General Relativity & Quantum Cosmology

In this talk we introduce loop quantum gravity and we apply the theory to the black hole singularity problem. The Schwarzschild black hole solution inside the event horizon coincides with the Kantowski-Sachs space-time and we can study a simple spherically symmetric mini-superspace model. We show the classical black hole singularity is controlled by the quantum theory and the space-time can be dynamically extended beyond the classical singularity. We consider a semiclassical analysis of the black hole in LQG and we focus our attention on the space-time structure. The semiclassical solution is regular everywhere and similar to the Reissner-Nordstrom metric. The LQG black hole metric interpolates between two asymptotically flat regions, the 'r=infinity' region and the 'r=0' region. The metric is self-dual in the sense it is invariant under the symmetry (r ->1/r) which relates small and large distances. We study the thermodynamics of the semiclassical solution.

I present a viewpoint on black hole thermodynamics according to which the entropy: derives from horizon 'degrees of freedom''; is finite because the deep structure of spacetime is discrete; is ``objective'' thanks to the distinguished coarse graining provided by the horizon; and obeys the second law of thermodynamics precisely because the effective dynamics of the exterior region is not unitary.

In this talk I will describe a topos formulation of consistent histories obtained using the topos reformulation of standard quantum mechanics put forward by Doering and Isham. Such a reformulation leads to a novel type of logic with which to represent propositions. In the first part of the talk I will introduce the topos reformulation of quantum mechanics. I will then explain how such a reformulation can be extended so as to include temporally-ordered collection of propositions as opposed to single time propositions. Finally I will show how such an extension will lead to the possibility of assigning truth values to temporal propositions.

The handling of the constraints on initial data is a major issue in most canonical formulations of general relativity. Since the 1960s unconstrained initial data for GR that living on null hypersurfaces has been known, but no canoncial formulation based on these data was developed due to conceptual and technical difficulties. I will explain how these dificulties have been overcome and outline the resulting canonical framework. I will also explain how this might be the ideal setting to attempt a proof of the Bousso entropy bound, or to incorporate the associated holographic principle in a quantization of gravity.

New spin foam models for gravity have been recently proposed to deal with the shortcomings of the Barrett-Crane model. In particular, they draw a closer connection between the Loop Quantum Gravity and the Spin Foam approaches to non perturbative quantum gravity. In this talk, I will present the construction for the case of Lorentzian signature and finite Immirzi parameter. An area operator can be defined and its spectrum agrees with the one defined in LQG. Finally, the amplitude is shown to be finite after a suitable regularization.

We give an overview of what we have called the \'LQG spinfoam models,\' that provide a spinfoam dynamics for LQG, for arbitrary values of the Barbero-Immirzi parameter in both Lorentzian and Euclidean signatures. The key motivation behind these models was to modify the Barrett-Crane model, by handling more carefully certain constraints, called simplicity constraints, which become second class in the quantum theory. As a result, the kinematics of the models exactly match those of LQG. The goal of this talk is to provide a broad picture of the significance of these models and their basic ideas.

I will review the present status of the black hole entropy computation in Loop Quantum Gravity within the isolated horizon framework. Starting from the recently discovered discretization effect, I will give an overview of the subsequent developments that have been obtained motivated by it. Through this further analysis of the problem I will present some new related results and the promising new open windows that they give rise to.

While analogue models for gravity have so far provided some insights on the kinematical aspects of general relativity, the emergence of gravitational dynamics is still unclear. In this talk I will present two models which aim at filling this gap. In the first one a BEC model is considered, to uncover the gravitational dynamics hidden in these systems. In particular, the emergence of a modified Newtonian dynamics and of the cosmological constant will be discussed. A second model will be used to show how Nordstrom theory of gravity emerges from a system defined in Minkowski spacetime, showing how background independent theories of gravity could emerge from background dependent models.

Using 2-dimensional CGHS black holes, I will argue that information is not lost in the Hawking evaporation because the quantum space-time is significantly larger than the classical one. I will begin with a discussion of the conceptual underpinnings of problem and then introduce a general, non-perturbative framework to describe quantum CGHS black holes. I will show that the Hawking effect emerges from it in the first approximation. Finally, I will introduce a mean field approximation to argue that, when the back reaction is included, future null infinity is `long enough\' to capture full information contained in pure states at past null infinity and that the S-matrix is unitary. There are no macroscopic remnants.

According to general relativity, space-time ends at singularities and classical physics just stops. In particular, the big bang is regarded as The Beginning. However, general relativity is incomplete because it ignores quantum effects. Through simple models, I will illustrate how the quantum nature of space-time geometry resolves the big bang singularity. Quantum physics does not stop there. Indeed, quantum space-times can be vastly larger than what general relativity had us believe, with unforeseen physical effects in the deep Planck regime.

It has long been thought that theories based on equations of motion possessing derivatives of order higher than second are not unitary. Specifically, they are thought to possess unphysical ghost states with negative norm. However, it turns out that the appropriate Hilbert space for such theories had not been correctly constructed, and when the theory is formulated properly [Bender and Mannheim, PRL 100, 110402 (2008). (arXiv:0706.0207 [hep-th]] there are no ghost states at all and time evolution is fully unitary. Unitarity can be established for theories based on both second and fourth order derivatives, and for theories based on fourth order derivatives alone. In this latter case the Hamiltonian is a non-diagonalizable, Jordan-block operator which possesses fewer eigenstates than eigenvalues. Despite the lack of completeness of the energy eigenstates, a consistent, unitary quantum mechanics for the theory can still be formulated [Bender and Mannheim, PRD 78, 025022 (2008). (arXiv:0807.2607 [hep-th]).] The implications of these results for the construction of a consistent theory of quantum gravity in four spacetime dimensions will be briefly discussed.