Quantum Horizons and Spacetime Non-commutativity

M.Martinis, V. Mikuta-Martinis
Ruder Boškovic Institute, Zagreb, Croatia

In standard General Relativity without Quantum Mechanics (QM) a fundamental instability against collapse implies the existence of black holes as stable solutions of Einstein's equations. The QM and the Quantum Field Theory (QFT) in the curved spacetime with classical event horizon are troubled by the singularity at the horizon. This problem may be solved by treating the black hole as a quantum state which implies that the energy of the black hole and its corresponding time do not commute at the horizon. In this picture we study the dynamics of a scalar field in the near-horizon region described by a static Klein-Gordon operator which is the Hamiltonian of the system. The explicite construction of the time operator near-horizon is given and its self-adjointness discussed.