# Is there a GR solution for the geocentric system?

It is possible to get the Schwartzschild metric assuming spherical symmetry, vacuum solution and Minkowski spacetime when $$r \to \infty$$.

Is it possible an analytic solution for a geocentric system? I mean, taking the apparent daily movement of the celestial bodies as real. So, the (apparent) trajectories of moon, sun and planets should be geodesics according to the metric.

I suppose it is necessary to assume that when $$r \to \infty$$ geodesics are circles, as the fixed stars does every night from an observer on earth. So it is not a Minkowski metric at infinity.

I don't know if the Godel solution is something like that.

• By geocentric system you mean the geocentric model? – Qmechanic Oct 17 at 20:48
• There isn’t even an exact solution for the two-body problem in GR. – G. Smith Oct 17 at 21:49
• Yes. It is known that it is possible to choose different frames, but I've never seen an example. – Claudio Saspinski Oct 17 at 21:49

There isn’t even an exact solution for the two-body problem in General Relativity, much less for the $$n$$-body problem, even if you take the center of mass as the origin. The $$n$$-body problem doesn’t have an exact solution in Newtonian gravity.
• Yes, you are right. I think about some conditions. When $r \to \infty => d\phi/dt = cte, dr/dt = \lambda r, d\theta/dt = 0$ – Claudio Saspinski Oct 17 at 22:20