Conclusion of Volume 1
In the first volume we have presented the theory of General Relativity comparing it at all times with the other Gauge Theories that describe non-gravitational interactions. We have also followed the complicated historical development of the ideas and of t
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Conclusion of Volume 1
In the first volume we have presented the theory of General Relativity comparing it at all times with the other Gauge Theories that describe non-gravitational interactions. We have also followed the complicated historical development of the ideas and of the concepts underlying both of them. In particular we have traced back the origin of our present understanding of all fundamental interactions as mediated by connections on principal fibre-bundles and emphasized the special status of Gravity within this general scheme. While recalling the historical development we have provided a, hopefully rigorous, exposition of all the mathematical foundations of gravity and gauge theories in a contemporary geometrical approach. In the last two chapters of Volume 1 we have considered relevant astrophysical applications of General Relativity that also provide some of the most accurate tests of its predictions. In Chap. 6 we considered stellar equilibrium and the mass-limits which combine General Relativity and Quantum Mechanics. In Chap. 7 we considered the emission of gravitational waves and the stringent tests of Einstein’s theory that come from the binary pulsar systems. The further historical and conceptual development of the theory is addressed in Volume 2 which covers the following topics: 1. 2. 3. 4. 5. 6. 7. 8. 9.
Extended Space-Times, Causal Structure and Penrose Diagrams. Rotating Black-Holes and Thermodynamics. Cosmology and General Relativity: From Hubble to WMAP. The theory of the inflationary universe. The birth of String Theory and Supersymmetry. The conceptual and algebraic foundations of Supergravity. An introduction to the Bulk-Brane dualism with a glance at brane solutions. An introduction to the Supergravity Bestiary. A bird-eye review of various type of solutions of higher dimensional supergravities.
P.G. Frè, Gravity, a Geometrical Course, DOI 10.1007/978-94-007-5361-7_8, © Springer Science+Business Media Dordrecht 2013
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Conclusion of Volume 1
Appendix A: Spinors and Gamma Matrix Algebra A.1 Introduction to the Spinor Representations of SO(1, D − 1) The spinor representations of the orthogonal and pseudo-orthogonal groups have different structure in various dimensions. Starting from the representation of the Dirac gamma matrices one begins with a complex representation whose dimension is equal to the dimension of the gammas. A vector in this complex linear space is named a Dirac spinor. Typically Dirac spinors do not form irreducible representations. Depending on the dimensions, one can still impose SO(1, D − 1) invariant conditions on the Dirac spinor that separate it into irreducible parts. These constraints can be of two types: (a) A reality condition which maintains the number of components of the spinor but relates them to their complex conjugates by means of linear relations. This reality condition is constructed with an invariant matrix C , named the charge conjugation matrix whose properties depend on the dimensions D. (b) A chirality condition constructed wit
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