  The current work considers solutions to the wave equation on asymptotically
flat, stationary, Lorentzian spacetimes in (1+3) dimensions. We investigate the
relationship between the rate at which the geometry tends to flat and the
pointwise decay rate of solutions. The case where the spacetime tends toward
flat at a rate of $|x|^{-1}$ was studied in \cite{tat2013}, where a $t^{-3}$
pointwise decay rate was established. Here we extend the result to geometries
tending toward flat at a rate of $|x|^{-\kappa}$ and establish a pointwise
decay rate of $t^{-\kappa-2}$ for $\kappa \in \mathbb{N}$ with $\kappa \ge 2$.
We assume a weak local energy decay estimate holds, which restricts the
geodesic trapping allowed on the underlying geometry. We use the resolvent to
connect the time Fourier Transform of a solution to the Cauchy data. Ultimately
the rate of pointwise wave decay depends on the low frequency behavior of the
resolvent, which is sensitive to the rate at which the background geometry
tends to flat.
