Geodesics are introduced and the geodesic equation analysed for the geometries introduced in chapter 2, using variation principles of classical mechanics. Geodesic motino on a sphere is described as well as the Coriolis effect and the Sagnac effect. Newtonian gravity is derived as the non-relativistic limit of geodesic motion in space-time. Geodesics in an expanding universe and heat death is described. Geodesics in Schwarzschild space-time are treated in detail: the precession of the perihelion of Mercury; the bending of light by the Sun; Shapiro time delay; black holes and the event horizon. Gravitational waves and gravitational lensing are also covered.

The simplest curved spacetimes of general relativity are the ones with the most symmetry, and the most useful of these is the geometry of empty space outside a spherically symmetric source of curvature – for example, a spherical star. This is called the Schwarzschild geometry. To an excellent approximation, this is the curved spacetime outside the Sun and therefore leads to the predictions of Einstein’s theory most accessible to experimental test. In this chapter, we explore the geometry of Schwarzschild’s solution, assuming it’s given. We will concentrate on predicting the orbits of test particles and light rays in the curved spacetime of a spherical star that exhibit some of the famous effects of general relativity – the gravitational redshift, the precession of the perihelion of a planet, the gravitational bending of light, and the time delay of light.