Peter Musgrave

Junctions and thin shells in general relativity using computer algebra: I. The Darmois - Israel formalism

We present the GRjunction computer algebra program which allows the study of non-null boundary surfaces and thin shells in general relativity. Implementing the Darmois - Israel thin-shell formalism requires a careful selection of definitions and algorithms to ensure that results are generated in a straightforward way. We have used the package to correctly reproduce a wide variety of examples from the literature. In this paper GRjunction is used to perform two new calculations: joining two Kerr solutions with differing masses and angular momenta along a thin shell in the slow rotation limit, and the calculation of the stress - energy of a Curzon wormhole. The Curzon wormhole has the interesting property that shells located at radius R < 2m have regions which satisfy the weak energy condition.

Junctions and thin shells in general relativity using computer algebra: II. The null formalism

We present extensions to the GRJunction computer algebra program, which allow the study of null boundary surfaces and thin shells in general relativity. We summarize the null formalism due to Barrabes and Israel and highlight those steps which differ from the timelike/spacelike cases. GRJunction has been used to verify a number of results from the literature. We then present two new results calculated with the aid of GRJunction . These are the junction of two Kerr - Newman solutions at a non-horizon straddling null shell in the slow rotation limit and the exact junction of two Kerr - Newman solutions at a horizon straddling shell.

Invariants of the Riemann tensor for class B warped product space-times

Abstract We use the computer algebra system GRTensorII to examine invariants polynomial in the Riemann tensor for class B warped product space-times — those which can be decomposed into the coupled product of two 2-dimensional spaces, one Lorentzian and one Riemannian, subject to the separability of the coupling ds2 = dsϵ12 (u,v) + C(xγ)2dsϵ22 (θ, φ). with C(xγ)2 = r(u,v)2w(θ, φ)2 and sig(ϵ1) = 0, sig(ϵ2) = 2ϵ ( ϵ = ± 1) for class B1 space-times and sig(ϵ1) = 2ϵ, sig(ϵ2) = 0 for class B2. Although very special, these spaces include many of interest, for example, all spherical, plane, and hyperbolic space-times. The first two Ricci invariants along with the Ricci scalar and the real component of the second Weyl invariant J alone are shown to constitute the largest independent set of invariants to degree five for this class. Explicit syzygies are given for other invariants up to this degree. It is argued that this set constitutes the largest functionally independent set to any degree for this class, and some physical consequences of the syzygies are explored.

Algorithms for computer algebra calculations in spacetime: I. The calculation of curvature

We examine the relative performance of algorithms for the calculation of curvature in spacetime. The classical coordinate component method is compared to two distinct versions of the Newman - Penrose tetrad approach for a variety of spacetimes, and distinct coordinates and tetrads for a given spacetime. Within the system GRTensorII, we find that there is no single preferred approach on the basis of speed. Rather, we find that the fastest algorithm is the one that minimizes the amount of time spent on simplification. This means that arguments concerning the theoretical superiority of an algorithm need not translate into superior performance when applied to a specific spacetime calculation. In all cases it is the simplification strategy which is of paramount importance. An appropriate simplification strategy can change an intractable problem into one which can be solved essentially instantaneously.

The regularity of static spherically cylindrically and plane symmetric spacetimes at the origin

We find the necessary and sufficient conditions for the regularity of all scalar invariants polynomial in the Riemann tensor at the origin of spherically, cylindrically and plane symmetric static spacetimes under the assumption that the metric functions are sufficiently smooth there. These conditions turn out to be simple enough to allow a check for regularity by inspection.

GRLite and GRTensorJ: Graphical user interfaces to the computer algebra system GRTensorII

Two open graphical user interfaces to the computer algebra system GRTensorII are described. Each can be run locally or via the Internet. These provide students and researchers in the area of General Relativity and related fields with advanced truly portable tools that reduce many complex calculations to elementary functions.

Scalar invariants of the radiating Kerr-Newman metric: a simple application of GRTensor

GRTensor is an interactive PC‐based program for tensor analysis primarily of interest for teaching and research in general relativity. It uses either M A P L E V or M A T H E M A T I C A as its algebraic engine. In this paper we use GRTensor to evaluate the Ricci and Weyl invariants for the radiating Kerr–Newman metric. This includes, as a special case, all nonvanishing invariants of the Kerr metric—the archetypical black hole solution in general relativity.