Par t 1
R EL AT IV IT Y, P AR TI CL E D Y NA MI CS , G R AV IT AT IO N,
A ND W AV E M O TI ON
FR AN K W. K. FIR K
Professor Emeritus of Physics
Throughout the decade of the 1990’s, I taught a one-year course of a specialized nature to
students who entered Yale College with excellent preparation in Mathematics and the
PhysicalSciences, and who expressed an interest in Physics or a closely related field. The
level of the course was that typified by the Feynman Lectures on Physics. My one-year
course was necessarily more restricted in content than the two-year Feynman Lectures.
The depth of treatment of each topic was limited by the fact that the course consisted of a
total of fifty-two lectures, each lastingone-and-a-quarter hours. The key role played by
invariants in the Physical Universe was constantly emphasized . The material that I
covered each Fall Semester is presented, almost verbatim, in this book.
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 linearoperators 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 Galileantransformation.
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 mechanicalenergy, 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 subjectof
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-energycollisions. 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 nomeasurable 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 inPhysics 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...