How can we combine metrics in a spacetime with multiple point masses?

[dman12]

Say you have two masses in an otherwise empty space and you can approximate the metric near each mass as a Schwarzschild (or whatever). Is there a way of combining or 'stitching together' these two metrics to give the overall metric of all spacetime?

I'm just interested to know how a test particle could move in a space where there are two (or more) point masses but don't know how to calculate the overall metric of such a space.

 

[Dale]

This is one of the big problems with GR in practice. The equations are not linear, so you cannot find a complicated solution by adding up simple solutions.

 

[ChrisVer]

This is one of the big problems with GR in practice. The equations are not linear, so you cannot find a complicated solution by adding up simple solutions.

Is it plausible to say that outside some region (enclosing the masses) the solution drops down to Schwarchzild metric with different mass parameter?
I think it is, since we describe the stars (objects with different mass pieces everywhere) like that..

However I don't know what happens in s-t between them...

 

[WannabeNewton]

I'm just interested to know how a test particle could move in a space where there are two (or more) point masses but don't know how to calculate the overall metric of such a space.

Calculations such as that are done using PPN or numerical relativity. It cannot be done analytically using the full theory.

C.f. http://lapth.cnrs.fr/pg-nomin/chardon/IRAP_PhD/NiceJune2012.pdf

 

[Dale]

Is it plausible to say that outside some region (enclosing the masses) the solution drops down to Schwarchzild metric with different mass parameter?

Sure. I think that once you are far enough out you can simply make the approximation that you are in the asymptotically flat region, perhaps with some small perturbation. Calculating that perturbation could be done with a linearized form of the equations.

I don't speak from experience, though, so I could be off.

 

[Nugatory]

Wikipedia says that there is no exact solution known: http://en.wikipedia.org/wiki/Two-body_problem_in_general_relativity

 

[ChrisVer]

Sure. I think that once you are far enough out you can simply make the approximation that you are in the asymptotically flat region, perhaps with some small perturbation. Calculating that perturbation could be done with a linearized form of the equations.

I don't speak from experience, though, so I could be off.

(Entering the PF it just came through my mind to express my problems about this)
What is the power of GR in determining how the rest planets affect the trajectory of Mercury then? does it do that without considering Scwarzchild metrics?

 

[Dale]

I think that is done using perturbative methods (linearized). Again, this is outside of my area of personal direct knowledge.

 

[pervect]

Say you have two masses in an otherwise empty space and you can approximate the metric near each mass as a Schwarzschild (or whatever). Is there a way of combining or 'stitching together' these two metrics to give the overall metric of all spacetime?

I'm just interested to know how a test particle could move in a space where there are two (or more) point masses but don't know how to calculate the overall metric of such a space.

You basically have to use approximation / perturbative methods, like we do in the solar system as other posters have already mentioned. The quickest answer is just "linearize Einstein's field equations". A bit longer but still short answer - see for instance http://arxiv.org/abs/astro-ph/0303376 for an overview of how one gets the currently recommended IAU 2000 solar system metric (click on pdf to get the whole paper and not just the abstract, assuming you want pdf and not, say, postscript). See section 3.2 in particular

Hilights: along with linearizing the problem (so you neglect high order terms of $c^{-n}$, you also adopt some coordinates that satisfy the "harmonic gauge condition", and you additionally assume that you're interested in the gravitational field only outside of bodies, and that the external gravity of said bodies can be described by their masses and multipole moments. (I think there are some papers by Dixon that talk about this part of the issue). Then you wind up by replacing Einstein's full nonlinear equation with an approximate linearized equation in a single vector potential called w.

$$\left( - \frac{1}{c^2}\frac{\partial^2}{\partial t^2} + \Delta \right) w \approx -4 \pi G \sigma$$
$$\Delta w^i \approx -4 \pi G \sigma^i$$

You also need boundary conditions to solve this, this is usually asymptotic flatness at infinity.

You can see the reference for the details of getting the metric back once you have solved for w. W is basically a gravitational vector potential similar to the electromagnetic vector potential. There are some related versions of the formalism discussed where you use a scalar potential u rather than the vector potential w. This was in fact done in the older IAU 1991 resolution.

This is good for most solar system experiments, there is already some discussion of a need for extensions though. Google for "Extension of the IAU metric to be considered for inner solar system experiments", for example.