  We derive precise late-time asymptotics for solutions to the wave equation on
spherically symmetric, stationary and asymptotically flat spacetimes including
as special cases the Schwarzschild and Reissner-Nordstrom families of black
holes. We also obtain late-time asymptotics for the time derivatives of all
orders and for the radiation field along null infinity. We show that the
leading-order term in the asymptotic expansion is related to the existence of
the conserved Newman-Penrose quantities on null infinity. As a corollary we
obtain a characterization of all solutions which satisfy Price's polynomial law
as a lower bound. Our analysis relies on physical space techniques and uses the
vector field approach for almost-sharp decay estimates introduced in our
companion paper. In the black hole case, our estimates hold in the domain of
outer communications up to and including the event horizon. Our work is
motivated by the stability problem for black hole exteriors and strong cosmic
censorship for black hole interiors.
