Interacting gravitational fields

[Drops of Fire]

How do gravitational fields interact with each other? Because the Earth is in the Sun's and the moon is in the Earth's and the sun's. I think ocean tides on Earth is an example of it, but is there anything more specific? What is happening to an area of space inside multiple gravitational fields?

 

[Dale]

In Newtonian gravity the equations are linear, so you simply add up the various individual fields to get the total field. In GR the equations are non-linear so you cannot simply add up the various individual field to get the total field. So even a simple two-body problem usually requires a numerical solution in GR.

 

[Jonathan Scott]

In Newtonian gravity the equations are linear, so you simply add up the various individual fields to get the total field. In GR the equations are non-linear so you cannot simply add up the various individual field to get the total field. So even a simple two-body problem usually requires a numerical solution in GR.

For "weak field" environments like the solar system, fields effectively add linearly for GR as well. For accurate calculations (such as Mercury's perihelion precession) you have to take into account not only the Newtonian field acting on objects at rest but also additional effects caused by motion through curved space, but these additional effects are still proportional to the Newtonian field and still effectively add up linearly for multiple sources.

In practice, you only need to take non-linearity into account in extreme environments such as neutron stars and black holes.

 

[Drops of Fire]

But what's happening to the space? If it warps one way, how to it react to warping another way?

 

[Dale]

As I said above, unless you are close to the Newtonian limit, you basically just have to numerically solve the EFE for the situation you have in mind. The nonlinearities make it difficult to make general statements of the kind you are asking.