Spinors and Space-Time: Volume 1, Two-Spinor Calculus and Relativistic Fields

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Cambridge University Press, Feb 5, 1987 - Mathematics - 472 pages
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This volume introduces and systematically develops the calculus of 2-spinors. This is the first detailed exposition of this technique which leads not only to a deeper understanding of the structure of space-time, but also provides shortcuts to some very tedious calculations. Many results are given here for the first time.
 

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Contents

The geometry of worldvectors and spinvectors
1
12 Null directions and spin transformations
8
13 Some properties of Lorentz transformations
24
14 Null flags and spinvectors
32
15 Spinorial objects and spin structure
41
16 The geometry of spinor operations
56
Abstract indices and spinor algebra
68
22 The abstractindex formalism for tensor algebra
76
49 Spinor form of commutators
242
410 Spinor form of the Bianchi identity
245
411 Curvature spinors and spincoefficients
246
412 Compacted spincoefficient formalism
250
413 Cartans method
262
414 Applications to 2surfaces
267
415 Spinweighted spherical harmonics
285
Fields in spacetime
312

23 Bases
91
24 The total reflexivity of G on a manifold
98
25 Spinor algebra
103
Spinors and worldtensors
116
32 Null flags and complex null vectors
125
33 Symmetry operations
132
34 Tensor representation of spinor operations
147
35 Simple propositions about tensors and spinors at a point
159
36 Lorentz transformations
167
Differentiation and curvature
179
42 Covariant derivative
190
43 Connectionindependent derivatives
201
44 Differentiation of spinors
210
45 Differentiation of spinor components
223
46 The curvature spinors
231
47 Spinor formulation of the EinsteinCartanSciamaKibble theory
237
48 The Weyl tensor and the BelRobinson tensor
240
52 EinsteinMaxwell equations in spinor form
325
53 The Rainich conditions
328
54 Vector Bundles
332
55 YangMills Fields
342
56 Conformal rescalings
352
57 Massless fields
362
58 Consistency conditions
366
59 Conformal invariance of various field quantities
371
510 Exact sets of fields
373
511 Initial data on a light cone
385
512 Explicit field integrals
393
diagrammatic notation
424
References
435
Subject and author index
445
Index of symbols
457
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About the author (1987)

Born in England, the son of a geneticist, Roger Penrose received a Ph.D. in 1957 from Cambridge University. Penrose then became a professor of applied mathematics at Birkbeck College in 1966 and a Rouse Ball Professor of Mathematics at Oxford University in 1973. Penrose, a mathematician and theoretical physicist, has done much to elucidate the fundamental properties of black holes. With Stephen Hawking, Penrose proved a theorem of Albert Einstein's general relativity, asserting that at the center of a black hole there must evolve a "space-time singularity" of zero volume and infinite density, in which the current laws of physics do not apply. He also proposed the hypothesis of "cosmic censorship," which claims that such singularities must possess an event horizon. In 1969 Penrose described a process for the extraction of energy from a black hole, as well as how rotational energy of the black hole is transferred to a particle outside the hole. In addition, Penrose has done much to develop the mathematics needed to unite general relativity, which deals with the gravitational interactions of matter, and quantum mechanics, which describes all other interactions.

Professor Wolfgang Rindler
Department of Physics
The University of Texas at Dallas
Richardson, TX 75083-0688
USA

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