Kirill Krasnov

A new spin foam model for 4D gravity

Starting from Plebanski formulation of gravity as a constrained BF theory we propose a new spin foam model for 4D Riemannian quantum gravity that generalizes the well-known Barrett–Crane model and resolves the inherent to it ultra-locality problem. The BF formulation of 4D gravity possesses two sectors: gravitational and topological ones. The model presented here is shown to give a quantization of the gravitational sector, and is dual to the recently proposed spin foam model of Engle et al which, we show, corresponds to the topological sector. Our methods allow us to introduce the Immirzi parameter into the framework of spin foam quantization. We generalize some of our considerations to the Lorentzian setting and obtain a new spin foam model in that context as well.

Barrett-Crane model from a Boulatov-Ooguri field theory over a homogeneous space

Boulatov and Ooguri have generalized the matrix models of 2d quantum gravity to 3d and 4d, in the form of field theories over group manifolds. We show that the Barrett–Crane quantum gravity model arises naturally from a theory of this type, but restricted to the homogeneous space S3=SO(4)/SO(3), as a term in its Feynman expansion. From such a perspective, 4d quantum space-time emerges as a Feynman graph, in the manner of the 2d matrix models. This formalism provides a precise meaning to the “sum over triangulations”, which is presumably necessary for a physical interpretation of a spin-foam model as a theory of gravity. In addition, this formalism leads us to introduce a natural alternative model, which might have relevance for quantum gravity.

Counting surface states in the loop quantum gravity

We adopt the point of view that (Riemannian) classical and (loopbased) quantum descriptions of geometry are macroand micro-descriptions in the usual statistical mechanical sense. This gives rise to the notion of geometrical entropy, which is defined as the logarithm of the number of different quantum states which correspond to one and the same classical geometry configuration (macro-state). We apply this idea to gravitational degrees of freedom induced on an arbitrarily chosen in space 2-dimensional surface. Considering an ‘ensemble’ of particularly simple quantum states, we show that the geometrical entropy S(A) corresponding to a macro-state specified by a total area A of the surface is proportional to the area S(A) = αA, with α being approximately equal to 1/16πl p . The result holds both for case of open and closed surfaces. We discuss briefly physical motivations for our choice of the ensemble of quantum states. ∗E-mail address: krasnov@phys.psu.edu

Minimal surfaces and particles in 3-manifolds

We consider 3-dimensional anti-de Sitter manifolds with conical singularities along time-like lines, which is what in the physics literature is known as manifolds with particles. We show that the space of such cone-manifolds is parametrized by the cotangent bundle of Teichmüller space, and that moreover such cone-manifolds have a canonical foliation by space-like surfaces. We extend these results to de Sitter and Minkowski cone-manifolds, as well as to some related “quasifuchsian” hyperbolic manifolds with conical singularities along infinite lines, in this later case under the condition that they contain a minimal surface with principal curvatures less than 1. In the hyperbolic case the space of such cone-manifolds turns out to be parametrized by an open subset in the cotangent bundle of Teichmüller space. For all settings, the symplectic form on the moduli space of 3-manifolds that comes from parameterization by the cotangent bundle of Teichmüller space is the same as the 3-dimensional gravity one. The proofs use minimal (or maximal, or CMC) surfaces, along with some results of Mess on AdS manifolds, which are recovered here in a different way, using differential-geometric methods and a result of Labourie on some mappings between hyperbolic surfaces, that allows an extension to cone-manifolds.

On Quantum Statistical Mechanics of a Schwarzschild Black Hole

Quantum theory of geometry, developed recently in the framework of non-perturbative quantum gravity, is used in an attempt to explain thermodynamics of Schwarzschild black holes on the basis of a microscopical (quantum) description of the system. We work with the formulation of thermodynamics in which the black hole is enclosed by a spherical surface B and a macroscopic state of the system is specified by two parameters: the area of the boundary surface and a quasilocal energy contained within it. To derive thermodynamical properties of the system from its microscopics we use the standard statistical mechanical method of Gibbs. Under a certain number of assumptions on the quantum behavior of the system, we find that its microscopic (quantum) states are described by states of quantum Chern-Simons theory defined by sets of points on B with spins attached. The level of the Chern-Simons theory turns out to be proportional to the horizon area of the black hole measured in Planck units. The statistical mechanical analysis turns out to be especially simple in the case when the entire interior of B is occupied by a black hole. We find in this case that the entropy contained within B, that is, the black hole entropy, is proportional to the horizon surface area.

