Calculating Spacetime Around Multiple Objects

[Sciencemaster]

TL;DR Summary

What metric would one use to describe spacetime around multiple massive objects?

In describing the spacetime around a massive, spherical object, one would use the Schwarzschild Metric. What metric would instead be used to describe the spacetime around multiple massive bodies? Say, for example, you want to calculate the Gravitational Time Dilation experienced by a rocket ship between two different stars (where everything in the system is stationary with respect to everything else for simplicity). How would you go about doing that?
Similarly, is there a metric that can calculate the warping of spacetime from arbitrary amounts of masses or around objects that aren't spherical (perhaps by approximating the object as a collection of point masses)?

 

[PeterDonis]

What metric would instead be used to describe the spacetime around multiple massive bodies?

No exact solution is known for this case. It can only be solved numerically.

 

[Ibix]

How would you go about doing that?

Get a big computer. And note that time dilation isn't strictly defined here since the spacetime is non-stationary.

Or approximate it as Newtonian and add the potentials.

 

[Orodruin]

To add to what has already been said: The reason you cannot just add upp the solutions for a lot of point masses is that — unlike Newtonian gravity — the Einstein field equations of general relativity are non-linear. This makes gravitation self-interacting and different solutions cannot be superpositioned.

 

[PeterDonis]

approximate it as Newtonian and add the potentials

This is the basis for multi-body numerical solutions in the weak field regime: assume all non-linear terms are negligible and add the contributions from each body. The most general scheme that I'm aware of for this is the Einstein-Infeld-Hoffmann equations:

https://en.wikipedia.org/wiki/Einstein–Infeld–Hoffmann_equations

 

[PAllen]

This is the basis for multi-body numerical solutions in the weak field regime: assume all non-linear terms are negligible and add the contributions from each body. The most general scheme that I'm aware of for this is the Einstein-Infeld-Hoffmann equations:

https://en.wikipedia.org/wiki/Einstein–Infeld–Hoffmann_equations

Actually, these equations are nonlinear, in that the acceleration of any body is affected by the acceleration of every other body. There is no ability to add solutions - an iterative procedure is needed for strict application of these equations. In practice, for solar system purposes, they are further approximated such that the accelerations in the RHS of the equations are computed via Newton.

 

[PeterDonis]

Actually, these equations are nonlinear, in that the acceleration of any body is affected by the acceleration of every other body.

Yes, as the equations are written, that's true. As the article I linked to notes, though, in practice, although the equations should be solved iteratively, the zero-order Newtonian acceleration of each body is used in the formulas for every other body and sufficient accuracy is gained that way. I think that amounts to linearizing the equations, although I'm not positive.

 

[PAllen]

Yes, as the equations are written, that's true. As the article I linked to notes, though, in practice, although the equations should be solved iteratively, the zero-order Newtonian acceleration of each body is used in the formulas for every other body and sufficient accuracy is gained that way. I think that amounts to linearizing the equations, although I'm not positive.

I also think that linearizes them. That method is used for the most accurate solar system extrapolations in current use, including modeling perihelion advance for all the planets. But the interesting thing about PPN methods is that they have proven quantitatively accurate (used to 3.5 order - the EIH equations are first order PPN) for predicting gravitational wave forms for inspiralling BH and neutron stars (up until the very last moments). This is considered surprising, because the relative speeds are a major fraction of light speed and the curvature is large. Clifford Will has argued that this unexpected success is evidence that the strong equivalence principle must be correct to very high precision. Which certainly creates tension with recent MOND papers claiming violations of the weak equivalence principle.