Can non-commutativity resolve the big-bang singularity?

Abstract

A possible way to resolve the singularities of general relativity is proposed based on the assumption that the description of space-time using commuting coordinates is not valid above a certain fundamental scale. Beyond that scale it is assumed that the space-time has noncommutative structure leading in turn to a resolution of the singularity. As a first attempt towards realizing the above programme a modification of the Kasner metric is constructed which is commutative only at large time scales. At small time scales, near the singularity, the commutation relations among the space coordinates diverge. We interpret this result as meaning that the singularity has been completely delocalized.

Gravity and the structure of noncommutative algebras

“…The essential ingredients in the definition of the map are the Leibniz rules and the assumption (2.1) on the structure of the differential calculus. Although there have been found [16,17,4,6] numerous particular examples, there is not yet a systematic discussion of either the range or kernel of the map. We have here to a certain extent alleviated this, but only in the context of perturbation theory around a vacuum and even then, only in the case of a high-frequency wave.…”

“…We would also like to know how many extensions there are and what their properties. There have been examples constructed [4,5,6] more-or-less ad hoc; we give here a more systematic analysis by restricting our considerations to the 'semi-classical' theory, retaining only contributions of first-order in the noncommutativity parameter. As a working hypothesis we shall suppose that there is one physical property, which at large scales manifests itself as gravity and at small scales as noncommutativity.…”

Noncommutative cosmologies

“…One noncommutative generalization of the Kasner space, that is, the frame and the momentum algebra, was introduced in [8]. Here we just briefly recall these results and derive the corresponding position algebra.…”

Non-singular Brans–Dicke collapse in deformed phase space

“…Cosmological scenarios within the framework of non-commutative geometry provide us the formulation of semiclassical approximations of quantum gravity allowing to deal with the cosmological constant problem [54,55]. More interestingly, non-commutativity can provide a reasonable groundwork for non-singular cosmological scenarios where big-bang/crunch singularities are dissolved [56]. In the context of Kantowski-Sachs cosmological model, non-commutativity has been introduced into the classical phase space and classical non-commutative equations of motion have established [57].…”