I'm working on a problem related to fluid perturbations of stars. I'm following this paper: http://arxiv.org/pdf/gr-qc/0008019.pdf . They start with the Einstein equation

$G_{\alpha \beta} = 8 \pi G T_{\alpha \beta}$ and then perturb the metric, as well as the stress-energy of the matter (which they assume to be a perfect fluid). The metric is basically rewritten as

$g_{\alpha \beta} = g^0_{\alpha \beta} + h_{\alpha \beta}$, where $h_{\alpha \beta}$ represents the perturbation. After working through the algebra, they arrive at a coupled set of differential equations that relate the metric perturbation to the fluid perturbations. 

My questions is: Is there any generic reason why $h'''_{\alpha \beta}$ would be zero? Assuming that $h_{\alpha \beta}$ only depends on $r$ and the primes represent derivatives with respect to $r$. 

If there is no general rule about this, does anyone know of specific situations where the third derivative of a perturbed metric does go to zero?

Thanks very much.