Introduction

A global solution of Einstein's equations is one in which an interior solution is matched smoothly onto an exterior solution, that is the metric and its derivatives are continuous at the boundary. Such global solutions can give considerable insight into the physical content of Einstein's equations and hence of general relativity. However, very few such global solutions are known partly because of the extreme scarcity of physically realistic interior solutions. One of the few global solutions known in general relativity is that of Van Stockum. Referring to this work Bonnor (1980a), who extended Van Stockum's results, says ‘In a fine paper, well ahead of its time, Van Stockum (1937) completely solved the problem of a rigidly rotating infinitely long cylinder of dust, including the application of adequate boundary conditions.’ In this chapter we shall consider in some detail the problem which Van Stockum solved.

In the first part of his paper Van Stockum found a class of exact interior solutions for rigidly notating neutral (uncharged) dust. By ‘dust’ here is meant pressureless matter. These solutions are axially symmetric but not necessarily cylindrically symmetric. They are quite distinct from the Van Stockum exterior rotating solutions discussed in Section 2.6. Like the exterior solutions the interior solutions are given in terms of a harmonic function. In the second part of his paper Van Stockum specialized his solution to cylindrical symmetry so that it described an infinitely long cylinder of rigidly rotating dust.