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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?

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  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.

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