Relativity 110f: Cosmology - Friedmann Equations Derivation + Universe Evolution Models (FINALE)
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- Опубликовано: 9 июл 2024
- Full relativity playlist: • Relativity by eigenchris
Powerpoint slide files: github.com/eigenchris/MathNot...
Leave me a tip: ko-fi.com/eigenchris
Links/Sources:
Sean Carroll's 15-page PDF on cosmology: preposterousuniverse.com/wp-c...
PDF with derivation of "3rd" Friedmann Equation based on Adiabatic expansion (page 19): www.uio.no/studier/emner/matn...
Article on Conservation of Energy in General Relativity: math.ucr.edu/home/baez/physic...
de Sitter's 1917 paper: academic.oup.com/mnras/articl...
Video on mathematics of de Sitter space: • The Cosmological Const...
Equation for de Sitter space: en.wikipedia.org/wiki/De_Sitt...
Equation for embedding Anti-de Sitter space: en.wikipedia.org/wiki/Anti-de...
PDF building up dS and AdS space from the basics: 3dhouse.se/ingemar/relteori/Ku...
Physics Stack exchange post about values of cosmological parameters: physics.stackexchange.com/que...
Chronology of the Universe: en.wikipedia.org/wiki/Chronol...
Dark Energy Survey (estimated cosmological parameters): en.wikipedia.org/wiki/Dark_En...
Video intro to Dark Energy Survey: • Exploring 7 billion li...
Dr Becky talks about Dark Energy Survey: • Dark matter: smooth or...
Cosmic Inflation: en.wikipedia.org/wiki/Inflati...)
"Gravity in a Nutshell" by A. Zee also has chapters on de Sitter and anti-de Sitter spacetimes (IX.10 and IX.11)
Sources for the estimations of cosmological parameters:
WMAP: arxiv.org/pdf/0803.0732.pdf
BOOMERanG: arxiv.org/pdf/astro-ph/000440...
Planck: en.wikipedia.org/wiki/Lambda-...
Other topics I didn't cover:
Killing Vectors: en.wikipedia.org/wiki/Killing...
Kerr Metric (rotating black holes): en.wikipedia.org/wiki/Kerr_me...
Einstein-Hilbert Action (Lagrangian formulation of GR): en.wikipedia.org/wiki/Einstei...
0:00 Introduction
1:04 Review of FLRW metric and Perfect Fluid
3:09 Friedmann Equations Derivation
7:04 "3rd" Friedmann Equation (conservation of energy)
8:25 Universes dominated by matter, radiation, dark energy
14:25 Einstein Static Universe
17:51 de Sitter / anti-de Sitter Universes
25:16 Cosmological parameters
28:02 Cosmological Models with Λ = 0
33:08 Cosmological Models with Λ ≠ 0
36:43 The model for our universe
39:17 Conclusion
One viewer pointed out some slight confusion at 8:14, in the 3rd term of the formula, where I wrote "β=μ".
What needs to be done in the 3rd term is: expand the two summations over β and μ manually. So when the two summations are expanded out over β and μ, there will be 16 terms (1 for each element of T^μ_β). Of these 16 terms, only 4 will stick around, because there are only 4 non-zero elements of the T^μ_β matrix. (These 4 non-zero elements correspond to the diagonal elements of the matrix... this is what I meant when I wrote "β=μ". I wasn't trying to tell you to take the trace of the matrix... I was trying to say that only the 4 diagonal elements of the 16 summation terms are non-zero.) Of these 4 terms, the Γ^0_00 term is zero, and the remaining three Γ terms are identical, so this is where the "3" comes from in the 3rd term. Sorry for the confusion!
The ending is the most beautiful thing I've seen in my entire physics career. Mathematical proofs in combination with empirical fact, logically ruling out all other possible models is making an undeniable case for the S shape cosmic evolution. Such colossal genius is truly awe inspiring to witness
Agreed
Could not agree more. Very well done Chris !
Eigenchris, I'm at a loss for words. The inevitable finale that has been building since Tensors for Beginners 0 has finally come. I am not sad, I feel complete. This feeling is hard to describe. I am content because I know I that I have grown as a person over the years, and here I am, July 22, 2022, witnessing what feels like the end of an era.
