99,99 €
Mehr erfahren.
- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
An introduction to semi-Riemannian geometry as a foundation for general relativity Semi-Riemannian Geometry: The Mathematical Language of General Relativity is an accessible exposition of the mathematics underlying general relativity. The book begins with background on linear and multilinear algebra, general topology, and real analysis. This is followed by material on the classical theory of curves and surfaces, expanded to include both the Lorentz and Euclidean signatures. The remainder of the book is devoted to a discussion of smooth manifolds, smooth manifolds with boundary, smooth manifolds with a connection, semi-Riemannian manifolds, and differential operators, culminating in applications to Maxwell's equations and the Einstein tensor. Many worked examples and detailed diagrams are provided to aid understanding. This book will appeal especially to physics students wishing to learn more differential geometry than is usually provided in texts on general relativity.
Sie lesen das E-Book in den Legimi-Apps auf:
Seitenzahl: 912
Veröffentlichungsjahr: 2019
Ähnliche
Table of Contents
Cover
Preface
Part I: Preliminaries
Chapter 1: Vector Spaces
1.1 Vector Spaces
1.2 Dual Spaces
1.3 Pullback of Covectors
1.4 Annihilators
Chapter 2: Matrices and Determinants
2.1 Matrices
2.2 Matrix Representations
2.3 Rank of Matrices
2.4 Determinant of Matrices
2.5 Trace and Determinant of Linear Maps
Chapter 3: Bilinear Functions
3.1 Bilinear Functions
3.2 Symmetric Bilinear Functions
3.3 Flat Maps and Sharp Maps
Chapter 4: Scalar Product Spaces
4.1 Scalar Product Spaces
4.2 Orthonormal Bases
4.3 Adjoints
4.4 Linear Isometries
4.5 Dual Scalar Product Spaces
4.6 Inner Product Spaces
4.7 Eigenvalues and Eigenvectors
4.8 Lorentz Vector Spaces
4.9 Time Cones
Chapter 5: Tensors on Vector Spaces
5.1 Tensors
5.2 Pullback of Covariant Tensors
5.3 Representation of Tensors
5.4 Contraction of Tensors
Chapter 6: Tensors on Scalar Product Spaces
6.1 Contraction of Tensors
6.2 Flat Maps
6.3 Sharp Maps
6.4 Representation of Tensors
6.5 Metric Contraction of Tensors
6.6 Symmetries of (0, 4)‐Tensors
Chapter 7: Multicovectors
7.1 Multicovectors
7.2 Wedge Products
7.3 Pullback of Multicovectors
7.4 Interior Multiplication
7.5 Multicovector Scalar Product Spaces
Chapter 8: Orientation
8.1 Orientation of ℝ
m
8.2 Orientation of Vector Spaces
8.3 Orientation of Scalar Product Spaces
8.4 Vector Products
8.5 Hodge Star
Chapter 9: Topology
9.1 Topology
9.2 Metric Spaces
9.3 Normed Vector Spaces
9.4 Euclidean Topology on ℝ
m
Chapter 10: Analysis in ℝ
m
10.1 Derivatives
10.2 Immersions and Diffeomorphisms
10.3 Euclidean Derivative and Vector Fields
10.4 Lie Bracket
10.5 Integrals
10.6 Vector Calculus
Part II: Curves and Regular Surfaces
Chapter 11: Curves and Regular Surfaces in ℝ
3
11.1 Curves inℝ
3
11.2 Regular Surfaces inℝ
3
11.3 Tangent Planes inℝ
3
11.4 Types of Regular Surfaces in ℝ
3
11.5 Functions on Regular Surfaces in ℝ
3
11.6 Maps on Regular Surfaces inℝ
3
11.7 Vector Fields Along Regular Surfaces in ℝ
3
Chapter 12: Curves and Regular Surfaces in
12.1 Curves in
12.2 Regular Surfaces in
12.3 Induced Euclidean Derivative in
12.4 Covariant Derivative on Regular Surfaces in
12.5 Covariant Derivative on Curves in
