) i RELATIVITY IN COMPLEX SPACETIME by KWUOR, JOHN BUERS m i aia THESIS HAS BBBH ac^ e p t e d f ^ HOB DBORTF. 0'~ 7 7 , U c E D IN THU WKfVERSrTY OF NAIROBI \ND A COl'V HAT - LIBRARY ONIVBKSIIY UKUAU ■ f. Or. Bex 30197 MAIBAX A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN THE DEPARTMENT OF PHYSICS, AT THE (UNIVERSITY OF NAIROBI."! *aN’- JULY 1998 UNIVERSITY OF NAIROBI LIBRARY DECLARATION This work is an original thesis prepared in fulfillment of the requirement for the Degree of Doctor of Philosophy (Ph.D.) in Physics at the University of Nairobi and has not been submitted before anywhere for assessment. AWUOR, J. BUERS x This thesis is submitted with our approval as University Supervisors: PROF. J. O. MALO DEPARTMENT OF PHYSICS, UNIVERSITY OF NAIROBI, P.0. BOX 30197, NAIROBI, KENYA. 5 Sign ...j."$l*i h f DR. SANT RAM DEPARTMENT OF PHYSICS, UNIVERSITY OF NAIROBI, P.O. BOX 30197, NAIROBI, KENYA. Sign. ^ > 1 2 DEDICATION . Dedicated to my parents: Henry Awuor & Helen Ong’ute whose complexions I bear in reality and in my imaginations. 3 \ ACKNOWLEDGEMENTS I wish to acknowledge my obligation and to convey my warmest thanks to those who knowingly and willingly have encouraged or aided me towards the preparation of this thesis. I am most grateful to my two supervisors, Prof. J.O. Malo and Dr. Sant Ram, for valuable discussions and guidance, to Prof. Roger Penrose (University of Oxford) and Prof. G.R. Pickett (Lancaster University) for their assistance with numerous relevant papers, and to Prof. G.F.R. Ellis (University of Cape Town) for making several encouraging and intriguing suggestions on spacetime characteristics and further references. The aim and content of this thesis are briefly described in the in­ troduction. My debt to those who have investigated on the same subject, notably, Johannes Kepler (1571-1630), Sir Isaac Newton (1642-1727), Max Planck (1858-1947), Albert Einstein (1879-1955), Niels Bohr (1885-1962) and Louis de Broglie (1892-1987), on whose inspiring formulations I have leant for justification, and as the basis of my confidence. Finally I am deeply grateful to the DAAD for granting me a most generous fellowship to ensure the success and completion of this research. Awuor J. Buers 4 ABSTRACT A general theory of Relativity has been developed on the basis of a complexified spacetime and its scale equivalence principle. The results are in agreement with the familiar Schwarzschild solution (as derived from Einstein’s relativity formulations) [Adler et eh, (1965)]1 and with the well confirmed Newtonian law of Gravitation. On the very large scale, a composite curvature spacetime cosmolog­ ical model is obtained in which an Euclidean horizon replaces the Event horizon in the determination of local scale evolution of spacetime. The complexified relativistic formulation calls to question the con­ stancy of G (Newton’s universal gravitational constant), the weak Cosmo­ logical Principle which prescribes a requirement of homogeneity in the dis­ tribution of cosmological matter [Narlikar, (1978)]2 , as well as the proper meaning of the black hole radius. Finally, some quantum considerations of gravity as well as of the Big Bang are propounded as plausible physical probabilities. 5 CONTENTS TITLE .................................................................. .1 DECLARATION ................................................. .2 DEDICATION .................................................... .3 ACKNOWLEDGEMENTS ............................... .4 ABSTRACT ........................................................ .5 CONTENTS ..................... .................................. .6 FIGURE CAPTIONS ........................................ .7 INTRODUCTION ............................................. 9 CHAPTER I COMPLEX SPACETIME ... 13 1.1 Basic Formulations ...................................... 13 1.2 Scale Equivalence Principle ........................ 14 1.3 The Weak Equivalence Principle ................ 15 CHAPTER II RELATIVITY ....................... 18 II. 1 Gravitational Acceleration ...................... 18 11.2 Special Relativity ........................................ 19 11.3 General Relativity ...................................... 22 11.4 The Schwarzschild Line-element ................ 24 11.5 Kepler’s Third Law ..................................... 25 11.6 Newton’s Gravitational L aw ....................... 26 CHAPTER III COSMOLOGY ................... 28 III. 1 A Composite Spacetime Curvature Model 28 111.2 The Observable Universe.......................... 32 111.3 The Fate of the Universe.......................... 34 CHAPTER IV APPLICATIONS ................ 36 IV. 1 The Variation of G .................................. 36 6 IV.2 Gravity as a Metric Phenomenon.................................................... 