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Free keywords:
General Relativity and Quantum Cosmology, gr-qc,Mathematical Physics, math-ph,Mathematics, Mathematical Physics, math.MP
Abstract:
We study the scalar wave equation on the open exterior region of an extreme
Reissner-Nordström black hole and prove that, given compactly supported data
on a Cauchy surface orthogonal to the timelike Killing vector field, the
solution, together with its $(t,s,\theta,\phi)$ derivatives of arbitrary order,
$s$ a tortoise radial coordinate, is bounded by a constant that depends only on
the initial data. Our technique does not allow to study transverse derivatives
at the horizon, which is outside the coordinate patch that we use. However,
using previous results that show that second and higher transverse derivatives
at the horizon of a generic solution grow unbounded along horizon generators,
we show that any such a divergence, if present, would be milder for solutions
with compact initial data.
