Cartesian tensors: an invitation to indices

LOCAL DIFFERENTIAL GEOMETRY consists in the first instance of an amplification and refinement of tensorial methods. In particular, the use of an index notation is the key to a great conceptual and geometrical simplification. We begin therefore with a transcription of elementary vector algebra in three dimensions. The ideas will be familiar but the notation new. It will be seen how the index notation gives one insight into the character of relations that otherwise might seem obscure, and at the same time provides a powerful computational tool.

The standard Cartesian coordinates of 3-dimensional space with respect to a fixed origin will be denoted xi (i = 1,2,3) and we shall write A = Ai to indicate that the components of a vector A with respect to this coordinate system are Ai . The magnitude of A is given by A · A = AiAi . Here we use the Einstein summation convention, whereby in a given term of an expression if an index appears twice an automatic summation is performed: no index may appear more than twice in a given term, and any ‘free’ (i.e. non-repeated) index is understood to run over the whole range.

NO SINGLE theoretical development in the last three decades has had more influence on gravitational theory than the discovery of the Kerr solution in 1963. The Kerr metric is a solution of the vacuum field equations. It is a generalization of the Schwarzschild solution, and represents the gravitational field of a special configuration of rotating mass, much as the external Schwarzschild solution represents the gravitational field of a spherical distribution of matter.

However, unlike the Schwarzschild case, no simple non-singular fluid ‘interior’ solution is known to match onto the Kerr solution. There is, nevertheless, no reason a priori why such a solution shouldn't exist.

Fortunately such speculations are in some respects beside the point, since the real interest in the Kerr solution for many purposes is its characterization of the final state of a black hole, after the hole has had the opportunity to ‘settle down’ and shed away (via gravitational radiation and other processes) eccentricities arising from the structure of the original body that formed the black hole.

To put the matter another way, suppose someone succeeded in exhibiting a good fluid interior for the Kerr metric. Well, that would be in principle very interesting; but there is no reason to believe that naturally occurring bodies (e.g. stars, galaxies, etc.) would tend to fall in line with that particular configuration.

IT IS INEVITABLE that with the passage of time Einstein's general relativity theory, his theory of gravitation, will be taught more frequently at an undergraduate level. It is a difficult theory—but just as some athletic records fifty years ago might have been deemed nearly impossible to achieve, and today will be surpassed regularly by well-trained university sportsmen, likewise Einstein's theory, now over seventy-five years since creation, is after a lengthy gestation making its way into the world of undergraduate mathematics and physics courses, and finding a more or less permanent place in the syllabus of such courses. The theory can now be considered both an accessible and a worthy, serious object of study by mathematics and physics students alike who may be rather above average in their aptitude for these subjects, but who are not necessarily proposing, say, to embark on an academic career in the mathematical sciences. This is an excellent state of affairs, and can be regarded, perhaps, as yet another aspect of the overall success of the theory.

A fresh look at anti-symmetric tensors

WE have introduced local differential geometry in a notation that makes great use of indices. This is the classical route and it does have a great deal of merit. There is a parallel development in an index free notation that is more generally used by pure mathematicians. The different approaches have their separate advantages and drawbacks: a calculation with indices may be cumbersome and sprawling; conversely an index-free notation may labour what is easily written with indices.

Light-cone geometry: the key to special relativity

WE HAVE SEEN how an index notation is strikingly helpful in the development of physical formulae for flat three-dimensional space. We found it convenient to work with a fixed Cartesian coordinate system, expressing the components of vectors and tensors with respect to that system. We know, nevertheless, as a matter of principle, that the general conclusions we draw are independent of the particular coordinatization chosen for the underlying space.

We now propose to formulate special relativity in essentially the same spirit. We shall regard space-time as a flat four-dimensional continuum with coordinates xa (a = 0,1,2,3). The points of space-time are called ‘events’, and we are interested in the relations of events to one another. Our purpose here is two-fold: first, to review some aspects of special relativity pertinent to that which follows later; and second, to develop further a number of index-calculus tools which are very useful in general relativity as well as special relativity.

