Ebook: Complex General Relativity

This book is written for theoretical and mathematical physicists and mat- maticians interested in recent developments in complex general relativity and their application to classical and quantum gravity. Calculations are presented by paying attention to those details normally omitted in research papers, for pedagogical r- sons. Familiarity with fibre-bundle theory is certainly helpful, but in many cases I only rely on two-spinor calculus and conformally invariant concepts in gravitational physics. The key concepts the book is devoted to are complex manifolds, spinor techniques, conformal gravity, ?-planes, ?-surfaces, Penrose transform, complex 3 1 – – space-time models with non-vanishing torsion, spin- fields and spin- potentials. 2 2 Problems have been inserted at the end, to help the reader to check his und- standing of these topics. Thus, I can find at least four reasons for writing yet another book on spinor and twistor methods in general relativity: (i) to write a textbook useful to - ginning graduate students and research workers, where two-component spinor c- culus is the unifying mathematical language.

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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. Original results include some geometrical properties of complex space-times with nonvanishing torsion, the Dirac operator with locally supersymmetric 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.
This book is written for students and research workers interested in classical gravity, quantum gravity and geometrical methods in field theory. It can also be recommended as a supplementary graduate textbook.

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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. Original results include some geometrical properties of complex space-times with nonvanishing torsion, the Dirac operator with locally supersymmetric 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.
This book is written for students and research workers interested in classical gravity, quantum gravity and geometrical methods in field theory. It can also be recommended as a supplementary graduate textbook.

Content:
Front Matter....Pages i-xii
Front Matter....Pages 1-1
Introduction to Complex Space-Time....Pages 2-16
Two-Component Spinor Calculus....Pages 17-29
Conformal Gravity....Pages 30-41
Front Matter....Pages 42-42
Twistor spaces....Pages 43-60
Penrose Transform for Gravitation....Pages 61-77
Front Matter....Pages 78-78
Complex Space-Times with Torsion....Pages 79-92
Spin-1/2 Fields in Riemannian Geometries....Pages 93-110
Spin-3/2 Potentials....Pages 111-135
Front Matter....Pages 136-136
Underlying Mathematical Structures....Pages 137-168
Back Matter....Pages 169-205

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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. Original results include some geometrical properties of complex space-times with nonvanishing torsion, the Dirac operator with locally supersymmetric 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.
This book is written for students and research workers interested in classical gravity, quantum gravity and geometrical methods in field theory. It can also be recommended as a supplementary graduate textbook.

Content:
Front Matter....Pages i-xii
Front Matter....Pages 1-1
Introduction to Complex Space-Time....Pages 2-16
Two-Component Spinor Calculus....Pages 17-29
Conformal Gravity....Pages 30-41
Front Matter....Pages 42-42
Twistor spaces....Pages 43-60
Penrose Transform for Gravitation....Pages 61-77
Front Matter....Pages 78-78
Complex Space-Times with Torsion....Pages 79-92
Spin-1/2 Fields in Riemannian Geometries....Pages 93-110
Spin-3/2 Potentials....Pages 111-135
Front Matter....Pages 136-136
Underlying Mathematical Structures....Pages 137-168
Back Matter....Pages 169-205
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