I was looking at a solution of the Einstein Field Equations - The Linearized Vacuum Einstein Equation. While deriving this, a flat-space wave operator (also called the d'Alembertian) is introduced to describe the curvature of nearly-flat spacetime, after considering a small perturbation parameter.
Why do we introduce this wave-operator and why isn't it included in the general EFE?
On the first place the general Einstein's Field Equations (EFE) are nonlinear, i.e. the quantity to be solved for, the metric tensor $g_{ik}$, appears in the EFE nonlinearly, in particular there are quadratical terms of the first derivatives of the metric tensor in the EFE. To be more specific the EFE can be written:
$R_{ik} =\kappa (T_{ik}-\frac{1}{2}g_{ik}T)$
where $R_{ik}$ is the Ricci-tensor, and $T_{ik}$ the energy-momentum tensor and its trace $T= \sum_i T^i_i$. The Ricci-tensor contains 2. derivatives of the metric tensor linearly, and 1. derivatives of the metric tensor quadratrically.
Therefore in its pure form the wave operator does not appear in the EFE. However, in the linear approximation $g_{ik} \approx \eta_{ik} + \psi_{ik}$ ($\eta_{ik}$ representing the flat Minkowski metric) those quadratically terms are neglected as being small, so upon evaluation of the Ricci-tensor with this approximation one obtains equations for $\psi_{ik}$, where the d'Alembertian of $\psi_{ik}$ appears on the left side of the equations. As solutions of this linearized equation system you get (assuming the corresponding boundary conditions) gravitational waves, discovered in 2015.
In strong gravitational fields the non-linear terms gain importance and can no longer be neglected. As far as I know the calculation of the EFE's contribution to the precession of the perihelion of Mercury requires the consideration of the nonlinear effects of EFEs. The nonlinearity of EFE's can be understood in the following very simplified way: The gravitational field couples to itself, because the field also represents field energy (non-localisable though) so it should couple to the field itself.
