# Can the N-body problem be solved numerically using the geodesic equation of mass-distorted spacetime?

I would like to write a program that solves the trajectories of objects (think rockets) that are influenced by mass of other objects (think planets). I saw that I can do this using Newton's laws, but I would like to confirm that another way is also possible.

If I could find a metric of a spacetime distorted by masses of some planets, then I could find Christoffel symbols and plug them into the geodesic equation to find ODEs describing the geodesics in this spacetime (for rockets). Than I could solve the regular N-body problem and recompute the metric. Is this right? If so, how can I build a metric of such a spacetime?

I've seen people find trajectories of photons in the Schwarzschild metric, and I consider it similar to my problem (except I want to have more than one object distorting the spacetime). I went through all questions with [differential-geometry] tag, but couldn't find anything related to my problem.

I'm very sorry about this question being not very specific. I know very little about GR and differential geometry, but I don't think it prevents me from using it as a tool.

Found a chapter "Multiple Black Hole Solutions" in http://www.aei.mpg.de/~gielen/black.pdf. It shows how to build a Reissner–Nordström metric describing a system of few black holes. I think, that this could be simplified into Schwarzschild metric by neglecting the charge of the black holes. If so, how can one obtain the metric for an object which is not a black hole?

• This is a perfectly fine question, but bear in mind that you're talking about solving the 1-body problem in a more or less arbitrary background, not the N-body problem. Mar 23, 2015 at 15:46
• @DavidZ yes, you are right that my question implies that the planets are stationary. Generally they could be moving, but I imagine that it would make the problem impossible to solve. Or wouldn't it? I'll try to improve my question. Mar 23, 2015 at 15:58
• In principle you can just solve the Einstein's equation. But bear in mind that this is a nonlinear differential equation (and Newton's are linear), which is numerically very challenging. If there is a need to go beyond Newton for accuracy, then you can probably take into account the GR correction perturbatively (i.e. weak gravitational field, non-relativistic motions). There are well-developed techniques for this kind of calculations, like post-Newtonian theory. For things rockets this should be a good starting point. Mar 23, 2015 at 17:47
• @MengCheng so basically, to find the metric of the spacetime distorted by few masses I have to solve EFE on my own? The suggestion about solving Newton equations and applying some corrections seems not bad, but I hoped that by using an existing metric I could walk around this. Mar 23, 2015 at 19:17
• @BartekChaber If the metric is fixed, then it is definitely a lot easier. However, this means that you are solving the problem of N light particles which do not disturb the metric in some background created by some other massive objects? And I don't think analytical solutions of Einstein equation is available if you have several (point-like) massive objects around. Approximate methods are probably still necessary. Mar 23, 2015 at 19:46