Non-linearity and self-coupling of gravity

Question

I have heard that non-linearity of Einstein's field equations has to do with the fact that gravity self-couples. What does non-linearity have to do with self-coupling?

Answer 1

1. In the action formalism a linear Euler-Lagrange (EL) equation corresponds to a quadratic action, i.e. an action which is quadratic in the dynamical field variables of the theory.

2. On the other hand, self-coupling or interaction terms in the action correspond to cubic or higher terms. Such terms leads to non-linear EL equations. See also this & this related Phys.SE posts.

3. In case of GR, the field variable is the metric tensor $g_{\mu\nu}$. The Einstein-Hilbert action is not quadratic in $g_{\mu\nu}$, which leads to self-coupling/interaction and non-linear EFE.