Pointwise decay for the wave equation on nonstationary spacetimes

Understand

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.