Complex general relativity (Q1910349)

The present monograph is a tour de force of the progress and promise of complex methods in general relativity. Notwithstanding the author's suggestion and that of the publisher, it is not really a graduate level textbook, but rather an elegant survey of the material with particular emphasis on the research interests of the author. It presupposes a familiarity with the two-volume treatise of \textit{R. Penrose} and \textit{W. Rindler}, Spinors and space-time, Vols. 1 and 2 [Cambridge University Press (1984; Zbl 0538.53024) and (1986; Zbl 0591.53002)], and for such readers it offers valuable insight on their presentation. The monograph is divided into four major parts with the indicated chapter headings: Part I -- Spinor Form of General Relativity: complex spacetime, two-component spinors; conformal gravity; Part II -- Holomorphic Ideas in General Relativity: twistor spaces, the Penrose transform for gravitation; Part III -- Torsion and Supersymmetry: complex spacetimes with torsion, spin 1/2 fields in Riemannian geometry, spin 3/2 potentials; Part IV -- Mathematical Foundations: underlying mathematical structures. In addition, the book includes a set of problems; brief appendices devoted to: Clifford algebras, the Rarita-Schwinger equations, fibre bundles, and sheaf theory; and a 95 item bibliography. Part III, with the author's application of complex methods to boundary value problems, involves original work and is the major contribution of the book. In effect, most of the monograph really serves as preliminaries for Part III. However, this is not a criticism since his presentation of such background material offers an engaging and fascinating tour of the complex manifold and tensor program. The author's style is clear and precise, and the monograph should provide very stimulating reading for anyone interested in the subject, and as such it is a most welcome addition to the literature.