It feels like when you beat a GTA game and you're left on free roam with no missions. I am not sad that it's over, but happy that it happened.
Thank you for these videos and sharing your knowledge. Nowhere on the internet can you find the rigor of GR explained this smoothly. Nowhere on the internet can you find this level of quality in digestible 30 minute lectures. I have poured tens of hours watching your videos (probably in the 100s when you count replays). Thank you for being the mentor we all needed. Thank you for this content. You are a true champion of knowledge in the modern era.
That, is poetry.
It’s great how you share this content for free
@@रोहित1 were you dropped on the head
I've gotten some nice tips, which don't hurt. But it's satisfying just helping other people understand stuff.
What a monumental series! After warming up on "Tensors for Beginners" and reviewing material learned many years ago with "Tensor Calculus," this series on Relativity has been more illuminating than any textbook or lecture I have ever experienced. There is nothing glossed over, no point left unexplored. With your invaluable help, I can finally comprehend the General Theory. Thank you.
Thank you so much for this utterly fantastic series. You've made General Relativity and Tensor calculus fun, engaging and much more manageable for a lot of people :D These series have helped me so much through my GR classes :)
An amazing series, there's nothing even close to this on RUclips. What a huge amount of work you have put into in, Tensors, Tensor Calculus, all the relativity videos and this the Finale. What happened to the fireworks at the end?? I guess that mathematicians regard the ending as much more beautiful than fireworks. 👑 Thank you so much.
Thanks for sticking with it all the way through!
eigenchris - the Sherpa that led us along Tensor Trail to the top of Mount Relativity. Thank You!
Thank you very much for this truly epic journey . I started with Tensors for beginners and followed right through,amassing several folders of dense notes and diagrams.The sense of achievement is immense and it is all down to you,Chris. I am now planning to get my head around "spinors".Can't wait !
I'm not sure if you've seen me mention this in other comments, but my next planned video series (after a couple months' break) is "Spinors for Beginners". If you search "eigenchris spinors", you'll find a few hours of spinor talks I gave last year, but they're pretty rough and disorganized. I plan to clean that up into a proper series.
Thanks Chris, as always, clear explanations and amazing tensor - gr course. enjoy your break!
This is such a great series, I'm sure I'll come back to most of the videos at least a few times. Excited to see what you have next :) Thank you!!
Thank you for putting the effort into this incredible series.
Thanks a lot. Your lessons helped me a lot to understand the LCDM model.
You're the best!
Great stuff. Thx for all the relativity videos, really helped to understand GR and its implications. 👍🏻😎🙏
That's an insane amount of work you've done, and there is no really a way for me to thank you.
All of tensor series was essential in building my intuition of how this world is described in polylinear way, and more important, what all of these numbers all about.
You teached me an uncountable quantity of things throught your courses, done with magnificent quality.
Lastly, You gave me much more, than my university did,
And I am much better mathematician in cause of that.
There is just so much I want to say to You, that I'm losing it.
That's truly inspiring, making me wanna to dig into things and explain them to other people. Your work is not letting this feeling to fade away.
Thank you, thank you so much.
I can't even express how much I love this series!!! I appreciate your effort and dedication to this topic extremely!!! I was trying to understand GR before, but I didn't succeeded. Now I know why and you're the reason I finally managed to understand this. I'm so thankful, you're amazing!!! Wish you all the bests!
That's great to hear. I'm glad the playlist was able to help you!
Since my days at school I was fascinated by the mystery of relativity. So I started to read books and lectures and tried to understand the math and philosophy behind it but it was somehow frustrating as I got stuck again and again. Then, by chance, I discovered this series and now one year later I think I understood the fundamentals of relativity. This is just great and gorgeous. Thanks a lot.
That's great to hear. I was in a similar position where I tried relativity over and over but it just didn't stick. After it finally clicked, I decided to make these videos to help save people who are in the same position I was.
Most EXCELLENT I've ever seen!
I have an assignment due tomorrow that asks me to derive Friedmann's equation, this is saving me so much, I can't imagine what it would be like without you. Thank you😭
Just finished the series. Science is scaringly interesting. Thank you sir
Hey! I'm a student of physics and I'm very happy to have discovered your channel. Great content dude!
