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Differential Forms and the Geometry of General Relativity

Tevian Dray (Oregon State University, Corvallis, USA)

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A K Peters
20 October 2014
Mathematics & Sciences; Relativity physics; Mathematical physics
Differential Forms and the Geometry of General Relativity provides readers with a coherent path to understanding relativity. Requiring little more than calculus and some linear algebra, it helps readers learn just enough differential geometry to grasp the basics of general relativity.

The book contains two intertwined but distinct halves. Designed for advanced undergraduate or beginning graduate students in mathematics or physics, most of the text requires little more than familiarity with calculus and linear algebra. The first half presents an introduction to general relativity that describes some of the surprising implications of relativity without introducing more formalism than necessary. This nonstandard approach uses differential forms rather than tensor calculus and minimizes the use of index gymnastics as much as possible.

The second half of the book takes a more detailed look at the mathematics of differential forms. It covers the theory behind the mathematics used in the first half by emphasizing a conceptual understanding instead of formal proofs. The book provides a language to describe curvature, the key geometric idea in general relativity.
By:   Tevian Dray (Oregon State University Corvallis USA)
Imprint:   A K Peters
Country of Publication:   United States
Dimensions:   Height: 229mm,  Width: 152mm,  Spine: 18mm
Weight:   582g
ISBN:   9781466510005
ISBN 10:   1466510005
Pages:   321
Publication Date:   20 October 2014
Audience:   College/higher education ,  College/higher education ,  Primary ,  Primary
Format:   Paperback
Publisher's Status:   Active
Spacetime Geometry Spacetime Line Elements Circle Trig Hyperbola Trig The Geometry of Special Relativity Symmetries Position and Velocity Geodesics Symmetries Example: Polar Coordinates Example: The Sphere Schwarzschild Geometry The Schwarzschild Metric Properties of the Schwarzschild Geometry Schwarzschild Geodesics Newtonian Motion Orbits Circular Orbits Null Orbits Radial Geodesics Rain Coordinates Schwarzschild Observers Rindler Geometry The Rindler Metric Properties of Rindler Geometry Rindler Geodesics Extending Rindler Geometry Black Holes Extending Schwarzschild Geometry Kruskal Geometry Penrose Diagrams Charged Black Holes Rotating Black Holes Problems General RelativityWarmup Differential Forms in a Nutshell Tensors The Physics of General Relativity Problems Geodesic Deviation Rain Coordinates II Tidal Forces Geodesic Deviation Schwarzschild Connection Tidal Forces Revisited Einstein's Equation Matter Dust First Guess at Einstein's Equation Conservation Laws The Einstein Tensor Einstein's Equation The Cosmological Constant Problems Cosmological Models Cosmology The Cosmological Principle Constant Curvature Robertson-Walker Metrics The Big Bang Friedmann Models Friedmann Vacuum Cosmologies Missing Matter The Standard Models Cosmological Redshift Problems Solar System Applications Bending of Light Perihelion Shift of Mercury Global Positioning Differential Forms Calculus RevisitedDifferentials Integrands Change of Variables Multiplying Differentials Vector Calculus Revisited A Review of Vector Calculus Differential Forms in Three Dimensions Multiplication of Differential Forms Relationships between Differential Forms Differentiation of Differential Forms The Algebra of Differential Forms Differential Forms Higher Rank Forms Polar Coordinates Linear Maps and Determinants The Cross Product The Dot Product Products of Differential Forms Pictures of Differential Forms Tensors Inner Products Polar Coordinates II Hodge Duality Bases for Differential Forms The Metric Tensor Signature Inner Products of Higher Rank Forms The Schwarz Inequality Orientation The Hodge Dual Hodge Dual in Minkowski 2-space Hodge Dual in Euclidean 2-space Hodge Dual in Polar Coordinates Dot and Cross Product Revisited Pseudovectors and Pseudoscalars The General Case Technical Note on the Hodge Dual Application: Decomposable Forms Problems Differentiation of Differential Forms Gradient Exterior Differentiation Divergence and Curl Laplacian in Polar Coordinates Properties of Exterior Differentiation Product Rules Maxwell's Equations I Maxwell's Equations II Maxwell's Equations III Orthogonal Coordinates Div, Grad, Curl in Orthogonal Coordinates Uniqueness of Exterior Differentiation Problems Integration of Differential FormsVectors and Differential Forms Line and Surface Integrals Integrands Revisited Stokes' Theorem Calculus Theorems Integration by Parts Corollaries of Stokes' Theorem Problems Connections Polar Coordinates II Differential Forms which are also Vector Fields Exterior Derivatives of Vector Fields Properties of Differentiation Connections The Levi-Civita Connection Polar Coordinates III Uniqueness of the Levi-Civita Connection Tensor Algebra Commutators Problems Curvature Curves Surfaces Examples in Three Dimensions Curvature Curvature in Three Dimensions Components Bianchi Identities Geodesic Curvature Geodesic Triangles The Gauss-Bonnet Theorem The Torus Problems Geodesics Geodesics Geodesics in Three Dimensions Examples of Geodesics Solving the Geodesic Equation Geodesics in Polar Coordinates Geodesics on the Sphere Applications The Equivalence Problem Lagrangians Spinors Topology Integration on the Sphere Appendix A: Detailed CalculationsAppendix B: Index Gymnastics Annotated Bibliography References

Reviews for Differential Forms and the Geometry of General Relativity

This is a brilliant book. Dray has an extraordinary knack of conveying the key mathematics and concepts with an elegant economy that rivals the expositions of the legendary Paul Dirac. It is pure pleasure to see far-reaching results emerge effortlessly from easy-to-follow arguments, and for simple examples to morph into generalizations. It is so refreshing to find a book that does not obscure the basics with unnecessary technicalities, yet can develop sophisticated formalism from very modest mathematical investments. -Paul Davies, Regents' Professor and Director, Beyond Center for Fundamental Concepts in Science; Co-Director, Cosmology Initiative; and Principal Investigator, Center for the Convergence of Physical Science and Cancer Biology, Arizona State University It took Einstein eight years to create general relativity by carefully balancing his physical intuition and the rather tedious mathematical formalism at his disposal. Tevian Dray's presentation of the geometry of general relativity in the elegant language of differential forms offers even beginners a novel and direct route to a deep understanding of the theory's core concepts and applications, from the geometry of black holes to cosmological models. -Jurgen Renn, Director, Max Planck Institute for the History of Science, Berlin


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