Ashtekar Connection Research Articles

On a global conformal invariant of initial data sets

A global conformal invariant Y of a closed initial data set is constructed. A spacelike hypersurface in a Lorentzian spacetime naturally inherits from the spacetime metric a differentiation , the so-called real Sen connection, which turns out to be determined completely by the initial data and induced on , and coincides, in the case of a vanishing second fundamental form , with the Levi-Civita covariant derivation of the induced metric is built from the real Sen connection in a similar way to how the standard Chern - Simons invariant is built from . The number Y is invariant with respect to changes of and corresponding to conformal rescalings of the spacetime metric. In contrast, the quantity Y built from the complex Ashtekar connection is not invariant in this sense. The critical points of our Y are precisely the initial data sets which are locally imbeddable into conformal Minkowski space.

Initial-data formulation of tetrad gravity utilizing York's extrinsic time approach

An ability to analyze the geometrodynamic degrees of freedom and initial-data formulation is central to the canonical quantization of gravity. In the metric theory of gravity York provided the most powerful technique to analyze the dynamic degrees of freedom and to solve the initial-data problem. In this paper we extend York's analysis to tetrad gravity. Such an extension is necessary for the quantization of gravity when coupled to a half-integer-spin field. We present a comparative analysis of the geometric information carried by (1) a 3-metric of an initial hypersurface and (2) the spacelike triad of a time-gauged tetrad. We apply the tetrad initial-data formulation to Ashtekar's connection variables, and provide a comparison with other alternative choices of canonical tetrad variables.

Geroch group in the Ashtekar formulation.

We study the Geroch group in the framework of the Ashtekar formulation. In the case of the one-Killing-vector reduction, it turns out that the third column of the Ashtekar connection is essentially the gradient of the Ernst potential, which implies that both quantities are based on the ``same'' complexification. In the two-Killing-vector reduction, we demonstrate the Ehlers and Matzner-Misner SL(2,R) symmetries, respectively, by constructing two sets of canonical variables that realize either of the symmetries canonically, in terms of the Ashtekar variables. The conserved charges associated with these symmetries are explicitly obtained. We show that the gl(2R) loop algebra constructed previously in the loop representation is not the Lie algebra of the Geroch group itself. We also point out that the recent argument on the equivalence to a chiral model is based on a gauge choice which cannot be achieved generically.

A connection approach to numerical relativity

We discuss a general formalism for numerically evolving initial data in general relativity in which the (complex) Ashtekar connection and the Newman--Penrose conformal scalars are taken as the dynamical variables. In the generic case three gauge constraints and twelve reality conditions must be solved. The analysis is applied to a Petrov type planar spacetime, where we find a spatially constant volume element to be an appropriate coordinate gauge choice.

REALITY CONDITIONS FOR LORENTZIAN AND EUCLIDEAN GRAVITY IN THE ASHTEKAR FORMULATION

Using Ashtekar variables, we analyze Lorentzian and Euclidean gravity in vacuum up to a constant conformal transformation. Keeping unaltered the symplectic structure in the full theory of complex gravity, we prove that the reality conditions are invariant under a Wick rotation of the time, and show that the compatibility of the algebra of commutators and constraints with the involution defined by the reality conditions restricts the possible values of the conformal factor to be either real or purely imaginary. In the first case, one recovers real Lorentzian general relativity. For purely imaginary conformal factors, the classical theory can be interpreted as real Euclidean gravity. The reality conditions associated with this Euclidean theory demand the hermiticity of the Ashtekar connection, but the densitized triad is represented by an anti-Hermitian operator. We also demonstrate that the Euclidean and Lorentzian sets of reality conditions lead to inequivalent quantizations of full general relativity. This conclusion also holds in the geometrodynamic formulation. As a consequence, it seems impossible to obtain Lorentzian physical predictions from the quantum theory constructed with the Euclidean reality conditions.

Jones polynomials for intersecting knots as physical states of quantum gravity

We find a consistent formulation of the constraints of quantum gravity with a cosmological constant in terms of the Ashtekar new variables in the connection representation, including the existence of a state that is a solution to all the constraints. This state is related to the Chern-Simons form constructed from the Ashtekar connection and has an associated metric in space-time that is everywhere nondegenerate. We then transform this state to the loop representation and find solutions to all the constraint equations for intersecting loops. These states are given by suitable generalizations of the Jones knot polynomial for the case of intersecting knots. These are the first physical states of quantum gravity for which an explicit form is known both in the connection and loop representations. Implications of this result are also discussed.

