Gravitational self-interaction 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? 
 A: 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.
A: The same thing happens in electromagnetism. And the first step in electromagnetism is to admit that the electromagnetic field exists and has particular values at every event. Then you note that an initial slice can be extended forwards or backwards by the Maxwell Equation. You can think of $$\frac{\partial \vec B}{\partial t}=-\vec \nabla\times \vec E$$ as evolving the magnetic part. And think of $$\frac{\partial \vec E}{\partial t}=\frac{1}{\epsilon_0}\left(-\vec J+\frac{1}{\mu_0}\vec \nabla\times \vec B\right)$$ as evolving the electric part.
So electromagnetic fields don't need something to force then to be nonzero, they have a particular value and they evolve in particular ways. That's exactly what happens with the metric. It has particular values. And just like Maxwell tells the electromagnetic field how to evolve, Einstein tells the metric how to evolve.
You literally can't object to Einstein and think Maxwell is fine. It's the exact same thing going on. Yes, the evolution equation is nonlinear and there are more of them (second order and for a symmetric rank two tensor). It just makes it harder to right them down, but it doesn't change what is going on.
A: A good example (assume e.g. black hole is accelerating and creates gravitational waves):
Step 1. Linearize general relativity (assume that space does not warp itself)
Step 2. Curvature creates gravitational waves
Step 3. Gravitational waves create curvature
Step 4. Re-do the calculation: Curvature creates gravitational waves (gravitational waves technically create gravitational waves, although the effect is miniscule)
