Canonical Quantization of Static and Rotating Black Hole as A Gravitational Atom

Abstract

Gravitational field is usually neglected in calculation of atomic energy levels as its effect is much weaker than the electromagnetic field, but that is not the case for a particle orbiting a black hole. In this work, canonical quantization of a particle under a gravitational field exerted by this tiny but very massive object, both static and rotating-is carried out. By using this method, very rare exact result of the particles quantized energy can be discovered. The presence of a very strong attractive field and the horizon make the energy complex valued and force the corresponding wave function to be a quasi-bound state. Moreover, by taking a small scale limit, the system becomes a gravitational atom in sense that hydrogenic atoms energy levels and wave functions can be recovered. Obtaining these exact solutions for fundamental black holes effectively empowers us to deal with more complex black holes such as cosmological black holes or black hole solutions emerging from modified gravity.