  In a seminal work, Hawking showed that natural states for free quantum matter
fields on classical spacetimes that solve the spherically symmetric vacuum
Einstein equations are KMS states of non-vanishing temperature. Although
Hawking's calculation does not include backreaction of matter on geometry, it
is more than plausible that the corresponding Hawking radiation leads to black
hole evaporation which is in principle observable.
  Obviously, an improvement of Hawking's calculation including backreaction is
a problem of quantum gravity. Since no commonly accepted quantum field theory
of general relativity is available yet, it has been difficult to reliably
derive the backreaction effect. An obvious approach is to use black hole
perturbation theory of a Schwarzschild black hole of fixed mass and to quantise
those perturbations. But it is not clear how to reconcile perturbation theory
with gauge invariance beyond linear perturbations.
  In a recent work we proposed a new approach to this problem that applies when
the physical situation has an approximate symmetry, such as homogeneity
(cosmology), spherical symmetry (Schwarzschild) or axial symmetry (Kerr). The
idea, which is surprisingly feasible, is to first construct the
non-perturbative physical (reduced) Hamiltonian of the reduced phase space of
fully gauge invariant observables and only then to apply perturbation theory
directly in terms of observables. The task to construct observables is then
disentangled from perturbation theory, thus allowing to unambiguosly develop
perturbation theory to arbitrary orders.
  In this first paper of the the series we outline and showcase this approach
for spherical symmetry and second order in the perturbations for
Einstein-Klein-Gordon-Maxwell theory. Details and generalisation to other
matter and symmetry and higher orders will appear in subsequent companion
papers.