12.6 Lie Bracket in
12.7 Orientation in
12.8 Gauss Curvature in
12.9 Riemann Curvature Tensor in
12.10 Computations for Regular Surfaces in
Chapter 13: Examples of Regular Surfaces
13.1 Plane in
13.2 Cylinder in
13.3 Cone in
13.4 Sphere in
13.5 Tractoid in
13.6 Hyperboloid of One Sheet in
13.7 Hyperboloid of Two Sheets in
13.8 Torus in
13.9 Pseudosphere in
13.10 Hyperbolic Space in
Part III: Smooth Manifolds and Semi‐Riemannian Manifolds
Chapter 14: Smooth Manifolds
14.1 Smooth Manifolds
14.2 Functions and Maps
14.3 Tangent Spaces
14.4 Differential of Maps
14.5 Differential of Functions
14.6 Immersions and Diffeomorphisms
14.7 Curves
14.8 Submanifolds
14.9 Parametrized Surfaces
Chapter 15: Fields on Smooth Manifolds
15.1 Vector Fields
15.2 Representation of Vector Fields
15.3 Lie Bracket
15.4 Covector Fields
15.5 Representation of Covector Fields
15.6 Tensor Fields
15.7 Representation of Tensor Fields
15.8 Differential Forms
15.9 Pushforward and Pullback of Functions
15.10 Pushforward and Pullback of Vector Fields
15.11 Pullback of Covector Fields
15.12 Pullback of Covariant Tensor Fields
15.13 Pullback of Differential Forms
15.14 Contraction of Tensor Fields
Chapter 16: Differentiation and Integration on Smooth Manifolds
16.1 Exterior Derivatives
16.2 Tensor Derivations
16.3 Form Derivations
16.4 Lie Derivative
16.5 Interior Multiplication
16.6 orientation
16.7 Integration of Differential Forms
16.8 Line Integrals
16.9 Closed and Exact Covector Fields
16.10 Flows
Chapter 17: Smooth Manifolds with Boundary
17.1 Smooth Manifolds with Boundary
17.2 Inward‐Pointing and Outward‐Pointing Vectors
17.3 orientation of Boundaries
17.4 Stokes's Theorem
Chapter 18: Smooth Manifolds with a Connection
18.1 Covariant Derivatives
18.2 Christoffel Symbols
18.3 Covariant Derivative on Curves
18.4 Total Covariant Derivatives
18.5 Parallel Ranslation
18.6 Torsion Tensors
18.7 Curvature Tensors
18.8 Geodesics
18.9 Radial Geodesics and Exponential Maps
18.10 Normal Coordinates
18.11 Jacobi Fields
Chapter 19: Semi‐Riemannian Manifolds
19.1 Semi‐Riemannian Manifolds
19.2 Curves
19.3 Fundamental Theorem of Semi‐Riemannian Manifolds
19.4 Flat Maps and Sharp Maps
19.5 Representation of Tensor Fields
19.6 Contraction of Tensor Fields
19.7 Isometries
19.8 Riemann Curvature Tensor
19.9 Geodesics
19.10 Volume Forms
19.11 orientation of Hypersurfaces
19.12 Induced Connections
Chapter 20: Differential Operators on Semi‐Riemannian Manifolds
20.1 Hodge Star
20.2 Codifferential
20.3 Gradient
20.4 Divergence of Vector Fields
20.5 Curl
20.6 Hesse Operator
20.7 Laplace Operator
20.8 Laplace–de Rham Operator
20.9 Divergence of Symmetric 2‐Covariant Tensor Fields
Chapter 21: Riemannian Manifolds
21.1 Geodesics and Curvature on Riemannian Manifolds
21.2 Classical Vector Calculus Theorems
Chapter 22: Applications to Physics
22.1 Linear Isometries on Lorentz Vector Spaces
22.2 Maxwell's Equations
22.3 Einstein Tensor
Part IV: Appendices
Appendix A: Notation and Set Theory
Appendix B: Abstract Algebra
B.1. Groups
B.2. Permutation Groups
B.3. Rings
B.4. Fields
B.5. Modules
B.6. Vector Spaces
B.7. Lie Algebras
Further Reading
Index
End User License Agreement
List of Tables
Chapter 8
Table 8.1.1. Orientation inℝ
2
Table 8.1.2. Orientation inℝ
3
Chapter 10
Table 10.6.1 Classical differential operators
