– part 1-relativity, particle dynamics, gravitation and wave motion
Essential Physics- part 1-relativity, particle dynamics, gravitation and wave motion
Author(s): Frank Firk
The first chapter contains key mathematical ideas, including some invariants of geometry and algebra, generalized coordinates, and the algebra and geometry of vectors. The importance of linear operators and their matrix representations is stressed in the early lectures. These mathematical concepts are required in the presentation of a unified treatment of both Classical and Special Relativity. Students are encouraged to develop a “relativistic outlook” at an early stage . The fundamental Lorentz transformation is developed using arguments based on symmetrizing the classical Galilean transformation. Key 4-vectors, such as the 4-velocity and 4-momentum, and their invariant norms, are shown to evolve in a natural way from their classical forms. A basic change in the subject matter occurs at this point in the book. It is necessary to introduce the Newtonian concepts of mass, momentum, and energy, and to discuss the conservation laws of linear and angular momentum, and mechanical energy, and their associated invariants. The discovery of these laws, and their applications to everyday problems, represents the high point in the scientific endeavor of the 17th and 18th centuries. An introduction to the general dynamical methods of Lagrange and Hamilton is delayed until Chapter 9, where they are included in a discussion of the Calculus of Variations. The key subject of Einsteinian dynamics is treated at a level not usually met in at the introductory level. The 4-momentum invariant and its uses in relativistic collisions, both elastic and inelastic, is discussed in detail in Chapter 6. Further developments in the use of relativistic invariants are given in the discussion of the Mandelstam variables, and their application to the study of high-energy collisions. Following an overview of Newtonian Gravitation, the general problem of central orbits is discussed using the powerful method of [p, r] coordinates.
Einstein’s General Theory of Relativity is introduced using the Principle of Equivalence and the notion of “extended inertial frames” that include those frames in free fall in a gravitational field of small size in which there is no measurable field gradient. A heuristic argument is given to deduce the Schwarzschild line element in the “weak field approximation”; it is used as a basis for a discussion of the refractive index of space-time in the presence of matter. Einstein’s famous predicted value for the bending of a beam of light grazing the surface of the Sun is calculated. The Calculus of Variations is an important topic in Physics and Mathematics; it is introduced in Chapter 9, where it is shown to lead to the ideas of the Lagrange and Hamilton functions. These functions are used to illustrate in a general way the conservation laws of momentum and angular momentum, and the relation of these laws to the homogeneity and isotropy of space. The subject of chaos is introduced by considering the motion of a damped, driven pendulum.