Brilliant.. very clear and very rigorous explained well.
This is the best gr playlist on RUclips, thank you so much for all your work. I'm so glad I found it, really allowed me to go into details after ScienceClic 's videos
I Generally don't comment but you are the one that need to be definitely appreciated for your great work and great explanation! Literally The Best video for relativity 😊
Wow this video is very good! Thank you very much!
Genius Chris is back!,🔥
It's like my favorite TV show got a good ending ;_;
Thank you so much. Your content is an invaluable resource..
Chris. You poopiehead, you've helped me a substantial amount, I've said this once before but I'll say it again. If it wasn't for you and w/ the help of a few Tensor & Holor Calculus Books I wouldn't have been able to sit here today in the position I'm in now. That being, being able to understand both General Relativity and, not just a conceptual level but from a mathematical stand point.
Since then I've been researching and/or learning QFT, I had already learned Quantum Mechanics, since then I've been wanting to integrate it from a relativistic-stand point. Check out Perry Moon's book on, "The Theory of Holors - A Generalization of Tensors." Great book.
Great Video Chris..... 👍👍.... Have a nice weekend.... 👍
Bravo, and thanks so much.
thanks, it was a fun journey
Jawdroping outstanding!
14:17 "A purely algebraic theory is required to describe reality." (Einstein, January, 1955).
Maybe GR was QG…
“The geometry of space in general relativity theory turned out to be another field, therefore the geometry of space in GR is almost the same as the gravitational field.” (Smolin).
However apparently, the gravitational field is space-time in the Planck system: F(G)/F(e)=Gm(pl)^2/e^2=1/α, that is, gravity~strong interaction*.
This assumption follows from the Schwarzschild solution: the gravitational radius (or Schwarzschild radius) is a characteristic radius defined for any physical body with mass: r(G)=2GM/c^2
Consequently: 2E(0)/r(G)=F(pl)=c^4/G=ε(pl)/r(pl): with indicating the mutual quantization of the mass (energy) and space-time: m(0)//m(pl)=r(G)/2r(pl)=n,where n-total number of quanta of the system; the tension vector flux: n=[(1/4π)(Gћc)^-½]gS ( const for all orbits of the system: n=0,1,2,3....).
Moreover, the parameter r(0)=r(G)-r(pl)=(2n-1)r(pl), defining the interval of the formation of the system, at n=0, when r=r(G)=0 (for example, the state of the "universe" before the Big Bang) turns out to be a quite definite quantity: r(0)=-r(pl).
In the area [(-rpl) - 0 - (+rpl)] there is an implementation of external forces, "distance": (-rpl)+(+rpl)=0 (≠2rpl).
On the Kruskal diagram of the hyperbole r=0 corresponds to the true Schwarzschild feature, the features V and VI are not even covered by the global (R, T)- space-time and correspond to the "absolute" vacuum; then the singular areas above and below the hyperbolas r=0 can be formally treated as the energy source (external forces).
That is, the frightening "true singularity" is actually a superconducting heterotrophic "window" between the proto-universe (the source) and physical bodies**.
P.S.
As a fundamental theory, GR has the ability with just one parameter: r(G)/r=k to predict, explain new physical effects, and amend already known ones.
Photon frequency shift in gravitational field Δw/w(0)=k; the angle of deflection of a photon from a rectilinear propagation path =2k, the Newtonian orbit of the planet shifts forward in its plane: during one revolution, a certain point of the orbit is shifted by an angle =3πk, for a circular orbit (eccentricity е=0); in the case of an elliptical orbit - for example, for perihelion displacement, the last expression must be divided by (1-e^2).
-------------------
*) - GR predicts a new physical effect: w/w(pl)=k; expression for gravitational radiation from a test body.
This is amenable to physical examination in laboratory conditions at present.
**) - From this, generally, from Einstein's equations, where the constant c^4/G=F(pl), one can obtain a quantum expression (as vibration field) for the gravitational potential: ф(G)=(-1/2)[Għ/с]^½ (w)=-[h/4πm(pl)]w.
