Ever since Newton introduced his theory of gravity, many famous physicists and mathematicians have worked on the problem of determining the properties of rotating bodies in equilibrium, such as planets and stars. In recent years, neutron stars and black holes have become increasingly important, and observations by astronomers and modelling by astrophysicists have reached the stage where rigorous mathematical analysis needs to be applied in order to understand their basic physics. This book treats the classical problem of gravitational physics within Einstein's theory of general relativity. It begins by presenting basic principles and equations needed to describe rotating fluid bodies, as well as black holes in equilibrium. It then goes on to deal with a number of analytically tractable limiting cases, placing particular emphasis on the rigidly rotating disc of dust. The book concludes by considering the general case, using powerful numerical methods that are applied to various models, including the classical example of equilibrium figures of constant density. Researchers in general relativity, mathematical physics and astrophysics will find this a valuable reference book on the topic. A related website containing codes for calculating various figures of equilibrium is available at www.cambridge. org/9780521863834.

Abstract. We give a detailed description of the recently developed multi-domain spectral method for constructing highly accurate general-relativistic models of rapidly rotating stars. For both "ordinary" and "critical" configurations, we show using representative examples, how the accuracy improves as the order of the approximation increases. As well as homogeneous fluid bodies, we also discuss models of polytropic and strange stars.

Abstract.A new spectral code for constructing general-relativistic models of rapidly rotating stars with an unprecedented accuracy is presented. As a first application, we reexamine uniformly rotating homogeneous stars and compare our results with those obtained by several previous codes. Moreover, representative relativistic examples corresponding to highly flattened rotating bodies are given.

In this Letter, we investigate uniformly rotating, homogeneous, and axisymmetric relativistic fluid bodies with a toroidal shape. The corresponding field equations are solved by means of a multidomain spectral method, which yields highly accurate numerical solutions. For a prescribed, sufficiently large ratio of inner to outer coordinate radius, the toroids exhibit a continuous transition to the extreme Kerr black hole. Otherwise, the most relativistic configuration rotates at the mass-shedding limit. For a given mass density, there seems to be no bound to the gravitational mass as one approaches the black hole limit and a radius ratio of unity.

By means of a highly accurate, multi-domain, pseudo-spectral method, we investigate the solution space of uniformly rotating, homogeneous and axisymmetric relativistic fluid bodies. It turns out that this space can be divided up into classes of solutions. In this paper, we present two new classes including relativistic core–ring and two-ring solutions. Combining our knowledge of the first four classes with post-Newtonian results and the Newtonian portion of the first ten classes, we present the qualitative behaviour of the entire relativistic solution space. The Newtonian disc limit can only be reached by going through infinitely many of the aforementioned classes. Only once this limiting process has been consummated can one proceed again into the relativistic regime and arrive at the analytically known relativistic disc of dust