We consider the black hole with charge, the Reissner–Nordstrom black hole. We describe the solution, and the BPS bound and its saturation, for extremal black holes, of mass = charge. We describe properties of the event horizon. Finally, we calculate the Penrose diagram of the Reissner–Nordstrom black hole, in the non-extremal and extremal cases.

We define Penrose diagrams, which keep the causal and topological properties of gravitational spacetimes, while moving infinity to a finite distance on the diagram. We use the examples of Minkowski space, in two dimensions and dimensions greater than two, then describe Anti-de Sitter spacetime in Poincaré coordinates (the Poincaré patch), and finally consider the Schwarzschild black hole.

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.