UNIVERSITY OF NAIROBI GENERAL RELATIVITY AND ITS APPLICATIONS BY DUKE MATOKE KENGURU A RESEARCH PROJECT SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE AWARD OF THE DEGREE OF MASTER OF SCIENCE IN APPLIED MATHEMATICS. UNIVERSITY OF NAIROBI, COLLEGE OF BIOLOGY AND PHYSICAL SCIENCES, SCHOOL OF MATHEMATICS, DEPARTMENT OF APPLIED MATHEMATICS. i DECLARATION This is my own research work and has not been presented for a degree award in any other university. SIGNATURE………………………………………DATE NAME: MATOKE DUKE KENGURU This project has been submitted with my approval as a university supervisors. SIGNATURE………………………………………DATE NAME: PROF. G. P. POKHARIYAL SIGNATURE………………………………………DATE NAME: Dr. MOINDI S. N. DEPARTMENT OF APPLIED MATHEMATICS SCHOOL OF MATHEMATICS UNIVERSITY OF NAIROBI ii DEDICATION I dedicate this work to my sons Kevin Kenguru Matoke, Justuce Nyarangi Matoke ; daughters F. Kemunto Matoke, Lilian Nyaboke Matoke and granddaughters Nicole Bosibori and Brianna Kemunto. Their care, patience, understanding and tolerance enabled me finish this work successfully. I thank you all and may God bless you in abundance. iii ACKNOWLEDGEMENT I would like to thank God most sincerely for seeing me through the of this course. I take this opportunity to thank my supervisor Prof. G. P. Pokhariyal for his availability, scholarly guidance, tireless efforts and interest to help me whenever I called upon him. I will forever be grateful. I am also grateful to the academic staff of the School of Mathematics, University of Nairobi, Dr. Ogana, Prof. Singh, Dr Nyandwi, Dr. Were, whose good teaching has enriched my interest in the academic pursuits. I thank Dr. Moindi for going through my project and making very relevant comments. Worth special mention is my classmates Violet and Fedli for the good company. I wish you all the best in life. I wish to acknowledge the support I received from the Dean Prof Gatere, Assistant Dean Mr Okioma, school Mathematics and workmates Mr W. Arisa Mr A. Motuka and everybody else who in one way or the other contributed to the success of this research work. May God bless you all. iv ABSTRACT Several Philosophers, Mathematicians and Physicists have in recent years tried to interpret general relativity. General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. Most of General relativity are Einstein’s predictions which were subject to interpretation. Mathematicians and Physicists have tried to test and apply the predictions. The aim of this study was to establish the tests , applications and future development of general relativity such as how general relativity can be reconciled with the laws of quantum physics to produce a complete and self-consistent theory of quantum gravity. The study has looked at the predictions, tests and their applications. The predictions have been tested by international observatory bodies such as, National Aeronautics and Space Administration (NASA). This predictions have found their applications in study of the universe. But some of the predictions have not been fully tested, their research is still on, such as, the production of gravitational waves. However, it is still an open question as to how the concepts of quantum theory can be reconciled with those of general relativity. Despite major efforts, no complete and consistent theory of quantum gravity is currently known, even though a number of promising candidates exist. iv `LIST OF ABREVIATIONS 1. LIGO: Laser interferometer Gravitational Observatory 2. ICRF: International Celestial Reference Frame 3. VLBI: Very Long Baseline Interferometer 4. GPS: Global Positioning System 5. LAGEOS: Laser Geodynamics Satellite 6. MGS: Mars Global Surveyor 7. LISA: Laser Interferometer Space Antenna 8. NASA: National Aeronautics and Space Administration 9. ESA: European Space Agency 10. GEO-600, TAMA-300 and VIRGO are Gravitational wave detectors in Germany, Japan and Italy respectively. v Contents DECLARATION ................................................................................................................................................ i DEDICATION................................................................................................................................................. iii ACKNOWLEDGEMENT.................................................................................................................................. iv ABSTRACT..................................................................................................................................................... iv LIST OF ABREVIATIONS.................................................................................................................................. v CHAPTER 1: Introduction ............................................................................................................................1 1.1 General Relativity..........................................................................................................................1 1.2 Predictions of General Relativity ..................................................................................................1 1.3 Problem Statement.......................................................................................................................2 Chapter 2: Evolution of General Relativity ...................................................................................................7 2.1. From classical mechanics to general relativity .............................................................................7 2.2. Geometry of Newtonian Gravity...................................................................................................7 2.3. Relativistic Generalization.............................................................................................................9 2.4. Einstein's Equations ....................................................................................................................10 CHAPTER 3: Definition and Tests of General Relativity..........................................................................13 3.1. Definition and Basic Properties...................................................................................................13 3.2. Model-Building............................................................................................................................14 3.3. Tests of General Relativity ..........................................................................................................15 3.3.1. Perihelion Precession of Mercury.......................................................................................16 3.3.2. Deflection of Light by the Sun.............................................................................................20 3.3.3. Gravitational Redshift of Light ............................................................................................21 3.4. Modern Tests ..............................................................................................................................23 3.4.1. Post-Newtonian Tests of Gravity ........................................................................................24 3.4.2. Gravitational Lensing ..........................................................................................................25 3.4.3. Gravitational Lensing ..........................................................................................................30 3.4.4. The Equivalence Principle ...................................................................................................32 3.4.5. Gravitational Redshift .........................................................................................................32 3.4.6. Frame-Dragging Tests .........................................................................................................34 3.4.7. Strong Field Tests................................................................................................................35 3.4.8. Gravitational Waves............................................................................................................37 vi 3.4.9. Cosmological Tests..............................................................................................................38 CHAPTER 4: Astrophysical Applications .....................................................................................................40 4.1. Gravitational Lensing ..................................................................................................................40 4.2. Gravitational Wave Astronomy...................................................................................................41 4.3. Black Holes and other Compact Objects.....................................................................................41 4.4. Cosmology...................................................................................................................................43 CHAPTER 5: Advanced Concepts.................................................................................................................46 5.1. Causal Structure and Global Geometry ......................................................................................46 5.2. Horizons ......................................................................................................................................46 5.3. Singularities.................................................................................................................................48