PROBLEMS OF GEOMETRIC FIELD THEORIES

Abstract

The thesis contains mainly three Chapters, English and Arabic summaries, two figures, nine tables and a list of references. Chapter 1: GEOMETRY AND FIELD THEORIES This Chapter reviews briefly three types of geometries usually used in constructing field theories. For each type we give a sample of theories constructed in its context, together with a critical review of problems and disadvantages of the geometry used and field theories given. We start by giving the main features of Riemannian geometry and theories constructed in its context, including the standard theory for gravity, GR. Then we give a brief account on the AP-geometry and a sample of theories written in this geometry. The third type of geometry given in this Chapter is Riemann-Cartan geometry together with Einstein’s unified field theory constructed in this geometry. This Chapter is terminated by a general discussion and aim of the work. Chapter 2: A SUGGESTED THEORY IN PAP-GEOMETRY This Chapter contains some details about a more wider geometry than both Riemannian and AP-geometries. Also, it is xix shown that this geometry is of the Riamann-Cartan type. The Chapter contains the derivation of a set of field equations using an action principle. The action used is constructed from the BB of the PAP-geometry. The equations of motion of the suggested theory is derived using the Bazanski approach (the path equation of the PAP-geometry). The Chapter is terminated by a discussion comparing the suggested field theory with GR. Chapter 3: EXTRACTION OF PHYSICS In this Chapter, three different methods are used to extract physics from the pure geometric objects of the suggested field theory. The first method comprises a comparison between the suggested theory and non-linear field theories. The second method used admits a comparison between the linearized form of the suggested theory and linear field theories. The comparisons fix the geometric objects responsible for, gravitational potential, electromagnetic potential, material and charge distributions and other physical objects. The third method is used in the transition phase from theory to physical applications. The Chapter contains the spherically symmetric application of the theory which gives the well known Schwarzschild exterior field as a unique solution of the field equations in the case of pure gravity in free space. The Chapter is terminated by a general discussion and some concluding remarks and suggestion for future work. xx Acknowledgements I would like to thank Professor Samia S. Elazab, Head of Mathematics Department, Faculty of Girls, for her continues encouragement during the period of research. I am greatly indebted to my supervisor Prof. M. I. Wanas for suggesting the points treated in this work and for his continuous encouragement and invaluable advices during every stage of this research. He willingly spent long time discussing and reviewing every aspect of this Thesis. He offered me the opportunity and resources to improve my knowledge. Without his guidance and leadership, this research would not have been realized in this form. I would like, also, to thank members of the Egyptian Relativity Group (ERG) for discussions and criticism, especially, Dr. M. Kahil for providing me with some relevant papers and Mr. W. El Hanafy for helping me in computer problems. I recognize the efforts done by the reviewers in reviewing the thesis. A new young entity was came to my life, an entity called me MAM, my lovely daughter Mariam. I wish that she will get all success and happiness in her life. Also, I would like to thank my family for their continuous scarifies and helping me in dealing with Ms. Mariam, to provide me necessary time for this work. Finally, I would like to express my appreciation to my husband for his unlimited devotion and continuous encouragement.