Final formula:ф(G)=-[w/w(pl)]c^2/2, where ф(G) - is Newtonian gravitational potential, r(n')=nλ/π=(n+n')2r(pl)l , the corresponding orbital radius, w - the frequency of the quanta of the gravitational field (space-time); - obviously, the quanta of the field are themselves quantized: λ=(1+n'/n)λ(pl) = 2πc/w, where n'/n - system gravity unpacking ratio, n'- the orbit number (n'=0,1,2,3…).
Obviously, on the horizon [r=r(rG), n'=0] the "door" is closed, however, the quanta [λ=λ(pl)] can go out singly through the "keyhole" and form the first and all subsequent orbits (n'=1,2, 3 ...) during the time t(0)=r/c=2nт, where т=1/w, т=((1+n'/n)т(pl), spending part of their energy on it each time. And it is this mechanism that provides the step-by-step formation of the gravitational field ( expansion of the space-time): of course, the quanta coming through the "window" are also rhythmically restored.
The phase velocity of evolution v' = r(pl)w: m(0)=(c/G)rv', where v'=v^2/c.
The angular momentum: L(p)=|pr|=n^2ћ [const for all orbits of the system; at n=1: L(p)=ћ] and moment of power: M(F)=dL(p)/dt(0)=nћw/2=-E(G),where t(0)=r/c.
Entropy (here: a measure of diversity/variety, not ugliness/disorder) of the system: S=πε(pl)r(t)=(n+n')k, where k is the Boltzmann constant. Obviously, with fundamental irreversibility, information is preserved (+ evolves): n=const for all orbits of the system.
Accordingly, m=m(pl)/(1+n'/n), where m=ħw/c^2, is the quantum of the full mass: M=n'm [
Thanks, master!
Thank you!
This is by far the best lectures note series on General Relativity, black holes, gravitational waves and cosmology!. Thank you for your time and dedication.
Is it possible in the future add video lectures on perturbation theory in the context of cosmology?
Thank you.
I appreciate the compliment, but I'm done making relativity videos. I spend 2.5 years making them and I'd rather move on to other things.
At 8:09, the last connection coefficient of the conservation of energy-momentum has a negative while the original derivation of the conversation of energy-momentum in the video of the energy-momentum tensor at 27:14 doesn't have any negative. Is there a Cristoffel symbol symmetry that when you swap an upper and lower index you get a negative sign?
Also at 8:09, the last connection coefficient seems to be different from the original derivation of the conversation of energy-momentum, the original has a summation over β of course before the lowering of index with a summation with the metric Tensor.
Though β in the derivation of the conversation of energy-momentum is alpha. My point is that there should be v instead of β in the last connection coefficient at 8:09.
nice job
Bravo!
Thx for sharing these topics! What is the next topics you would like to cover?
I'm planning on doing a "Spinors for Beginners" later this year.
@@eigenchris Your presentation in relativity topic is excellent!
@eigenchris bro I have seen your full lecture series and I am grateful to you for presenting such a complex topic in such a simple way. Bro I have to do cosmological perturbation theory for my project work can you please tell me from where I can understand this topic clearly. It would be really helpful if you will answer me as soon as possible.
Sorry, I can't offer any help. I didn't know "cosmological perturbation theory" was a topic in the first place.
Consider talking about global techniques, those are interesting.
Well, thanks for the work.
Now I get what they are talking about the holographic principle and only with respect to negative DeSitter space. But still the sine function has a negative square root, how is that any less problematic than a a K= 0, or 1?. Like so many others I am grateful beyond words for your hard and painstaking work?
at 6:48 there should be , end of equation, (p+3P/c^2)
Great Videos! Looks to me that the Perfect Fluid scenario (min 2:31) can be applied to free vacuum space. If so, then can we have a scenario, where instead of the vacuum energy density is constant, it varies with the scale factor? Since density is mass/energy divided by volume, it seems sensible to me that vacuum energy density just like ordinary matter decreases with the (scale factor) cubed. Why is the cosmological constant scenario so hardwired into the models?
The cosmological constant has a constant energy density by definition (as I show later in the video, its density is constant and doesn't depend on the scale factor). I don't understand why this is the case. I don't think anyone does. At this point it's an unsolved problem in physics. There are alternative forms of dark energy whose density grows with the scale factor ("Phantom Energy") and decreases with the scale factor ("Quintessence") but right now the Cosmological Constant is the mainstream accepted form of dark energy.
