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

The problem of describing the quantum behavior of gravity, and thus understanding quantum spacetime, is still open. Loop quantum gravity is a well-developed approach to this problem. It is a mathematically well-defined background-independent quantization of general relativity, with its conventional matter couplings. Today research in loop quantum gravity forms a vast area, ranging from mathematical foundations to physical applications. Among the most significant results obtained so far are: (i) The computation of the spectra of geometrical quantities such as area and volume, which yield tentative quantitative predictions for Planck-scale physics. (ii) A physical picture of the microstructure of quantum spacetime, characterized by Planck-scale discreteness. Discreteness emerges as a standard quantum effect from the discrete spectra, and provides a mathematical realization of Wheeler’s “spacetime foam” intuition. (iii) Control of spacetime singularities, such as those in the interior of black holes and the cosmological one. This, in particular, has opened up the possibility of a theoretical investigation into the very early universe and the spacetime regions beyond the Big Bang. (iv) A derivation of the Bekenstein-Hawking black-hole entropy. (v) Low-energy calculations, yielding n-point functions well defined in a background-independent context. The theory is at the roots of, or strictly related to, a number of formalisms that have been developed for describing background-independent quantum field theory, such as spin foams, group field theory, causal spin networks, and others. I give here a general overview of ideas, techniques, results and open problems of this candidate theory of quantum gravity, and a guide to the relevant literature.

Loop Quantum Gravity

Applications of Riemannian quantum geometry to cosmology have had notable successes. In particular, the fundamental discreteness underlying quantum geometry has led to a natural resolution of the big bang singularity. However, the precise mathematical structure underlying loop quantum cosmology and the sense in which it implements the full quantization program in a symmetry reduced model has not been made explicit. The purpose of this paper is to address these issues, thereby providing a firmer mathematical and conceptual foundation to the subject.

Mathematical structure of loop quantum cosmology

The theory of gravitational lensing is reviewed from a spacetime perspective, without quasi-Newtonian approximations More precisely, the review covers all aspects of gravitational lensing where light propagation is described in terms of lightlike geodesics of a metric of Lorentzian signature It includes the basic equations and the relevant techniques for calculating the position, the shape, and the brightness of images in an arbitrary general-relativistic spacetime It also includes general theorems on the classification of caustics, on criteria for multiple imaging, and on the possible number of images The general results are illustrated with examples of spacetimes where the lensing features can be explicitly calculated, including the Schwarzschild spacetime, the Kerr spacetime, the spacetime of a straight string, plane gravitational waves, and others

https://link.springer.com/content/pdf/10.12942%2Flrr-2004-9.pdf

Gravitational Lensing from a Spacetime Perspective

I used to keep the entire spin networks literature in a small folder on my shelf. The recent explosion of interest in the subject has made this impossible; more is probably now written every week than the contents of my folder in 1991. This brings the need to consolidate the subject and collect together results and formulae in one coherent place. Kauffman and Lins' book is a splendid contribution, as the well thumbed reference copy can testify. The renewed interest in the subject came from two areas. Firstly there was the q-revolution which spawned the representation theory of the q-deformation of SU(2) and its application to 3-manifold theory by combinatorial techniques. Secondly, spin networks came to be applied in quantum field theories in places where discrete methods began to seem more fundamental. There are many overlaps between these, and many of the tools developed for specific 3-manifold purposes have, in fact, a much wider application. Perhaps recognizing this, the authors made the first half of the book a review of the recoupling theory of SU(2), leaving the application to 3-manifolds strictly to the second part. The development of the recoupling theory makes no reference to SU(2) itself, as the invariant theory can be developed entirely by the means of diagrams. This is a continuation of the theme of the spin networks and strand networks developed by Penrose, extended to the q-deformed case. The use of diagrams was originally conceived as an aid to understanding algebraic formulae; now the connection runs deeper. We now understand that a planar diagram for the recoupling of representations of SU(2) is an indispensible part of the theory. Without it, the fine detail of the formulae become impossible to keep track of in a coherent way. Thus basing the theory on diagrams is probably the most useful and certainly the quickest way of grasping the subject. The book starts with the Jones - Wenzl projectors, which are the diagrammatic version of the projection onto irreducible representations of SU(2). Then it quickly moves on to calculating the value of q-spin networks with these irreducible representations on the edges. Particular networks are calculated in detail, namely the diagram and the tetrahedron. This then allows the properties of the 6j symbol to be developed. Having all the formulae in one place, and they have proved remarkably reliable, makes this book invaluable, as the literature is plagued by slightly different versions of the same formulae caused by differences in convention. One unfortunate departure from the conventions established in the physics literature is that the curly bracket symbol for the 6j symbol has been assigned a different normalization, which makes the formulae related to associativity simple but not symmetrical under permutations of the tetrahedron. Most of the recoupling theory is developed for general q, and the particular case of q as a root of unity. However, the reader interested only in classical spin networks need not be lost; one can put q = 1 everywhere and still enjoy reading the book. The calculations of q-spin network evaluations are long but tantalizingly similar to the formulae for the classical q = 1 case. General formulae are known only for the classical case. Chapter 8 describes the chromatic method for q = 1 due to Penrose, leaving the generalization to the q-deformation as an unsolved problem. The combinatorics of the chromatic method are somewhat delicate to describe, and the informal method of presentation, which serves the book well elsewhere, is stretched to the limit at this point. The second part of the book is much more specialized. There are reviews of the Turaev - Viro invariant, the Witten - Reshetikhin - Turaev invariant and the `shadow world' description of graph invariants due to Kirillov and Resheatikhin. All of these have been condensed and simplified from their original presentations, but some of the theory, for example, of the topology, has been omitted. The culmination of the book is a series of tables of the WRT invariants for some 3-manifolds, based on the formulae given in the first part of the book.

