R A d'Inverno

Classifying geometries in general relativity I: standard forms for symmetric spinors

This is the first in a series of papers concerning a project to set up a computer database of exact solutions in general relativity which can be accessed and updated by the user community. In this paper, we briefly discuss the Cartan-Karlhede invariant classification of geometries and the significance of the standard form of a spinor. We then present algorithms for putting the Weyl spinor, Ricci spinor and general spinors into standard form.

Classifying geometries in general relativity: III. Classification in practice

This is the third in a series of papers concerning a project to set up a computer database of exact solutions in general relativity which can be accessed and updated by the user community. In this paper, we describe how the Cartan-Karlhede method for classifying a geometry is accomplished in practice. We give as an example the classification of the Edgar-Lugwig conformally flat pure radiation metrics.

2+2 decomposition of Ashtekar variables

We derive a 2 t 2 formulation of Ashtekar variables, focusing on the important special case of a double null foliation of spacetime. The significance of the 2 + 2 formalism in general is that it isolates the me gravitational degrees of freedom. The derivation mirrors that of a recent paper by Coldberg PZ d in which Ashtekars 3 t I canonical formulation of Einsteins theory of gravitation is reformulated in term of a time parameter whose level sets are null hypersurfaces. In this paper we restdct anention to the Lagrangian formulation. PACS numbers: 0420C

Upper bounds for the Karlhede classification of type D vacuum spacetimes

The authors show how the upper bound on the order of covariant differentiation of the Weyl spinor required to perform the Karlhede classification of a type D vacuum spacetime may be shown to be three, using a direct calculation in the GHP formalism.

Cauchy-characteristic matching for a family of cylindrical solutions possessing both gravitational degrees of freedom

This article is part of a long-term programme to develop Cauchy-characteristic matching (CCM) codes as investigative tools in numerical relativity. The approach has two distinct features: (a) it dispenses with an outer boundary condition and replaces this with matching conditions at an interface between the Cauchy and characteristic regions, and (b) by employing a compactified coordinate, it proves possible to generate global solutions. In this paper CCM is applied to an exact two-parameter family of cylindrically symmetric vacuum solutions possessing both gravitational degrees of freedom due to Piran, Safier and Katz. This requires a modification of the previously constructed CCM cylindrical code because, even after using Geroch decomposition to factor out the z-direction, the family is not asymptotically flat. The key equations in the characteristic regime turn out to be regular singular in nature.

Numerical relativity on a transputer array

The results of a feasibility study on using a transputer array to obtain a numerical solution of the characteristic initial value problem of general relativity are presented. Codes have been developed for two cases: gravity only, and gravity and matter. The testing of the codes is, so far, very limited, but timing results etc. are presented thus giving an indication of the expected performance.

Hamiltonian analysis of the double null 2+2 decomposition of Ashtekar variables

We derive a canonical analysis of a double null 2+2 Hamiltonian description of general relativity in terms of complex self-dual 2-forms and the associated SO(3) connection variables. The algebra of first class constraints is obtained and forms a Lie algebra that consists of two constraints that generate diffeomorphisms in the 2-surface, a constraint that generates diffeomorphisms along the null generators and a constraint that generates self-dual spin and boost transformations.

Hamiltonian analysis of the double null 2+2 decomposition of general relativity expressed in terms of self-dual bivectors

In this paper, we obtain a 2 + 2 double null Hamiltonian description of general relativity using only the (complex) SO(3) connection and the components of the complex densitized self-dual bivectors ?A. We carry out the general canonical analysis of this system and obtain the first class constraint algebra entirely in terms of the self-dual variables. The first class algebra forms a Lie algebra and all the first class constraints have a simple geometrical interpretation.

Combining Cauchy and characteristic numerical evolutions in curved coordinates

In numerical relativity there would seem to be considerable advantages in combining Cauchy and characteristic methods in different regions of spacetime. Numerical experiments demonstrate, for the model problem of the scalar wave equation in flat space using general null coordinates, that there are no unforseen problems with the interface between the two regions.

Classifying geometries in general relativity: II. Spinor tools

This is the second in a series of papers concerning a project to set up a computer database of exact solutions in general relativity which can be accessed and updated by the user community. In this paper, we discuss the choice of computer algebra platform and the general relativity application package. The derivative operators needed in the Cartan-Karlhede classification algorithm and the behaviour of symmetric spinors under frame rotations are then presented.