An Introduction to Special Relativity and Its Applications

The following sections are included:

• PREFACE

• TABLE of CONTENTS

In thinking about the laws of physics we implicitly assume, without further discussion, that these laws are the same everywhere (space is homogeneous), that there are no preferred directions in space (space is isotropic) and also that they are the same at all times. It cannot therefore matter where we take the origin of our coordinates, how we orient their axes, or when we set our clocks to zero, the laws of physics will still be expressed by the same formulae. Classical, Galilean relativity further asserts that all velocity is only relative so that there is no universal frame of reference at absolute rest. Although acceleration has an absolute significance, the laws of dynamics are the same when expressed in terms of two frames of reference which differ only by a uniform relative velocity. An object dropped in a moving train falls vertically to the floor, though relative to the ground it is, of course, moving at an angle to the vertical…

The following sections are included:

• Vectors and Coordinates

• The Lorentz Transformation

• Simple Kinematic Results

• General Axes

• Relative Velocity

• Successive Lorentz Transformations

• The Invariant Interval

• Past and Future: The Light Cone

• Proper Time

• Discussion

• Problems

In the last chapter we looked at three basic kinematic effects: time dilation, the Lorentz contraction and the relativistic rules for combining velocities. We now look at some slightly more complicated effects.

Newton’s first law states that in the absence of external influences a body moves with a constant velocity along a straight line. This is a purely kinematic law. Dynamics begins when we try to describe how external influences cause deviations from Newton’s first law.

The following sections are included:

• Mass and Momentum

• Energy and Mass

• Force and Work

• Summary

• Problems

The following sections are included:

• Radiation Pressure

• The Transformation of Energy

• The Limiting Velocity

• The Zero Momentum Frame

• The Centre of Momentum Frame and Collisions

• Elastic Scattering and the Compton Effect

• Radiative Recoil

• Centre of Mass Motion

• Force

• Forced Particle Motions

• Summary

• Problems

We now need a better and more compact notation for dealing with relativistic equations, something analogous to the use of vectors in classical physics. Although vector notation improves the appearance of the equations of classical physics, when we want numerical results we still have to revert to the use of a coordinate system. In relativity we are much more closely tied to the use of coordinate systems but we can still introduce a notation which improves the appearance of relativistic equations and, in the conventional jargon, makes their invariance under Lorentz transformations manifest…

Electromagnetism as Maxwell finally formulated it in 1864 was in fact, though this could hardly have been recognised at the time, an entirely relativistic theory. Some 40 years would elapse before Lorentz derived the coordinate transformation that left Maxwell’s equations invariant, and before Einstein formulated the principle of relativity. The close relation between these developments and Maxwell’s equations is well worth exploring…

In this chapter we intend to exhibit the new insights into the structure and applications of electromagnetism that can be obtained by considering the theory in a relativistic context. Although we already know that Maxwell’s equations must be invariant under Lorentz transformations, we must first confirm this by expressing all the electrodynamic equations in a manifestly covariant form…

The usual classical force F does not form part of a 4-vector without modification but we can, as long as we are only discussing the force acting on a single particle, use the proper time associated with the particle and express the equation of motion as dp_i/dτ = F_i in terms of the Minkowski 4-force F_i = γ(v) (F, iF.v/c). However, since this introduces the velocity v of the particle on which the force acts, it would obviously be an impossible procedure for dealing with a system of interacting particles, each with its own velocity and proper time. The first sections of this chapter are devoted to devising a technique for dealing with this problem based on the fact that, unlike the force, the force density, i.e. the force acting per unit volume on matter, is the spatial component of a 4-vector.

Concepts associated with the principle of least action such as conjugate variables, the Hamiltonian and path integrals play an important role in classical and quantum mechanics; it therefore seemed sensible to end with a short treatment of the principle and how it is modified by relativity…

The following sections are included:

• APPENDIX
  • Volume
  • Force and Force Density
  • Density and Flux
  • Volume and 3-Surfaces
  • Invariants of a Second-Rank Tensor
  • The Electromagnetic Energy-Momentum Tensor
  • Retarded Potentials
  • Linearity of the Lorentz Transformation
  • Free particles and Newton’s First Law of Motion
  • Some Physical Constants

• REFERENCES and BIBLIOGRAPHY

• ANSWERS TO PROBLEMS

• NAME INDEX

• SUBJECT INDEX