Space-Time Non-Commutativity near Horizon of a Black Hole

We consider dynamics of a quantum scalar field, minimally coupled to classical gravity, in the near-horizon region of a Schwarzschild black-hole. It is described by a static Klein-Gordon operator which in the near-horizon region reduces to a scale invariant Hamiltonian of the system. This Hamiltonian is not essentially self-adjoint, but it admits a one-parameter family of self-adjoint extension. The time-energy uncertainty relation, which can be related to the thermal black-hole mass fluctuations, requires explicit construction of a time operator near-horizon. We present its derivation in terms of generators of the affine group. Matrix elements involving the time operator should be evaluated in the affine coherent state representation. PACS: 03.65 Fd, 02.30.Tb, 04.70.-s, 04.62.+v, 04.60.-m Preprint submitted to Elsevier Science 7 February 2008