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Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics

The goal of this review is to present an introduction to loop quantum gravity—a background-independent, non-perturbative approach to the problem of unification of general relativity and quantum physics, based on a quantum theory of geometry. Our presentation is pedagogical. Thus, in addition to providing a bird's eye view of the present status of the subject, the review should also serve as a vehicle to enter the field and explore it in detail. To aid non-experts, very little is assumed beyond elements of general relativity, gauge theories and quantum field theory. While the review is essentially self-contained, the emphasis is on communicating the underlying ideas and the significance of results rather than on presenting systematic derivations and detailed proofs. (These can be found in the listed references.) The subject can be approached in different ways. We have chosen one which is deeply rooted in well-established physics and also has sufficient mathematical precision to ensure that there are no hidden infinities. In order to keep the review to a reasonable size, and to avoid overwhelming non-experts, we have had to leave out several interesting topics, results and viewpoints; this is meant to be an introduction to the subject rather than an exhaustive review of it.

Background independent quantum gravity: A Status report

An improved Hamiltonian constraint operator is introduced in loop quantum cosmology. Quantum dynamics of the spatially flat, isotropic model with a massless scalar field is then studied in detail using analytical and numerical methods. The scalar field continues to serve as ''emergent time'', the big bang is again replaced by a quantum bounce, and quantum evolution remains deterministic across the deep Planck regime. However, while with the Hamiltonian constraint used so far in loop quantum cosmology the quantum bounce can occur even at low matter densities, with the new Hamiltonian constraint it occurs only at a Planck-scale density. Thus, the new quantum dynamics retains the attractive features of current evolutions in loop quantum cosmology but, at the same time, cures their main weakness.

Quantum Nature of the Big Bang: Improved dynamics

Loop quantum cosmology (LQC) is the result of applying principles of loop quantum gravity (LQG) to cosmological settings. The distinguishing feature of LQC is the prominent role played by the quantum geometry effects of LQG. In particular, quantum geometry creates a brand new repulsive force which is totally negligible at low spacetime curvature but rises very rapidly in the Planck regime, overwhelming the classical gravitational attraction. In cosmological models, while Einstein's equations hold to an excellent degree of approximation at low curvature, they undergo major modifications in the Planck regime: for matter satisfying the usual energy conditions, any time a curvature invariant grows to the Planck scale, quantum geometry effects dilute it, thereby resolving singularities of general relativity. Quantum geometry corrections become more sophisticated as the models become richer. In particular, in anisotropic models, there are significant changes in the dynamics of shear potentials which tame their singular behavior in striking contrast to older results on anisotropies in bouncing models. Once singularities are resolved, the conceptual paradigm of cosmology changes and one has to revisit many of the standard issues—e.g. the 'horizon problem'—from a new perspective. Such conceptual issues as well as potential observational consequences of the new Planck scale physics are being explored, especially within the inflationary paradigm. These considerations have given rise to a burst of activity in LQC in recent years, with contributions from quantum gravity experts, mathematical physicists and cosmologists. The goal of this review is to provide an overview of the current state of the art in LQC for three sets of audiences: young researchers interested in entering this area; the quantum gravity community in general and cosmologists who wish to apply LQC to probe modifications in the standard paradigm of the early universe. In this review, effort has been made to streamline the material so that each of these communities can read only the sections they are most interested in, without loss of continuity.

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Loop Quantum Cosmology: A Status Report

The Petrov type D equation on genus $>0$ sections of isolated horizons

Generic black holes in vacuum-de Sitter / Anti-de Sitter spacetimes are studied in quasi-local framework, where the relevant properties are captured in the intrinsic geometry of the null surface (the horizon). Imposing the quasi-local notion of stationarity (null symmetry of the metric up to second order at the horizon only) we perform the complete classification of all the so called special Petrov types of these surfaces defined by the properties (structure of principal null direction) of the Weyl tensor at the surface. The only possible types are: II, D and O. In particular all the geometries of type O are identified. The condition distinguishing type D horizons, taking the form of a second order differential equation on certain complex invariant constructed from the Gaussian curvature and the rotation scalar, is shown to be an integrability condition for the so called near horizon geometry equation. The emergence of the near horizon geometry in this context is equivalent to the hyper-suface orthogonality of both double principal null directions. We further formulate a no-hair theorem for the Petrov type D axisymmetric null surfaces of topologically spherical sections, showing that the space of solutions is uniquely parametrized by the horizon area and angular momentum.

The Petrov type D isolated null surfaces

A quantum representation of holonomies and exponentiated fluxes of a $U(1)$ gauge theory that contains the Pullin-Dittrich-Geiller (DG) vacuum is presented and discussed. Our quantization is performed manifestly in a continuum theory, without any discretization. The discretness emerges on the quantum level as a property of the spectrum of the quantum holonomy operators. The new type of a cylindrical consistency present in the DG approach, now follows easily and naturally. A generalization to the non--Abelian case seems possible.

Continuum approach to the BF vacuum: The U(1) case

(This short article is a continuation of a longer, review work, in the same volume of Proceedings, by Ashtekar, Marolf and Mour\~ao [gr-qc/9403042]. All the details and other results are to be found in joint papers of the author with Abhay Ashtekar.) The projective limit technics derived for spaces of connections are extended to a new framework which for the associated projective limit plays a role of the differential geometry. It provides us with powerfull technics for construction and studding various operators. In particular, we introduce the commutator algebra of `vector fields', define a divergence of a vector field and find for them a quantum representation. Among the vector fields, there are operators which we identify as regularised Rovelli-Smolin loop operators linear in momenta. Another class of operators which comes out naturally are Laplace operators.

Differential Geometry for the Space of Connections Modulo Gauge Transformations

A distinguished (invariant) Bondi-like coordinate system is defined in the spacetime neighbourhood of a non-expanding horizon of arbitrary dimension via geometry invariants of the horizon. With its use, the radial expansion of a spacetime metric about the horizon is provided and the free data needed to specify it up to given order are determined in spacetime dimension $4$. For the case of an electro-vacuum horizon in $4$-dimensional spacetime the necessary and sufficient conditions for the existence of a Killing field at its neighbourhood are identified as differential conditions on the horizon data and data on null surface transversal to the horizon.

Jerzy Lewandowski

Papers

The Petrov type D equation on genus $>0$ sections of isolated horizons

The Petrov type D isolated null surfaces

Continuum approach to the BF vacuum: The U(1) case

Differential Geometry for the Space of Connections Modulo Gauge Transformations

Neighbourhoods of Isolated Horizons and their stationarity