The concept of general covariance

We begin this chapter by introducing the idea of a field in physics (to be distinguished from the ‘field’ in algebra that mathematicians talk about). The idea was popularized by Michael Faraday in the context of the electric and magnetic fields. Figure 4.1 shows what happens when iron filings are sprinkled in the vicinity of a bar magnet. The filings get distributed in a pattern somewhat like that in this figure. Faraday called these curves lines of force. If we imagine a magnetic pole placed anywhere on one of these lines, it will move along that line, being guided by the magnetic force on it. The lines of force therefore represent the ‘magnetic field’ B both in strength and direction at any point in the vicinity of the magnet. In short the magnet generates a ‘field’ of B vectors all around it, representing the force exerted by it on another magnetic pole.

We generalize the concept of a vector field by defining a vector function of spacetime variables, so that at each point a vector is defined. This idea may further be generalized by having tensor fields as functions of spacetime coordinates. Thus we could argue that equations of physics involve fields related by partial differential equations. If we additionally require that these equations do not change their form under changes of coordinates (thereby being the same for all observers), then they should be represented by tensor fields.