Complex General RelativityThis 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. |
Contents
| 3 | |
TWOCOMPONENT SPINOR CALCULUS | 17 |
CONFORMAL GRAVITY | 30 |
TWISTOR SPACES | 43 |
PENROSE TRANSFORM FOR GRAVITATION | 61 |
COMPLEX SPACETIMES WITH TORSION | 79 |
SPIN FIELDS IN RIEMANNIAN GEOMETRIES | 93 |
SPIN POTENTIALS | 111 |
UNDERLYING MATHEMATICAL STRUCTURES | 137 |
PROBLEMS FOR THE READER | 169 |
Sheaf Theory | 185 |
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Common terms and phrases
analysis boundary conditions boundary-value problem chapter complex conjugation complex manifold complex space-time conformal gravity conformal rescaling conformally invariant coordinates corresponding covariant derivative curved space-time D’Eath defined definition denote differential equations Dirac operator Einstein spaces Esposito field strengths field theory finds following Penrose gauge freedom geometrical global Hence holomorphic functions hypersurface hypersurface twistors implies integrability condition ISBN isomorphic Lorentzian mathematical metric Minkowski space-time Moreover non-vanishing null geodesics null twistors numbers obtained one-form one-loop Penrose and Rindler Penrose transform projective twistor space quantum cosmology real Riemannian Ricci identities Ricci tensor Riemann curvature scalar secondary potentials sheaf solution spin spinor fields spinor form spinor Ricci identities studied supersymmetry symmetric theorem three-sphere torsion trace-free twistor space twistor theory two-component spinor two-form two-spinor unprimed vanish vector bundle vector space virtue Ward Weyl equation Weyl spinor φα