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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 flat FRW model coupled to the massless scalar field according to the improved, background scale independent version of Ashtekar, Pawlowski and Singh is considered. The core of the theory is addressed directly: the APS construction of the quantum Hamiltonian is analyzed under the assumption that cosmological constant $\Lambda\le 0$. We prove the essential self-adjointness of the operator whose square-root defines the quantum Hamiltonian operator and therefore, provide the explicit definition. If $\Lambda 0$ being some constants) plus a trace class operator.

The flat FRW model in LQC: the self-adjointness

We derive all the axi-symmetric, vacuum and electrovac extremal isolated horizons. It turns out that for every horizon in this class, the induced metric tensor, the rotation 1-form potential and the pullback of the electromagnetic field necessarily coincide with those induced by the monopolar, extremal Kerr-Newman solution on the event horizon. We also discuss the general case of a symmetric, extremal isolated horizon. In particular, we analyze the case of a two-dimensional symmetry group generated by two null vector fields. Its relevance to the classification of all the symmetric isolated horizons, including the non-extremal once, is explained.

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Extremal Isolated Horizons: A Local Uniqueness Theorem

Isolated Horizons and their Applications

We characterize a general solution to the vacuum Einstein equations which admits isolated horizons. We show that it is a nonlinear superposition (in a precise sense) of the Schwarzschild metric with a certain free data set propagating tangentially to the horizon. This proves Ashtekar's conjecture about the structure of spacetime near the isolated horizon. The same superposition method applied to the Kerr metric gives another class of vacuum solutions admitting isolated horizons. More generally, a vacuum spacetime admitting any null, non-expanding, shear- free surface is characterized. The results are applied to show that, generically, the non-rotating isolated horizon does not admit a Killing vector field and a spacetime is not spherically symmetric near a symmetric horizon. PACS numbers: 0470B, 0420 The quantum geometry considerations applied to black hole entropy (1) led Ashtekar et al to a new approach to black hole mechanics. The idea is to consider a null surface which locally has the properties of the Schwarzschild horizon, but is not necessarily infinitely extendible, so the spacetime metric in a neighbourhood is not necessarily that of Schwarzschild. Such a surface was called a non-rotating isolated horizon (NRIH). The number of degrees of freedom describing a spacetime admitting a NRIH is much larger than that describing a static black hole (see below). In a series of works the laws of the black hole thermodynamics and mechanics were extended to this case (2, 3). In this letter we completely characterize a general solution to the Einstein vacuum equations which admits an isolated horizon and, in particular, a NRIH. For that purpose, we use Friedrich's characteristic Cauchy problem defined on null surfaces (4). (The idea of constructing solutions to Einstein's equations starting with data defined on a null surface was first formulated by Newman (5).) The null Cauchy problem formulation gives rise to our superposition method: given a local solution to the Einstein vacuum equations and the data it defines on a null surface, a new solution can be constructed from the null surface data and certain new data freely defined on a transversal null surface. We show that a general solution which admits a NRIH is given by the superposition of the data defined by the Schwarzschild metric on the horizon and the data defined freely on a transversal null surface. This result is then applied to prove that a generic NRIH does not admit a Killing vector field. Even though there are vector fields defined on the horizon which Lie annihilate the metric tensor (7), none of them, generically, can be extended to a neighbourhood. The statement concerns the null vector fields as well as the spacelike vectors generating symmetries of the internal geometry induced on the two-dimensional cross sections of the NRIH.

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Spacetimes admitting isolated horizons

Do the SU(2) intertwiners parametrize the space of the Engle, Pereira, Rovelli, Livine (EPRL) solutions to the simplicity constraint? What is the complete form of the partition function written in terms of this parametrization? We prove that the EPRL map is injective in the general n-valent vertex case for the Barbero–Immirzi parameter less than 1. We find, however, that the EPRL map is not isometric. In the consequence, a partition function can be defined either using the EPRL intertwiners Hilbert product or the SU(2) intertwiners Hilbert product. We use the EPRL one and derive a new, complete formula for the partition function. Next, we view it in terms of the SU(2) intertwiners. The result, however, goes beyond the SU(2) spin-foam models' framework and the original EPRL proposal.

https://iopscience.iop.org/article/10.1088/0264-9381/27/16/165020/pdf

Jerzy Lewandowski

Papers

The flat FRW model in LQC: the self-adjointness

Extremal Isolated Horizons: A Local Uniqueness Theorem

Isolated Horizons and their Applications

Spacetimes admitting isolated horizons

The EPRL intertwiners and corrected partition function