Relativity 110c: Cosmology - FLRW Tensor/Coefficient Derivations
HTML-код
- Опубликовано: 25 июл 2024
- Full relativity playlist: • Relativity by eigenchris
Powerpoint slide files: github.com/eigenchris/MathNot...
Leave me a tip: ko-fi.com/eigenchris
Some other sources for the tensor formulas are:
- Sean Carroll GR Cosmology notes: preposterousuniverse.com/wp-c...
- Section 2.9 of the Catalogue of Spacetimes (note they they take the g_tt metric term to be c^2 instead of 1, so the locations of c in the formulas are different: arxiv.org/pdf/0904.4184.pdf
0:00 Intro + Notation
2:48 Connection Coefficients
10:08 Ricci Tensor
19:04 Ricci Scalar
Calculating in GR is like doing accounting in physics.
Yeah physicists were like: let's just use our regular math for GR, and only at the end did they realize
Thanks a lot. I am doing self study on GR and I was only able to get the first Ricci tensor on my own, but I needed your video to get the other 3. You did a fantastic job of organizing the work. I could not get it without your help. Please keep doing videos that show all the mathematical steps. I greatly appreciate it.
Glad you found it helpful. The description has a link to a pdf called the "Catalogue of Spacetimes", and it has the tensors/coefficients for a lot of different spacetime geometries.
Absolutely heroic detail on these calculations
Simple enough and clear to everyone, and one does not need another reference after this
Thank you for your amazing work.Much appreciated!
Great video. Always amazes me how compact the EFE are and how long and annoying calculations get :).
Lol that's tensor calculus for ya!
Yeah... and this is in a relatively "simple" case where the Ricci tensor is diagonal.
@@eigenchris hey, is there a metric for a flat gravitating disc? I tried to derive it myself but you know, it hurt my head real bad
Not only rare but it a sufficient for value that physicist's assignment as diamond of physics of universe.
It is better than any class lecture, your quantum on GR that matters 😀
Like your video even after finishing it never end
Endless thanks to you and your channel.
Why we need Ricci sensor? Is this step is truncate any information.
Old is gold
Nice videoes on GTR. These videoes are helping me a lot in my GTR course.
hi
looking forward to your other parts, d,e,f of this section. great details and explanations.
Thanks for all your videos.....! I'm waiting your next videos.
Well, yeah, obviously, this is just the simple part, right? Holy mackrel, that's very impressive work! Wow! Thanks again!
To be honest this was one of the easier videos, because all these calculations are well-documented so there wasn't much room for error. Other videos tend to involve a lot of fact-checking or trying to figure out calculations that are harder to track down sources for, which makes their production time balloon. Kind of in that situation right now with 110d. Hope I can work it out soon.
that's very impressive work! Wow! Thanks again!😍
That was quick! thanks!
On a 3rd viewing of 110c @7:45 last line middle term should partial w/r 2 not 1 as shown but the end is correctly shown. Again, fantastic work.
Awesome thank you so much...genius..you are the best..you made my life easier..no one is as amazing as you😍😍😍😍😍
A detail at 16:03, the minus sign shown for the cos^2 term is a plus sign. However, your rigor is exemplary.
Wow. Thanks.
I don't know if I missed it in last video, but can k depend on time?
Is it being constant a derived property or a one chosen because we limited the scope to those 3 possible metrics?
As I understand this is a chosen value +1, 0 or -1 depending on what we assume - it's introduced only so we can write three metric in one.
k doesn't depend on time. It just indicates the choice of which geometry we're using (closed, flat, open). The scale factor a(t) obviously depends on time. This more or less plays the role of the "radius" of a closed universe and the equivalent curvature paramter for an open universe.
Thank you for the video! I just have one question… what are the Riemann tensor components then?
Sir please explain the kerr-neumann matric!!!!
One thing I've noticed is that, while the FLRW metric is spacially homogeneous and isotropic, it DOES differentiate between frames of reference (in that it's possible to know which time axis is the "correct" one). I suppose this isn't too much of an assumption since the CMB is known to have a particular velocity, but it is worth noting I think.
