The Lorentz transformation between frames of reference is determined and presented in a matrix form. The transformation of velocities is derived with an example presented in the form of the relativistic rocket equation. The Doppler effect is discussed. Relativistic momentum and energy are presented. The quantum nature of particles and light is discussed. The four-vector and tensor notation are introduced.

In this chapter we extend our review of mechanics to include Einstein’s special theory of relativity. We will see that our previous Newtonian framework is a useful description of the mechanical world only when speeds are much less than that of light. We also use this chapter to introduce index notation and general technical tools that will help us throughout the rest of the book. Then, in the following chapter, we will show how relativity provides insights for an entirely different formulation of mechanics -- the so-called variational principle.

In 1905, Einstein published his paper on the special theory of relativity. Although Lorentz and Poincaré had introduced a number of the essential feature of the special theory, Einstein's' paper went much deeper and showed that we live in four-dimensional space-time. A consequence of this approach was the derivation of the mass-energy relation E = mc2. The route to the special theory of relativity was not straightforward, but Einstein's approach was quickly adopted and the mass energy relation confirmed by the Cockcroft and Walton experiment of 1932. In this chapter, the theory of special relativity is developed through the use of four-vectors, providing a robust framework for carrying out calculations in special relativity.