The de Donder gauge is often used to simplify the linearised equations of motion of general relativity. If the metric is linearised as $g_{ab} = \bar g_{ab} + \gamma_{ab}$, then the de Donder gauge reads
$\nabla^a(\gamma_{ab} - \frac{1}{2}\bar g_{ab}\gamma) = 0$.
The partial differential equation for the gauge transformation vector $v^a$ is $ \nabla^b\nabla_b v_a + R_a^b v_b = \nabla^a(\gamma_{ab} - \frac{1}{2}\bar g_{ab}\gamma)$.
In chapter 7.5 of Wald, I read that this equation can always be solved because it is of the form
$g^{ab}\nabla_a\nabla_b \phi_i + \sum_j (A_{ij})^a\nabla_a \phi_j + \sum_j B_{ij}\phi_j + C_i$.
Theorem 10.1.2 of Wald says that in a globally hyperbolic spacetime this equation has a well posed initial value formulation on any spacelike Cauchy surface.
In stead of de Donder gauge, I want to use a similar gauge:
$\nabla^a(\gamma_{ab} - n \bar g_{ab}\gamma) = 0$.
The partial differential equation changes to
$ \nabla^b\nabla_b v_a + (1 -2n)\nabla_a\nabla_b v^b + R_a^b v_b = \nabla^a(\gamma_{ab} - n\bar g_{ab}\gamma)$.
This equation is not covered by theorem 10.1.2 of Wald. My question is: is the existence of a solution for this equation guaranteed in an AdS background when $n=1$?
