  In recent work of Allen at. al., heuristic and numerical arguments were put
forth to suggest that boundary value problems for black hole evolution, where
an appropriate Sommerfeld radiation condition is imposed, would fail to produce
Price law tails. The interest in this issue lies in its possible implications
for numerical relativity, where black hole evolution is typically studied in
terms of such boundary formulations. In this note, it is shown rigorously that
indeed, Price law tails do not arise in this case, i.e. that Sommerfeld (and
more general) radiation conditions lead to decay faster than any polynomial
power. Our setting is the collapse of a spherically symmetric self-gravitating
scalar field. We allow an additional gravitationally coupled Maxwell field. The
proof also applies to the easier problem of a spherically symmetric solution of
the wave equation on a Schwarzschild or Reissner-Nordstrom background. The
method relies on previous work of the authors.
