Algebraic computing in general relativity

The purpose of this paper is to bring to the attention of potential users the existence of algebraic computing systems, and to illustrate their use by reviewing a number of problems for which such a system has been successfully used in General Relativity. In addition, some remarks are included which may be of help in the future design of these systems.     

Using camal for algebraic computations in general relativity

CAMAL is a collection of computer algebra systems developed in Cambridge, England for use mainly in theoretical physics. One of these was designed originally for general relativity calculations, although it is often used in other fields. In a recent paper Cohen, Leringe, and Sundblad compared six systems for algebraic computations applied to general relativity available in Stockholm. Here we give similar information for CAMAL, and by using the same tests we add CAMAL to the comparison.     

Local and global algebraic structures in general relativity

The extent to which the well-known pointwise algebraic canonical forms used for the energy-momentum tensor, the Weyl tensor, etc., can be regarded as smooth relations over some open subset of (possibly the whole of) space-time is investigated.     

An algebraic classification of neutrino fields in general relativity

An algebraic classification of neutrino fields in curved space-times is found. Several classes are simplified by assuming a positive energy density. These classes then correspond algebraically to null and non-null electromagnetic fields. The uniqueness of the neutrino fields for these classes is discussed.     

Algebraic computing and the Newman-Penrose formalism in general relativity

The curvature tensor of space-time can be described most concisely by giving the components of the Weyl and Ricci tensors relative to a complex null tetrad. The Newman-Penrose equations provide a simple and direct algorithm for calculating these components. This paper describes a computer program, written in the symbolic manipulation language CAMAL, which performs this calculation. Comparisons are made with the classical tensorial method of calculation, and some applications are discussed.     

The gauge in general relativity

The view is taken that the field equations of General Relativity, without a definition of congruence of length and time intervals at different events, are without physical content. The possibility is explored that the customary Einstein field equations are to be used but with a different congruence definition than is customary. When these resulting equations are, in turn, expressed with the customary congruence, they comprise a new set of field equations physically not equivalent to either Einstein's or Brans-Dicke's formulations of general relativity. Similarities with Einstein's and Brans-Dicke's formulations are discussed, and the possibility of experimental confirmation of these new equations is also briefly considered.     

Elucidation of covariant proofs in general relativity: example of the use of algebraic software in the shear-free conjecture in MAPLE

In this paper we explore the use of a new algebraic software package in providing independent covariant proof of a conjecture in general relativity. We examine the proof of two sub-cases of the shear-free conjecture \(\displaystyle \sigma =0 {\displaystyle \, =>}\, {\displaystyle \omega }\,{\displaystyle \varTheta } =0\) by Senovilla et al. (Gen. Relativ. Gravit 30:389–411, 1998): case 1: for dust; case 2: for acceleration parallel to vorticity. We use TensorPack, a software package recently released for the Maple environment. In this paper, we briefly summarise the key features of the software and then demonstrate its use by providing and discussing examples of independent proofs of the paper in question. A full set of our completed proofs is available online at http://www.bach2roq.com/science/maths/GR/ShearFreeProofs.html. We are in agreeance with the equations provided in the original paper, noting that the proofs often require many steps. Furthermore, in our proofs we provide fully worked algebraic steps in such a way that the proofs can be examined systematically, and avoiding hand calculation. It is hoped that the elucidated proofs may be of use to other researchers in verifying the algebraic consistency of the expressions in the paper in question, as well as related literature. Furthermore we suggest that the appropriate use of algebraic software in covariant formalism could be useful for developing research and teaching in GR theory.     

The program ORTOCARTAN for algebraic calculations in relativity

The program ORTOCARTAN can calculate the curvature tensors (Riemann, Ricci, Einstein and Weyl) from a given orthonormal tetrad representation of the metric tensor. It was first announced in 1981, but since then has undergone several extensions and transplants onto other computers. This article reviews the current status of the program from the point of view of a user. The following topics are discussed: the problems that the program can be applied to, the special features of the algorithms that make the program powerful, the technical requirements to run the program and two simple examples of applications.     

