2nd order perturbation of a charged rotating body What is the 2nd order perturbation to the flat Minkowski metric $\eta_{ab}$ caused by a charged, rotating body? In particular, if we take a metric $\eta_{ab} + h_{ab} $, what $h_{ab}$ satisfies the Einstein equations for a source of mass $\epsilon m$ and charge $\epsilon e$ up to second order in $\epsilon$?
I know that the second order perturbation by a charged, non-rotating body is given by
$u_{ab}=\frac{1}{2r^2} \left[ (3m^2 -q^2)\eta_{ab} -(m^2 +3e^2)\nabla_a t \nabla_b t\right]$
(see eqn 61 here: https://arxiv.org/abs/1608.04359)
[where the first order perturbation is
$h_{ab}=\frac{2M}{r}(\eta_{ab}+2\nabla_a t \nabla_b t)$
and we are working in spherical coordinates]
but what is the equivalent formulation when rotation is included? Or even for a rotating but uncharged mass?
Further detail - I'm looking for something like eqn (3) and (4)
here
but (without necessarily including the quadrupole moment $q$) with non-zero charge $e$.
 A: One particularly simple example of a field of a rotating charged object is the Kerr-Newman metric in Kerr-Schild coordinates (see also Scholarpedia entry). This is an exact solution to Einstein equations and if you do not assume anything about the magnitude of the black hole angular momentum, the Kerr-Schild form gives you what you are looking for (if you assume angular momentum$\sim \epsilon$, you can just expand the expressions to second order in $a$).
However, notice that the Kerr-Schild expression will not agree with what you have in the OP when $a=0$. This is because you have not specified your coordinates or equivalently gauge conditions on $h^{\mu\nu}$. Whether this matters depends really on the application you are looking for. 
An additional point to consider is the fact that the second-order perturbation is not the general external field of a rotating body because a general rotating body will have a different quadrupole (the $\sim a^2$ term). In fact, any finite body made of ordinary matter will have a larger quadrupole at given $a$. You can use the appropriately expanded Manko et al. metric discussed in the paper "An all-purpose metric for the exterior of any kind of rotating neutron star" by Pappas & Apostolatos (2013) for the exterior of a generic rotating object.
