Question

How to approach a 2-body problem in General Relativity?

Answer 1

I am aware that closed form solution for 2-body problem does not exist. What I am looking for is a numerical solution. I was thinking on how to start to approach this very problem. I am aware of lagrangian formulation of general relativity where we cook up a lagrangian of different interactions to form the action integral. Pure gravitational action which would be something like

A=−m∫dsA=−m∫ds

Now since ds2=gμνdxμdxνds2=gμνdxμdxν, if we let the metric tensor gμνgμν to be a non-flat one (say) schwarzchild or kerr then minimizing the action would lead us to the geodesic equation for a particle in that spacetime.

This formulation literally means that to incorporate the effects of gravity in any situation just instead of flat metric use a non-flat metric. Which means for a 2-body problem what I need is a metric that is specifically calculated for such a situation. So how to even begin to solve such a problem?

Computer simulations of relativistic bodies under mutual interaction with each other have been created. They must have formulated nn-body interaction in some numerical way. So how do they do it for a general case?

Edit: I looked up there is a metric for 2-body named "Curzon-Chazy metric" but it's not for a general case.