Complex General Relativity

Complex General Relativity

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English

Description

This volume introduces the application of two-component spinor calculus and fibre-bundle theory to complex general relativity. A review of basic and important topics is presented, such as two-component spinor calculus, conformal gravity, twistor spaces for Minkowski space-time and for curved space-time, Penrose transform for gravitation, the global theory of the Dirac operator in Riemannian four-manifolds, various definitions of twistors in curved space-time and the recent attempt by Penrose to define twistors as spin-3/2 charges in Ricci-flat space-time. Results include some geometrical properties of complex space-times with non-vanishing torsion, the Dirac operator with locally super-symmetric boundary conditions, the application of spin-lowering and spin-raising operators to elliptic boundary value problems, and the Dirac and Rarita-Schwinger forms of spin-3/2 potentials applied in real Riemannian four-manifolds with boundary.

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Published 01 January 1995
Reads 6
EAN13 0306471183
License: All rights reserved
Language English

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T A B L E O F C O N T E N T S
P R E F A C E. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . xi
PART I: SPINOR FORM OF GENERAL RELATIVI.T .Y . . . . . . . . . . . . .1
1. I N T R O D U C T I O N T O C O M P L E X S P A C E  T I M E. . . . . . . . . . . . . . 2 1.1 From Lorentzian SpaceTime to Complex SpaceTime . . . . . . . . . . . . . . . . 3 1.2Complex Manifolds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 An Outline of This Work . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.T W O  C O M P O N E N T C A L C U L U SS P I N O R . . . . . . . . . . . . . . . 17 2.1. . . . . . . . . . . . . . . . . . . . . . . . . . . TwoComponent Spinor Calculus 18 2.224. . . . . . . . . . . . . . . . . . . Curvature in General Relativity 2.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Petrov Classification
3. C O N F O R M A L G R A V I T Y30. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1CSpaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Einstein Spaces . . . 33 3.3Complex SpaceTimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.4Complex Einstein Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.540. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conformal Infinity
PART I I : HOLOMORPHI C I DEAS I N GENERAL RELATI VI T. .Y. 42
4. T W I S T O R S P A C E S43. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.145. . . . . . . . . . . . . . . . . . . . . . . . . Planes in Minkowski SpaceTime 4.250Surfaces and Twistor Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.353Geometrical Theory of Partial Differential Equations . . . . . . . . . . . . .
5. PENROSE TRANSF ORM F OR GRAVI TATI ON. . . . . . . 61 5.1 AntiSelfDual SpaceTimes . . . . . . . . . . . . . . . . . . . . . . . . 62 5.2 Beyond AntiSelfDuality . . . . . . . . . . . . . . . . . . . . . . . . . 68
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3 5.3Twistors as Spin – Charges. 2
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6. C O M P L E X S P A C E  T I M E S W I T H T O R S I O N. . . . . . . . . . .7 9 6.1Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8 0 8 2 6.2 Frobenius’ Theorem for Theories with Torsion . . . . . . . . . . . . . . . . . 6.3 Spinor Ricci Identities for ComplexU4Theory . . . . . . . . . . . . . . . . . . . 8 6 6.4 Integrability Condition forSurfaces. . . . . . . . . . . . . . . . . . . . . . . .90 6.5 Concluding Remarks.. . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 0
1 9 3 7. S P I N – FIELDS IN RIEMANNIAN GEOMETRIES. . . . . . . . . 2 7.1 Dirac and Weyl Equations in TwoComponent Spinor Form . . . . . . . . . . . 94 7.2Boundary Terms for Massless Fermionic Fields . . . . . . . . . . . . . . . . . . . . 95 7.3 SelfAdjointness of the BoundaryValue Problem . . . . . . . . . . . . . . . . . 100 7.4 Global Theory of the Dirac Operator . . . . . . . . . . . . . . . . . . . . . . . 106
3 8 . SPIN – POTENTIALS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 2 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 8.2 SpinLowering Operators in Cosmology . . . . . . . . . . . . . . . . . . . . . 113 8.3 SpinRaising Operators in Cosmology . . . . . . . . . . . . . . . . . . . .115 3 8.4Dirac’s Spin Potentials in Cosmology117. . . . . . . . . . . . . . . . . . . . . 2 8.5 Boundary Conditions in Supergravity . . . . . . . . . . . . . . . . . . . .121 8.6 RaritaSchwinger Potentials and Their Gauge Transformations . . . . . .124 8.7 Compatibility Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .125 126 8.8 Admissible Background FourGeometries . . . . . . . . . . . . . . . . . . . . . 8.9 Secondary Potentials in Curved Riemannian Backgrounds . . . . . . . . . .128 8.10 Results and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . 130
P A R T I V : MATHEMATI CAL FOUNDATI ONS. . . . . . . . . . . . .136
. . . . . . . .137 9 . U N D E R L Y I N G M A T H E M A T I C A L S T R U C T U R E S 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .138
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9.2140Local Twistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Global Null Twistors . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 9.4Hypersurface Twistors. . . . . . . . . . . . . . . . . . . .145 9.5Asymptotic Twistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 9.6Penrose Transform . . . . . . . . . . . . . . . . . . . . . . . . . . .152 9.7Ambitwistor Correspondence157. . . . . . . . . . . . . . . . . . . . . . . 9.8Radon Transform. . . . . . . . . . . . . . . . . . . . .159 9.9 Massless Fields as Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . 159 9.10162 Quantization . . . . . . . . . . . . . . . . . . . . . . of Field Theories
P R O B L E M S F O R T H E R E A D E R. . . . . . . . . . . . . . . . . . . . . . 169
APPENDIX A:Clifford Algebras
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APPENDIX B:RaritaSchwinger Equations
APPENDIX C:Fibre Bundles
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APPENDIX D:Sheaf Theory . 185. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B I B L I O G R A P H Y. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .187
SUBJECT INDEX. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
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