List of Illustrations
Chapter 4
Figure 4.8.1 Light cone: Diagram for Example 4.8.1
Figure 4.8.2 Diagram for Example 4.8.1
Chapter 9
Figure 9.2.1 Diagram for Theorem 9.2.3
Chapter 11
Figure 11.2.1. Diagram for Example 11.2.3
Figure 11.2.2. Diagram for Example 11.2.4
Figure 11.2.3. Diagram for Theorem 11.2.8
Figure 11.2.4. Diagram for Theorem 11.2.11
Figure 11.2.5. Stereographic projection: Diagram for Example 11.2.13
Figure 11.4.1. Diagram for Theorem 11.4.4
Figure 11.6.1. Differential map
Chapter 12
Figure 12.7.1. Möbius band: Diagram for Example 12.7.9
Figure 12.8.1. Gauss map
Figure 12.8.2. Diagram for Theorem 12.8.5
Chapter 13
Figure 13.3.1. Con
Figure 13.4.1.
Figure 13.5.1. Trc
Figure 13.6.1. One,
Figure 13.7.1. Two, ℋ
2
Figure 13.8.1. Tor
Chapter 14
Figure 14.2.1 Bump function
Figure 14.2.2 Diagram for Theorem 14.2.6
Chapter 16
Figure 16.10.1. Diagram for Example 16.10.2
Chapter 17
Figure 17.1.1. Interior chart and boundary chart
Figure 17.2.1. Inward‐pointing and outward‐pointing tangent vectors
Chapter 18
Figure 18.9.1. Exponential map
Figure 18.9.2. Differential map of exponential map
Figure 18.11.1. Geodesic variation
Chapter 19
Figure 19.5.1. Commutative diagram for Theorem 19.5.3
Figure 19.11.2 Diagram for Theorem 19.11.4
Chapter 22
Figure 22.1.1.
and L
m
Guide
Cover
Table of Contents
Begin Reading
Pages
xiii
xiv
1
3
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
302
303
271
272
273
274
275
276
278
279
280
281
282
283
284
285
286
277
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
335
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
609
611
612
613
614
615
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
627
Semi-Riemannian Geometry
The Mathematical Language of General Relativity
Stephen C. Newman
University of Alberta Edmonton, Alberta, Canada
Copyright
This edition first published 2019
© 2019 John Wiley & Sons, Inc.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions.
The right of Stephen C. Newman to be identified as the author of this work has been asserted in accordance with law.
Registered Office
John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USA
Editorial Office
111 River Street, Hoboken, NJ 07030, USA
For details of our global editorial offices, customer services, and more information about Wiley products visit us at www.wiley.com.
Wiley also publishes its books in a variety of electronic formats and by print-on-demand. Some content that appears in standard print versions of this book may not be available in other formats.
Limit of Liability/Disclaimer of Warranty
While the publisher and authors have used their best efforts in preparing this work, they make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives, written sales materials or promotional statements for this work. The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further information does not mean that the publisher and authors endorse the information or services the organization, website, or product may provide or recommendations it may make. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for your situation. You should consult with a specialist where appropriate. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.
Library of Congress Cataloging-in-Publication Data
Names: Newman, Stephen C., 1952‐ author.