The regimen of general relativity

New developments in our understanding of the electron-atom collision process have been made possible by combining the use of highly monochromatic electron beams and intense CO₂ lasers. This paper reviews such experiments and discusses possible future progress in what is a new field in atomic collision physics.     

The charged line-mass in general relativity

The two exterior solutions for a charged line-mass are examined. In both cases the mass per unit length is negative.     

The similarity hypothesis in general relativity

Self-similar models are important in general relativity and other fundamental theories. In this paper we shall discuss the “similarity hypothesis”, which asserts that under a variety of physical circumstances solutions of these theories will naturally evolve to a self-similar form. We will find there is good evidence for this in the context of both spatially homogenous and inhomogeneous cosmological models, although in some cases the self-similar model is only an intermediate attractor. There are also a wide variety of situations, including critical phenomena, in which spherically symmetric models tend towards self-similarity. However, this does not happen in all cases and it is important to understand the prerequisites for the conjecture.     

The Ernst equation in general relativity

Stationary axisymmetric gravitational fields are governed by the Ernst equation. Its internalSU(1, 1) symmetry gives rise to a linear problem and to Bäcklund transformations which map known solutions into new ones. The main features of the solution generating techniques are outlined, the generalization to Einstein-Maxwell fields is discussed and some applications are given.Invited talk presented at the International Conference Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 23–27, 1986.This paper summarizes some common work with G. Neugebauer. I would like to thank him for many stimulating discussions.     

The boost problem in general relativity

We show that any asymptotically flat initial data for the Einstein field equations have a development which includes complete spacelike surfaces boosted relative to the initial surface. Furthermore, the asymptotic fall off is preserved along these boosted surfaces and there exists a global system of harmonic coordinates on such a development. We also extend former results on global solutions of the constraint equations. By virtue of this extension, the constraint and evolution parts of the problem fit together exactly. Several theorems are given which concern the behaviour in the large of general classes of linear and quasilinear differential systems. This paper contains in addition a systematic exposition of the functional spaces employed.     

The Huggins term in general relativity

It is shown that any symmetric tensor density of covariant rank 2 which is divergence-free and is a concomitant of the metric tensor, a vector field, and their first and second derivatives will be entirely independent of the vector field. From this we show that there is no suitable generalization of the Huggins term for the theory of general relativity.     

The curvature problem in general relativity

This essay investigates the relationships among the metric, the connection, the curvature, and the covariant curvature derivatives in general relativity. The extent to which the connection or the curvature together, possibly with certain curvature derivatives, determines the metric is considered, as well as other related problems. Some topological aspects of the problem are included and some applications to holonomy and symmetry groups are given.This article received honorable mention from the Gravity Research Foundation for 1987.     

Spinors,algebraic geometry,and the classification of second-order symmetric tensors in general relativity

Two approaches to the problem of classifying second-order symmetric tensors in space-time given by Ludwig and Scanlon and by Penrose are discussed. Ludwig and Scanlon use both spinor and tensor algebra in their approach, whereas Penrose uses spinors and the properties of certain curves in complex projective 3-space. These approaches yield essentially identical classifications, and this paper points out the connections between them in detail and tabulates the results.     

The principle of relativity, kinematics and algebraic relations

Based on the relativistic principle and the postulate of universal invariant constants (c, l), all kinematic symmetries can be set up as the subsets of the Umov-Weyl-Fock-Hua transformations for the inertial motions. These symmetries are connected to each other via combinations rather than via contractions and deformations.     

The conservation of matter in general relativity

It is shown that in classical general relativity, if space-time is nonempty at one time, it will be nonempty at all times provided that the energy momentum tensor of the matter satisfies a physically reasonable condition. The apparent contradiction with the quantum predictions for the creation and annihilation of matter particles by gravitons is discussed and is shown to arise from the lack of a good energy momentum operator for the matter in an unquantised curved space-time metric.