Is it possible to derive curvature of spacetime from EFEs to create a computer simulation of the solar system? [duplicate]

Question

Background

I am a professional scientist (biologist), statistician and a computer programmer, but an amateur physicist so pardon me if this question appears silly. In order to understand general relativity (a truly inspiring theory, btw), I would like to create a computer simulation of the solar system using gravity as defined by the general relativity theory (not Newtonian gravity). I am thinking about following steps for the process:

1. Introduce the XYZ locations, masses, and tangential velocities for celestial bodies
2. Calculate the curvature of spacetime introduced by these celestial bodies at time t0.
3. Remove the curvature caused by each celestial body for their own path, but include all other curvatures.
4. Increase time to t1, iterate the new position of celestial bodies based on conditions set in step 1 and 3 (and those in general relativity).
5. Calculate steps 2 and 3 for the new position. Repeat 4. Etc.

Question

After using half a day online, I am still unsure whether Einstein Field Equations (EFEs) let me derive the new XYZ locations in step 4. Does General relativity let me do calculations outlined above using computational power in a modern laptop?

My background in physics and maths fails me here (this is part of the fun in learning something new). If doing the above is theoretically possible and not too computationally intensive, figuring out how to do each step is just details. I might ask such details in separate questions, but here I am after whether my understanding is completely off (never worked with partial differential equations before).

Answer 1

Yes it can be done, but it is a remarkably difficult computation to do using the full theory. The Einstein equation looks simple as it is usually written, but it is actually a set of ten simultaneous non-linear partial differential equations. Using computers to solve for the motion of bodies in this way is generally referred to as numerical relativity but (a) you'll need a supercomputer and (b) the mathematical sophistication required is if anything even greater than using analytical methods. You aren't going to be doing this without a thorough understanding of general relativity, and you aren't going to be doing it on your home computer at all.

As mentioned in the comments, the Solar system is well described by the weak field approximation to GR, sometimes referred to as linearised gravity, or an equivalent approach such as the post-Newtonian formalism. These are computationally vastly simpler and for planetary systems are accurate to within experimental error. It is only for extreme systems like merging black holes that we would resort to the full theory.