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2 results

  • 9 - Higher-Dimensional Descriptions

  • Kirill Krasnov, University of Nottingham
  • Book:
    Formulations of General Relativity
    Published online:
    06 November 2020
    Print publication:
    26 November 2020, pp 304-359

  • Summary

    This Chapter describes the geometry of twistor space of a 4-dimensional manifold. We motivated the twistor space as a geometrical construction that realisesthe action of the conformal group in 4D as the direct analog of that in 2D. This explains why the coordinates of a 4D space can be naturally put into a 2x2 matrix. We describe both the complexified version of the twistor space, as well as treat all the 3 possible signatures in detail. We then specialise to the case of Euclidean twistors, and describe how the twistor space can be interpreted as the total space of the bundle of almost complex structures of a 4D Riemannian manifold. Quaternionic Hopf fibration and its relation to the Euclidean twistor space is desccribed. We then describe the geometry of 3-forms in seven dimensions, and describe two different G2 structures on the 7-sphere. We end with a description of a lift of the usual twistor construction of integrable almost complex structures into seven dimensions. This is based on the notion of nearly parallel G2 structures.

Formulations of General Relativity
  • Gravity, Spinors and Differential Forms
  • Kirill Krasnov
  • Published online:
    06 November 2020
    Print publication:
    26 November 2020
  • This monograph describes the different formulations of Einstein's General Theory of Relativity. Unlike traditional treatments, Cartan's geometry of fibre bundles and differential forms is placed at the forefront, and a detailed review of the relevant differential geometry is presented. Particular emphasis is given to general relativity in 4D space-time, in which the concepts of chirality and self-duality begin to play a key role. Associated chiral formulations are catalogued, and shown to lead to many practical simplifications. The book develops the chiral gravitational perturbation theory, in which the spinor formalism plays a central role. The book also presents in detail the twistor description of gravity, as well as its generalisation based on geometry of 3-forms in seven dimensions. Giving valuable insight into the very nature of gravity, this book joins our highly prestigious Cambridge Monographs in Mathematical Physics series. It will interest graduate students and researchers in the fields of theoretical physics and differential geometry.