Geometry And Quantum Features Of Special Relativity
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Geometry and Quantum Features of Special Relativity
This second edition of "The Geometry of Special Relativity - a Concise Course" offers more than just corrections and enhancements. It includes a new chapter on four-velocities and boosts as points and straight lines of hyperbolic geometry. Quantum properties of relativistic particles are derived from the unitary representations of the Poincaré group. Notably, the massless representation is related to the concept of a Hopf bundle. Scattering theory is developed analogously to the non-relativistic case, relying on proper symmetry postulates. Chapters on quantum fields, reflections of charge, space, and time, and the necessary gauge symmetry of quantized vector fields complete the foundation for evaluating Feynman graphs. An extended appendix covers more than a dozen additional topics. The first half of this edition refines the first edition, using simple diagrams to explain time dilation, length contraction, and Lorentz transformations based on the invariance of the speed of light. The text derives key results of relativistic physics and resolves apparent paradoxes. Following a presentation of the action principle, Noether's theorem, and relativistic mechanics, the book covers the covariant formulation of electrodynamics and classical field theory. The groups of rotations and Lorentz transformations are also examined as a transition to relativistic quantum physics. This text is aimed at graduate students of physics and mathematics seeking an advanced introduction to special relativity and related topics. Its presentation of quantum physics aims to inspire fellow researchers.
The Geometry of Special Relativity
The Geometry of Special Relativity provides an introduction to special relativity that encourages readers to see beyond the formulas to the deeper geometric structure. The text treats the geometry of hyperbolas as the key to understanding special relativity. This approach replaces the ubiquitous γ symbol of most standard treatments with the appropriate hyperbolic trigonometric functions. In most cases, this not only simplifies the appearance of the formulas, but also emphasizes their geometric content in such a way as to make them almost obvious. Furthermore, many important relations, including the famous relativistic addition formula for velocities, follow directly from the appropriate trigonometric addition formulas. The book first describes the basic physics of special relativity to set the stage for the geometric treatment that follows. It then reviews properties of ordinary two-dimensional Euclidean space, expressed in terms of the usual circular trigonometric functions, before presenting a similar treatment of two-dimensional Minkowski space, expressed in terms of hyperbolic trigonometric functions. After covering special relativity again from the geometric point of view, the text discusses standard paradoxes, applications to relativistic mechanics, the relativistic unification of electricity and magnetism, and further steps leading to Einstein’s general theory of relativity. The book also briefly describes the further steps leading to Einstein’s general theory of relativity and then explores applications of hyperbola geometry to non-Euclidean geometry and calculus, including a geometric construction of the derivatives of trigonometric functions and the exponential function.
Elements of General Relativity
Author: Piotr T. Chruściel
language: en
Publisher: Springer Nature
Release Date: 2020-03-19
This book provides an introduction to the mathematics and physics of general relativity, its basic physical concepts, its observational implications, and the new insights obtained into the nature of space-time and the structure of the universe. It introduces some of the most striking aspects of Einstein's theory of gravitation: black holes, gravitational waves, stellar models, and cosmology. It contains a self-contained introduction to tensor calculus and Riemannian geometry, using in parallel the language of modern differential geometry and the coordinate notation, more familiar to physicists. The author has strived to achieve mathematical rigour, with all notions given careful mathematical meaning, while trying to maintain the formalism to the minimum fit-for-purpose. Familiarity with special relativity is assumed. The overall aim is to convey some of the main physical and geometrical properties of Einstein's theory of gravitation, providing a solid entry point to further studies of the mathematics and physics of Einstein equations.