Gen Relativ Gravit (2010) 42:1341–1343 DOI 10.1007/s10714-010-0955-y BOOK REVIEW

Reinhard Meinel, Marcus Ansorg, Andreas Kleinwächter, Gernot Neugebauer and David Petroff: Relativistic figures of equilibrium Cambridge University Press, 2008, 218p., GBP 73.00, USD 129.00. ISBN-13: 9780521863834 Eva Hackmann · Claus Lämmerzahl Received: 15 February 2010 / Accepted: 24 February 2010 / Published online: 14 March 2010 © Springer Science+Business Media, LLC 2010

One of the most difficult and advanced problems in General Relativity is to find the gravitational field of an isolated object like a star or a planet, a galaxy, a body with an accretion disc, or a compact body like a neutron star. This problem is of great theoretical importance and also essential for a deeper insight into astrophysical phenomena. Here the book Relativistic figures of equilibrium by R. Meinel, M. Ansorg, A. Kleinwächter, G. Neugebauer and D. Petroff gets involved. The authors address in depth the difficult question of solutions of Einstein’s field equations with different models of localized matter using highly advanced analytical as well as numerical methods. Even for the idealized matter models treated in this book the mathematical challenge is immense. The results of this book are of importance for a better understanding of the state and of the dynamics of astrophysical objects, which will influence, among others, future high precision observations and the calculation of gravitational wave templates. The long lasting experience of the authors from the “Theoretisch-Physikalisches Institut” of the Friedrich Schiller University Jena, who are well-known experts in the field of analytic as well as numerical methods in gravity theory, is summarized in this advanced book. Divided into three main chapters, the book first introduces the problem and the basic notions and equations. The second chapter explains and applies analytical methods for limiting cases with a focus on the rigidly rotating disc of dust solution of Einstein’s field equations found by two of the authors [G. Neugebauer and R. Meinel, Phys. Rev. Lett. 75, 3046 (1995)], and the third one gives a numerical treatment of the general case of three-dimensional rotating bodies.

E. Hackmann (B) · C. Lämmerzahl ZARM, University of Bremen, Am Fallturm, 28359 Bremen, Germany e-mail: [email protected] C. Lämmerzahl e-mail: [email protected]

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In the first chapter the equations and the physical properties of the stationary axisymmetric perfect fluid bodies treated in this book are presented. Different choices of the energy–momentum tensor and of the equation of state are discussed, ranging from homogeneous fluids over a completely degenerate ideal gas of neutrons and strange quark matter to the disc of dust. Furthermore, limiting cases of stationary and axisymmetric perfect fluid bodies are described, namely the Newtonian, the static, the disc, and the mass-shedding limit as well as the transition to a black hole. The second chapter is devoted to the analytic treatment of some of the limiting cases presented in the previous chapter. First, the Newtonian limit, i.e. Maclaurin spheroids, with its special cases of a sphere and a disc, and second the static limit, i.e. Schwarzschild spheres, are considered. As these cases are well known, the authors keep their explanations short. As a third limiting case the rigidly rotating disc of dust is discussed at length. The problem is reduced to a boundary value problem of the Ernst equation which can be tackled using the “inverse method”. This ansatz is based on a corresponding linear problem which is solved by an integration along the symmetry axis, the disc, and infinity. In turn, the complex Ernst potential is calculated first on the disc and then in the whole spacetime using the solution of Jacobi’s inversion problem and hyperelliptic theta functions. Although the solution steps are explained in some detail, this is an advanced topic and a reader not familiar with the inverse method or with theta functions will it find difficult to follow. The same holds for the subsequent discussion of the mathematical properties of the disc of dust solution including some reformulation and computations of the involved theta functions as well as the explicit representation of the resulting metric on the disc, on the axis of the disc, and on the symmetry axes. Also the physical properties are discussed in detail. Exact expressions for physical quantities as mass, angular momentum and surface mass-density are derived and the Newtonian limit is considered. In addition, the multipole moments of the rigidly rotating disc of dust are calculated and the appearance of an ergosphere depending on the surface mass-density is explained. An interesting part is also the analysis of test particle motion inside the plane of the disc with a detailed study of circular motion, which also provides some elucidating figures. To complete the discussion of the rigidly rotating disc of dust, the Kerr black hole limit for vanishing (Weyl-)coordinate radius of the disc is also considered. Finally, the methods presented so far are being applied to derive the Kerr metric. It is shown that under certain conditions the extreme Kerr solution is the only axisymmetric space–time with a degenerate horizon. The general case of perfect fluids in equilibrium is too complicated for an analytic treatment and, thus, considered from a numerical point of view in the third chapter. After presenting the pseudo-spectral methods, which were used in the following for the ordinary differential equation of second order with boundary condition, the partition of the space–time in multiple subdomains depending on the spheroidal or toroidal configuration is explained. In the remainder of the chapter the methods that have been established this way, are applied to different matter models, starting with homogeneous bodies. On the basis of bifurcation points of the Newtomian limit and the study of the other limiting cases—i.e. the static, the infinite central pressure, the mass-shedding, the extreme Kerr black hole, and the two-body limit—the general case of a single homogeneous body is divided in infinitely, but countably, many classes the first few of which are examined in detail. Building on that, polytropic configurations,

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a completely degenerate Fermi gas of neutrons, and strange matter are discussed in a similar way, maintaining a class structure. The presentation of the classes of solutions is very clear and based on a great number of explanatory figures. The chapter closes with a discussion of central masses surrounded by a fluid ring, including unfamiliar features like a negative Komar mass of a central black hole or a disc of dust. A brief discussion of stability properties of perfect fluid configuration in equilibrium with respect to small perturbations in chapter 4 and a number of appendices with calculational details and supplementary material completes this book. It should also be mentioned that the computer codes used to calculate figures of equilibrium are available at http://www.tpi.uni-jena.de/gravity/relastro/rfe/. In summary, this is an advanced and specialized book presenting the state of the art in this field, with emphasis on the authors’ own expertise and approaches. It contains an extensive discussion of the analytical handling of limiting cases on the one hand and the numerical treatment of the general case on the other, thus providing a complete picture of the subject. Although some aspects could have been more elaborated, the book is pleasant to read. This book is a standard reference for this subject every researcher in relativity theory should know.

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