I am trying to understand how they find an exact solution of Einstein equation and the field equations for the matter fields. My problem is that in GR the coordinate system cannot be chosen independently from matter from the very begining like in Newtonian mechanics. For example, in Schwarzschild's solution they kind of decide on the coordinates and the expected form of metric based on the desired symmetries of spacetime (spherically symmetric, static etc.) and then workout the Einstein equations. There are no field equations for the matter fields since there is no matter in this case. I wonder how the procedure goes in numerical solution of Einstein eq + matter field equations. That will help me probably understand how they find the exact solutions. As a first naive guess, I can imagine I write a completely arbitrary smooth metric in some arbitrary coordinates (which is 10 arbitrary smooth functions of 4 coordinates). Then I can calculate the Einstein tensor (the left side of einstein equation) and that will give me the corresponding energy momentum tensor (for all spacetime points) of the matter fields creating that metric. That will be a valid solution, only in general the obtainted energy mom. tensor won't resemble any known matter even remotely. So what kind of matter (what type of energy mom. tensor) they assume in numerical GR (my guess is pressureless ideal fluid i. e. 'dust')? How do they choose the coordinates of the metric when there won't be any symmetry to exploit like the Shwartzschild case? The Einstein eqs. are second order in metric derivatives. I guess they specify the metric and the first derivatives on a spatial hypersurface. How do they choose the surface and the derivatives? Please explain or refer me to article/book. I want to understand the procedure so that I could program a computer do it if I wanted to.