going!

[@StratosFair]

This channel deserves to blow up

[@lIlIlIlIlIlIlIlIlI]

Thank you so very much for this series. Your efforts are an enlightenment to the understanding of general relativity for many. I will cherish this series for the rest of my life. Thank you.

[@wedothework4105]

I could listen to you folks explain to me how I tie my shoes in the morning! . Please keep the content coming!

[@kka107]

You made general theory of relativity accessible to engineers like me who understand vector calculus and linear algebra. I deeply thank you for that.

[@r.i.p.volodya]

These videos are superb! I wish wish wish I'd had access to them when I was doing my Masters in Theoretical Physics! 👏👏👏

[@naysay02]

I didn't understand a thing and yet felt as if I did. Great work making math visually intuitive, thank you!

[@anthonyp4949]

What a brilliant series! This is the most intuitive explanation of GR math I’ve come across. Well done!

[@mskEduTech]

One of the best GR series video.

[@bjarnivalur6330]

Thank you, Click, this series has been incredibly convenient and easy to follow. I'm looking forward to your next ventures with great excitement.

[@onelivingsoul2962]

These videos are for those who doesn't like to die out in the domains of mathematics used in General Relativity, you kept these rather simple for those science enthusiasts who like to venture into these Uncharted territories of physics while still avoiding rigorous mathematical calculations

[@ARBB1]

Unbelievably, someone managed to make a good pop-sci series about GR without memeing the thing to heat death; top tier work Alessandro

[@canyadigit6274]

I wouldn’t even call this pop-sci. It gets rigorous. Pop sci would just show the concepts.

[@proximacentauri4106]

Great job, that was perfectly splendid.

[@Geddyfleaharris]

Fantastic series! As a non-physicist interested reading about relativity, black holes etc., the visualizations were great. I love the look of the math but at my age I am probably not going to be learning it any time soon! Thanks for the series!

[@quantummind1187]

Please make more series like these! Absolutely amazing work!

[@sevisymphonie5666]

The cosmic constant in the field equation would also be worth another video. Why does it exist and what problems does it pose? For this channel it is no problem to go into detail. Just as things that are actually not simple have been communicated very well and understandably here. Actually, I had expected the constant to be the 8th part of this General Relativity series.

[@giorgosmanolakos9253]

Thank you for the great course! I am actually awed by the level of your understanding of general relativity to teach it like this. Congratulations.

[@listocheck]

An absolute masterpiece, a must-watch for any physics undergrad

[@aniketeuler6443]

Sir your efforts in this series were really fantastic and I can't wait to see that the next upcoming series is on which topic ☺

[@aresgalamatis7022]

This series is the best teaser trailer I got for starting my GR module this year. Now, I pumped to dive into all the PDEs and co/contra-variant indexes of tensors. Also fluid mechanics helped me a lot to understand the explanation of the energy-momentum tensor (part 6, I think) :D

[@alejandrotobon1149]

What a great series of videos!! 👏🏼👏🏼😊 excellent explanations, clear, concise and deep as well!!

[@0dWHOHWb0]

Thanks for making these I'm just skimming through them for now, but at some point I'll make the time to learn these things properly with the help of this series

[@aliazizi8743]

These videos are the best that I have ever seen. They are masterpiece. Thanks a looooooooot

[@luudest]

Please, more examples! Such great illustrations and explanations!

[@MisterWizardo]

Really nice explained and animated, love it. I would also love to watch a series like this about quantum Physics.

[@vansh2637]

An absolute masterpiece

[@carknower]

I’m curious as to who your audience is. I’m just an economist/business grad who enjoys math and science. Sometimes I find myself doing the math joust for fun and to verify my understanding but I don’t go much further than that. All in all, I really enjoy your videos.

[@Inkognitohaloramics]

This video series is incredible. Thank you for making it.

[@MoshkitaTheCat]

Stunning…. Please don’t stop making those priceless videos.

[@stefenleo9863]

Thanks a lot for your video... You are the future noble prize physician

[@flmbray]

This was such a great series! I can't wait to see what you guys do next.

[@johnwilr]

Thank you very much for this excellent work, it gives an intuitive understanding of maths applied in this field. I really love it!

[@hoodio]

a collab between you and 3blue1one would be too powerful

[@tensevo]

Great series of lectures, the content, visuals, narration and subtle sound design are all "on point". Well done on 100k and please keep up the good work.

[@OneEyedAnaconda]

Loved the series. Can you make a video on Anti-deSitter Space and other types of curvature? I really want you guys to make videos on Hawking Radiation and Holographic Principle. And ADS-CFT correspondence.

[@brianmckay1256]

At long last i get to see the beauty of GR. Simply amazing series. Thank you

[@karkunow]

Great addition to Ch19 of Road to Reality book!

[@alHailHale]

Light has no proper time, to itself. As soon as it is born, it dies, traveling everywhere it will ever be at that moment. This trips me up and it is so hard for me to wrap my mind around that.

[@johnnychinstrap]

At 12:01 of the video you make a few assumptions . Can you please show your work and list the assumptions above. I am not sure I am making the same assumptions to solve that equation. If you could post the math for this entire excellent series that would be much appreciated.

[@Mattef72]

This is so great, now I will go back and watch Susskinds lectures

[@ΠΡΟΔΡΟΜΟΣΠΑΠΑΔΟΠΟΥΛΟΣ-ζ9ρ]

Congratulations! I know the subject. I celebrated all your 8 wonderful videos as memory of lovers of pioneers of truth. The flow of energy-momentum tensor seems to me as the flow of information from some mysterious Platonic world, ie flow of geometry of pure mathematics, where classical matter is something like shadow of inertia flow defined from temperature in entropy change or in quantum observation...?... I'll send your videos to friends, is very special work. I hope you becomes a pioneer, you deserves it.