When will you post yours next vedios on spinors
Hi Eigenchris, do you have any recommendations on books that I can use while watching the series? It would be awesome to have more material! Thanks a lot!
I didn't really follow any particular book, but I used these as references, particularly for the applications of GR such as cosmology.
Some free sources:
- Sean Carroll's online GR course notes on his website "preposterous universe"
- "Exploring Black Holes" by Edwin F Taylor (textbook has a free PDF download link)
Other texts used for reference:
- Relativity, Gravitation, and Cosmology by Lambourne
- Introduction to General Relativity by Ryder
- Gravitation and Cosmology by Weinerg
- Gravity in a Nutshell by A. Zee
- Gravitation by Misner, Thorne and Wheeler (very heavy... more like a reference book, probably not great for beginners)
@@eigenchris Oh wow these are some awesome resources! Honestly I should have commented this in the very first video in the series for all to see :/ I plan on purchasing Relativity, Gravitation and Cosmology by Lambourne. Thank you so much for your help and reply, these videos are so much helpful and please keep more advanced topics coming! Btw really enjoyed the momentum video xD
@@jaechoi1344 As I said at the end of this video, this is planned to be my last GR video in the series. I still have to go back an insert a couple I skipped, but after that I'm taking a break, then moving to spinors.
Thanks for all your efforts. In Min 2:36 -does the negative sign of the pressure in the EM tensor for perfect fluid have any absolute significance? Looks to me that by choosing different values of Equation of State, one can make it have any sign. In other words, is the negative sign of pressure an intrinsic & integral part of a perfect fluid?
The minus signs there are because I'm writing the tensor using "mixed indices" (one lower index, one upper index) using the mostly-minus metric convention. If using both indices lowered, all diagonal terms will have + signs. Standard fluids will have positive pressure, but "dark energy" that causes the universe to have accelerated expansion will have a negative p-value for pressure. So the signs are not arbitrary. But you have to keep in mind there's the sign for p itself (either normal fluid or dark-energy) and there are signs due to the metric raising indices. Also note if you use the mostly-plus metric convention instead, the mixed-index tensor will have + signs in front of the pressure p terms and the - sign in front of the energy density (rho*c^2) instead.
Thanks @@eigenchris makes sense
Sure wish you were a physics professor somewhere so I could go there to study physics.
eiginchris, I was just wondering would you be interested in describing how computers today use metrics & EFE to solve real problems in GR?
I don't know anything about that topic, unfortunately.
Hello Dear Eigenchris 🧡, I want to tip a small amount. But I am from Bangladesh. Here Paypal is not available. is there any other way?
Hi Chris I have a question. At 31:40 you say that the universe will reach a maximum size, because of a' -> +0 and a'' -> -0. But if a(t) ~ t^1/2 it fits the conditions but tends to infinity when time increases. So why does the universe approach to maximum size after all?
The case at 31:40 is only for a flat universe (k=0) with zero cosmological constant. We live in a flat universe with a positive cosmological constant, and that's causing expansion to win over re-collapse. The values I give for a(t) toward the end of the video are just approximations assuming the universe falls neatly into one of the 3 categories I showed (energy-dominated, matter-dominated, vacuum dominated). Does that answer your question?
@@eigenchris I mean if the universe is flat and the cosmological constant equals zero. The function a(t)=x^1/2 can correspond to the condition that the first and second derivative will be zeroed at infinity. And here for t that tends to infinity the function a(t) will also tend to infinity. And here there will be no maximum size beyond which the universe cannot expand any further.
Dear Professor,
Thank you very much for your classes.
I have noticed that the Friedmann or FLRW equation,
like the Einstein-De Sitter model, generated
the Friedmann equation from the Einstein equation
without the cosmological constant, after
Einstein removed it and before dark energy was discovered.
Were you aware of this, or can you provide your opinion
on the matter?
Thank you.
Sorry, I'm not sure I understand the question. You can get the Friedmann Equations with or without the cosmological constant. Can you explain what you mean?
@@eigenchris I meant that after Einstein set lambda to zero in his equation, the FLWR equation without the cosmological constant began to emerge. Also, the Einstein- De Sitter model no longer had the constant, until the found dark energy. Thanks
You will not cover FLRW Geodsics ?
