# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Advanced general relativity. (English) Zbl 0752.53048
Cambridge Monographs on Mathematical Physics. Cambridge etc.: Cambridge University Press. viii, 228 p. (1990).

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)