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?

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    $\begingroup$ 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. $\endgroup$
    – David Z
    Mar 23, 2015 at 15:46
  • $\begingroup$ @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. $\endgroup$ Mar 23, 2015 at 15:58
  • $\begingroup$ 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. $\endgroup$
    – Meng Cheng
    Mar 23, 2015 at 17:47
  • $\begingroup$ @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. $\endgroup$ Mar 23, 2015 at 19:17
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    $\begingroup$ @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. $\endgroup$
    – Meng Cheng
    Mar 23, 2015 at 19:46

1 Answer 1


The problem of two fixed centers, aka Euler's three body problem, is integrable in in Newtonian mechanics. It is chaotic in general relativity.

This problem goes way back. (As indicated by the name, it goes back to Euler.)

Contopoulos, G., and H. Papadaki. "Newtonian and relativistic periodic orbits around two fixed black holes." Celestial Mechanics and Dynamical Astronomy 55.1 (1993): 47-85.

  • $\begingroup$ Thanks, but what about N bodies? Can I derive the metric for N (even fixed) bodies using for example gravitational potential (mentioned by @aandreev)? $\endgroup$ Mar 23, 2015 at 23:36
  • $\begingroup$ What makes you think N bodies is easier? That makes no sense. $\endgroup$ Mar 24, 2015 at 0:34
  • $\begingroup$ I didn't say it will be easier, but I would like to solve a problem with N bodies. $\endgroup$ Mar 24, 2015 at 8:28

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