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General relativity and gravitation. One hundred years after the birth of Albert Einstein. Volumes 1, 2. (English) Zbl 0537.00011

A Publication of the International Society on General Relativity and Gravitation. New York-London: Plenum Press. Vol. 1: XIX, 598 p. $ 62.50; Vol. 2: XVIII, 540 p. $ 62.50 (1980).
These two volumes present an attempt to cover the significant activities in general relativity and gravitation by a series of review articles and research surveys. This has been done very successfully. The editors organized a comprehensive survey of the state of the art in this field of science. Their two-volume work can be of benefit to the advanced student as well as to the experienced researcher. The wide spectrum of topics discussed in both volumes may be characterized by some key-words: quantization and general relativity, gauge theories, supergravity, asymptotics, the twistor approach - complexification of general relativity and its applications, singularity theory, cosmology, gravitational wave detection etc. In the following each contribution to these two volumes will be described separately. In most cases the mathematical point of view will be emphasized. John Stachel (Einstein and the rigidly rotating disk, 1-15); Heinrich Hora (Eintein’s photon distribution for blackbodies and the discovery of the laser, 17-21): These first two articles have an historical flavor and therefore should only be indicated by their titles.
James Isenberg, James Nester (Canonical gravity, 23-97): Dirac’s canonical analysis (i.e. a mixture of action variational techniques and the \(3+1\) decompositions) is described and applied to gravity. Other field theories are studied from this point of view as well. This survey is complemented by many bibliographical hints.
Yvonne Choquet-Bruhat, James W. Yorkjr. (The Cauchy problem, 99- 172): Results on the Cauchy problem for Einstein’s equations are described or surveyed respectively. Special attention is given to the question of stability. Initial value problems are studied in detail and for certain examples. A comprehensive bibliography is given. Dieter R. Brill, Pong Soo Jang (The positive mass conjecture, 173-193): This is a report on progress made toward the proof of the positive mass conjecture. Several approaches and examples are discussed. Claudio Teitelboim (The Hamiltonian structure of space-time, 195-225): Recent developments are reviewed concerning the following problem: Find a way to clearly exhibit the four-dimensional Riemannian structure of the general theory of relativity without loosing track of the dynamically important degrees of freedom. Peter G. Bergmann, Arthur Komar (The phase space formulation of general relativity and approaches toward its canonical quantization, 227- 254); P. C. W. Davies (Quantum fields in curved space, 255-286): Both articles mainly are of interest for theoretical physicists. Andrzej Trautmann (Fiber bundles, gauge fields, and gravitation, 287-308); Yuval Ne’eman (Gravity, groups and gauges, 309-328); Friedrich W. Hehl, Jürgen Nitsch, Paul von der Heyde (Gravitation and the Poincaré gauge field theory with quadratic Lagrangian, 329-355): This is a series of contributions on gauge theories. It begins with a theoretical introduction, followed by several examples and applications in the second paper, and ends with a reformation of gravitational theory different from the Riemannian point of view.
S. Deser (From gravity to supergravity, 357-392): Thisis a very motivating introduction to the subject mentioned in the title and a comprehensive survey on the research in this rapidly developing area of mathematical physics. Sergio Benenti, Mauro Francavigli (The theory of separability of the Hamilton-Jacobi equation and its applications to general relativity, 393-439): Direct methods for the study of geodesics of a Lorentz manifold and their first integrals are developed. In this context recent research on the separability of Hamiltonian systems is reported and exhibited in some detail. About 100 references are given.
