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Symmetries And Curvature Structure In General Relativity : World Scientific Lecture Notes in Physics - Graham S. Hall

Symmetries And Curvature Structure In General Relativity

World Scientific Lecture Notes in Physics

Hardcover Published: 28th December 1994
ISBN: 9789810210519
Number Of Pages: 440

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This is a text on classical general relativity from a geometrical viewpoint. Introductory chapters are provided on algebra, topology and manifold theory, together with a chapter on the basic ideas of space-time manifolds and Einstein's theory. There is a detailed account of algebraic structures and tensor classification in general relativity and also of the relationships between the metric, connection and curvature structures on space-times. The latter includes chapters on holonomy and sectional curvature. An extensive study is presented of symmetries in general relativity, including isometries, homotheties, conformal symmetries and affine, projective and curvature collineations. Several general properties of such symmetries are studied and a preparatory section on transformation groups and on the properties of Lie algebras of vector fields on manifolds is provided.

Industry Reviews

"This book is strongly recommended to everyone who wants to get a mathematically rigorous as well as readable presentation of the underlying pseudo-Riemannian manifolds of General Relativity Theory."Zentralblatt MATH"... it is a complete and thorough review of the subject from a consistent point of view. It covers some most important mathematical aspects of Einstein's theory of general relativity. Written by a leading authority on the topic, this monograph is sure to become a standard reference in this field."Mathematical Reviews"The book is a thorough and in depth exposition of its subject written in a consistent way ... This will make it a pleasant read not only for students and researchers in physics but also in mathematics. It is a safe bet that this monograph soon will become a standard reference in the field."Monatshefte fur Mathematik"Overall this book presents a wealth of results on the geometry of four-dimensional space-times in a well organized and carefully presented form. Most of them are perhaps rather technical or specialized but they will certainly be useful, in particular, for workers on exact solutions of Einstein's equations."General Relativity and Gravitation

Prefacep. v
Introductionp. 1
Geometry and Physicsp. 1
Preview of Future Chaptersp. 5
Algebraic Conceptsp. 11
Introductionp. 11
Groupsp. 11
Vector Spacesp. 15
Dual Spacesp. 22
Forms and Inner Productsp. 23
Similarity, Jordan Canonical Forms and Segre Typesp. 28
Lie Algebrasp. 38
Topologyp. 41
Introductionp. 41
Metric Spacesp. 42
Topological Spacesp. 45
Basesp. 49
Subspace Topologyp. 51
Quotient Spacesp. 52
Product Spacesp. 53
Compactness and Paracompactnessp. 53
Connected Spacesp. 56
Covering Spaces and the Fundamental Groupp. 58
The Rank Theoremsp. 61
Manifold Theoryp. 63
Introductionp. 63
Calculus on R[superscript n]p. 64
Manifoldsp. 65
Functions on Manifoldsp. 68
The Manifold Topologyp. 70
The Tangent Space and Tangent Bundlep. 73
Tensor Spaces and Tensor Bundlesp. 75
Vector and Tensor Fieldsp. 77
Derived Maps and Pullbacksp. 81
Integral Curves of Vector Fieldsp. 83
Submanifoldsp. 85
Quotient Manifoldsp. 92
Distributionsp. 92
Curves and Coveringsp. 97
Metrics on Manifoldsp. 99
Linear Connections and Curvaturep. 104
Grassmann and Stiefel Manifoldsp. 116
Lie Groupsp. 119
Topological Groupsp. 119
Lie Groupsp. 121
Lie Subgroupsp. 122
Lie Algebrasp. 124
One Parameter Subgroups and the Exponential Mapp. 127
Transformation Groupsp. 131
Lie Transformation Groupsp. 131
Orbits and Isotropy Groupsp. 133
Complete Vector Fieldsp. 135
Groups of Transformationsp. 138
Local Group Actionsp. 140
Lie Algebras of Vector Fieldsp. 142
The Lie Derivativep. 144
The Lorentz Groupp. 147
Minkowski Spacep. 147
The Lorentz Groupp. 150
The Lorentz Group as a Lie Groupp. 158
The Connected Lie Subgroups of the Lorentz Groupp. 163
Space-Times and Algebraic Classificationp. 169
Space-Timesp. 169
Electromagnetic fieldsp. 171
Fluid Space-Timesp. 172
The Vacuum Casep. 172
Bivectors and their Classificationp. 173
The Petrov Classificationp. 184
Alternative Approaches to the Petrov Classificationp. 191
The Classification of Second Order Symmetric Tensorsp. 202
The Anti-Self Dual Representation of Second Order Symmetric Tensorsp. 208
Examples and Applicationsp. 215
The Local and Global Nature of Algebraic Classificationsp. 221
Holonomy Groups and General Relativityp. 227
Introductionp. 227
Holonomy Groupsp. 227
The Holonomy Group of a Space-Timep. 234
Vacuum Space-Timesp. 245
Examplesp. 248
The Connection and Curvature Structure of Space-Timep. 255
Introductionp. 255
Metric and Connectionp. 255
Metric, Connection and Curvaturep. 259
Sectional Curvaturep. 270
Retrospectp. 282
Affine Vector Fields on Space-Timep. 285
General Aspects of Symmetriesp. 285
Affine Vector Fieldsp. 287
Subalgebras of the Affine Algebra; Isometries and Homothetiesp. 291
Fixed Point Structurep. 296
Orbit Structurep. 311
Space-Times admitting Proper Affine Vector Fieldsp. 323
Examples and Summaryp. 333
Conformal Symmetry in Space-timesp. 341
Conformal Vector Fieldsp. 341
Orbit Structurep. 345
Fixed Point Structurep. 349
Conformal Reduction of the Conformal Algebrap. 352
Conformal Vector Fields in Vacuum Space-Timesp. 358
Other Examplesp. 359
Special Conformal Vector Fieldsp. 363
Projective Symmetry in Space-timesp. 371
Projective Vector Fieldsp. 371
General Theorems on Projective Vector Fieldsp. 375
Space-Times Admitting Projective Vector Fieldsp. 381
Special Projective Vector Fieldsp. 389
Projective Symmetry and Holonomyp. 391
Curvature Collineationsp. 397
Introductionp. 397
Curvature Collineationsp. 397
Some Techniques for Curvature Collineationsp. 400
Further Examplesp. 408
Bibliographyp. 413
Indexp. 421
Table of Contents provided by Rittenhouse. All Rights Reserved.

ISBN: 9789810210519
ISBN-10: 9810210515
Series: World Scientific Lecture Notes in Physics
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 440
Published: 28th December 1994
Country of Publication: SG
Dimensions (cm): 22.86 x 15.49  x 2.54
Weight (kg): 0.73

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