Will the 2-Body Problem in General Relativity Require Both Bodies' Metrics?

[quantum123]

For a 2 body problem in GR, will the metric be that due to one body, or both bodies?

 

[cesiumfrog]

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. If you think the curvature of spacetime depends on both bodies, you're going to have to find a metric that satisfies the field equation AND physically describes two masses with your initial conditions. This is quite difficult (although as a bonus it will also tell you what kind of gravitational wave signal is produced) and its a good bet there are computers working on this problem right now (hopefully the people take a break for Christmas).

 

[Jheriko]

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 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).

 

[Chris Hillman]

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).

 

[quantum123]

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.
You mean in GR, we need to consider the metric of the two body in order to derive the geodesic of anyone body?

 

[Chris Hillman]

Hi, quantum123, I am not sure whether you are talking to me or not:

You mean in GR, we need to consider the metric of the two body in order to derive the geodesic of anyone body?

I can't make sense of this question as stated, but here are some thoughts which might help:

1. by "derive the geodesic of anyone body" you probably mean "solve the geodesic equations in a spacetime model to find the world line of a test particle", whose mass is assumed to be too small to disturb the ambient gravitational field,

2. by "the metric of the two body" you probably mean "an exact vacuum solution of the EFE modeling two bodies forming a gravitationally bound isolated system" (so that this spacetime should be asymptotically flat but not axisymmetric, and not stationary since the two objects should very gradually "spiral" in upon each other),

3. gtr is a relativistic and nonlinear classical field theory, whereas Maxwell's theory of electromagnetism is a relativistic and linear classical field theory,

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, because in the first case we assume the mass of the test particle is too small to disturb the field of the large object, so in the first case we have the one body problem and in the second the two body problem,

5. the EFE implies that the world lines of test particles are timelike geodesics, and even (with due care) something similar for sizeable bodies; see the "Einstein-Infeld-Hoffman procedure".

 

[Jheriko]

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).

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 separated by a rod in some way.

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

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.

 

[George Jones]

Just out of curioisity, is this because even a zero mass particle will effect the gravitational field?

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.

Contrast this with the motion of a test particle near a massive body whose metric (e.g., Schwarzschild, Kerr, etc.) is already known. The motion of the test particle is found by solving the geodesic equation of the known metric.

 

[Stingray]

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.

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.

It is possible to show that in many cases of interest, the self-field has an insignificant effect, but that is certainly not true in general.

You mean in GR, we need to consider the metric of the two body in order to derive the geodesic of anyone body?

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.

But there are other issues in GR that don't exist in Maxwell theory. Perhaps the most conceptually important is that any motion you calculate only has meaning when supplemented with knowledge of the local geometry. You can't say where an object is going with any real meaning if you don't understand concepts of distance, angle, etc. Those concepts are defined by the full physical metric.

Beyond this, (as Chris mentioned) GR is a nonlinear theory. The metric perturbation induced by object B in the vicinity of A depends (a bit) on the metric perturbation of A itself.

Anyway, as in EM, there are interesting regimes where all of these problems can be justifiably ignored or dealt with to a good degree of approximation. The self-fields have no effect on bulk motion in the Newtonian limit, and that can of course be recovered from GR when the appropriate approximations are made.