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.

A black hole can be rightfully thought as the most extreme manifestation of gravity – and thus of curvature! Besides being a unique source of puzzles and paradoxes for scientists, they have also been the inspiration for endless and breathtaking adventures in science-fiction novels and movies. This chapter will, therefore, explain the concept of black hole by making use of two different mechanical equivalents that have many points in common with black holes. In this way, it will become clear what is an event horizon and why it represents a one-way membrane, which can be entered, but from within which nothing can exit, not even light. Similarly, we will introduce the concept of spacetime singularity and explain why this is a problem that worries us physicists most, and for which we have not found any satisfactory solution yet. We will see that black holes are beautiful manifestations of nature and are not more monstrous than an erupting volcano.