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The current work considers solutions to the wave equation on asymptoticallyflat, stationary, Lorentzian spacetimes in (1+3) dimensions. We investigate therelationship between the rate at which the geometry tends to flat and thepointwise decay rate of solutions. The case where the spacetime tends towardflat 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 geometriestending toward flat at a rate of $|x|^{-\kappa}$ and establish a pointwisedecay 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 thegeodesic trapping allowed on the underlying geometry. We use the resolvent toconnect the time Fourier Transform of a solution to the Cauchy data. Ultimatelythe rate of pointwise wave decay depends on the low frequency behavior of theresolvent, which is sensitive to the rate at which the background geometrytends to flat.

The effect of metric behavior at spatial infinity on pointwise wave decay in the asymptotically flat stationary setting