Could anyone explain to me what I have misunderstood/missed when trying to understand this paper on GR perturbation?
The paper is http://arxiv.org/pdf/0704.0299v1.pdf
In equation 25 for $R_{00}$,
$$R_{00}= -\frac{1}{2} [\nabla^2 h_{00}+\partial_i h_{00}\partial^i h_{00} -h^{ij}\partial_i\partial_j h_{00}]$$
which if we jump to section V:C where they calculate $h_{00}$ to $\mathcal{O}(4)$, I would have thought this (which equals the LHS of equation 45) gives
$$-[\nabla^2 (U+(1/2)h_{00}^{[4]})+2\partial_i U\partial^i U +2U\nabla^2 U] = -\nabla^2 U-\nabla^2 h_{00}^{[4]} -\nabla^2 U^2$$ right? I am pretty sure I have misunderstood something because when using the well-established equations in the appendix (A2), since there is no contribution to the $\rho v^2$ and $\rho U$ terms from the LHS and they have opposite signs on the RHS of equation 45, how does one get the familiar result - equation 46 - for $h_{00}$ where the coefficents of $\Phi_1, \Phi_2$ are of the same sign? Thank you.
The equation you have written is not correct, that is $$ -[\nabla^2 (U+(1/2)h_{00}^{[4]})+2\partial_i U\partial^i U +2U\nabla^2 U] \ne -\nabla^2 U-\nabla^2 h_{00}^{[4]} -\nabla^2 U^2. $$ Rather, it should be $$ -[\nabla^2 (U+(1/2)h_{00}^{[4]})+2\partial_i U\partial^i U +2U\nabla^2 U] = -(1+2U)\nabla^2 U-\frac{1}{2}\nabla^2 h_{00}^{[4]} -\nabla^2 U^2-2|\nabla U|^2. $$