Does superposition work for the metric tensor of two bodies?

[tarquinius]

Hello there. I would like to find the metric tensor produced by the existence of two massive bodies. Does the principle of superposition work for metrics as well? The first idea I got was to add the two metrics for each separate body in order to obtain the resulting one. Is this approach valid? I should be grateful for any help.

 

[Bill_K]

tarquinius, If you really mean two free particles, the solution will be quite complicated since they will accelerate toward each other, emitting gravitational radiation, and eventually coalesce. The problem can only be solved numerically.

Otherwise a static Weyl solution exists that has two point masses held a fixed distance apart by a nonphysical strut.

 

[tarquinius]

And does this Weyl solution work the way I have described above? Would it be possible to just add the metric of each body in order to obtain the resulting one?

 

[pervect]

You can't add the solutions together, at least not in strong fields, because the equations aren't linear.

 

[tarquinius]

OK. That's what I expected. I just wanted to be sure that the resulting metric can't be produced in such a simply way before starting deriving the tensor from the field equation.

 

[Bill_K]

Here, courtesy of Google books, is a description of the Weyl solutions. They comprise all static axially symmetric vacuum solutions, and are described by an axially symmetric solution U of Laplace's equation in a flat 3-space. Although Laplace's equation is linear, and you can superpose solutions in that sense, there are other terms in the metric which depend on U in a nonlinear fashion.