Loop quantum black hole

In this paper we consider the Kantowski-Sachs space-time in Ashtekar variables and the quantization of this space-time starting from the complete loop quantum gravity theory. The Kanthowski-Sachs space-time coincides with the Schwarzschild black hole solution inside the horizon. By studying this model we can obtain information about the black hole singularity and about the dynamics across the point r = 0. We studied this space-time in ADM variables in two previous papers where we showed that the classical black hole singularity disappears in quantum theory. In this work we study the same model in Ashtekar variables and we obtain a regular space-time inside the horizon region and that the dynamics can be extend further the classical singularity. Introduction In this work we study the space-time inside the horizon of a Schwazschild black hole using the connection variables introduced by Ashtekar [1]. We start the quantization program from the complete theory of “loop quantum gravity” [2] and we reduce the theory to consider an homogeneous but anisotropic minisuperspace model. We studied the singularity problem in two previous papers [3] and [4]. There we used the same non Schrödinger quantization procedure as in the work of V. Husain and O. Winkler on quantum cosmology. This formalism was introduced by Halvorson [13] and also by A. Ashtekar, S. Fairhust and J. Willis [10]. In the previous papers we used ADM formulation of general relativity and we obtained that the quantum minisuperspace model for the black hole is singularity free; in fact in the first very simple model [3] we showed that the operator 1/r and so the curvature invariant Rμνρσ Rμνρσ = 48MGN/r 6 are not divergent in the quantum theory. In the second paper [4] we considered a general two dimensional minisuperspace with space section of topology R × S (the Kantowski-Sachs spacetime [6]) and we obtained that the inverse volume operator and the curvature invariant are singularity free in r = 0 in quantum gravity. In both models we have that at the quantum level the Hamiltonian constraint acts like a difference operator. A quantum extension on the other side of the classical singularity is straightforward. We acknowledge Ioannis Raptis’ s work on a mathematical approach to the singularity problem in general relativity [7]. The paper is organized as follows : in the first section we introduce the classical theory, in particular we recall the invariant connection 1-form for the Kantowski-Sachs sapec-time. We introduce the holonomies and the form of the Hamiltonian constraint and the inverse volume operator in terms of such holonomies. We also express the quantity 1/r in terms of holonomies. This quantity is 1 interesting because it is connected with the Schwarzschild curvature invariant Rμνρσ Rμνρσ in the metric formulation of general relativity. In the second section we quantize the system. In particular we calculate explicitly the spectrum of the inverse volume operator, the spectrum of the operator 1/r and the solution of the Hamiltonian constraint in the dual to the kinematical Hilbert space. 1 Classical theory In this section first of all we summarize the fundamental Hamiltonian formulation of “loop quantum gravity” [2] then we recall the formulation of the Kantowski-Sachs space-time in Ashtekar variables [9]. At this point we define the holonomies for the connection and we rewrite the classical Hamiltonian constraint and the inverse volume operator in terms of holonomies. 1.1 Loop quantum gravity preliminary The classical starting point of loop quantum gravity (LQG) is the Hamiltonian formulation of general relativity. Historically in the ADM Hamiltonian formulation of the Einstein theory the fundamental variables are the three-metric qab of the spatial section Σ of a foliation of the 4−dmanifold M ∼= R×Σ and the extrinsic curvature Kab. In LQG the fundamental variables are the Ashtekar variables: they consist of an SU(2) connection Aa and the electric field E a i , where a, b, c, · · · = 1, 2, 3 are tensorial indexes on the spatial section and i, j, k, · · · = 1, 2, 3 are indexes in the su(2) algebra. The density weighted triad E i is define in terms of the triad e i a by the relation E a i = 1 2ǫ abc ǫijk e j b e k c . And the metric can be expressed in terms of the triad by qab = e i a e j b δij . We can express also the metric in terms of the weighted triad as q q = E i E b j δ ij , q = √ det(qab). (1) The relation between the variables (Aa, E a i ) and the ADM variables (qab,Kab) is Aa = Γ i a + γ KabE b jδ ij (2) where γ is the Immirzi parameter and Γa is the spin connection, being the solution of Cartan’s equation: ∂[ae i b] + ǫ i jk Γ j [ae k b] = 0. The action in the new variables is S = 1 k ∫