*(English)*Zbl 0752.53048

The study of gravitational radiation from isolated sources is one of the most important fields in general relativity. In particular, this field leads to the study of asymptotically flat spacetimes, which further relates general relativity to the rest of physics. Among the principal investigators of this theory, are Bondi, Dirac, Geroch, Goldberg-Sacks and Newman-Penrose. In modern context, the basic mathematical structures, used in this field, are the spinor calculus and the Newman-Penrose formalism.

The book under review is an introduction to the theoretical aspects of the gravitational radiation and asymptotically flat spacetimes. In Chapter 1, the author provides the necessary information on the local differential geometry needed in the rest of the book. Chapter 2 is devoted to the study of 2-component spinors (Dirac 4-spinor is presented in the appendix) and the related Newman-Penrose formalism. This is supported by several examples, in particular, reference to the zero-rest mass fields. Chapter 3 is the core of the subject, where the description of the gravitational field is presented in the arena of an asymptotically flat spacetime. The key idea of this chapter, is to rescale the metric by a conformal factor going to zero at infinity. In Chapter 4, the author presents (mainly for physicists) an introduction to the characteristic initial value problem and its relationship with the theory of singularity. The objective is to make the book self-contained and reasonably well informative.

The book is very well written and has quite a collection of references on the subject. It has a systematic and natural mainstream development of all the four chapters. This book should be a valuable contribution to an important field in general relativity.

##### MSC:

53Z05 | Applications of differential geometry to physics |

53C27 | Spin and Spin${}^{c}$ geometry |

83C30 | Asymptotic procedures (general relativity) |

83C60 | Spinor and twistor methods in general retativity; Newman-Penrose formalism |

83-01 | Textbooks (relativity) |