I'll cover that in 110d, which I'm working on right now. I mentioned in a community post that I skipped it because of some stuff I was struggling to understand.
Which universe is your favorite?
mine is k = 0, lambda = 0, rho = constant :P
Hello Eigengris. I finished your tensor calculus playlist and I finished your general relativity playlist. Now quantum field theory. I will do research in areas such as unified field theory, string theory, just like the beginner's tensor, you made it in the tensor calculus playlist, I would appreciate if you make a playlist about spinor calculus like this
Greetings from Turkey 🇹🇷😁😃👍
I plan on doing a spinor playlist later this year. If you search "spinors eigenchris", you'll find some talks I gave last year on spinors, but they are not very organized.
@@eigenchris first of all, thank you very much for the feedback, yes I watched your video last year, but it would be much better if there was a more comprehensive and qualified playlist based more on lie algebras and lie groups, clifford algebra or spinors for us to understand physics because I watched your tensor calculation video and You explained it really well and we can understand it better with some graphics, but unfortunately I couldn't find a channel about the topics I mentioned. Since I know you, I think this playlist would be very good. And thank you very much. But do you plan to shoot a series about lie groups? (By the way, I can write complicated sometimes, my English is not very good)
@@Oylesinebiri58 I don't plan on doing a series specifically on Lie groups, but I will have at least a couple videos on Lie groups and Lie algebras as part of my spinors series. And yeah, I plan to make it more clear and organized with better diagrams.
@@eigenchris okey, thank you
"With dark energy, we gain energy as space expands": does this imply that the source of dark energy is external to the expanding universe? For example, what if Cosmos is a black hole, whose event horizon is the source of gravitons traveling towards the expanding universe causing it to accelerate??! This scenario would explain why the universe is gaining energy as it expands. Great explanations btw!!
I don't think dark energy is understood very well at this point (at least, I don't understand it), so I don't think anyone knows at this point. On a mathematical level, the idea that energy needs to "come from somewhere" is rooted in conservation of energy. This doesn't apply globally in GR, so I don't think trying to "balance" energy by finding a source for the exact amount gained makes sense.
Please put forward some notes on more advance topics of general relativity
Chris, I am waiting for your quantum machenics videos! When you will uplode?
And also one thing, will you make quantum field theory videos?
I plan to do a series on spinors, which has applications in QFT, but I don't plan on doing a full QFT series, because I don't understand QFT.
Wow, energy density of our universe amounts to just over 1.6 protons per cubic metre. Puts into perspective how empty our universe is.
You're missing a type of universe - one that space is expanded by the gravity force, but not expanding. It's a matter of moving what changes with per meter per sec and change just change per meter. This seems like a much more likely scenario, that the gravitational bodies in the universe displace space, but the space is already displaced, and not being displaced. This still stretches space tangential to the sources, and iight which passes past gravitational objects has to go through more space per time than it would without gravitational displacements. Something that's static, but still stretched...this conserves energy too.
What would be the mathematical explanation for that word salad?
The first 2 Assumptions do not explain our universe though…. And those assumptions are self contradictory.
How about this Thought experiment, if you remove the observation of difference, you as a observer of the universe where everything “looks” the same in all directions to you, would disappear from those forces. The missing mass of space you once occupied would cause a missing mass in the surrounding areas; hence the universe would then be unevenly distributed.
How can the dark energie then stay equal, when the universe is expanding? +1 - (-1 × -1 × -1 x -1 x -1 x -1 x -1 x -1) = 0 ... (Dark energie with -1 value stay's 1)
The cosmological constant is like a dark energy density per unit space. This stays constant overtime. But as the size of the universe increases, the total dark energy also increases.
@@eigenchris Only until all mass (+1) is transformed into dark energie (-1). This will keep the universe in a state where it can conserve its total energie. I think black holes are (one of) the creators of dark energie as they leak. The Universe however will never have value 0... Because with absolute nothing you can not explain this universe when it is a closed system. The 0 - (-0 x -0) = 0 not 1... The fundamental equations do not alow this. They are the first fundamental mesure tools of each other in nature, that already creates a own Heisenberg explenation. I do not belief that the universe was absoluut homogene at its birth. The set of universal equations that explain this universe, can be seen as messuring tools of each other and i doubt you will ever find 1 formula to explain nature. I think that it is imposseble for nature to explain "nothing" without "something" and that this "something (energy/mass)" , most stay to do so like time only can run in 1 direction and the absolute speed of light can not exist because of no absoluut vaccuum.
