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**Spinors and space-time. Vol. 2: Spinor and twistor methods in space-time geometry.**
*(English)*
Zbl 0591.53002

Cambridge Monographs on Mathematical Physics. Cambridge etc.: Cambridge University Press. IX, 501 p.; £45.00; $ 89.50 (1986).

After having waited twenty years for the first volume (1984; Zbl 0538.53024) of this work, the mathematical community is now fortunate in being treated to volume two after only a year. The complete work now provides an extensive and excellent treatment of spinor calculus and its many applications to the study of space-time geometry. The authors adopt the ”two-component spinor formalism”, i.e. explicitly use the (special to dimension 4) identification of the double cover of the Lorentz group as SL(2,\({\mathbb{C}})\). I find this to be a natural and helpful simplification of the general theory of spinors. It is extremely useful now to have a standard reference with all the formulae and conventions recorded once and for all. This is the definitive account.

These books, however, go much further than this. Not only do they stand as important reference works, but also they are highly readable. This property is heightened by the superb drawings. Mathematics is rather difficult to write down so as not to lose the ”true train of thought” (Felix Klein). The true train of thought is preserved in these volumes. This is particularly stimulating when one is confronted with the original thinking of Roger Penrose.

Whilst volume one is concerned with straight spinor analysis in 4- dimensional Lorentzian geometry, volume two now goes on to utilize this structure in the theory of twistors. After a summary of volume one, the first real chapter discusses twistors for Minkowski space. This theory culminates in the descriptions of massless fields (the Penrose transform) and Yang-Mills fields (the Ward correspondence) in which the field equations are absorbed into the geometry in a remarkable way.

The next chapter discusses null congruences (an important motivation in the original development of twistor theory with a view to extending twistor theory to curved space-times. It is discovered that such congruences are governed by the Weyl curvature and the following chapter describes the Petrov classification of the Weyl tensors and their geometric significance. Similarly for the Ricci tensor (giving 41 different types !).

The final chapter of volume two is concerned with the asymptotic structure of space-time. It is studied by means of a ”conformal infinity”. Again there is strong input from twistor theory.

In summary, volume two is a worthy successor to volume one. Together they constitute an absolute goldmine of information and geometric insight.

These books, however, go much further than this. Not only do they stand as important reference works, but also they are highly readable. This property is heightened by the superb drawings. Mathematics is rather difficult to write down so as not to lose the ”true train of thought” (Felix Klein). The true train of thought is preserved in these volumes. This is particularly stimulating when one is confronted with the original thinking of Roger Penrose.

Whilst volume one is concerned with straight spinor analysis in 4- dimensional Lorentzian geometry, volume two now goes on to utilize this structure in the theory of twistors. After a summary of volume one, the first real chapter discusses twistors for Minkowski space. This theory culminates in the descriptions of massless fields (the Penrose transform) and Yang-Mills fields (the Ward correspondence) in which the field equations are absorbed into the geometry in a remarkable way.

The next chapter discusses null congruences (an important motivation in the original development of twistor theory with a view to extending twistor theory to curved space-times. It is discovered that such congruences are governed by the Weyl curvature and the following chapter describes the Petrov classification of the Weyl tensors and their geometric significance. Similarly for the Ricci tensor (giving 41 different types !).

The final chapter of volume two is concerned with the asymptotic structure of space-time. It is studied by means of a ”conformal infinity”. Again there is strong input from twistor theory.

In summary, volume two is a worthy successor to volume one. Together they constitute an absolute goldmine of information and geometric insight.

Reviewer: M.Eastwood

### MSC:

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

53C27 | Spin and Spin\({}^c\) geometry |

53B50 | Applications of local differential geometry to the sciences |

83C30 | Asymptotic procedures (radiation, news functions, \(\mathcal{H} \)-spaces, etc.) in general relativity and gravitational theory |

53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |