  A well-known open problem in general relativity, dating back to 1972, has
been to prove Price's law for an appropriate model of gravitational collapse.
This law postulates inverse-power decay rates for the gravitational radiation
flux on the event horizon and null infinity with respect to appropriately
normalized advanced and retarded time coordinates. It is intimately related
both to astrophysical observations of black holes and to the fate of observers
who dare cross the event horizon. In this paper, we prove a well-defined (upper
bound) formulation of Price's law for the collapse of a self-gravitating scalar
field with spherically symmetric initial data. We also allow the presence of an
additional gravitationally coupled Maxwell field. Our results are obtained by a
new mathematical technique for understanding the long-time behavior of large
data solutions to the resulting coupled non-linear hyperbolic system of
p.d.e.'s in 2 independent variables. The technique is based on the interaction
of the conformal geometry, the celebrated red-shift effect, and local energy
conservation; we feel it may be relevant for the problem of non-linear
stability of the Kerr solution. When combined with previous work of the first
author (gr-qc/0307013) concerning the internal structure of charged black
holes, which assumed the validity of Price's law, our results can be applied to
the strong cosmic censorship conjecture for the Einstein-Maxwell-real scalar
field system with complete spacelike asymptotically flat spherically symmetric
initial data. Under Christodoulou's C^0 formulation, the conjecture is proven
to be false.
