Non-commutative black holes of various genera in the connection formalism

Abstract

We consider black hole interiors of arbitrary genus number within the paradigm of non-commutative geometry. The study is performed in two ways: One way is a simple smearing of a matter distribution within the black hole. The resulting structure is often known in the literature as a "model inspired by non-commutative geometry". The second method involves a more fundamental approach, in which the Hamiltonian formalism is utilized and a non-trivial Poisson bracket is introduced between the configuration degrees of freedom, as well as between the canonical momentum degrees of freedom. This is done in terms of connection variables instead of the more common ADM variables. Connection variables are utilized here since non-commutative effects are usually inspired from the quantum theory, and it is the connection variables that are used in some of the more promising modern theories of quantum gravity. We find that in the first study, the singularity of the black holes can easily be removed. In the second study, we find that introducing a non-trivial bracket between the connections (the configuration variables) may delay the singularity, but not necessarily eliminate it. However, by introducing a non-trivial bracket between the densitized triads (the canonical momentum variables) the singularity can generally be removed. In some cases, new horizons also appear due to the non-commutativity.