Degeneracy in loop variables

The small algebra of loop functionals, defined by Rovelli and Smolin, on the Ashtekar phase space of general relativity is studied. Regarded as coordinates on the phase space, the loop functionals become degenerate at certain points. All the degenerate points are found and the corresponding degeneracy is discussed. The intersection of the set of degenerate points with the real slice of the constraint surface is shown to correspond precisely the Goldberg-Kerr solutions. The evolution of the holonomy group of Ashtekar's connection is examined, and the complexification of the holonomy group is shown to be preserved under it. Thus, an observable of the gravitational field is constructed.

Ashtekar formulation of general relativity and loop-space nonperturbative quantum gravity: a report

The formulation of general relativity discovered by Ashtekar (1986, 1987) and the recent results obtained in non-perturbative quantum gravity using loop-space techniques are reviewed. The new formulation is based on the choice of a set of Lagrangian (and Hamiltonian) variables, instead of the spacetime metric. In terms of these new variables, the dynamical equations are remarkably simplified and a structural identity between general relativity and the Yang-Mills theories is revealed. The formalism has proven to be useful in numerous problems in gravitational physics. In quantum gravity, the new formalism has overcome long-standing difficulties and led to unexpected results. A nonperturbative approach to quantum theory has been constructed in terms of the Wilson loops of the Ashtekar connection. This approach, denoted as loop-space representation, has led to the complete solution of the quantum diffeomorphism constraint in terms of knot states, to the discovery of an infinite-dimensional class of solutions to the quantum gravitational dynamics, and to certain surprising indications on the existence of a discrete structure of spacetime around the Planck length. These results are presented in a compact self-contained form. The basic Ashtekar formalism is presented and its applications are outlined. The loop-space representation and the non-perturbative knot states of quantum gravity are described in detail, with particular regard to their physical interpretation and to the information they may provide on the microstructure of spacetime.

Intersecting N loop solutions of the hamiltonian constraint of quantum gravity

Following the introduction by Ashtekar of a new set of canonical variables for the gravitational field and a new representation of quantum gravity based on them, Jacobson and Smolin presented a large class of solutions to the quantum hamiltonian constraint of general relativity based on the holonomy of the Ashtekar connection along simple loops and two intersecting loops. For all these solutions the metric is degenerate everywhere, including the point of intersection. This motivated Husain to extend the results to consider the case of three intersecting loops. However, the metric was degenerate as well on the three-loop solutions found since the loops were only allowed to have two independent tangent vectors at the point of intersection. We have developed a computer algebra code capable of generating solutions for an arbitrary number of loops. We explicitly present new four- and five-loop solutions. These are the first solutions known to have three independent tangent vectors at the intersection point. In spite of this they fail, however, to have a nondegenerate metric. We present a general argument that shows that the same situation arises for an arbitrary finite number of loops. Implications of this degeneracy for the interpretation of the connection and loop space representation of quantum gravity are also discussed.

Intersecting-loop solutions of the hamiltonian constraint of quantum general relativity

Recently, using Ashtekar's new canonical formulation of general relativity, a large class of solutions of the quantum hamiltonian constraint were found by Jacobson and Smolin. These solutions are the traces of the holonomy for Ashtekar's connection that are based on differentiable loops. In this paper, additional solutions of the quantum hamiltonian constraint are described. These new solutions also involve the holonomy, but are based on multiple loops that intersect at a point. It is shown that no solutions exist for three intersecting loops that have more than two linearly independent tangent vectors at the intersection point.

Nonperturbative quantum geometries

Using the self-dual representation of quantum general relativity, based on Ashtekar's new phase space variables, we present an infinite dimensional family of quantum states of the gravitational field which are exactly annihilated by the hamiltonian constraint. These states are constructed from Wilson loops for Ashtekar's connection (which is the spatial part of the left handed spin connection). We propose a new regularization procedure which allows us to evaluate the action of the hamiltonian constraint on these states. Infinite linear combinations of these states which are formally annihilated by the diffeomorphism constraints as well are also described. These are explicit examples of physical states of the gravitational field — and for the compact case are exact zero eigenstates of the hamiltonian of quantum general relativity. Several different approaches to constructing diffeomorphism invariant states in the self dual representation are also described. The physical interpretation of the states described here is discussed. However, as we do not yet know the physical inner product, any interpretation is at this stage speculative. Nevertheless, this work suggests that quantum geometry at Planck scales might be much simpler when explored in terms of the parallel transport of left-handed spinors than when explored in terms of the three metric.