Formulations of General Relativity : Gravity, Spinors and Differential Forms

Contents
Preface
Introduction
1. Aspects of Differential Geometry
	1.1 Manifolds
	1.2 Differential Forms
	1.3 Integration of Differential Forms
	1.4 Vector Fields
	1.5 Tensors
	1.6 Lie Derivative
	1.7 Integrability Conditions
	1.8 The Metric
	1.9 Lie Groups and Lie Algebras
	1.10 Cartan’s Isomorphisms
	1.11 Fibre Bundles
	1.12 Principal Bundles
	1.13 Hopf Fibration
	1.14 Vector Bundles
	1.15 Riemannian Geometry
	1.16 Spinors and Differential Forms
2. Metric and Related Formulations
	2.1 Einstein–Hilbert Metric Formulation
	2.2 Gamma–Gamma Formulation
	2.3 Linearisation
	2.4 First-Order Palatini Formulation
	2.5 Eddington–Schrödinger Affine Formulation
	2.6 Unification: Kaluza–Klein Theory
3. Cartan’s Tetrad Formulation
	3.1 Tetrad, Spin Connection
	3.2 Einstein–Cartan First-Order Formulation
	3.3 Teleparallel Formulation
	3.4 Pure Connection Formulation
	3.5 MacDowell–Mansouri Formulation
	3.6 Dimensional Reduction
	3.7 BF Formulation
4. General Relativity in 2+1 Dimensions
	4.1 Einstein–Cartan and Chern–Simons Formulations
	4.2 The Pure Connection Formulation
5. The ‘Chiral’ Formulation of General Relativity
	5.1 Hodge Star and Self-Duality in Four Dimensions
	5.2 Decomposition of the Riemann Curvature
	5.3 Chiral Version of Cartan’s Theory
	5.4 Hodge Star and the Metric
	5.5 The ‘Lorentz’ Groups in Four Dimensions
	5.6 The Self-Dual Part of the Spin Connection
	5.7 The Chiral Soldering Form
	5.8 Plebański Formulation of GR
	5.9 Linearisation of the Plebański Action
	5.10 Coupling to Matter
	5.11 Historical Remarks
	5.12 Alternative Descriptions Related to Plebański Formalism
	5.13 A Second-Order Formulation Based on the 2-Form Field
6. Chiral Pure Connection Formulation
	6.1 Chiral Pure Connection Formalism for GR
	6.2 Example: Page Metric
	6.3 Pure Connection Description of Gravitational Instantons
	6.4 First-Order Chiral Connection Formalism
	6.5 Example: Bianchi I Connections
	6.6 Spherically Symmetric Case
	6.7 Bianchi IX and Reality Conditions
	6.8 Connection Description of Ricci Flat Metrics
	6.9 Chiral Pure Connection Perturbation Theory
7. Deformations of General Relativity
	7.1 A Natural Modified Theory
8. Perturbative Descriptions of Gravity
	8.1 Spinor Formalism
	8.2 Spinors and Differential Operators
	8.3 Minkowski Space Metric Perturbation Theory
	8.4 Chiral Yang–Mills Perturbation Theory
	8.5 Minkowski Space Chiral First-Order Perturbation Theory
	8.6 Chiral Connection Perturbation Theory
9. Higher-Dimensional Descriptions
	9.1 Twistor Space
	9.2 Euclidean Twistors
	9.3 Quaternionic Hopf Fibration
	9.4 Twistor Description of Gravitational Instantons
	9.5 Geometry of 3-Forms in Seven Dimensions
	9.6 G_2-Structures on S^7
	9.7 3-Form Version of the Twistor Construction
10. Concluding Remarks
References
Index