G. S. Hall

Symmetries and curvature structure in general relativity

Introduction topological spaces groups and linear algebra manifold theory transformation groups the Lorentz group general relativity theory space-time holonomy curvature structure in general relativity affine symmetries in space-time conformal symmetries in space-time curvature collineations sectional curvature structure.

Curvature collineations and the determination of the metric from the curvature in general relativity

It is shown that for a very general class of space-times, the componentsRbcda of the curvature tensor determine the metric components up to a constant conformal factor. This general class contains most of those cases which are usually considered to be interesting from the point of view of Einsteins general relativity theory. The connection between the above result and the existence of proper curvature collineations is given.

Three-dimensional space-times

A real version of the Newman-Penrose formalism is developed for (2+1)-dimensional space-times. The complete algebraic classification of the (Ricci) curvature is given. The field equations of Deser, Jackiw, and Templeton, expressing balance between the Einstein and Bach tensors, are reformulated in triad terms. Two exact solutions are obtained, one characterized by a null geodesic eigencongruence of the Ricci tensor, and a second for which all the polynomial curvature invariants are constant.

Ricci and matter collineations in space-time

A discussion of Ricci and matter collineations (mainly the former) is presented. A mathematical description of their dimensionality, differentiability, extendibility etc. is given. Examples of Ricci collineations are constructed particularly in decomposable space-times.

Affine collineations in space-time

The existence of affine collineations in space‐time is discussed and the types of space‐time admitting proper affine collineations is displayed. The close connection between such space‐times and their holonomy structure and local decomposability is established. Affine collineations with fixed points are also considered as is the problem of extending local affine collineations to the whole of space‐time.

The principle of equivalence and projective structure in spacetimes

This paper discusses the extent to which one can determine the spacetime metric from a knowledge of a certain subset of the (unparametrized) geodesics of its Levi-Civita connection, that is, from the experimental evidence of the equivalence principle. It is shown that, if the spacetime concerned is known to be vacuum, then the Levi-Civita connection is uniquely determined and its associated metric is, generically, uniquely determined up to a choice of units of measurement, by the specification of these geodesics. A special case arises here concerning pp-waves and is dealt with. It is further demonstrated that if two spacetimes share a similar subset of unparametrized geodesics they share all geodesics (that is they are projectively related). Furthermore, if one of these spacetimes is assumed vacuum then their Levi-Civita connections are again equal (and so the other metric is also a vacuum metric) and the first result above is recovered.

Algebraic Determination of the Metric from the Curvature in General Relativity

The general solution for a symmetric second-order tensorX of the equationXe(aRebcd=0 whereR is the Riemann tensor of a space-time manifold, andX is obtained in terms of the curvature 2-form structure ofR by a straightforward geometrical technique, and agrees with that given by McIntosh and Halford using a different procedure. Two results of earlier authors are derived as simple corollaries of the general theorem.

Projective equivalence of Einstein spaces in general relativity

There has been some recent interest in the relation between two spacetimes which have the same geodesic paths, that is, spacetimes which are projectively equivalent (sometimes called geodesically equivalent). This paper presents a short and accessible proof of the theorem that if two spacetimes have the same geodesic paths and one of them is an Einstein space then (either each is of constant curvature or) their Levi-Civita connections are identical. It also clarifies the relationship between their associated metrics. The results are extended to include the signatures (+ + + +) and (− − + +), and some examples and discussion are given in the case of dimension n > 4. Some remarks are also made which show how these results may be useful in the study of projective symmetry.

Holonomy groups and spacetimes

A study is made of the possible holonomy group types of a spacetime for which the energy-momentum tensor corresponds to a null or non-null electromagnetic field, a perfect fluid or a massive scalar field. The case of an Einstein space is also included. The techniques developed are, in addition, applied to vacuum and conformally flat spacetimes and contrasted with already known results in these two cases. Examples are given.

Physical structure of the energy-momentum tensor in General Relativity

The algebraic classification of second-order symmetric tensors based on Segré type is used to give a systematic description of energy-momentum tensors in General Relativity. The uniqueness of the physical interpretation of a given energy-momentum tensor is discussed algebraically and a brief description of their “inheritance of symmetry” properties is also given.