Hubert F. Goenner (Local isometric embedding of Riemannian manifolds and Einstein’s theory of gravitation, 441-468): This is a survey on the essential results on local isometric embeddings of space-times. To a minor extent the problem of global isometric embeddability is discussed too. The topics are selected with respect to possible applications in physics. (144 reference). For the last four items of volume I the description by their titles will be sufficient: Joshua N. Goldberg (Invariant transformations, conservation laws and energy-momentum, 469-489); R. A. D’Inverno (A review of algebraic computing in general relativity; 491-537); A. H. Taub (High frequency gravitational waves, two-timing and averaged Lagrangians, 539-555); S. Ferrara, P. van Nieuwenhuizen (Supergravity: an odyssey through space- time and superspace, 557-585). E. T. Newman, K. P. Tod (Asymptotically flat space-times, 1-36): In this first article of volume II an outline of the development of asymptotically flat space-times is given and some directions of related research are indicated. The NK tetrad is established and the spin coefficient equations are written in this tetrad. Asymptotic conditions are discussed giving asymptotic solutions to these equations. The Bondi-Metzner-Sachs group is reviewed and further methods of solution are presented.
Abhay Ashtekar (Asymptotic structure of the gravitational field at spatial infinity, 37-69): The ideas involved and the results having been obtained in one of the newer approaches for the subject mentioned in the title are given with special reference to the work of the author and R. O. Hansen. Other approaches are described on the base of this discussion. No detailed proofs or calculations are given.
Jeffrey Winicour (Angular momentum in general relativity, 71-96): State of the art report with 117 references. F. J. Tipler, C. J. S. Clarke, G. F. R. Ellis (Singularities and horizons - a review article, 97-206): This is an excellent report on the history and the current development of research concerning the two most interesting features in the large scale structure of space-times. For detailed exhibitions and comprehensive descriptions of the subject the reader is referred to the literature. Almost 500 references are compiled by the authors. Edward J. Flaherty jr. (Complex variables in relativity, 207- 239); C. P. Boyer, J. D. Finley III, F. J. Plebański (Complex general relativity, \({\mathfrak H}\), and \({\mathfrak HH}\) spaces - a survey of one approach, 241-281); R. Penrose, R. S. Ward (Twistors for flat and curved space-time, 283-328): These three chapters are devoted to the use of methods from complex analysis in relativity. The first paper describes the use of spinors in general relativity in connection with null tetrad methods and the role of holomorphic functions of complex variables in this context. In the second one the method of complexifying dynamical laws is discussed. The three wellknown approaches lead to the same class of geometrical objects, the \({\mathfrak H}\) spaces. Here the complexification of the field equation is chosen as the main approach. After a detailed study of \({\mathfrak H}\) spaces the \({\mathfrak HH}\) spaces are introduced and investigated as a generalization. In the third paper applications of twistor theory to special and general relativity are given, leading again to Newman’s \({\mathfrak H}\) space at the end.
E. P. T. Liang, R. K. Sachs; (Cosmology, 329-357): This is a brief survey of current cosmology, with particular reference to Einstein’s ideas: historical survey, geometry of the universe, cosmological models, closedness of the universe, beginning of the universe, content of the universe, current questions. The remaining articles of volume II are of minor interest for mathematicians and therefore only their titles are mentioned here: J. C. Miller, D. W. Sciama (Gravitational collapse to the black hole state, 359-391); L. P. Grishchuk, A. G. Polnarev (Gravitational waves and their interaction with matter and fields, 393-434); J. Weber (The search for gravitational radiation, 435-467); Irwin I. Shapiro (Experimental tests of the general theory of relativity, 469-490); W. Israel, J. M. Stewart (Progress in relativistic thermodynamics and electrodynamics of continuous media, 491- 525).
Reviewer: Bernd Wegner

MSC:

00Bxx Conference proceedings and collections of articles
53B50 Applications of local differential geometry to the sciences
53C80 Applications of global differential geometry to the sciences
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53D50 Geometric quantization
35Q99 Partial differential equations of mathematical physics and other areas of application
32Q99 Complex manifolds
53C05 Connections (general theory)
22E70 Applications of Lie groups to the sciences; explicit representations
83Exx Unified, higher-dimensional and super field theories
83D05 Relativistic gravitational theories other than Einstein’s, including asymmetric field theories
83-06 Proceedings, conferences, collections, etc. pertaining to relativity and gravitational theory
81T20 Quantum field theory on curved space or space-time backgrounds

Biographic References:

Einstein, Albert