39 IV.3 The Black Hole Radius....................................................................... 41 CONCLUSION ............................................................................................ 43 APPENDICES ............................................................................................. 44 Appendix A: Quantum Big Bang Probability ......................................... 44 Appendix B: Quantum G ravity................*...............................................47 Appendix C: A Composite Spacetime Curvature Model ........................52 REFERENCES ............................................................................................59 FIGURE CAPTIONS a w Fig 1. ‘Semi-Rigid’ ^isc (SRD) model comprising concentric rings ... .21 Fig 2. Lorentzian dynamics on a SRD model ........................................ 24 Fig 3. Effective gravity profile for the CSC m odel................................. 31 i w o m i v n Fig 4. Energy curves on the complex velocity plane of the CSC model.33 Fig 5. The metric curvature about a finite mass in the CSC model ... 40 Fig 6. Quantum probability density for the Big Bang origin..................46 4 t f V t i ® n Fig 7. Quantum gravity profile in correspondence with classical gravity. ......................................................................................... 50 7 ....it must be conceded that a theory has an important advantage if its basic concepts and fundamental hypotheses are ‘close to experience'....... Yet more and more, as the depth of our knowledge increases, we must give up this advantage in our quest for logical simplicity and uniformity in the foundations of physical theory.... ALBERT EINSTEIN - Scientific American, 1950. 8 INTRODUCTION Throughout the history of physics, many attempts have been made to explain physical phenomena by geometrical arguments. Geometry grew in ancient Greece as the science of plane and solid figures [McKenzie, (I960)]3 . The construction of a figure was perceived to bring out points, which one then naturally regards as pre-existing - at least potentially - in a continuous medium (called space) that provides so to speak, the ma­ terial from which the figures are made. Understandably, geometry came to be conceived of as the study of this medium, the science of points and their relation in space. In other words, space was considered as the repos­ itory of all points required for carrying out the admissible construction of geometry. iJNfVFRSfTY OF /VA CHIROMO LI&RARr Later, space came to be identified as the divine presence in which matter is placed, thus setting the stage for natural philosophy and ge­ ometrical principles mainly attributed to Newton and based on Euclid’s geometric construction. In the 19th century, bold yet entirely plausible variations of the established principles and methods of geometric thinking gave rise to a whole crop of systems, that furnished either generalizations and extensions of, or alternatives to, the geometry of Euclid [Torretti, (1983)]4 . The choice of a physical geometry, among the many possibili­ ties offered by mathematics, was either a matter of fact, to be resolved by experimental reasoning, or a mere matter of agreement. In the third quarter of the century, Bernhard Riemann (1826-1862) sought to extend the two dimensional Gaussian geometry and to classify physical space within the vast genus of structured sets or ‘manifolds’ of which it is an instance. This was the basis of Riemann’s theory of metric manifolds which Albert Einstein adopted as the right approach to phys­ ical geometry, in his theory of gravitation or, as he preferred to call it, the General Theory of Relativity [Einstein, (1920)]5 . However, a local­ ized approximation to General Relativity, known as the Special Theory of 9 Relativity, which had earlier been developed by Einstein as a form of a non-Newtonian physics, was only properly interpreted on the basis of non- Euclidean geometry discovered by Hermann Minkowski, in which time and space are unified as a single chronogeometric structure [Einstein, (1920)]5 . This, so called, Minkowski geometry defines a uniquely affine and rather special case of the Riemann geometry. Minkowski’s significant contribution towards the formal development of the theory of relativity was his recognition that the four-dimensional space-time continuum on which Albert Einstein had founded his relativis­ tic electrodynamics of moving bodies was none other than the theory of in­ variants of a definite group of linear transformations, namely, the Lorentz group. This, he achieved by a replacement of the usual real time co­ ordinate by a proportional magnitude of imaginary time. Under these conditions, the natural laws satisfying