Loop Quantum Black Hole Extensions Within the Improved Dynamics

Abstract

We continue our investigation of an improved quantization scheme for spherically symmetric loop quantum gravity. We find that in the region where the black hole singularity appears in the classical theory, the quantum theory contains semi-classical states that approximate general relativity coupled to an effective anisotropic fluid. The singularity is eliminated and the space-time can be continued into a white hole space-time. This is similar to previously considered scenarios based on a loop quantum gravity quantization.

Towards a quantum notion of covariance in spherically symmetric loop quantum gravity

“…Notice that g F tx changes sign at the bounce since ðE x 0 Þ 0 is positive and ðE x −1 Þ 0 is negative. This does not introduce singularities in the curvature, as we have shown explicitly in [6], where we proved that it is of order Planck at the bounce.…”

“…Let us now address the covariance of the framework at the bounce that replaces the singularity in [6]. It is important to remark that the bounce occurs at a point that may be identified in a way that is invariant under changes of foliation and radial coordinates and is given by the infimum of jE x j j.…”

“…For instance in principle it could be possible to choose different polymerizations for the shift and the spatial metric as we did in [3] that would not lead to a quantum covariant formulation. The covariant version of the improved quantization appears in [6].…”

Towards a quantum notion of covariance in spherically symmetric loop quantum gravity

“…One way of doing this is to introduce a function F (x j ) such that sin (ρ j K ϕ,j ) = F (x j ) with F (x j ) ∈ [−1, 1] ∀x j and therefore, with the notation F (x j ) ≡ F j ∈ [−1, 1]. Each choice of F j leads to a different foliation, for instance F (x j ) = ρ j r S / E x j leads to ingoing Painlevé-Gullstrand form of the metric [6] and F (x j ) = ρ j r S / E x j (1 + r S / E x j ) to ingoing Eddington-Finkelstein coordinates [3]. 3 Note that as we explained for K ϕ,j , F (x j ) it can either be a c-number function or an operator, function of the Dirac observables, and should be treated accordingly.…”

“…Let us now address the covariance of the framework at the bounce that replaces the singularity in [6]. It is important to remark that the bounce occurs at a point that may be identified in a way that is invariant under changes of foliation and radial coordinates and is given by the infimum of |E x j |.…”