The Schwarzschild Solution

Abstract

In this chapter we study the Schwarzschild metric, corresponding to the gravitational field of a spherically symmetric body of mass M. Unlike Minkowski space-time, it contains a preferred class of observers, called stationary observers, who are not moving with respect to the central mass. We compute their proper time in terms of the time coordinate t and use it to obtain the exact redshift formula (“gravity delays time”). We also explain how by measuring distances between them stationary observers are led to the conclusion that space is curved (“gravity curves space”). Next we write the differential equations for the geodesics and see how they differ subtly from the Newtonian differential equations for free-falling motion. This leads to orbits that are approximately ellipses but whose axes slowly rotate. This effect, which had been observed for the planet Mercury, was the first triumph of the general theory of relativity. The differential equations for the null geodesics (light rays) also predict two new effects: a bending of the light rays passing near the spherically symmetric body (whose experimental confirmation in 1919 led to a wide acceptance of the theory), and a delay in the time of arrival of the light, known as the Shapiro delay (experimentally confirmed in 1966). Finally, we analyze the surface $r=2M,$ where the Schwarzschild metric is not defined. We show that this is a problem with the choice of coordinates, and not the space-time itself, and introduce the so-called Painlevé coordinates, which remove this problem. The surface $r=2M,$ however, remains special: it marks the boundary of a region from which nothing, not even light, can escape–a black hole.