Simple spin networks as Feynman graphs

We show how spin networks can be described and evaluated as Feynman integrals over an internal space. This description can, in particular, be applied to the so-called simple SO(D) spin networks that are of importance for higher-dimensional generalizations of loop quantum gravity. As an illustration of the power of the new formalism, we use it to obtain the asymptotics of an amplitude for the D simplex and show that its oscillatory part is given by the Regge action.

Plebański formulation of general relativity: a practical introduction

We give a pedagogical introduction into an old, but unfortunately not commonly known formulation of GR in terms of self-dual two-forms due to in particular Jerzy Plebański. Our presentation is rather explicit in that we show how the familiar textbook solutions: Schwarzschild, Volkoff–Oppenheimer, as well as those describing the Newtonian limit, a gravitational wave and the homogeneous isotropic Universe can be obtained within this formalism. Our description shows how Plebański formulation gives quite an economical alternative to the usual metric and frame-based schemes for deriving Einstein equations.

Deformations of the constraint algebra of Ashtekar's Hamiltonian formulation of general relativity.

In [1] Ashtekar described a new Hamiltonian formulation of general relativity (GR) in which the canonically conjugate phase-space variables are a densitized triad σ̃ and a (complexified) SU(2) connection Aa. Here spatial and “internal” indices are denoted by lower-case Latin letters from the beginning and from the middle of the alphabet, respectively. The constraints of general relativity take an amazingly simple form in this formulation:

Quantum Geometry and Black Holes

In his Ph.D. thesis, Bekenstein suggested that, for a black hole in equilibrium, a multiple of its surface gravity should be identified with its temperature and a multiple of the area of its event horizon should be identified with its thermodynamic entropy [1]. In this reasoning, he had to use not only general relativity but also quantum mechanics. Indeed, without recourse to the Planck’s constant, ℏ, the identification is impossible because even the physical dimensions do not match. Around the same time, Bardeen, Carter and Hawking derived laws governing the mechanics of black holes within classical general relativity [2]. These laws have a remarkable similarity with the fundamental laws of thermodynamics. However, the derivation makes no reference to quantum mechanics at all and, within classical general relativity, a relation between the two seems quite implausible: since nothing can come out of black holes and since their interiors are completely inaccessible to outside observers, it would seem that, physically, they can only have zero temperature and infinite entropy. Therefore the similarity was at first thought to be purely mathematical. This viewpoint changed dramatically with Hawking’s discovery of black hole evaporation in the following year [3]. Using an external potential approximation, in which the gravitational field is treated classically but matter fields are treated quantum mechanically, Hawking argued that black holes are not black after all! They radiate as if they are black bodies with a temperature equal to 1/27π times the surface gravity. One can therefore regard the similarity between the laws of black hole mechanics and those of thermodynamics as reflecting physical reality and argue that the entropy of a black hole is given by 1/4-th its area. Thus, Bekenstein’s insights turned out to be essentially correct (although the precise proportionality factors he had suggested were modified).

Non-metric gravity: II. Spherically symmetric solution, missing mass and redshifts of quasars

We continue the study of the non-metric theory of gravity introduced by Krasnov (2006 Preprint hep-th/0611182) and obtain its general spherically symmetric vacuum solution. It respects the analog of the Birkhoff theorem, i.e. the vacuum spherically symmetric solution is necessarily static. As in general relativity, the spherically symmetric solution is seen to describe a black hole. The exterior geometry is essentially the same as in the Schwarzschild case, with power-law corrections to the Newtonian potential. The behaviour inside the black-hole region is different from the Schwarzschild case in that the usual spacetime singularity gets replaced by a singular surface of a new type, where all basic fields of the theory remain finite but metric ceases to exist. The theory does not admit arbitrarily small black holes: for small objects, the curvature on the would-be horizon is so strong that non-metric modifications prevent the horizon from being formed. The theory allows for modifications of gravity of a very interesting nature. We discuss three physical effects, namely (i) correction to Newtons law in the neighborhood of the source, (ii) renormalization of effective gravitational and cosmological constants at large distances from the source and (iii) additional redshift factor between spatial regions of different curvature. The first two effects can be responsible, respectively, for the observed anomaly in the acceleration of the Pioneer spacecraft and for the alleged missing mass in spiral galaxies and other astrophysical objects. The third effect can be used to propose a non-cosmological explanation of high redshifts of quasars and gamma-ray bursts.