Einstein's Apple

327 Pages·2012·1.91 MB·English

Einstein’s
Apple
Homogeneous Einstein Fields
by
Engelbert L. Schu¨cking
Eugene J. Surowitz
March 7, 2012
Copyright c 2011 by
(cid:13)
Engelbert L. Schu¨cking
and Eugene J. Surowitz
Einstein’s Apple – Chapter – March 7, 2012
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EINSTEIN’S APPLE – CHAPTER – March 7, 2012 iii
This work is dedicated to
Alex Harvey
who inspired our effort to put all the pieces together.
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EINSTEIN’S APPLE – CHAPTER – March 7, 2012 v
PREFACE
The role of homogeneous gravitational fields in the formulation of the
equivalence principle and in the foundation of Einstein’s theory of gravita-
tion is well known. However, the original treatment of these concepts was
done in terms of Newtonian gravity and for small velocities. We believe,
therefore, that it is necessary to treat homogeneous fields relativistically in
Einstein’s theory of gravitation. In this book we discuss how this can be
done for manifolds with simply transitive isometry groups and we mention
possible applications.
Since the results presented here are far from complete, we are aware of
the preliminary character of our investigation. What we have tried to do is
to study the concept of homogeneous fields in Riemannian manifolds from
different points of view in an exploratory spirit. This leads to a certain
amount of repetition in the different chapters which we hope the reader
will excuse. Included are some straightforward calculations that we would
expect to readily find in the literature but appear to be confined to papers
in obscure forms or are not in the common bibliographies and texts.
We would like to acknowledge the assistance and support that we have
have received from various individuals during the gestation of this work.
We would especially like to thank Istv´an Ozsv´ath, Alex Harvey, and Mal-
colm MacCallum. Peter Bergmann should be mentioned for his lifelong
efforts to clarify the content of Einstein’sTheory of Gravitation. New York
University provided the facilities where most of the work was carried out.
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CONTENTS
Preface v
Table of Contents vii
List of Figures ix
0. “The Happiest Thought of My Life” 1
I. Accelerated Frames 13
II. Torsion and Telemotion 28
III.Inertialand Gravitational Fieldsin Minkowski
Spacetime 38
IV. The Notion of Torsion 47
V. Homogeneous Fields on Two-dimensional
Riemannian Manifolds 60
VI. Homogeneous Vector Fields in N-dimensions 79
VII. Homogeneous Fields on Three-dimensional
Spacetimes: Elementary Cases 94
VIII. Proper Lorentz Transformations 110
IX. Limits of Spacetimes 135
X. Homogeneous Fields in Minkowski Spacetimes 161
XI. Euclidean Three-dimensional Spaces 181
XII. Homogeneous Fields in Arbitrary
Dimension 207
XIII. A Diagram for Bianchi A-types 224
XIV. Summary 234
Appendix A. Conventions and Notations 239
Appendix B. A Non-trivial Global Frame Bundle 250
AppendixC. Geodesics ofthe Poincar´eHalf-Plane 254
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Appendix D. Determination of Homogeneous
Fields in Two-dimensional Riemannian Spaces 263
Appendix E. Space Expansion 266
Appendix F. The Reissner-Nordstrom Isotropic
Field 268
Appendix G. The Cremona Transformation 273
Appendix H. Hessenberg’s “Vectorial Foundation
of Differential Geometry” 276
Appendix K. Gravitation Is Torsion 280
Appendix R. References 285
Appendix X. Index 296
Appendix N. Notation and Conventions 315
EINSTEIN’S APPLE – CHAPTER – March 7, 2012 ix
FIGURES
1.A.1 Born Motion Vectors 15
1.C.1 A Homogeneous Gravitational Field
in Minkowski Plane 22
1.C.2 A Homogeneous Gravitational Field
in Minkowski Plane 24
1.D.1 The Newtonian Gravitational Field 26
2.A.1 Cartan’s Torsion 29
2.B.1 The Gravitational Frequency Shift 32
2.C.1 Fermi’s Torsion 35
4.C.1 Einstein’s Torsion 51
4.D.1 The Notion of Torsion (1) 54
4.D.2 The Notion of Torsion (2) 56
4.D.3 The Notion of Torsion (3) 57
4.D.4 The Notion of Torsion (4) 58
5.F.1 Two-Dimensional Space of Constant
Curvature Embedding 70
5.I.1 Hyperboloid of the Steady State Model 77
8.D.1 Pseudosphere Cross-section 119
8.D.2 Two Circles in Velocity Space 121
8.D.3 Case II: The Vector ξ Is Spacelike 123
8.D.4 Klein-Beltrami Map I 124
8.D.5 Klein-Beltrami Map II 125
8.E.1 Lorentz Transformation Null Rotations 129
8.F.1 Observable Transverse Velocity 132
8.F.2 Apparent Transverse Velocity 133
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13.D.1 Stereographic Projections 229
13.E.1 Cremona Projection for Birational Map 232
13.F.1 Gnomic Projection 233
C.1 Geodesics of the Poincar´e Plane 256
C.2 Pencil of Geodesics in the Poincar´e Plane 257
C.3 Smith Chart 260
G.1 The Cremona Transformation 273
G.2 Bianchi Class A Degeneracy Pattern 274