THE AIM OVER the next two chapters is to construct a solution of Einstein's equations with sources that will provide a model for the large scale features of the universe. First, we must find a reasonable form for the metric and energy-moment urn tensor consistent with the observed symmetries of the universe. Then we shall be led to specific cosmological models by the imposition of the Einstein equations.

We are thinking of the average features of the universe on the scale of tens of millions of light years and we may regard the basic building blocks as clusters of galaxies. The first observational fact about the universe that we must use is that the observed distribution of the clusters of galaxies is isotropic to a high degree. If we assume that our position is in no particular way privileged, we must assume the universe is isotropic about every point, which leads to an assumption of homogeneity.

We must distinguish a preferred class of observers, namely those that actually see the universe as isotropic. Thus our cosmological model admits a preferred time-like vector field ua, tangent to the world lines of the preferred or ‘fundamental’ observers.

WE PROCEED to solve the geodesic equations in the Schwarzschild solution and use the solution to describe the classical tests of general relativity. These are the precession of the perihelion of planetary orbits and the bending of light by the sun, effects that arise from the small differences between orbits in Newtonian gravitation and orbits, i.e. geodesies, in general relativity.

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.