Solving Einstein Equations to obtain Schwarzschild solution

Question

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.

Answer 1

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.

Answer 2

@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.