Main Question or Discussion Point
For a 2 body problem in GR, will the metric be that due to one body, or both bodies?
I think this is called "the restricted two-body problem". Since it has the additional constraint that the second object be massless (at least to a good approximation).If one body isn't very massive, you use the metric due to the other body, and just have to integrate the geodesic equation for the first.
I can't make sense of this question as stated, but here are some thoughts which might help:You mean in GR, we need to consider the metric of the two body in order to derive the geodesic of any one body?
Doesn't the double Kerr vacuum solution use an "ideal rod" to keep the two masses apart? Wikipedia mentions something about cables... but I seem to remember reading about the bodies being seperated by a rod in some way.There is an axisymmetric "double Kerr vacuum solution" (two bodies) which can be written down (using a lot of space) in closed form, but its interpretation is trickier than the Kerr vacuum solution (one body).
Just out of curioisity, is this because even a zero mass particle will effect the gravitational field? So as the mass of an object tends towards zero its path does not tend towards the path through the field from a non-field generating test particle? All though I can imagine the contribution to be minimal.. this makes sense to me considering that an electromagnetic field, for instance, can create a gravitational attraction.4. the problem of determing the motion of a test particle orbiting an isolated object is not the same as the problem of finding an exact solution describing two objects forming a gravitationally bound isolated system
Here's what I think Chris is saying. Consider a situation in which two massive bodiles (e.g., black holes, neutron stars, etc.) orbit their "centre of mass". In this case, both bodies make substantial contributions to the stress-energy-momentum tensor of the system, so describinging the "dynamics" and "orbital motion" of the system involves solving the Einstein field equation for the appropriate spacetime metric.Jheriko said:Just out of curioisity, is this because even a zero mass particle will effect the gravitational field?
To do things correctly, you do have to use the self-field. It's only in a first (though usually excellent) approximation that the self-fields can be neglected when computing bulk equations-of-motion. A piece of charge cannot "know" which portion of the field around it is due to matter attached to it versus matter that isn't. So the relevant field must be the full physically-measurable one sourced by all objects in combination.In classical electrodynamics, when we have 2 electric charge, we consider the motion of one charge due to the electric field of another and not the resultant field of both of them, otherwise we will be using the "self" field.
First of all, objects in GR do not always move on geodesics. In general, that is not the correct way to compute an object's motion (though it will often be close). Regardless, there is a known exact procedure to calculate the center-of-mass motion if the metric is known. As in electromagnetism, the metric used in these equations must be the full one.You mean in GR, we need to consider the metric of the two body in order to derive the geodesic of any one body?