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Differential Forms and the Geometry of General Relativity [Kõva köide]

4.67/5 (12 hinnangut Goodreads-ist)
(Oregon State University, Corvallis, USA)
  • Formaat: Hardback, 321 pages, kõrgus x laius: 229x152 mm, kaal: 770 g, 7 Tables, black and white; 87 Illustrations, black and white
  • Ilmumisaeg: 20-Oct-2014
  • Kirjastus: A K Peters
  • ISBN-10: 1466510005
  • ISBN-13: 9781466510005
Teised raamatud teemal:
Differential Forms and the Geometry of General RelativitySuurem pilt
  • Formaat: Hardback, 321 pages, kõrgus x laius: 229x152 mm, kaal: 770 g, 7 Tables, black and white; 87 Illustrations, black and white
  • Ilmumisaeg: 20-Oct-2014
  • Kirjastus: A K Peters
  • ISBN-10: 1466510005
  • ISBN-13: 9781466510005
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781466510005.html
Märksõnad:
Dray begins by introducing general relativity to advanced undergraduate and beginning graduate students of physics or mathematics. He describes some of the surprising implications of relativity without introducing more formalism than necessary. Taking a nonstandard path, he uses differential forms rather than tensor calculus, and tries to minimize the use of index gymnastics as much as possible. In the other, separate but intertwined part of the book, he looks in detail at the mathematics of differential forms. These provide the theory behind the mathematics used in the first half, he says, but does so by emphasizing conceptual understanding rather than formal proofs. This second half provides a language to describe curvature, he says, the key geometric idea in general relativity. Annotation ©2015 Ringgold, Inc., Portland, OR (protoview.com)

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.

Arvustused

"In this book, the author outlines an interesting path to relativity and shows its various stages on the way The author inserts suggestive pictures and images, which make the book more attractive and easier to read. The book addresses not only specialists and graduate students, but even advanced undergraduates, due to its interactive structure containing questions and answers." Zentralblatt MATH 1315

"the presentation is very far from the definition-theorem-proof-example style of a traditional mathematics text; rather, we meet important ideas several times, and they are developed further with each new exposure. This is a pedagogical decision which seems to me to be sound, as it allows the students understanding of the ideas to develop." Robert J. Low, Mathematical Reviews, June 2015

"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 Drays 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 theorys core concepts and applications, from the geometry of black holes to cosmological models." Jürgen Renn, Director, Max Planck Institute for the History of Science, Berlin

