Showing papers by "Jerzy Lewandowski published in 2016"

Loop quantum gravity coupled to a scalar field

Abstract: We consider the model of gravity coupled to the Klein-Gordon time field. We do not deparametrize the theory using the scalar field before quantization, but quantize all degrees of freedom. Several new results for loop quantum gravity are obtained: (i) a Hilbert space for the gravity-matter system and a nonstandard representation of the scalar field thereon is constructed, (ii) a new operator for the scalar constraint of the coupled system is defined and investigated, (iii) methods for solving the constraint are developed. Commutators of the new quantum constraint operators correspond to the quantization of the Poisson bracket. This, however, poses problems for finding solutions. Hence the states we consider---and perhaps the whole setup---still needs some improvement. As a side result we describe a representation of the gravitational degrees of freedom in which the flux is diagonal. This representation is related to the BF theory vacuum of Dittrich and Geiller.

Spacetimes foliated by nonexpanding and Killing horizons: Higher dimension

Abstract: The theory of non-expanding horizons (NEH) geometry and the theory of near horizon geometries (NHG) are two mathematical relativity frameworks generalizing the black hole theory. From the point of view of the NEHs theory, a NHG is just a very special case of a spacetime containing an NEH of many extra symmetries. It can be obtained as the Horowitz limit of a neighborhood of an arbitrary extremal Killing horizon. An unexpected relation between the two of them, was discovered in the study of spacetimes foliated by a family of NEHs. The class of 4-dimensional NHG solutions (either vacuum or coupled to a Maxwell field) was found as a family of examples of spacetimes admitting a NEH foliation. In the current paper we systematically investigate geometries of the NEHs foliating a spacetime for arbitrary matter content and in arbitrary spacetime dimension. We find that each horizon belonging to the foliation satisfies a condition that may be interpreted as an invitation for a transversal extremal Killing horizon to exist. Assuming the existence of a transversal extremal Killing horizon, we derive all the spacetime metrics satisfying the vacuum Einstein's equations.

Coherent 3 j -symbol representation for the loop quantum gravity intertwiner space

Abstract: We introduce a new technique for dealing with the matrix elements of the Hamiltonian operator in loop quantum gravity, based on the use of intertwiners projected on coherent states of angular momentum. We give explicit expressions for the projections of intertwiners on the spin coherent states in terms of complex numbers describing the unit vectors which label the coherent states. Operators such as the Hamiltonian can then be reformulated as differential operators acting on polynomials of these complex numbers. This makes it possible to describe the action of the Hamiltonian geometrically, in terms of the unit vectors originating from the angular momentum coherent states, and opens up a way towards investigating the semiclassical limit of the dynamics via asymptotic approximation methods.

The algebra of observables in Gaußian normal spacetime coordinates

Abstract: We discuss the canonical structure of a spacetime version of the radial gauge, i.e. Gausian normal spacetime coordinates. While it was found for the spatial version of the radial gauge that a “local” algebra of observables can be constructed, it turns out that this is not possible for the spacetime version. The technical reason for this observation is that the new gauge condition needed to upgrade the spatial to a spacetime radial gauge does not Poisson-commute with the previous gauge conditions. It follows that the involved Dirac bracket is inherently non-local in the sense that no complete set of observables can be found which is constructed locally and at the same time has local Dirac brackets. A locally constructed observable here is defined as a finite polynomial of the canonical variables at a given physical point specified by the Gausian normal spacetime coordinates.

When Isolated Horizons met Near Horizon Geometries

Abstract: There are two mathematical relativity frameworks generalizing the black hole theory: the theory of isolated horizons (IH) and the theory of near horizon geometries (NHG). We outline here and discuss the derivation of the NHG from the theory of IH by composing spacetimes from IH. The simplest but still quite general class of solutions to Einstein's equations of this type defines spacetimes foliated by Killing horizons emanating from extremal horizons. That derivation, clearly being a link between the two frameworks, seems to be unknown to the NHG researchers and is hardly acknowledged in reviews on the IH. This lecture was a contribution to the Mathematical Structures session of the 2nd LeCosPA International Symposium "Everything about Gravity" celebrating the centenary of Einstein's General Relativity on December 14-18, 2015 in Taipei.

Observer's observables. Residual diffeomorphisms

Abstract: We investigate the fate of diffeomorphisms when the radial gauge is imposed in canonical general relativity. As shown elsewhere, the radial gauge is closely related to the observer's observables. These observables are invariant under a large subgroup of diffeomorphisms which results in their usefulness for canonical general relativity. There are, however, some diffeomorphisms, called residual diffeomorphisms, which might be "observed" by the observer as they do not preserve her observables. The present paper is devoted to the analysis of these diffeomorphisms in the case of the spatial and spacetime radial gauges. Although the residual diffeomorphisms do not form a subgroup of all diffeomorphisms, we show that their induced action in the phase space does form a group. We find the generators of the induced transformations and compute the structure functions of the algebras they form. The obtained algebras are deformations of the algebra of the Euclidean group and the algebra of the Poincare group in the spatial and spacetime case, respectively. In both cases the deformation depends only on the Riemann curvature tensor and in particular vanishes when the space or spacetime is flat.