Title: Semi‐Riemannian geometry : the mathematical language of general relativity /
Stephen C. Newman (University of Alberta, Edmonton, Alberta, Canada).
Description: Hoboken, New Jersey : Wiley, [2019] | Includes bibliographical references and index. |
Identifiers: LCCN 2019011644 (print) | LCCN 2019016822 (ebook) | ISBN 9781119517542 (Adobe PDF) |
ISBN 9781119517559 (ePub) | ISBN 9781119517535 (hardcover)
Subjects: LCSH: Semi‐Riemannian geometry. | Geometry, Riemannian. | Manifolds (Mathematics) |
Geometry, Differential.
Classification: LCC QA671 (ebook) | LCC QA671 .N49 2019 (print) | DDC 516.3/73‐dc23
LC record available at https://lccn.loc.gov/2019011644
Cover design: Wiley
Dedication
To Sandra
Preface
Physics texts on general relativity usually devote several chapters to an overview of semi‐Riemannian geometry. Of necessity, the treatment is cursory, covering only the essential elements and typically omitting proofs of theorems. For physics students wanting greater mathematical rigor, there are surprisingly few options. Modern mathematical treatments of semi‐Riemannian geometry require grounding in the theory of curves and surfaces, smooth manifolds, and Riemannian geometry. There are numerous books on these topics, several of which are included in Further Reading. Some of them provide a limited amount of material on semi‐Riemannian geometry, but there is really only one mathematics text currently available that is devoted to semi‐Riemannian geometry and geared toward general relativity, namely, Semi‐Riemannian Geometry: With Applications to Relativity by Barrett O'Neill. This is a classic, but it is pitched at an advanced level, making it of limited value to the beginner. I wrote the present book with the aim of filling this void in the literature.
There are three parts to the book. Part I and the Appendices present background material on linear algebra, multilinear algebra, abstract algebra, topology, and real analysis. The aim is to make the book as self‐contained as possible. Part II discusses aspects of the classical theory of curves and surfaces, but differs from most other expositions in that Lorentz as well as Euclidean signatures are discussed. Part III covers the basics of smooth manifolds, smooth manifolds with boundary, smooth manifolds with a connection, and semi‐Riemannian manifolds. It concludes with applications to Lorentz vector spaces, Maxwell's equations, and the Einstein tensor. Not all theorems are provided with a proof, otherwise an already lengthy volume would be even longer.
The manuscript was typed using the WYSIWYG scientific word processor EXP®, and formatted as a camera‐ready PDF file using the open‐source TEX‐LATEX typesetting system MiKTeX, available at https://miktex.org. Figure 19.5.1 was prepared using the TEX macro package diagrams.sty developed by Paul Taylor. I am indebted to Professor John Lee of the University of Washington for reviewing portions of the manuscript. Any remaining errors or deficiencies are, of course, solely my responsibility.
I am most interested in receiving your comments, which can be emailed to me at [email protected]. A list of corrections will be posted on the website https://sites.ualberta.ca/˜sn2/. Should the email address become unavailable, an alternative will be included with the list of corrections. On the other hand, if the website becomes inaccessible, the list of corrections will be stored as a public file on Google Drive that can be searched using “Corrections to Semi‐Riemannian Geometry by Stephen Newman”.
Allow me to close by thanking my wife, Sandra, for her unwavering support and encouragement throughout the writing of the manuscript. It is to her, with love, that this book is dedicated.
Part IPreliminaries
Differential geometry rests on the twin pillars of linear algebra‐multilinear algebra and topology‐analysis. Part I of the book provides an overview of selected topics from these areas of mathematics. Most of the linear algebra presented here is likely familiar to the reader, but the same may not be true of the multilinear algebra, with the exception of the material on determinants. Topology and analysis are vast subjects, and only the barest of essentials are touched on here. In order to keep the book to a manageable size, not all theorems are provided with a proof, a remark that also applies to Part II and Part III.
