We are now ready to go to the heart of general relativity, to learn how matter generates gravity. This subject is usually left out of discussions of general relativity below the level of an advanced university course. The reason is mathematics, not physics: Einstein formulated his field equations, his gravity-generating equations, using the language of differential geometry. This is the mathematical discipline that deals with curvature, and it is far from elementary. The physical ideas that Einstein expressed in this mathematical language are simply too important, however, to pass over. In this chapter we whittle down the mathematics to a form that is as close as possible to the algebra we used in our earlier chapters on Newton's gravity. This allows us to share in Einstein's thinking, to see what general relativity really predicts about the world we live in.
In this chapter: we study the equations that show how matter generates gravity in general relativity. We identify four properties of matter and gravity that act as sources of gravity, and we show how these different sources produce different gravitational effects. Using only little algebra, we compute the curvature of space and get the observed deflection of light as it passes the Sun. We show how special relativity and the curvature of time lead to something called the dragging of inertial frames. We examine the special properties of the cosmological constant as a source of gravity.