The concept to the metric is introduced. Various geometries, both flat and curved, are described including Euclidean space; Minkowski space-time; spheres; hyperbolic planes and expanding space-times. Lorentz transformations and relativistic time dilation in flat space-time is discussed as well as gravitational red-shift and the Global Positioning System. Hubble expansion and the cosmological red-shift are also explained.

This chapter (and the next one) covers some basic mathematics needed to describe four-dimensional curved spacetime geometry. Much of this is a generalization of the concepts introduced in Chapter 5 for flat spacetime. Coordinates are a systematic way of labeling the points of spacetime. The choice of coordinates is arbitrary as long as they supply a unique set of labels for each point in the region they cover, but for a particular problem, one coordinate system may be more useful than another. We then define the metric for a general geometry and explain common conventions. We show how to compute lengths of curves, areas, three-volumes, and four-volumes for a given metric. Concepts such as wormholes, extra dimensions, the Lorentz hyperboloid, and null spaces are introduced.