Equations of Motion in General Relativity

Abstract

Devoted exclusively to the problem of motion in general relativity, this book by H. Asada, T. Futamase, and P. A. Hogan is highly welcome to close up a gap in the book sector presenting a concise account of theoretical developments and results on gravitational equations of motion achieved since the discovery of the binary neutron star system PSR 1913+16 in 1974.For the most part, the book is concerned with the development and application of the important post-Newtonian approximation (PNA) framework which allows for highly efficient approximate analytic solutions of the Einstein field equations for many-body systems in terms of a slow-motion and weak-field ordering parameter. That approximation scheme is shown to be applicable also to the external motion of strongly self-gravitating objects if their internal dynamics is frozen in (strong field point particle limit) and the external conditions fit.Relying on the expertise of the authors, the PNA framework is presented in a form which, at the 1PNA level, had become famous through the work by Einstein, Infeld and Hoffmann in 1938; therein, surface integrals over gravitational field expressions in the outside-body regime play a crucial role. Other approaches which also succeeded with the highest achieved PNA level so far are mentioned too, if not fully exhaustively with respect to the highest, the 3.5PNA level which contains the inverse power of the speed of light to the seventh order. Regarding the 3PNA, the reader gains a clear understanding of how the equations of motion for binary systems with compact components come about. Remarkably, no deviation from four-dimensional space-time is needed. Various explicit analytic expressions are derived for binary systems: the periastron advance and the orbital period at the 2PNA, the orbital decay through gravitational radiation reaction at the 2.5PNA, and effects of the gravitational spin–orbit and spin–spin couplings on the orbital motion. Also the propagation of light-like signals in gravitational fields of binary systems is treated, which is important for a precise interpretation of pulsar observation measurements.Based on original research by the authors, a detailed presentation is given of a mathematical scheme which makes feasible the treatment of small black holes in background space-times. Using that approach, the equations of motion of small charged black holes are derived in vacuum Einstein–Maxwell space-times without encountering infinities, showing up electromagnetic radiation reaction, background field, and tail forces. At this stage, the book defines various issues to be tackled in future research within the given formalism, such as a physical understanding of the very structure of the tail contribution or a more detailed calculation of the motion of a small Schwarzschild black hole in an external vacuum gravitational field.A unique chapter is devoted to the choreographic three-body solution of the 1PNA dynamics, also mentioning the 2PNA dynamics, in the form of figure-eight configurations. The emitted gravitational waves are shown, and a discussion of their observability in future gravitational wave astronomy is given.Four appendices support and complement the main part of the book. Here, the far zone non-contribution to the equations of motion at 3PNA is carefully investigated, forces resulting from the extendedness of objects are presented in analytic form, null geodesic congruences with their optical scalars are discussed, and the perturbed vacuum Einstein–Maxwell field equations are given.The book delivers a very readable account of the problem of motion in general relativity. It covers the state of the art up to the years 2006/8 and presents a plethora of interesting and important topics and results. Whenever appropriate, connection with observation is made. A reader with good post-introductory knowledge of the theory of general relativity should find easy access to the book, and will surely benefit from the mathematical and conceptual issues elaborated on therein. Being of observational relevance too, the book fits into every library which provides a collection of advanced books in physics-applied mathematics, gravitational physics, astronomy, or astrophysics.