The consequences of taking the total active gravitational mass of the universe phasewise constant together with a decaying vacuum energy in the background of Robertson-Walker space-time are investigated. The model so determined admits a contracted Ricci-collineation along the fluid flow vectorν
 i. It is geometrically closed but ever-expanding and does not possess the initial singularity, horizon, entropy, monopole or cosmological constant problems of the standard big bang cosmology. Estimates of the present matter; radiation and vacuum energy densities, the age of the universe and the present values of the deceleration parameter and the scale factor are also obtained.

A Model of the universe with decaying vacuum energy

Ricci collineations for the Ricci tensor which is constructed from a general spherically symmetric and static metric are classified for all possibilities of Rab(r) (such that Rab≠0 for a=b). It turns out that the only collineations admitted by this tensor can be ten, six, or four and there does not appear any case in between.

Collineations of the Ricci tensor

Necessary and sufficient conditions are derived for space‐time to admit a spacelike conformal motion with symmetry vector parallel to a unit spacelike vector field na. These conditions are expressed in terms of the shear and expansion of the spacelike congruence generated by na and in terms of the four‐velocity of the observer employed at any given point of the congruence. It is shown that either the expansion or the rotation of this spacelike congruence must vanish if Dna/dp =0, where  p denotes arc length measured along the integral curves of na, and also that there exist no proper spacelike homothetic motions with constant expansion. Propagation equations for the projection tensor and the rotation tensor are derived and it is proved that every isometric spacelike congruence is rigid. Fluid space‐times are studied in detail. A relation is established between spacelike conformal motions and material curves in the fluid: if a fluid space‐time admits a spacelike conformal Killing vector parallel to na and ...

Spacelike conformal Killing vectors and spacelike congruences

This paper treats matter field space-times admitting timelike conformai motions and timelike members of the family of contracted Ricci collineations (FCRC). The physical properties of these timelike symmetries in relation to the time development of relativistic matter field space-times are developed in terms of a number of specific theorems. Insofar as possible, the similarities and differences of the timelike conformal motions and the FCRC are discussed in some detail. Special applications are given that illustrate the possible value of the present considerations and related conservation expressions in relation to the Cauchy problem of matter field space-times admitting timelike symmetry properties.

On certain timelike symmetry properties and the evolution of matter field space-times that admit them

The spacetime homogeneous Godel-type spacetimes which have four classes of metrics are studied according to their matter collineations. The results obtained are compared with Killing vectors and Ricci collineations. It is found that these spacetimes have an infinite number of matter collineations in the degenerate case, i.e. det(Tab) = 0, and do not admit proper matter collineations in the non-degenerate case, i.e. det(Tab) ≠ 0. The degenerate case has the new constraints on the parameters m and w which characterize the causality features of the Godel-type spacetimes.

Matter collineations of spacetime homogeneous Gödel-type metrics

This paper focuses attention on the important role of Ricci collineations admitted by certain relativistic matter fields and the related conservation laws that may be admitted in the corresponding Riemannian space-times. Accordingly, one of the main objects of the present investigation is to show that Ricci collineations provide an invariant classification scheme for certain types of matter fields. These results are elaborated in terms of a number of theorems connecting particular symmetry properties with special properties of the eigenvalues and eigenvectors of the matter tensor characterizing the given matter fields. Finally, consideration is given to conservation laws relating to particular properties of the timelike principal-curve congruence of the given matter field and to the fact that certain symmetries may be admitted. Some of the results obtained are applied to the Robertson-Walker cosmological metrics. It is shown that a special class of metrics, belonging to the cosmological model, admits a particular nondegenerate Ricci collineation and the corresponding conservation law relating to matter conservation. Moreover, brief consideration is given to the appropriate specialization of these results which could be of interest at the level of ordinary relativistic hydrodynamics and plasma physics in flat space-time.

Relativistic matter fields admitting Ricci collineations and elated conservation laws

This paper further investigates and interprets the structure of Einstein-Maxwell space-times in terms of the infinitesimal holonomy groups (IHG) of theC
∞ Riemannian connection. In particular, this investigation provides a fundamental physical classification of Einstein-Maxwell space-times in terms of the IHG group structure of the Riemann curvature tensor. It will be shown that the Maxwell fieldsF
μλ and *F
μλ of a given Einstein-Maxwell space-time define a representation of a subalgebra of the Lie algebra of the IHG. The main results of this paper are stated in the form of two theorems and a corollary. Also the results obtained here indicate that physical insight could be gained by studying more general gauge fields in terms of the IHG of the relevant Einstein-Yang-Mills space-times.

The infinitesimal holonomy group structure of EEnstein-Maxwell space-times

The central role of the Riemann curvature tensor in Einstein's gravitational theory, including the significance of its invariant decomposition in spacetime (with respect to the duality and trace operations) into essentially local and non-local parts, is considered in the light of some of the thoughts and contributions of Cornelius Lanczos (1893–1974). Also, in this context, the fundamental role of the embedded two-space elements, specified by simple bivectors, in defining the Riemannian curvatures is briefly investigated. In particular, it is shown that the formulation of a bivector form of geodesic deviation for the Riemannian curvatures elucidates some of the insights of Weyl and Lanczos. Finally, some of Lanczos' thoughts relating to the nature of the essentially local and non-local parts of the Riemann tensor are discussed. These considerations are discussed in terms of Lanczos' discovery (1962) of the existence (in all Riemannian spacetimes) of a third order tensor Hijkthat can be regarded as the potential of the non-local (Weyl) part of the Riemann tensor.

L. K. Norris

Papers

Relativistic matter fields admitting Ricci collineations and elated conservation laws

The infinitesimal holonomy group structure of EEnstein-Maxwell space-times

Elucidation of some of lanczos' thoughts concerning spacetime curvature and the local and non-local parts of the riemann tensor