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Scientific paper
Aug 2009
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2009cqgra..26p9001m&link_type=abstract
Classical and Quantum Gravity, Volume 26, Issue 16, pp. 169001 (2009).
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Scientific paper
Compact fluid bodies in equilibrium under its own gravitational field are abundant in the Universe and a proper treatment of them can only be carried out using the full theory of General Relativity. The problem is of enormous complexity as it involves two very different regimes, namely the interior and the exterior of the fluid, coupled through the surface of the body. This problem is very challenging both from a purely theoretical point of view, as well as regarding the obtaining of realistic models and the description of their physical properties. It is therefore an excellent piece of news that the book 'Relativistic Figures of Equilibrium' by R Meinel, M Ansorg, A Kleinwächter, G Neugebauer and D Petroff has been recently published. This book approaches the topic in depth and its contents will be of interest to a wide range of scientists working on gravitation, including theoreticians in general relativity, mathematical physicists, astrophysicists and numerical relativists.
This is an advanced book that intends to present some of the present-day results on this topic. The most basic results are presented rather succinctly, and without going into the details, of their derivations. Although primarily not intended to serve as a textbook, the presentation is nevertheless self-contained and can therefore be of interest both for experts on the field as well as for anybody wishing to learn more about rotating self-gravitating compact bodies in equilibrium. It should be remarked, however, that this book makes a rather strong selection of topics and concentrates fundamentally on presenting the main results obtained by the authors during their research in this field.
The book starts with a chapter where the fundamental aspects of rotating fluids in equilibrium, including its thermodynamic properties, are summarized. Of particular interest are the so-called mass-shedding limit, which is the limit where the body is rotating so fast that it is on the verge of starting losing material, and the black hole transition, where rotating fluids are seen to approach black holes for suitable limits of their parameters.
As the authors themselves mention, one of the emphasis of this book is placed 'on the rigorous treatment of simple models instead of trying to describe real objects with their many complex facets...'. After discussing constant density models both in Newtonian theory (the Maclaurin spheroids) and in the non-rotating relativistic case (the Schwarzschild interior model), the book concentrates on the so-called rigidly rotating disc of dust. Chapter two is mainly devoted to deriving this model and presenting its physical properties. The derivation is based in the so-called inverse scattering method of integrable systems and on a thorough knowledge of the theory of integration on Riemann surfaces. The details, which take up about one fifth of the whole length, are difficult to follow for any reader without a previous mastering of the techniques involved. For the expert, however, this part of the book is very useful because it brings together all the steps required for the complete determination of the solution.
After the derivation of the disc of dust, the physical properties of the resulting one-parameter family of solutions are described, including its multipole moment structure, the existence of ergospheres, the Newtonian limit or the motion of test particles. Of particular interest is the transition from the disc of dust to the extreme black hole configuration corresponding to the limit when the parameter describing the fluid approaches its upper end.
After this chapter devoted to exact models, the book looks at the problem from a completely different point of view, namely by using numerical methods. This tool has proven to be fundamental for a proper study of this physical problem. This book concentrates on the so-called pseudo-spectral methods and the use of multidomains adapted to the different regions of the spacetime with qualitatively different behaviours. The presentation of the main ideas behind this method is very clear and accessible even to the non-expert. The book then is devoted to presenting both qualitative and quantitative results for a number of models with different equations of state. The case treated more in depth is the constant density case, but results for polytropic equations of state as well as a degenerate ideal gas of neutrons and strange quark matter are also presented. The emphasis is put on the exploration of the parameter space for a fixed equation of state. This is done by studying the various limiting cases involved, namely the non-rotating limit, the Newtonian limit, the mass-shedding limit, the infinite central pressure limit, the transition from one rotating body to several bodies, the black hole limit and the disc limit. The emerging picture in the constant density case is a division of the parameter space into an infinite number of classes, all connected through the Maclaurin spheroids and approaching the limiting case of a Maclaurin disc of dust, which in turn is the Newtonian limit of the relativistic disc of dust. Although the phase space of solutions differs for other equations of state, the main feature of having classes of solutions remains. Despite the inherent complexity and variety of possible behaviours, the authors manage to describe the results in a very lucid manner, and the resulting picture emerges very clearly. The presentation also includes many well-chosen figures, which clarify greatly the understanding of the results and makes this chapter very informative indeed. Furthermore, the book has a related webpage (http://www.tpi.uni-jena.de/gravity/relastro/rfe/) where the source codes for calculating various figures of equilibrium are publicly available.
Besides considering single fluids, configurations where a central and very compact object is surrounded by a ring of fluid are also treated to some extent. The central object may be a Newtonian point mass, a black hole or a rotating disc of dust. Special emphasis is put in studying the Komar mass of the central object, which is shown to be negative in several circumstances.
The book ends with a very brief description of stability of rotating configurations and a number of appendices summarizing some of the more technical material needed for the main body.
In summary, this book is a very valuable tool for anybody wishing to learn more about relativistic rotating bodies in equilibrium. The combination of exact analytic results and numerical methods makes it of particular interest, as both aspects are important in this field and their combined use gives rise to a much deeper understanding of the subject. The book contains many results and in general it is pleasant to read, the most arid part being the derivation of the disk of dust solution. The book is perhaps excessively brief at some places, but overall it is an excellent reference on this topic.
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