Actually, I've changed my mind. It is a somewhat large assumption to assume that at a cosmological scale all points in space will have the same time axis at which they are locally homogeneous and isotropic.
@@neopalm2050 Yeah, that's an issue I've basically ignored in the past couple videos. There's such a thing as the "cosmic rest frame", which is the frame in which you don't see any doppler shift when observing the CMB (you see the same frequencies in all direction, instead of seeing blueshift on one half of the sky and redshift on the other half). In the cosmic rest frame, geodesics that point forward in time will remain at fixed spatial coordinates, so basically the FLRW coordinate system is "expanding along with spacetime" (for this reason these are also called "comoving coordinates"). To be honest I'm still working through all the details for the justification for why we can pick the time axis that we do, but my plan is to explain it in the next video.
Thank you very much for this material, Professor Eigenchris.
A small observation/question for minute 5:21 :
\Gamma ^{1}_{00}=\dfrac{1}{2}g^{11}\left ( \partial_0 g_{1 0} + \partial_0 g_{10} - \partial_1 g_{00}
ight ) is correct? [In words: positive partial derivatives have index 1]
Hey eigenchris, which topic would you touch after covering general theory of relativity?
I was thinking of doing a series on spinors.
@@eigenchris continuum mechanics some time maybe?
@@abcxyz9723 I've never studied continuum mechanics. Sorry!
I understand the math and am comfortable with computations although boring! Is there a software which given a metric can compute the connection coefficients, the Ricci tensor and scalar?
There may be software, although I'm not familiar with it. There's a PDF online called the "Catalogue of Spacetimes" which has a lot of the coefficients and tensors for many different spacetimes and coordinate systems: arxiv.org/abs/0904.4184
@@eigenchris Thanks for the link! Based on the link you provided I found that there is indeed a GRTensorII package coupled with Maple which does most of these computations.
arxiv.org/pdf/cs/0409006.pdf
❤❤❤❤❤
Hie Chris u did'nt tell me about Bianchi Identity Isotropic Solution about Cosmology
Sorry, I don't understand your question. I brought up the Bianchi Identity in my Tensor Calculus 23 video. Is this related to cosmology somehow?
@@eigenchris Yes i am saying that, Can u gave me your id or something so i can sand the pdf of Bianchi type-I Cosmological Models With Perfect Fluid in General Relativity
Excuse me teacher! How to get Einstein's paper for general relativity in English?
The main consolidation of general relativity is in this paper: einsteinpapers.press.princeton.edu/vol6-trans/158
You can see more of his 1914-1917 papers in English on this website: einsteinpapers.press.princeton.edu/vol6-trans/
You forgot the minus sign on the right side (6:45)
Great job, but I'm glad I didn't have to take notes.
:)
Hi eigenchris have you seen my email I sent you a few weeks ago?
Hi Idan. I took a look at it but I admit I didn't understand much of it. I don't know what a "Von Neumann Universe" is, and I'm not sure where you got the GM = tc^3 from.
Hie Idan Sagiv Can u give me mail of Eigen Chris
@@eigenchris
Surely you know Chris. This is the shape of the universe (FLRW Metric Derivation, the fourth option) The volume is of Horn Torus. And the equation GM = tc^3 also can be derived using a simple calculation (see last page in the paper).
Listen, we are both aware that it requires dedication and a true willingness to delve into the depths of things.
I can't make you speak about topics you don't want to talk about or appreciate them.
But I'm sure you'll find the time to look into it. It just floats in a very natural way from relativity.
Thank you for your time :)
@@himanshuchaudhary5796
Are you asking me to send you someone else's email when his ״presence״ is near you? Just ask him, I'm sure he will meet you halfway.. It will not be ethical on my part to publish his email here in public. I respect his privacy.
Imagine I would bring someone your email .. how would you feel ?! (#RhetoricalQuestion)
I'm sure you'll understand. Have a wonderful day :)
I am Sorry Idan Sagiv and Eigenchris
First:)