Malcolm MacCallum

A class of homogeneous cosmological models

Einsteins field equations are studied under the assumptions that (1) the source of the gravitational field is a perfect fluid, and (2) there exists a group of motions simply transitive on three-surfaces orthogonal to the fluid flow vector. There are two classes of solutions; these are studied in detail. Three special families of solutions examined include all analytic solutions of the field equations obeying (1) and (2) of which the authors are aware. The relation of these solutions to various vacuum solutions is indicated.

Twistor theory: An approach to the quantisation of fields and space-time

Abstract Twistor theory offers a new approach, starting with conformally-invariant concepts, to the synthesis of quantum theory and relativity. Twistors for flat space-time are the SU(2,2) spinors of the twofold covering group O(2,4) of the conformal group. They describe the momentum and angular momentum structre of zero-rest-mass particles. Space-time points arise as secondary concepts corresponding to linear sets in twistor space. They, rather than the null cones, should become “smeared out” on passage to a quantised gravitational theory. Twistors are represented here in two-component spinor terms. Zero-rest-mass fields are described by holomorphic functions on twistor space, on which there is a natural canonical structure leading to a natural choice of canonical quantum operators. The generalisation to curved space can be accomplished in three ways; i) local twistors, a conformally invariant calculus, ii) global twistors, and iii) asymptotic twistors which provide the framework for an S -matrix approach in asymptotically flat space-times. A Hamiltonian scattering theory of global twistors is used to calculate scattering cross-sections. This leads to twistor analogues of Feynman graphs for the treatment of massless quantum electrodynamics. The recent development of methods for dealing with massive (conformal symmetry breaking) sources and fields is briefly reviewed.

A class of homogeneous cosmological models. III. Asymptotic behaviour

The behaviour near to and far from an initial singularity in a broad subclass of the models studied in previous papers [1–3] is examined. The influence of the matter on the evolution at these times is discussed. The singularity types for the various models, which are mostly of cigar or oscillatory nature, are found. It is discovered that among these models, only those of the same Bianchi type as a Robertson-Walker model can become “approximately Robertson-Walker” in a sense defined in the paper. Qualitative conclusions concerning black-body isotropy, the Hubble relation, helium abundance and horizon structure are given.

NONADIABATIC CHARGED SPHERICAL GRAVITATIONAL COLLAPSE

We present a complete set of the equations and matching conditions required for the description of physically meaningful charged, dissipative, spherically symmetric gravitational collapse with shear. Dissipation is described with both free-streaming and diffusion approximations. The effects of viscosity are also taken into account. The roles of different terms in the dynamical equation are analyzed in detail. The dynamical equation is coupled to a causal transport equation in the context of Israel-Stewart theory. The decrease of the inertial mass density of the fluid, by a factor which depends on its internal thermodynamic state, is reobtained, with the viscosity terms included. In accordance with the equivalence principle, the same decrease factor is obtained for the gravitational force term. The effect of the electric charge on the relation between the Weyl tensor and the inhomogeneity of the energy density is discussed.

Gravitational waves in general relativity XIV. Bondi expansions and the ‘polyhomogeneity’ of ℐ

The structure of polyhomogeneous space-times (i.e. space-times with metrics which admit an expansion in terms of r-j logi r) constructed by a Bondi-Sachs type method is analysed. The occurrence of some log terms in an asymptotic expansion of the metric is related to the non-vanishing of the Weyl tensor at ℐ. The validity in this more general context of various results from the standard treatment of ℐ, including the Bondi mass loss formula, the peeling-off of the Riemann tensor and the Newman-Penrose constants of motion, is considered.

Causality and the speed of sound

A usual causal requirement on a viable theory of matter is that the speed of sound be at most the speed of light. In view of various recent papers querying this limit, the question is revisited here. We point to various issues confronting theories that violate the usual constraint.

Variational principles and spatially-homogeneous universes, including rotation

The validity of imposing spatial homogeneity on the variations in the usual action principle for Einsteins equations is studied. It is proved that with this procedure the standard and ADM Lagrangians give correct Einstein equations if and only if the space belongs to Class A of Ellis and MacCallum [1], i.e., the structure constants of the simply transitive group satisfy Cfgf=0. The possibility of overcoming this difficulty in the Class B spaces is examined.

Shearfree cylindrical gravitational collapse

We consider diagonal cylindrically symmetric metrics, with an interior representing a general nonrotating fluid with anisotropic pressures. An exterior vacuum Einstein-Rosen spacetime is matched to this using Darmois matching conditions. We show that the matching conditions can be explicitly solved for the boundary values of metric components and their derivatives, either for the interior or exterior. Specializing to shearfree interiors, a static exterior can only be matched to a static interior, and the evolution in the nonstatic case is found to be given in general by an elliptic function of time. For a collapsing shearfree isotropic fluid, only a Robertson-Walker dust interior is possible, and we show that all such cases were included in Cockes discussion. For these metrics, Nolan and Nolan have shown that the matching breaks down before collapse is complete, and Tod and Mena have shown that the spacetime is not asymptotically flat in the sense of Berger, Chrusciel, and Moncrief. The issues about energy that then arise are revisited, and it is shown that the exterior is not in an intrinsic gravitational or superenergy radiative state at the boundary.

A Rational Approach to the Prelle-Singer Algorithm

We present an approach to computing the Darboux polynomials required in the Prelle?Singer algorithm which avoids algebraic extensions of the constant field, and describe a partial implementation in REDUCE in which the leading terms of the polynomials are obtained by a modified version of the method described by Christopher and Collins.

Physical interpretation of vacuum solutions of Einstein's equations. Part II. Time-dependent solutions

The study of interpretations of the vacuum solutions of Einsteins field equations is continued by considering some well known time-dependent solutions. Among these are metrics representing accelerating particles, cylindrical and plane waves and cosmological solutions.