Gravity An Introduction to Einstein's General Relativity

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.