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I went to this site to find the solution. However, I have a few questions about where these equations come from.

  1. In the category of Assumptions and Notations, what equation gives you $\partial_tg_{\mu\nu}=0$ and what does it represent?
  2. In General Relativity the equation is $G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}$ however in this website it uses $R_{ab}-\frac{R}{2}g_{ab}=0$ so how did they get that equation?
  3. Under the category "diagnosing the metric" they eventually get this equation:$$ds^2=g_{11}dr^2+g_{22}d\theta^2+g_{33}d\phi^2+g_{44}dt^2$$is this just Einstein's tensor simplified? and if so, then what is $d$?
Qmechanic
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Dude Hexafandra
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  • There are plenty other better sites with that issue. Do you mean Schwarzschild solution for perfect fluid sphere? Maybe you will find helpful my contribution to this topic here https://physics.stackexchange.com/a/679431/281096 . – JanG May 15 '22 at 17:55
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    A lot of these questions are very standard GR ones, so I'm wondering what stage you're at & why you're looking for the derivation? For the first two question, the answers are literally stated on the Wiki page you linked (i.e. static spacetime and vacuum solution with $T_{ab}=0$). – Eletie May 15 '22 at 17:57
  • Related: https://physics.stackexchange.com/q/21705/2451 and links therein. – Qmechanic May 15 '22 at 18:21
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    If you're asking what $d$ is, it seems like you might be new to calculus in general, and General Relativity is a very advanced topic. It might be best to focus on learning about it from a conceptual standpoint. Diving right into the math will leave you chasing all kinds of notation and terminology that will obscure the main ideas. – RC_23 May 15 '22 at 18:26
  • But to answer your 3rd point, that equation is called the line element of the metric of spacetime, and is essentially the Pythagorean theorem for how to measure distance in a curved spacetime. Einstein's equations allow us to use the matter and energy distribution $T_{\mu\nu}$ in a region to find those $g_{\mu\nu}$ coefficients. This might help https://physics.stackexchange.com/a/669265/313823 – RC_23 May 15 '22 at 18:32

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