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E-raamat: Formulations of General Relativity: Gravity, Spinors and Differential Forms

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(University of Nottingham)
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Formulations of General Relativity: Gravity, Spinors and Differential FormsSuurem pilt
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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.

Arvustused

'The work is mathematically rigorous and complete. Researchers working in fields such as quantum gravity will find this a very useful reference. Postgraduate students will also find it a helpful adjunct to the usual books on general relativity.' A. Spero, Choice ' in this book Krasnov deals mainly with chiral formulations of four-dimensional general relativity, a topic that usually is not present in most (if not all) of the relevant textbooks, thereby bringing to the forefront the importance of Cartan's calculus of differential forms, the geometry of fibre bundles, but also spinors and (finally) twistors.' Theophanes Grammenos, MathSciNet

Muu info

Documenting the different formulations of general relativity, this book reveals valuable insight into the nature of the gravitational force.
Preface xiii
Introduction 1(7)
1 Aspects of Differential Geometry
8(70)
1.1 Manifolds
8(4)
1.2 Differential Forms
12(6)
1.3 Integration of Differential Forms
18(2)
1.4 Vector Fields
20(5)
1.5 Tensors
25(3)
1.6 Lie Derivative
28(5)
1.7 Integrability Conditions
33(1)
1.8 The Metric
34(4)
1.9 Lie Groups and Lie Algebras
38(11)
1.10 Cartan's Isomorphisms
49(2)
1.11 Fibre Bundles
51(4)
1.12 Principal Bundles
55(7)
1.13 Hopf Fibration
62(3)
1.14 Vector Bundles
65(5)
1.15 Riemannian Geometry
70(3)
1.16 Spinors and Differential Forms
73(5)
2 Metric and Related Formulations
78(11)
2.1 Einstein-Hilbert Metric Formulation
78(2)
2.2 Gamma-Gamma Formulation
80(3)
2.3 Linearisation
83(3)
2.4 First-Order Palatini Formulation
86(1)
2.5 Eddington-Schrodinger Affine Formulation
87(1)
2.6 Unification: Kaluza-Klein Theory
88(1)
3 Cartan's Tetrad Formulation
89(36)
3.1 Tetrad, Spin Connection
91(13)
3.2 Einstein-Cartan First-Order Formulation
104(1)
3.3 Teleparallel Formulation
105(2)
3.4 Pure Connection Formulation
107(2)
3.5 MacDowell-Mansouri Formulation
109(3)
3.6 Dimensional Reduction
112(2)
3.7 BF Formulation
114(11)
4 General Relativity in 24-1 Dimensions
125(7)
4.1 Einstein-Cartan and Chern-Simons Formulations
125(4)
4.2 The Pure Connection Formulation
129(3)
5 The `Chiral' Formulation of General Relativity
132(60)
5.1 Hodge Star and Self-Duality in Four Dimensions
133(1)
5.2 Decomposition of the Riemann Curvature
133(4)
5.3 Chiral Version of Cartan's Theory
137(3)
5.4 Hodge Star and the Metric
140(11)
5.5 The `Lorentz' Groups in Four Dimensions
151(9)
5.6 The Self-Dual Part of the Spin Connection
160(3)
5.7 The Chiral Soldering Form
163(8)
5.8 Plebanski Formulation of GR
171(3)
5.9 Linearisation of the Plebanski Action
174(6)
5.10 Coupling to Matter
180(2)
5.11 Historical Remarks
182(1)
5.12 Alternative Descriptions Related to Plebanski Formalism
183(4)
5.13 A Second-Order Formulation Based on the 2-Form Field
187(5)
6 Chiral Pure Connection Formulation
192(58)
6.1 Chiral Pure Connection Formalism for GR
192(19)
6.2 Example: Page Metric
211(7)
6.3 Pure Connection Description of Gravitational Instantons
218(5)
6.4 First-Order Chiral Connection Formalism
223(1)
6.5 Example: Bianchi I Connections
224(8)
6.6 Spherically Symmetric Case
232(5)
6.7 Bianchi IX and Reality Conditions
237(4)
6.8 Connection Description of Ricci Flat Metrics
241(6)
6.9 Chiral Pure Connection Perturbation Theory
247(3)
7 Deformations of General Relativity
250(5)
7.1 A Natural Modified Theory
250(5)
8 Perturbative Descriptions of Gravity
255(49)
8.1 Spinor Formalism
256(6)
8.2 Spinors and Differential Operators
262(12)
8.3 Minkowski Space Metric Perturbation Theory
274(1)
8.4 Chiral Yang-Mills Perturbation Theory
275(5)
8.5 Minkowski Space Chiral First-Order Perturbation Theory
280(15)
8.6 Chiral Connection Perturbation Theory
295(9)
9 Higher-Dimensional Descriptions
304(56)
9.1 Twistor Space
306(12)
9.2 Euclidean Twistors
318(11)
9.3 Quaternionic Hopf Fibration
329(6)
9.4 Twistor Description of Gravitational Instantons
335(2)
9.5 Geometry of 3-Forms in Seven Dimensions
337(6)
9.6 G2-Structures on S7
343(12)
9.7 3-Form Version of the Twistor Construction
355(5)
10 Concluding Remarks
360(5)
References 365(4)
Index 369
Kirill Krasnov is Professor of Mathematical Physics at the University of Nottingham. Since receiving his Ph.D. from Pennsylvania State University, he has worked at the University of California, Santa Barbara, and the Max Planck Institute for Gravitational Physics in Potsdam.
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