Black hole interior from loop quantum gravity

In this paper we calculate modifications to the Schwarzschild solution by using a semiclassical analysis of loop quantum black hole. We obtain a metric inside the event horizon that coincides with the Schwarzschild solution near the horizon but that is substantially different at the Planck scale. In particular we obtain a bounce of the S sphere for a minimum value of the radius and that it is possible to have another event horizon close to the r = 0 point. Introduction Quantum gravity, the theory that wants reconcile general relativity and quantum mechanics, is one of major problem in theoretical physics today. General relativity tells as that because also the space-time is dynamical, it is not possible to study other interactions on a fixed background. The background itself is a dynamical field. Among the quantum gravity theories, the theory called “loop quantum gravity” [1] is the most widespread nowadays. This is one of the non perturbative and background independent approaches to quantum gravity (another non perturbative approach to quantum gravity is called “asymptotic safety quantum gravity” [2]). In the last years the applications of loop quantum gravity ideas to minisuperspace models lead to some interesting results to solve the problem of space-like singularity in quantum gravity. As shown in cosmology [3], [4] and recently in black hole physics [5], [6], [7], [8] it is possible to solve the cosmological singularity problem and the black hole singularity problem by using the tools and ideas developed in full loop quantum gravity theory. In the other well known approach to quantum gravity, the called “asymptotic safety quantum gravity”, authors [9], using the GN running coupling constant obtained in “asymptotic safety quantum gravity”, have showed that non perturbative quantum gravity effects give a much less singular Schwarzschild metric and that for particular values of the black hole mass it is possible to have the formation of another event horizon. In this paper we study the space-time inside the event horizon at the semiclassical level using a constant polymeric parameter δ (see the paper [10] for an analysis of the black hole interior using a non constant polymeric parameter). We consider the Hamiltonian constraint obtained in [8]; in particular we study the Hamiltonian constraint introduced in the first paper of reference [8], where the authors have taken the general version of the constraint for real values of the Immirzi parameter γ. This paper is organized as follows. In the first section we briefly recall the Schwarzschild solution inside the event horizon (r < 2MGN ) of [8]. In the second section we introduce the Hamiltonian constraint in terms of holonomies and then the relative trigonometric form solving the Hamilton equations of motion. In the third section we give the metric form of the solution and we discuss the new physics suggested by loop quantum gravity. 1 1 Schwarzschild solution inside the event horizon in Ashtekar variables We recall the classical Schwarzschild solution inside the event horizon [8]. For the homogeneous but non isotropic Kantowski-Sachs space-time the Ashtekar’s connection and density triad are (after the fixing of a residual global SU(2) gauge symmetry on the spherically reduced phase space [8]) A = cτ3dx+ bτ2dθ − bτ1 sin θdφ+ τ3 cos θdφ, E = pcτ3 sin θ ∂ ∂x + pbτ2 sin θ ∂ ∂θ − pbτ1 ∂ ∂φ . (1) The components variables in the phase space can be read From the symmetric reduced connection and density triad we can read the components variables in the phase space: (b, pb), (c, pc). The Poisson algebra is: {c, pc} = 2γGN , {b, pb} = γGN . Following papers [8] we recall that the classical Hamiltonian constraint in terms of the components variables is CH = − 1 2γGN [ (b + γ) pb b + 2c pc ]