Today, someone asked me why "the warped space-time warps itself" (he read it in Kip Thorne's: The Science of Interstellar). I guess this is related to the gravitational self-interaction. But I don't really understand the gravitational self-interaction. Why the curvature in general relativity influences itself?
[Physics] Gravitational self-interaction
curvaturegeneral-relativitygravityinteractionsspacetime
Best Answer
There are a million ways to answer the "why" here, but here is the simplest way to see that there has to be a gravity-gravity interaction in GR:
we have two things baked into the theory:
1) locally, we can only move at the speed of light, which means that we can only travel at the speed of light, as measured by the metric tensor
2) we can transmit signals with gravitational waves
So, let's set up a gravitational field, somehow. This will bake in some energy into the metric tensor, and create a non-trivial spacetime geometry.
Now, send a localized gravitational wave with a small energy relative to the curvature through this geometry. It will travel, to first approximation, along a null geodesic in the background geometry. This is a different path than it would travel in the absence of the spacetime curvature, obviously. Well, here you go -- the gravitational field is interacting with itself.