Please have some vedios on kerr metric it is difficult topic you explain it just in a shallow way that can no one other
I'm taking a break from making videos. You can try looking at Robert Davie's youtube videos on the Kerr metric.
Thanks for your videos
Seems K=0 and K=-1 would have positive and negative curvatures over regions with stress-energy-momentum and those devoid of such. Perhaps positive curvature is even more confined to galaxies or solar systems, and exhibits less confinement of positive curvature as K=1 becomes the case. If the curvature was positive everywhere the universe would have a limited size as the curvature closes on itself. Though Gaussian curvature is not correct I can't get away from picturing the open and closed universes in this way. Tidal Ricci tensor has the negative Gaussian curvature flavor when over time eigenvectors have eigenvalues of both acceleration and deceleration? I suppose for a positive curvature sphere, excess enclosed hypervolume and crossections must be correct generally, so deficits for negative curvature.
Chris, What will be your next videos?
I can't wait!
I plan on doing a "Spinors for Beginners" series.
@@eigenchris In quantum machenics?
@@asrafali8093 I'll be talking about spin in quantum mechanics a bit, but the objects called "spinors" are mathematical objects that can be used in both classical mechanics and quantum mechanics. The idea that spinors are quantum-only is something I'd like to explain is wrong.
Please have some vedios on Kerr metric it is very difficult topic you can explain in a very shallow that can no one other
I'm a bit tired of making relativity videos so I probably won't make a Kerr metric video anytime soon. You can check out Robert Davie's channel for videos on the Kerr metric.
Good Day Chris, in your Video 8:14 for both Christoffel Symbol we have 3. (1/c).(a dot/a)... Could you please explain where this 3 come from?
As seen in the very top box, Γ^1_10 = Γ^2_20 = Γ^3_30 = (1/c)*(a-dot/a). So for the summation Γ^μ_μ0, this summation will just be a sum of the 3 above Christoffel symbols, which are all equal, so you can just put a "3" in front of the value.
Looking again though... I'm a bit confused at what I did in the 3rd term... give me a moment to figure it out.
Good Day Chris, i try to calculate again but still can not get the right result... Why In your 3 term Trace of energy Momentum tensor T^u_u = - p, instead of [(rho. c^2) - 3p] ?
@@steffenleo5997 Yeah, I think I've made a mistake there... I'll try to find the correct derivation and get back to you.
Hi again. Sorry for the long delay. I think what needs to be done in the 3rd term is: expand the two summations manually. So when the two summation is expanded out over β and μ, there will be 16 terms (1 for each element of T^μ_β). Of these 16 terms, only 4 will stick around, because there are only 4 non-zero elements of the T^μ_β matrix. (These 4 non-zero elements correspond to the diagonal elements of the matrix... this is what I meant when I wrote "β=μ". I wasn't trying to tell you to take the trace of the matrix... I was trying to say that only the 4 diagonal elements of the 16 summation terms are non-zero.) Of these 4 terms, the Γ^0_00 term is zero, and the remaining three Γ terms are identical, so this is where the "3" comes from in the 3rd term. Sorry for the confusion!!
I have a doubt 🤔....how to apply curl to a higher dimensional vector field ? 🤔💖
"Curl" in the standard vector calculus sense is only defined in 3 dimensions. There may be other operations from exterior calculus that behave like a kind of curl in higher dimensions. Is there a specific part of the video making you ask this question?
@@eigenchris well I don't know about exterior algebra 🤓 I couldn't tell whether there is a specific video or not but can you give some suggestions about curl like operators to infinite dimensional vector field may be ?....and im just now going through your tensors for beginners playlist 😇 it's so useful and hard to find these maths in youtube like this beautiful explanation 💗
Thanks eigenchris, I think I was able to follow your explanations in principle but not the maths. I'm still trying to work out if this is your real speaking voice or a mechanised one? A a bit like Stephen Hawking's voicebox.
