  In this article we study the pointwise decay properties of solutions to the
wave equation on a class of stationary asymptotically flat backgrounds in three
space dimensions. Under the assumption that uniform energy bounds and a weak
form of local energy decay hold forward in time we establish a $t^{-3}$ local
uniform decay rate for linear waves. This work was motivated by open problems
concerning decay rates for linear waves on Schwarzschild and Kerr backgrounds,
where such a decay rate has been conjectured by R. Price. Our results apply to
both of these cases.
