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Filmed as received. 4 Xerox University Microfilms 300 North Zeeb Road Ann Arbor, Michigan 48106 74-17,764 CORUM, James Frederic, 1943- AN INVESTIGATION OF THE ANHOLONOMIC NATURE OF THE ROTATING LORENTZ TRANSFORMATION WITH APPLICATIONS TO ELECTRODYNAMICS. The Ohio State University, Ph.D., 1974 Engineering, electrical University Microfilms, A XEROX Company , Ann Arbor, Michigan © 1974 JAMES EREDERIC CORUM ALL RIGHTS RESERVED AN INVESTIGATION OF THE ANHOLONOMIC MATURE OF THE ROTATING LORENTZ TRANSFORMATION WITH APPLICATIONS TO ELECTRODYNAMICS DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School o f The Ohio State U niversity by James Frederic Corum, B.S., M.Sc. ******** The Ohio State U niversity 1974 Reading Committee: Approved By C.V. Heer U.H. Gerlach H.C. Ko Adviser Department of Electrical Engineering ACKNOWLEDGMENTS The author wishes to express his sincere appreciation to the following people: ................ Professor Ulrich H. Gerlach, of the Department of Mathematics, (who provided an introduction to Cartan calculus, and the vortex of “Modern R elativity"), for his infinite office hours and for numerous discussions concerning those mystical derivations which are known ubique et ab omnibus*, and are so "triflin g " that no one bothers to exhibit them in the literature. ................ Professor C.V. Heer, o f the Department o f Physics, (whose lectures on Landau and Lifshitz provided an introduction to the structure of classical general relativity), for his stimulating and perceptive comments. ................ Professor H.C. Ko, of the Department of Electrical Engineering, whose perceptive insight and penetrating inquisitiveness in the area of relativistic electrodynamics has been so inspiring over the last seven years. ................ Dr. L. Glaser o f B a tte l!e Memorial In s titu te - Columbus, for consultation on the relativistic expansion of elleptic integrals. * everywhere and by everybody ................ Mr. Chiwei Chuang for several discussions and for his kind interest in this paper. ................ Dr. and Mrs. Fred T. Corum, fo r th e ir encouragement in th is academic venture. And lastly, Linda - "Die du mir Jugend Und Freud' und Mut" (Goethe) VITA August 15, 1943 . Born - Natick, Massachusetts 1965 .................................... B.S.E.E., Lov/ell Technological Institute, Lowell, Massachusetts 1966 .................................... Electronic Engineer, National Security Agency, Washington, D.C. 1967 ....... M.Sc., The Ohio State University, Columbus, Ohio 1968 .................................... Research A ssistant, The Ohio State U n ive rsity Radio Observatory 1970 - 1974 ....................... Instructor, Department of Physics and Mathematics, Ohio In s titu te o f Technology, Columbus, Ohio FIELDS OF STUDY Major Field: Electrical Engineering Studies in Radio Astronomy. Professors H.C. Ko and John D. Kraus Studies in Electromagnetism. Professor R.G. Kouyoumjian Studies in Antennas and Propagation. Professors C.H. Walter and J. Richmond Studies in Communications. Professor W. Davis Studies in Applied Mathematics. Professors H. Colson and S. Drobot Studies in General R e la tiv ity . Professors C.V. Heer and U.H. Gerlach TABLE OF CONTENTS Page ACKNOWLEDGMENTS .................................................................................. i VITA .................................................................................................... H i LIST OF FIGURES.......................................................................... v ii Chapter I . INTRODUCTION......................................................... 1 I I . MATHEMATICAL TOOLS ............................................ 8 ] . A rithm etic space 2. Topological spaces 3. Manifolds 4. Curves on the manifold 5. Tangent vectors 6. Basis vectors 7. Differential forms 8. Anholonomic frames 9. Affine connection 10. Covariant d e riva tive 11. Interpretation of transformations 12. Anholonomic transform ations 13. Anholonomic coordinates 14. Lie derivatives 15. The object of anholonomity iv page I I I . ROTATIONAL WORLD LINES ..................................... 44 IV. CALCULATION OF THE ANHOLONOMIC OBJECT . 53 1. Spherical polar coordinates 2. Cylindrical polar coordinates V. INTERPRETATION OF THE TRANSFORMATION . 56 VI. THE SAGNAC EFFECT.............................................. 65 V II. THOMAS PRECESSION............................................... 71 V III. ANHOLONOMIC FIELD THEORY ............................. 75 IX. INVARIANT EQUATIONS ......................................... 80 1. Charge Invariance 2. Invariance of Maxwell's Equations X. THE FIELDS ARISING FROM ROTATING SPHERICAL CHARGE DISTRIBUTIONS........................ ..... 84 1. Charge Rotates 2. Observer Rotates a. Method o f Anholonomic Frames b. Method o f Galilean Frames c. Method of "Instantaneous (M iller) Frames" XI. SCHIFF'S PARADOX ................................................. 118 X II. A POINT CHARGE AND AN OBSERVER IN RELATIVE ROTATION .............................................. 122 1. Observer a t Rest - Charge Rotates 2. Charge at Rest - Observer Rotates X III. CONCLUSIONS.......................................................... 137 V APPENDIX Page A. Summary of Cartan's Notation ................... 140 B. Summary of Classical Tensor Computations 141 C. Metrical Conventions ................................. 142 D. Maxwell's Equations . .................. 150 1. Cylindrical Coordinates 2. Spherical Coordinates BIBLIOGRAPHY................................................................... 153 Vi LIST OF ILLUSTRATIONS Page Fig. 1. A point in R ..................................................... 9 Fig. 2. A manifold M .......................... .......................... 12 Fig. 3. A differential curve in M . ...................... 14 Fig. 4. Tangent vectors and d iffe re n tia l forms. 22 Fig. 5. Vectors and forms in polar coordinates. 23 Fig. 6. Points on the manifold p, q* e M . 26 Fig. 7. Geometrical interpretation of the Lie d e r iv a tiv e ......................................................... 35 Fig. 8. An alternative interpretation of the anholonomic object .......................................... 38 Fig. 9. Basis vectors in polar coordinates . 41 Fig. 10. Differential forms in polar coordinates . 42 Fig. 11. Space-time diagram of a rotating observer 49 Fig. 12. World view of an inertial observer . 57 Fig. 13. Local tetrad frame of an inertial o b s e rv e r ............................................................. 58 Fig. 14. World view of inertial observer .... 59 Fig. 15. Local tetrad frames fo r anholonomic o b s e rv e r ............................................................. 60 Fig. 16. Local tetrads of galilean observer . 61 Fig. 17. Local tetrads of "instantaneous" observer ................................................. 62 Fig. 18. World line of a rotating observer . 66 Fig. 19. Thomas p re ce ssio n ............................................ 72 Fig. 20. A Charged s p h e r e ............................................ 84 v n* • page F1g. 21. Magnetic flux arising from rotation . 88 Fig. 22. A spherical charge distribtution . 118 Fig. 23. Rotating o b se rve rs ........................................... 123 Fig. 24. D e fin itio n of w o r k .................... 124 Fig. 25. Calculations in special relativity . 138 Fig. 26. Calculations in anholonomic theory . 139 v i i i I . INTRODUCTION At the turn of the century, discussions of inertial motion led to the group of space-time transformations called the Lorentz Trans formations. Carried out in rectangular

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