  The first article in a two-part series (the second article being
[arXiv:2205.13197]) assumes a weak local energy decay estimate holds and proves
that solutions to the linear wave equation with variable coefficients in
$\mathbb R^{1+3}$, first-order terms, and a potential decay at a rate depending
on how rapidly the vector fields of the metric, first-order terms, and
potential decay at spatial infinity. We prove results for both stationary and
nonstationary metrics. The proof uses local energy decay to prove an initial
decay rate, and then uses the one-dimensional reduction repeatedly to achieve
the full decay rate.
