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Complex General Relativity

The statement is made by the author that Complex General Relativity is written for theoretical and mathematical physicists and mathematicians to describe recent developments in the area of complex general relativity and their application to classical and quantum gravity. It succeeds in doing this well in the area of complex relativity that deals with spinor and twistor techniques and discusses themes mainly first developed by Roger Penrose and co-authors. It is a textbook for lectures at a graduate level and, as such, there are 28 problems at the end of the book covering material throughout the book. But it is more than a textbook written for a wide audience and is a handy reference book for researches in the areas covered. The author sets out to (i) present a self-consistent mathematical theory of complex spacetimes with non-vanishing torsion and (ii) to present the first application to boundary-value problems in cosmology of the Penrose formalism for spin-3/2 potentials. The key concepts discussed are complex manifolds, spinor techniques, conformal gravity, -planes, -surfaces, Penrose transform, complex spacetime models with non-vanishing torsion, spin-1/2 fields and spin-3/2 potentials. The work on spin-3/2 potentials is well developed and interesting to read; it includes a discussion of various cosmological applications. It is backed up by an extensive chapter on mathematical foundations which contains various definitions of the concept of a twistor, namely local twistors, global null twistors, hypersurface twistors and asymptotic twistors. A number of summaries of more mathematical topics are left to the appendices. The one disappointment with the book is that although it is called Complex General Relativity it really only covers one area of this large field. It does not discuss, for example, any of the work of Ted Newman and others, particularly in the 1970s and 1980s, about H-spaces and related problems, nor the work of Plebañski, Ivor Robinson and co-workers about the same time on the hyperheavenly wave equation and problems of exact solutions, nor the work of Mark Hickman and the reviewer in the 1980s along similar lines. It really only covers the twistor approach of Roger Penrose and co-workers. Perhaps the title of the book could have been slightly narrower and included the word `twistor' rather than the wider title Complex General Relativity.