Singularity problem and phase-space noncanonical noncommutativity

Abstract

The Wheeler-DeWitt equation arising from a Kantowski-Sachs model is considered for a Schwarzschild black hole under the assumption that the scale factors and the associated momenta satisfy a noncanonical noncommutative extension of the Heisenberg-Weyl algebra. An integral of motion is used to factorize the wave function into an oscillatory part and a function of a configuration space variable. The latter is shown to be normalizable using asymptotic arguments. It is then shown that on the hypersufaces of constant value of the argument of the wave function's oscillatory piece, the probability vanishes in the vicinity of the black hole singularity.

Noncanonical phase-space noncommutative black holes

“…We have shown that the Hawking temperture and entropy can be recovered for a non-vanishing value of η 0 , namely η 0 = 0.487. It is worth pointing out that one of the main features of this model is the regularization of the Schwarzschild singularity [1], a feature that can be extended for the KS singularity as well. The regularization of the two types of singularities is due to the eigenstates of the assymptotic behaviour of the obtained wave functions, which are square integrable, even when they are associated to a continuous set of eigenstates [1].…”

“…It is worth pointing out that one of the main features of this model is the regularization of the Schwarzschild singularity [1], a feature that can be extended for the KS singularity as well. The regularization of the two types of singularities is due to the eigenstates of the assymptotic behaviour of the obtained wave functions, which are square integrable, even when they are associated to a continuous set of eigenstates [1]. Actually, this is a property shared by all Hamiltonians with a potential that behaves like V (z) ≈ −Kz 2+ε for some K, ε > 0 as z → ∞ [8].…”

“…We consider a noncanonical phase-space NC extension of a Kantowski Sachs (KS) cosmological model to sudy the interior of a Schwarzschild BH [1,2]. In a Schwarzschild BH, for r < 2M, time and radial coordinates are interchanged (r ↔ t) and space-time is described by the metric:…”

“…In Ref. [1], the noncommutative Wheeler-deWitt (NCWDW) equation of a KS cosmological model was obtained and is used here to study the interior of a Schwarzschild BH. When ε = 0 we recover the canonical NC relations as in Refs.…”

“…[1]. The NCWDW equation is obtained considering the canonical quantization of the KS Hamiltonian, and the fact that the operator = µP β c + η 2µΩ c commutes with the Hamiltonian (it corresponds to a constant of motion of the classical problem), and writing the wave function as…”