  Applying an expansion in spherical harmonics, turns the black hole with its
microstates into something about as transparent as the hydrogen atom was in the
early days of quantum mechanics. It enables us to present a concise description
of the evolution laws of these microstates, linking them to perturbative
quantum field theory, in the background of the Schwarzschild metric. Three
pieces of insight are obtained: One, we learn how the gravitational back
reaction, whose dominant component can be calculated exactly, turns particles
entering the hole, into particles leaving it, by exchanging the momentum- and
position operators; two, we find out how this effect removes firewalls, both on
the future and the past event horizon, and three, we discover that the presence
of region II in the Penrose diagram forces a topological twist in the
background metric, culminating in antipodal identification. Although a cut-off
is required that effectively replaces the transverse coordinates by a lattice,
the effect of such a cut-off minimizes when the spherical wave expansion is
applied. This expansion then reveals exactly how antipodal identification
restores unitarity - for each partial wave separately. We expect that these
ideas will provide new insights in some highly non-trivial topological
space-time features at the Planck scale.
