Introduction

Having acquainted ourselves with the trials and tribulations of working in non-Euclidean spacetimes we are now prepared for the next step, that of describing physics in such curved spacetimes. For we recall from Chapter 2 that the Einstein programme for general relativity consists of replacing the Newtonian perception of gravitation as a force by the notion that its effect makes the geometry of spacetime ‘suitably non-Euclidean’. What we mean by ‘suitably’ will be clear in the next two chapters. But given that the geometry is non-Euclidean we first need to know how the rest of physics is described in it.

For example, how do we describe the motion of a particle under a non-gravitational force? How do we write Maxwell's equations? What is the role of energy-momentum tensors? Can a dynamical action principle be written in curved spacetime? Such questions need our attention before we turn to the basic issue of how gravity actually leads to curved spacetime.

To this end we will introduce a concept that Einstein took as a basic principle in formulating general relativity. It is known as the principle of equivalence.

The principle of equivalence

Let us go back to the purely mathematical result embodied in the relations shown in Section 4.6 and attempt to describe their physical meaning. These relations tell us that special (locally inertial) coordinates that behave like the coordinates (t, x, y, z) of special relativity exist in the neighbourhood of any point P in spacetime.