List of Figures and Tables xiii
Preface xvii
Acknowledgments xxi
How to Read This Book xxiii
I Spacetime Geometry 1(64)
1 Spacetime
3(8)
1.1 Line Elements
3(3)
1.2 Circle Trigonometry
6(1)
1.3 Hyperbola Trigonometry
7(1)
1.4 The Geometry of Special Relativity
8(3)
2 Symmetries
11(8)
2.1 Position and Velocity
11(2)
2.2 Geodesics
13(1)
2.3 Symmetries
14(1)
2.4 Example: Polar Coordinates
15(1)
2.5 Example: The Sphere
16(3)
3 Schwarzschild Geometry
19(22)
3.1 The Schwarzschild Metric
19(1)
3.2 Properties of the Schwarzschild Geometry
20(1)
3.3 Schwarzschild Geodesics
21(2)
3.4 Newtonian Motion
23(2)
3.5 Orbits
25(4)
3.6 Circular Orbits
29(2)
3.7 Null Orbits
31(3)
3.8 Radial Geodesics
34(1)
3.9 Rain Coordinates
35(4)
3.10 Schwarzschild Observers
39(2)
4 Rindler Geometry
41(8)
4.1 The Rindler Metric
41(1)
4.2 Properties of Rindler Geometry
41(2)
4.3 Rindler Geodesics
43(2)
4.4 Extending Rindler Geometry
45(4)
5 Black Holes
49(16)
5.1 Extending Schwarzschild Geometry
49(2)
5.2 Kruskal Geometry
51(4)
5.3 Penrose Diagrams
55(3)
5.4 Charged Black Holes
58(1)
5.5 Rotating Black Holes
59(2)
5.6 Problems
61(4)
II General Relativity 65(62)
6 Warmup
67(12)
6.1 Differential Forms in a Nutshell
67(5)
6.2 Tensors
72(1)
6.3 The Physics of General Relativity
73(2)
6.4 Problems
75(4)
7 Geodesic Deviation
79(8)
7.1 Rain Coordinates II
79(1)
7.2 Tidal Forces
80(3)
7.3 Geodesic Deviation
83(1)
7.4 Schwarzschild Connection
84(1)
7.5 Tidal Forces Revisited
85(2)
8 Einstein'5, Equation
87(14)
8.1 Matter
87(2)
8.2 Dust
89(2)
8.3 First Guess at Einstein's Equation
91(2)
8.4 Conservation Laws
93(2)
8.5 The Einstein Tensor
95(2)
8.6 Einstein's Equation
97(1)
8.7 The Cosmological Constant
98(1)
8.8 Problems
99(2)
9 Cosmological Models
101(18)
9.1 Cosmology
101(1)
9.2 The Cosmological Principle
101(2)
9.3 Constant Curvature
103(2)
9.4 Robertson—Walker Metrics
105(2)
9.5 The Big Bang
107(1)
9.6 Friedmann Models
108(1)
9.7 Friedmann Vacuum Cosmologies
109(1)
9.8 Missing Matter
110(1)
9.9 The Standard Models
111(2)
9.10 Cosmological Redshift
113(2)
9.11 Problems
115(4)
10 Solar System Applications
119(8)
10.1 Bending of Light
119(3)
10.2 Perihelion Shift of Mercury
122(3)
10.3 Global Positioning
125(2)
III Differential Forms 127(128)
11 Calculus Revisited
129(6)
11.1 Differentials
129(1)
11.2 Integrands
130(1)
11.3 Change of Variables
131(1)
11.4 Multiplying Differentials
132(3)
12 Vector Calculus Revisited
135(8)
12.1 A Review of Vector Calculus
135(3)
12.2 Differential Forms in Three Dimensions
138(1)
12.3 Multiplication of Differential Forms
139(1)
12.4 Relationships between Differential Forms
140(1)
12.5 Differentiation of Differential Forms
141(2)
13 The Algebra of Differential Forms
143(14)
13.1 Differential Forms
143(1)
13.2 Higher-Rank Forms
144(1)
13.3 Polar Coordinates
145(1)
13.4 Linear Maps and Determinants
146(1)
13.5 The Cross Product
147(2)
13.6 The Dot Product
149(1)
13.7 Products of Differential Forms
149(1)
13.8 Pictures of Differential Forms
150(4)
13.9 Tensors
154(1)
13.10 Inner Products
155(1)
13.11 Polar Coordinates II
156(1)
14 I-lodge Duality
157(18)
14.1 Bases for Differential Forms
157(1)
14.2 The Metric Tensor
157(2)
14.3 Signature
159(1)
14.4 Inner Products of Higher-Rank Forms
159(2)
14.5 The Schwarz Inequality
161(1)
14.6 Orientation
162(1)
14.7 The Hodge Dual
162(2)
14.8 Hodge Dual in Minkowski 2-space
164(1)
14.9 Hodge Dual in Euclidean 2-space
164(1)
14.10 Hodge Dual in Polar Coordinates
165(1)
14.11 Dot and Cross Product Revisited
166(1)
14.12 Pseudovectors and Pseudoscalars
167(1)
14.13 The General Case
167(1)
14.14 Technical Note on the Hodge Dual
168(2)
14.15 Application: Decomposable Forms
170(1)
14.16 Problems
171(4)
15 Differentiation of Differential Forms
175(16)
15.1 Gradient
175(1)
15.2 Exterior Differentiation
175(1)
15.3 Divergence and Curl
176(1)
15.4 Laplacian in Polar Coordinates
177(1)
15.5 Properties of Exterior Differentiation
178(2)
15.6 Product Rules
180(1)
15.7 Maxwell's Equations I
181(1)
15.8 Maxwell's Equations II
181(1)
15.9 Maxwell's Equations III
182(2)
15.10 Orthogonal Coordinates
184(1)
15.11 Div, Grad, Curl in Orthogonal Coordinates
185(2)
15.12 Uniqueness of Exterior Differentiation
187(1)
15.13 Problems
188(3)
16 Integration of Differential Forms
191(12)
16.1 Vectors and Differential Forms
191(1)
16.2 Line and Surface Integrals
192(2)
16.3 Integrands Revisited
194(1)
16.4 Stokes' Theorem
194(2)
16.5 Calculus Theorems
196(2)
16.6 Integration by Parts
198(1)
16.7 Corollaries of Stokes' Theorem
199(1)
16.8 Problems
200(3)
17 Connections
203(12)
17.1 Polar Coordinates II
203(2)
17.2 Differential Forms That Are Also Vector Fields
205(1)
17.3 Exterior Derivatives of Vector Fields
205(1)
17.4 Properties of Differentiation
206(1)
17.5 Connections
206(1)
17.6 The Levi-Civita Connection
207(1)
17.7 Polar Coordinates III
208(1)
17.8 Uniqueness of the Levi-Civita Connection
209(1)
17.9 Tensor Algebra
210(1)
17.10 Commutators
211(1)
17.11 Problems
212(3)
18 Curvature
215(20)
18.1 Curves
215(1)
18.2 Surfaces
216(1)
18.3 Examples in Three Dimensions
217(2)
18.4 Curvature
219(2)
18.5 Curvature in Three Dimensions
221(1)
18.6 Components
222(1)
18.7 Bianchi Identities
223(1)
18.8 Geodesic Curvature
224(2)
18.9 Geodesic Triangles
226(2)
18.10 The Gauss—Bonnet Theorem
228(3)
18.11 The Torus
231(3)
18.12 Problems
234(1)
19 Geodesics
235(10)
19.1 Geodesics
235(1)
19.2 Geodesics in Three Dimensions
236(1)
19.3 Examples of Geodesics
237(1)
19.4 Solving the Geodesic Equation
238(2)
19.5 Geodesics in Polar Coordinates
240(1)
19.6 Geodesics on the Sphere
241(4)
20 Applications
245(10)
20.1 The Equivalence Problem
245(2)
20.2 Lagrangians
247(2)
20.3 Spinors
249(1)
20.4 Topology
250(2)
20.5 Integration on the Sphere
252(3)
A Detailed Calculations 255(22)
A.1 Coordinate Symmetries
255(2)
A.2 Geodesic Deviation: Details
257(2)
A.3 Schwarzschild Curvature
259(2)
A.4 Rain Curvature
261(3)
A.5 Components of the Einstein Tensor
264(1)
A.6 Divergence of the Einstein Tensor
265(2)
A.7 Divergence of the Metric in Two Dimensions
267(1)
A.8 Divergence of the Metric
268(1)
A.9 Robertson—Walker Curvature
269(2)
A.10 Birkhoff's Theorem
271(3)
A.11 The Stress Tensor for a Point Charge
274(3)
B Tensor Notation 277(4)
B.1 Invariant Language
277(1)
B.2 Components
278(1)
B.3 Index Gymnastics
279(2)
Annotated Bibliography 281(4)
References 285(2)
Index 287
Tevian Dray