In the previous chapter we obtained and studied the Schwarzschild solution of the vacuum Einstein equations and arrived at the BH concept. However, many of the general features of BHs that we discussed, such as the no-hair conjecture, make reference to BHs in the presence of matter fields. In this chapter we are going to initiate the study and construction of BH solutions of the Einstein equations in the presence of matter fields, starting with the simplest ones: massless scalar and vector fields.

The (unsuccessful) search for BH solutions of gravity coupled to a scalar field will allow us to deepen our understanding of the no-hair conjecture.

The (successful) search for BH solutions of gravity coupled to a vector field will allow us to find the simplest BH solution different from Schwarzschild's: the Reissner–Nordström (RN) solution. Simple as it is, it has very interesting features, in particular, the existence of an extreme limit with a regular horizon and zero Hawking temperature that will be approached with positive specific heat, as in standard thermodynamical systems. Later on we will relate some of these properties to the unbroken supersymmetry of the extreme RN (ERN) solution, which will allow us to reinterpret it as a self-gravitating supersymmetric soliton interpolating between two vacua of the theory.

The ERN BH is the archetype of the more complicated self-gravitating supersymmetric solitons that we are going to encounter later on in the context of superstring low-energy effective actions (actually, one of our goals will be to recover it as a superstring solution) and many of its properties will be shared by them. Furthermore, the four-dimensional Einstein–Maxwell system exhibits electric–magnetic duality in its simplest form.