Relativity 109b: Gravitational Waves - Linearized Gravity / Weak Gravity

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0:00 Introduction to Linearized Gravity
2:26 Inverse Metric
5:03 Connection Coefficients
6:15 Riemann Tensor
7:39 Ricci Tensor
8:53 Ricci Scalar
9:58 Einstein Tensor
13:02 Summary

[AhmedMohamed-zuvv]

General relativity is such a huge deep subject, but it feels so good when you feel something click and you get some pieces of the puzzle. Your channel was a great way to do that, and it deserves 10 times more subscriptions honestly.

[EGOPON]

Thank you so much. I was so confused that books do not explicitly say that the derivative of h_\mu\nu is assumed to be small, yet, they assume it in calculations. I am so relieved that someone explicitly says it.

[caiociardelli]

Thank you so much for the wonderful work you have been doing! I found your channel just a few weeks ago and I am amazed by how well you explain the topics. In my opinion, everyone should be teaching GR like you do. I'm a seismologist but always wanted to learn GR and now I can! Thanks a lot!

[beagle]

Thanks a bunch, Chris. Having just finished taking notes on this video, I have to say that it was another amazing epic! I can't thank you enough, you are a star!

[daGama]

Thanks for the video. I am currently in a GR class and the only things that are helping me are your videos and a DG class I took last semester

[longsarith]

You always come with an amazing video. I sincerely appreciate it!

[mikefields]

Hei there, first of all thanks, you are gonna save my course! Second, on the notes I got from my professor is shown that we can get rid of the small derivatives of h because they are actually a gauge transformation for h, and we can just not write them then, since h is not uniquely fixed for a given source. For showing this you grab infinitesimal coord transformations and gets rid of a term from the Killing equation assuming that the transformation is small. Hmu if you want it for reference.

[davidsantosromacho]

You're so good man, i love your videos.

[rahulpatel-migb]

"At 9:12, the index is raised for \( h^{\alpha}_{\sigma} \) and not on the partial derivative \( \partial_{\nu} \) in the second term."

[BakedAlaska]

13:32 Wondering if LIGO backs up ‘ small’ assumptions?

[matthias]

13:31 Why can’t the Alpha and Beta summation indices in the Einstein Tensor not be relabeled consistently with the other summation indices? I seem to be missing something here

[dixieflatline]

Hello! I think that if h is very small but its variations aren't, then h could become not so small and we'd have a contradiction!
Other than that thank you so very much for this series of videos! A lot clearer than my GR class, you're saving my life! I particularly appreciate that you give the calculations with all their gory details.

[badtaste]

Nice video. You got yourself a subscriber.

[physicslover]

Respectful Sir,
Thank you so much your video explains concept in a very nice and simple way...Thank you so much Sir...
Thanking All,
All Sincerely,
Shreyansh

[paulbuchinger]

Thank you very much, this was a great help

[FuadRaedJubranHaddad]

in 3:32 the eta is before the k but in the next step the order is changed. why is that possible?

[Lennon]

This is a late comment.
When we derived the kappa coefficient of Einstein field equation, kappa = 8*pi*G/c^4. We assume low velocity limit, weak gravity limit and metric tensor is time independent (partial dg/dt). But here in gravitational wave, the metric perturbation h is dependent on time from gravitational wave equation.

[sebastiandierks]

Thanks, was just doing these calculations myself last week, along my reading of Sean Carroll's "Spacetime and Geometry". Will you also derive the non-covariant form of these equations after decomposing the metric perturbation h into phi, w_i, psi and s_ij? (This is similar to decomposing the field strengh tensor F_mu, nu into electric and magnetic fields in EM, you lose the covariance but you gain more insight into the physical degrees of freedom.)

Thanks for mentioning explicitly the assumptions on h, especially that derivatives are also small. This was exactly one of my questions as Carroll didn't write it explicitly and just because a function is small, its derivative is not necessarily.

[CallOFDutyMVP]

So Linearized Gravity is a "halfway" approximation between the full field Equations and Riemann Normal Coordinates at a point (where all Christoffel Symbols are zero)? Also, what is the physical interpretation of partial derivative operators with a raised index? Is it just to sum with the Einstein Notation? At first glance I thought they were antiderivatives.

[joydeepsarkar]

If we use an epsilon in front of h_{ij} in the definition of g_{ij} and claim that epsilon<<1 and epsilon ^2 =0, then we don't have to bother about the combinations of "h" which will become<<1 and zero.