http://iopscience.iop.org/0264-9381/13/12/024/pdf/0264-9381_13_12_024.pdf

Temperley - Lieb Recoupling Theory and Invariants of 3-Manifolds

(November 29, 1999)We propose a deﬁnition of an exact lens equation without reference to a background spacetime,and construct the exact lens equation explicitly in the case of Schwarzschild spacetime For theSchwarzschild case, we give exact expressions for the angular-diameter distance to the sources aswell as for the magniﬁcation factor and time of arrival of the images We compare the exactlens equation with the standard lens equation, derived under the thin-lens-weak-ﬁeld assumption(where the light rays are geodesics of the background with sharp bending in the lens plane, andthe gravitational ﬁeld is weak), and verify the fact that the standard weak-ﬁeld thin-lens equationis inadequate at small impact parameter We show that the second-order correction to the weak-ﬁeld thin-lens equation is inaccurate as well Finally, we compare the exact lens equation with therecently proposed strong-ﬁeld thin-lens equation, obtained under the assumption of straight pathsbut without the small angle approximation, ie, with allowed large bending angles We show thatthe strong-ﬁeld thin-lens equation is remarkably accurate, even for lightrays that take several turnsaround the lens before reaching the observerI INTRODUCTION

Spacetime perspective of Schwarzschild lensing

We compute the complete spectrum of the area operator in the loop representation of quantum gravity, using recoupling theory. This result extends previous derivations, which did not include the `degenerate' sector, and agrees with the recently computed spectrum of the connection-representation area operator.

https://iopscience.iop.org/article/10.1088/0264-9381/13/11/008/pdf

The complete spectrum of the area from recoupling theory in loop quantum gravity

We find a one-parameter family of variables which recast the 3+1 Einstein equations into first-order symmetric-hyperbolic form for any fixed choice of gauge. Hyperbolicity considerations lead us to a redefinition of the lapse in terms of an arbitrary factor times a power of the determinant of the 3-metric; under certain assumptions, the exponent can be chosen arbitrarily, but positive, with no implication of gauge-fixing.

First-order symmetric hyperbolic Einstein equations with arbitrary fixed gauge.

We establish rigorous results about the Newtonian limit of general relativity by applying to it the theory of different time scales for non-linear partial differential equations as developed in [4, 1, 8]. Roughly speaking, we obtain a priori estimates for solutions to the Einstein's equations, an intermediate, but fundamental, step to show that given a Newtonian solution there exist continuous one-parameter families of solutions to the full Einstein's equations — the parameter being the inverse of the speed of light — which for a finite amount of time are close to the Newtonian solution. These one-parameter families are chosen via aninitialization procedure applied to the initial data for the general relativistic solutions. This procedure allows one to choose the initial data in such a way as to obtain a relativistic solution close to the Newtonian solution in any a priori given Sobolev norm. In some intuitive sense these relativistic solutions, by being close to the Newtonian one, have little extra radiation content (although, actually, this should be so only in the case of the characteristic initial data formulation along future directed light cones). Our results are local, in the sense that they do not include the treatment of asymptotic regions; global results are admittedly very important — in particular they would say how differentiable the solutions are with respect to the parameter — but their treatment would involve the handling of tools even more technical than the ones used here. On the other hand, this local theory is all that is needed for most problems of practical numerical computation.

/pdf/on-the-newtonian-limit-of-general-relativity-3wndrvr3w4.pdf

On the Newtonian limit of general relativity

We reformulate the Einstein equations as equations for families of surfaces on a four‐manifold. These surfaces eventually become characteristic surfaces for an Einstein metric (with or without sources). In particular they are formulated in terms of two functions on R4×S2, i.e., the sphere bundle over space–time, one of the functions playing the role of a conformal factor for a family of associated conformal metrics, the other function describing an S2’s worth of surfaces at each space–time point. It is from these families of surfaces themselves that the conformal metric, conformal to an Einstein metric, is constructed; the conformal factor turns them into Einstein metrics. The surfaces are null surfaces with respect to this metric.

http://cds.cern.ch/record/276826/files/9502028.pdf

Simonetta Frittelli

Papers

Spacetime perspective of Schwarzschild lensing

The complete spectrum of the area from recoupling theory in loop quantum gravity

First-order symmetric hyperbolic Einstein equations with arbitrary fixed gauge.

On the Newtonian limit of general relativity

GR via characteristic surfaces