It's my real voice. I'm just a very monotone person.
This seems like a bit of a rude comment?
You have already shared alot with us in free, but can I request something? Please can you make a summery video of the playlist for us, please...!
Thank you so much... 🙌
Like a sort of "table of contents"? I might do that after I finish the 110d video.
@@eigenchris yes, & thanks, waiting for further uploads from your side🧩
Eignchris please…please…please I request you to make some vedios on Kerr metric
I might at some point. But probably not soon. Have you seen Robert Davie's playlist on the Kerr metric.
Could you talk a little bit of perturbation theory? Thank teacher.
Sorry but I don't know much about that. So I won't be making videos on it.
nice video but I claim bs on this one: 37:15
You don't know what k is. It's "Almost 0".
The space constant term, which is the length of the spatial constant term Λ, is Λ=π-e,
norm s= √|π+e|√|π-e|√±1 √±1 √±i√±i=.
In 0.423310825130748,
given the infinite dimension of The non-commutative Gauge Theory of Yang-Mills theory,
both π and e [ Energy ] are constants of infinite small limits.
The constant of infinite small curve distance, π,
the constant e of infinite linear distance.
E^+∞ is Hyperinflation !
In that case,
π^+∞ is Super rotation.
The Einstein equation is correct!
The gauge field was the space constant term λ.
God doesn't roll the dice!
The enormous number that can be measured by human intelligence is -1+360^360,360^360,+1+360^360.
A 3-bit AI computer is required for the ± signed integer part of the decimal number, and the mantissa is up to 1+360^360 bits.
Einstein is exactly the stone of the sage, the mysterious dazzling glitter of diamonds.
A Black Hole is
a ς(GL|Gauge Group) of Gluons, and the Space Constant Term λ creates a Guge Field for General Relativity.
If Lim ∞→0, then π≒1 or π≒0,
(8πT/c^4)^-1 × (8πT/ c^4)=
√4√|+1|√|-1|√|+i|√|-i|=2ηab,
E8×E8
-(1/2)(Fμν)^-1 ×(1/2)(Fμν) =
E8×E8
-η(s+1)Γ(s+1)/ζ(s+1)Γ(s+1) ,
e^πi +1=0 卍 e^iπ-1=0,
0≡0,
,
E8×E8= (E^πi +1)×(E^iπ-1),
E8×E8=
E^πi +1=0 卍 E^iπ-1=0,
2π≡,
≒/,
≒
,
e^πi+1=0 exchanges information with 8 elements.
It is 8 elements in total of e,^,π,i,+,1,=,0. In other words,
it corresponds to 8 of Gluon. Apart from the two elements of the arithmetic symbol ^,= ,
you get 6 quarks.
Gluon acts like a soap bubble in a weak gravitational field, and the inner space of a soap bubble is a black hole. There are various colors on the surface of soap bubbles. The two-dimensional space of the soap bubble surface becomes the quark Yang-Mills field. Soap bubbles will burst. The right-handed system remembers as the magnetic line of force of the left hand black hole.
★★★★★★★★★★★☆
The number e of the Napier may be 2
Yo put a high pass filter on the audio. Cut the syllabance. If your recorder takes vsts use spitfish. Sounds like a snake being spaghettified
Sorry, I don't know what syllabance is, or vsts, or spitfire. Can you point out a part of the video where a high pass filter would help? Or is it the entire thing?
@@eigenchris yes apply to your whole audio (heard in other vids of yours too). Sibilance is the sound of whistling winds and hissing snakes. It's the “s” sound we hear produced in “s,” “ci,” and some “z” words - for example snake, cistern, and zip. You get rid of these in audio production with a high pass filter. A Vst is a external addon/program for audio production. I think obs (the screen recording software) accepts vsts (in which spitfish is a good vst to addon), but if you have audacity you can get the same effect simply adding a high pass filter (cut off the high frequencies). better than just using a equalizer on the highs because it preserves presence/clarity of voice when tuned in for your voice. Hope that was written coherent enough (not rewriting). best and thanks for the content bro. (And yes no sound in space ,..-for the most part)
What an amazing thing you have done... respect