  Getting the mathematical rules for quantised black holes correctly is far
from straightforward. Many earlier treatises got it not quite correctly. The
general relativistic transformation linking the distant observer (who only
detects particles outside the hole) with the observations of a local observer
(who falls into a black hole) must map the quantum states in a one-to-one way.
This does not come out right if one follows text book rules. Here it is
advocated that demanding very strict logic leads to new insights, such as the
non-triviality of space-time topology near a black hole. This way one may
attempt to make up for the lack of direct experimental evidence concerning
gravitation at the Planck scale. It is noted that this approach does not
require assumptions such as string theories or AdS/CFT conjectures. All we need
to assume is the validity of quantum field theory wherever the Schwarzschild
metric is regular, combined with the requirement that only those general
coordinate transformations apply that map pure quantum states one-to-one onto
pure quantum states.
