Singularities and Quantum Gravity

Abstract

Abstract.Although there is general agreement that a removal of classical gravitational singularities is not only a crucial conceptual test of any approach to quantum gravity but also a prerequisite for any fundamental theory, the precise criteria for non-singular behavior are often unclear or controversial. Often, only special types of singularities such as the curvature singularities found in isotropic cosmological models are discussed and it is far from clear what this implies for the very general singularities that arise according to the singularity theorems of general relativity. In these lectures we present an overview of the current status of singularities in classical and quantum gravity, starting with a review and interpretation of the classical singularity theorems. This suggests possible routes for quantum gravity to evade the devastating conclusion of the theorems by different means, including modified dynamics or modified geometrical structures underlying quantum gravity. The latter is most clearly present in canonical quantizations which are discussed in more detail. Finally, the results are used to propose a general scheme of singularity removal, quantum hyperbolicity, to show cases where it is realized and to derive intuitive semiclassical pictures of cosmological bounces. OVERVIEWPhysical theories are always idealizations without which the complexity of nature would be too great to fathom. Theoretical physics is, mostly very successfully, based on assumptions needed to formulate equations, find solutions and use them to describe, explain and further investigate physical phenomena.Sometimes, however, these assumptions may not be general enough for all purposes. When they are violated, the theory breaks down which mathematically appears as the development of singularities. An example is given by the use of continuous fields rather than discrete atomic structures in condensed matter physics. When fields vary too strongly on small length scales, such as in shock waves, singularities can occur in continuous field equations even though the basic, discrete physical description remains valid. Usually, deviations between solutions and observations increase before a mathematical singularity is reached. It is then clear that the approximate description can no longer be trusted beyond a certain point. But observations are not always available in such regimes where singularities are approached and an interpretation of mathematical singularities becomes more difficult. This is the case especially for gravity where observations of strong field regimes are lacking.Singularities in general relativity therefore play a special and dual role. First, the classical importance of singularities can be questioned since there are always assumptions behind special solutions or general theorems leading to singularities. But classical singularities in general relativity also provide an excellent chance to derive implications for the structure of space-time described by general relativity. When the theory breaks down, lesson...

Lattice refining loop quantum cosmology, anisotropic models, and stability

“…derived from the original coefficients (24). The structure of this difference equation is quite different from the original one: not only is it of higher order, but now only one value of the wave function appears at each level of ω, rather than combinations of values at different values of µ.…”

“…This feature of lattice refinements was not mimicked in the first formulations of loop quantum cosmology [20,21,14,22,15] since the main focus was to understand small-volume effects such as classical singularities [23,24]. In this context, lattice refinements appear irrelevant because only a few action steps of the Hamiltonian, rather than long evolution, are sufficient to probe a singularity.…”

Quantum gravitational collapse and Hawking radiation in2+1dimensions

“…Singularity avoidance has been claimed from loop quantum gravity [5] and has been shown to hold for collapsing dust shells in 3+1 dimensions [30]. The quantum collapse of shells in 2+1 dimensions was recently examined in [31] and the results suggest that at least a portion of the collapsing shell rebounds, reemerging from the horizon in a finite proper time, while the remaining portion of the shell is lost to Hawking radiation.…”

“…Ideally, one would like to have a quantum model that is able to predict what the final state of collapse will be, in particular whether or not and under what conditions the formation of a singularity may be avoided [1,5]. The model should also explain some features of the endstates.…”

Fermions in loop quantum cosmology and the role of parity

“…Loop quantum cosmology [4] provides a general mechanism for fundamental singularity resolution [5,6]. Commonly in quantum cosmology, when volume is used as an intrinsic measure of time, evolution must stop at the classical singularity where the volume vanishes.…”