Abstract. Data on kinematics, spatial distributions, and galaxy morphology in different density regimes within individual galaxy clusters show that many clusters are not in a stationary state but are still in the process of forming.

INTRODUCTION

Paradigms for galaxy clusters are changing. As in all tearing away from secure positions (Kuhn 1970) the process is controversial, yet continuing. Most papers in this volume suggest directions that will probably lead to even stronger new ideas about cluster cosmogony. We are concerned in this review with physical properties that have relevance for the question of whether clusters of galaxies are generally stationary, changing only slowly in a crossing time or if they are dynamically young. We examine if parts of a cluster may still be forming, falling onto an old dense core that would have been the first part of a density fluctuation to collapse even if all galaxies in a cluster are the same age, having formed before the cluster. During the 1930's the stationary nature of clusters seemed beyond doubt. A suggestion that they are dynamically young would have been too radical even for Zwicky who was the model of prophetic radicals. Rather, Zwicky (1937) took the stationary state to be given in making his calculation of a total mass, following an earlier calculation by Sinclair Smith (1936). The justification was that rich clusters such as Coma (1257 +2812; or Abell A1656), Cor Bor (1520 +2754; A2065), Bootis (1431 +3146; A1930), and Ursa Major No.2 (1055 +5702; A1132), known already to Hubble (1936) and to Humason (1936), appear so regular.

Abstract. Past X-ray surveys have shown that clusters of galaxies contain hot gas. Observations of this hot gas yield measurements of the fundamental properties of clusters. Results from a recent study of the X-ray luminosity function of local Abell clusters is described. Future surveys are discussed and the potential for studying the evolution of clusters is analyzed.

INTRODUCTION

The systematic study of clusters began with the surveys of Abell (1958) and Zwicky et al. (1968) who each created well defined catalogues according to specific definitions of the object class. In particular Abell defined clusters as overdensities of galaxies within a fixed physical radius around a center, classifying such objects as a function of their apparent magnitude (distance) and of their overdensity (“richness”).

The first X-ray survey of the sky by the UHURU X-ray satellite showed that “rich” nearby clusters were powerful X-ray sources (Gursky, et al. 1971, Kellogg et al. 1972). Subsequent spectroscopic studies detected X-ray emission lines of highly ionized iron and demonstrated that the X-ray emission was produced by thermal radiation of a hot gas with temperatures in the range of 30 to 100 million degrees (Mitchell et al. 1976, Serlemitsos, et al. 1977).

With the launch of the HEAO1 and the Einstein Observatories, surveys of significant samples of nearby clusters demonstrated that as a class, clusters of galaxies are bright X-ray sources with luminosities between 1042 and 1045 ergs/sec (Johnson, et al. 1983, Abramopoulos and Ku 1983, and Jones and Forman 1984).

Abstract. This contribution reviews the X-ray properties of clusters of galaxies and includes a brief summary of the X-ray characteristics of early-type galaxies and compact, dense groups. The discussion of clusters of galaxies emphasizes the importance of X-ray observations for determining cluster substructure and the role of central, dominant galaxies. The X-ray images show that substructure is present in at least 30% of rich (Abell) clusters and, hence that many rich clusters whose other properties are those of dynamically young systems, suggests that most cluster classification systems which utilize a property related to dynamical evolution, require a second dimension related to the dominance of the central galaxy. X-ray surveys of rich clusters show that central, dominant galaxies are twice as common as optical classifications suggest. The evidence for mass deposition (“cooling flows”) around central, dominant galaxies is reviewed. Finally, the implications of X-ray gas mass and iron abundance measurements for understanding the origin of the intracluster medium are discussed.

HOT GAS IN GALAXIES, GROUPS, AND CLUSTERS

Hot gas has been been found to be commonly associated with both individual early-type galaxies and with the poor and rich clusters in which they lie. Although this presentation will concentrate on the hot gas in rich clusters, we briefly describe the characteristics of individual galaxies and groups, as well as clusters since their evolution and present epoch properties are interrelated. Recent reviews of X-ray properties of clusters of galaxies include Forman and Jones (1982) and Sarazin (1986).