Finsler Geometry, Relativity and Gauge Theories

ISBN-13: 9789027719607 / Angielski / Twarda / 1985 / 370 str.

G. S. Asanov

Opis

The methods of differential geometry have been so completely merged nowadays with physical concepts that general relativity may well be considered to be a physical theory of the geometrical properties of space-time. The general relativity principles together with the recent development of Finsler geometry as a metric generalization of Riemannian geometry justify the attempt to systematize the basic techniques for extending general relativity on the basis of Finsler geometry. It is this endeavour that forms the subject matter of the present book. Our exposition reveals the remarkable fact that the Finslerian approach is automatically permeated with the idea of the unification of the geometrical space-time picture with gauge field theory - a circumstance that we try our best to elucidate in this book. The book has been written in such a way that the reader acquainted with the methods of tensor calculus and linear algebra at the graduate level can use it as a manual of Finslerian techniques orientable to applications in several fields. The problems attached to the chapters are also intended to serve this purpose. This notwithstanding, whenever we touch upon the Finslerian refinement or generalization of physical concepts, we assume that the reader is acquainted with these concepts at least at the level of the standard textbooks, to which we refer him or her.

Recenzje

`... Asanov's book is an important contribution to the literature and should benefit both experts and novices in applications of Finsler geometry.'
Mathematics Abstracts, 576:7 (1986)

Spis treści

A. Motivation and Outline of the Book.- B. Introduction to Finsler Geometry.- 1/Primary Mathematical Definitions.- 1.1. Concomitants of the Finslerian Metric Function.- 1.2. The Indicatrix.- 1.3. The Group of Invariance of the Finslerian Metric Function.- Problems.- Notes.- 2/Special Finsler Spaces.- 2.1. S3-like Finsler Spaces.- 2.2. Spaces with Quadratic Dependence of the Finslerian Metric Tensor on the Unit Tangent Vectors.- 2.3. Properties of the Berwald-Moór Metric Funcion.- 2.4. 1-Form Finsler Spaces.- 2.5. The Randers Metric Function.- 2.6. The Kropina Metric Function.- 2.7. C-Reducible Finsler Spaces.- Problems.- Notes.- C. Basic Equations.- 3/Implications of the Invariance Identities.- 3.1. Invariance Identities.- 3.2. Construction of the Connection Coefficients with the Help of the Invariance Identities.- 3.3. Fundamental Tensor Densities Associated with Direction-Dependent Scalar Densities.- 3.4. Choice of the Finslerian Scalar Density L = JK.- Problems.- Notes.- 4/Finslerian Approach Based on the Concept of Osculation.- 4.1. Formulation of Gravitational Field Equations in Terms of the Fundamental Tensor Densities.- 4.2. Application to Non-Gravitational Fields.- 4.3. Derivation of the Finslerian Equations of Motion of Matter from the Gravitational Field Equations.- 4.4. Significance of the Auxiliary Vector Field from the Viewpoint of the Clebsch Representation.- 4.5. Static Gravitational Field.- 4.6. Reduction of the Gravitational Lagrangian Density to First-Order Form in the 1-Form Case.- 4.7. Conservation Laws for the Gravitational Field in the 1-Form Case.- Problems.- Notes.- 5/Parametrical Representation of Physical Fields. The Relevance to Gauge Theories.- 5.1. Application of the Parametrical Representation of the Indicatrix.- 5.2. The Emergence of Gauge Fields.- 5.3. Finslerian Representation of Gauge Fields and Tensors.- 5.4. Gauge-Covariant Derivatives of Spinors and Isospinors.- 5.5. Linear Gauge Transformations. Finslerian Geometrization of Isotopic Invariance.- 5.6. Example of Nonlinear Internal Symmetry.- 5.7. Use of the Parametrical Concept of Osculation.- Problems.- Notes.- D. Additional Observations.- 6/Classical Mechanics from the Finslerian Viewpoint.- 6.1. Parametrically Invariant Extension of the Lagrangian.- 6.2. The Hamilton-Jacobi Equation for Homogeneous Lagrangians.- 6.3. The Generalized Hamilton-Jacobi Theory Based on the Clebsch Representation of the Canonical Momenta Field.- Problems.- Notes.- 7/Finslerian Refinement of Special Relativity Theory.- 7.1. Allowance for the Dependence of Space-Time Scales on the Directions of Motion of Inertial Frames of Reference.- 7.2. Finslerian Extension of the Special Principle of Relativity.- 7.3. Three Types of Velocities. The Fundamental Kinematic Relation.- 7.4. Finslerian Kinematics.- 7.5. Proper Finslerian Kinematic Effects.- 7.6. Finslerian Kinematics as a Consequence of the Equations of Motion of Matter.- Problems.- Notes.- Concluding Remark.- Appendix A Direction-Dependent Connection and Curvature Forms.- Problems.- Notes.- Appendix B/ General Gauge Field Equations Associated with Curved Internal Space.- B. 1. Introduction.- B. 2. The Parametrical Representation.- B. 3. Associated Gauge Tensors.- B. 4. Identities Satisfied by the Gauge Tensors.- B. 5. Variational Principle for the Parametrical Gauge Fields.- B. 6. General Gauge-Covariant Physical Field Equations.- B. 8. Implications of Metric Conditions.- B. 9. Specification of the Internal Metric Tensor.- B.10. Transition to the Parametrical Finslerian Limit.- B.11. Proper Finslerian Gauge Transformations.- B.12. Flat Internal Space.- Problems.- Note.- Solutions of Problems.- List of Publications on Finsler Geometry.- Biographies.