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I am a newbie in the field of general relativity. Recently my prof shared a paper with me which talked about solving differential equations with some novel method. He asked me to solve Einstein equation to obtain a black hole solution. However, seeing the form of einstein equations, i cannot see how will they form a system of differential equations. My model is built for system of ODEs. I can plug any number of ODEs and it will give me the solution.

Could you please help me visualise this on how can i obtain a system of differential equations from this einstein equation and how will they look like. Will the system be ordinary or would it be partial DEs. Because right now my model is built for system of ODEs. Is there any way i can plug these in my system to obtain the desired solution.

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    $\begingroup$ Have you tried looking at the derivations of the solution in typical textbooks? That may give you some hints. $\endgroup$
    – Anders Sandberg
    Commented Dec 4, 2021 at 23:50
  • $\begingroup$ Well, initially i thought i do not need to dive into the intricacies of the equation as my work is only related to computing, but now i am giving it second thoughts. But will i be able to understand it without having knowledge about tensor calculus. $\endgroup$
    – unstableEquilibrium
    Commented Dec 5, 2021 at 0:04
  • $\begingroup$ The derivations show how to pick apart the tensor equation into a bunch of PDEs, that can then be turned into a set of Odes. $\endgroup$
    – Anders Sandberg
    Commented Dec 5, 2021 at 0:10

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To get ODEs from the vacuum EFE you should impose that the metric is spherically symmetric and that its components only depends on the radial coordinate $r$. See also e.g. this Phys.SE post.

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  • $\begingroup$ You can see it clearly In physics.stackexchange.com/a/679431/281096 $\endgroup$
    – JanG
    Commented Dec 5, 2021 at 7:11
  • $\begingroup$ @Harsh, is your solution system numerical or symbolic? If you need the derivation of ODEs from EFE you may want to see Tolmans's paper "Static solutions of Einstein's field equation for fluid spheres", authors.library.caltech.edu/4362/1/TOLpr39.pdf . $\endgroup$
    – JanG
    Commented Dec 5, 2021 at 8:16
  • $\begingroup$ Actually I have a machine learning model for solving a system of ODEs which can solve any number of coupled ODEs. I just want a system of coupled equations out of Einstein's equation. I just want to test this new ML method for these equations as this will help me in my later project. $\endgroup$
    – unstableEquilibrium
    Commented Dec 5, 2021 at 8:42
  • $\begingroup$ Then, take that three ODEs from my first link. There are four functions there, of which one is given, one variable and one parameter. It is a parametric ODE system. If your model cannot deal with parameters, take them as numbers. $\endgroup$
    – JanG
    Commented Dec 5, 2021 at 12:31
  • $\begingroup$ Thanks a lot @JanGogolin , Just one question though, these differential equations correspond to a spherically symmetric spacetime with a schwarzschild black hole at origin? $\endgroup$
    – unstableEquilibrium
    Commented Dec 6, 2021 at 6:24
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@Harsh, that with black hole is a little bit more complicated. The equations describe static spherically symmetric spacetime for the case of constant energy density sphere. The solution depends on parameter $\alpha=r_{S}/R$, where $r_{S}$ is Schwarzschild radius and $R$ radius of the sphere. By the way, the name radius is a shortcut for the curvature radius. Now, as long as $\alpha < 8/9$, all solution functions are finite, whereas for $\alpha=8/9$, the pressure function diverges at origin and the so-called singularity emerges there. It is an established view to see it as the begin of gravitational collapse which however is non-static. In consequence, the equation system you are solving is no more valid. The static Schwarzschild black hole solution arises first at $\alpha=1$, when $r_S=R$ is reached. I would suggest, you should test your system for static part of the model, before collapse.

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