Singularity avoidance in Bianchi I quantum cosmology

Abstract

We extend recent discussions of singularity avoidance in quantum gravity from isotropic to anisotropic cosmological models. The investigation is done in the framework of quantum geometrodynamics (Wheeler-DeWitt equation). We formulate criteria of singularity avoidance for general Bianchi class A models and give explicit and detailed results for Bianchi I models with and without matter. The singularities in these cases are big bang and big rip. We find that the classical singularities can generally be avoided in these models. I. INTRODUCTIONA major issue in any quantum theory of gravity is the fate of the classical singularities. So far, such a theory is not available in final form, although various approaches exist in which this question can be sensibly addressed [1,2]. It is clear that such an investigation cannot yet be done at the level of mathematical rigor comparable to the singularity theorems in the classical theory (see e.g. [3]). Nevertheless, focusing on concrete approaches and concrete models, one can state criteria of singularity avoidance and discuss their implementation. This is what we shall do here.We restrict our analysis of singularity avoidance to quantum geometrodynamics, with the Wheeler-DeWitt equation as its central equation [1]. Although this may not be the most fundamental level of quantum gravity, it is sufficient for addressing the issue of singularity avoidance. Quantum geometrodynamics follows directly from general relativity by rewriting the Einstein equations in Hamilton-Jacobi form and formulating quantum equations that yield the Hamilton-Jacobi equations in the semiclassical (WKB) limit. It thus makes as much sense to addressing singularity avoidance here than it does to addressing it in quantum mechanics at the level of the Schrödinger equation. Singularity avoidance has also been discussed in loop quantum gravity [2,4,5], with various results, but we will not consider this here.Singularity avoidance was already addressed by DeWitt in his pioneering paper on canonical quantum gravity [6]. He suggested to impose the condition Ψ → 0 for the quantum-gravitational wave functional Ψ when approaching the region of a classical singularity. The wave functional is effectively defined on the configuration space of all threedimensional geometries, also called superspace [1,7]. The "DeWitt criterion" of vanishing wave function then means that Ψ must approach zero when approaching a singular three-geometry (which itself is not part of superspace, but can be envisaged as its boundary). It is important to emphasize that this criterion is a sufficient but not a necessary one: singularities can be avoided for non-vanishing or even diverging Ψ (recall the ground state solution of the Dirac equation for the hydrogen atom, which diverges).DeWitt had in mind cosmological singularities such as big bang or big crunch. The DeWitt criterion applies, of course, also to the singularities that classically arise from gravitational collapse. In simple models of quantum geometrodynamics, their avoidance can be...

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“…will be retrieved, with, e.g., {q k } ≡ a, φ j . Equation 25is aiming at matching minisuperspace covariance (see [22]). Moreover, the Lévy index was coined within an Euclidean setting whereas a (Lorentzian) minisuperspace may now require and 'mix' different α's, per minisuperspace variable, q i , q j , allowing for the several path components, now in the configuration space, parametrized by an affine term, e.g., τ.…”

“…Extending the framework of Lévy paths and fractional (quantum) mechanics to relativistic settings, it could impose to adequately import the results from the explorations in [23] and alike. However, if bearing intrinsic minisuperspace covariance [22], would then unitarity be regained? Let us just mention that non-unitarity also emerges in discussions about semiclassical quantum gravity [19], namely from a Wheeler-DeWitt expansion towards obtaining a Schrödinger equation (as well a WKB-like "many-fingered' functional time) in the presence of quantum gravitational corrections, whereby that covariance is lost.…”

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