As we discussed in the Black Holes section charged black holes are described by the Reissner Nordstrøm (RN) solution and neutral black holes by the Schwarzschild (SS) solution in 'normal' General Relativity. We also presented the 'No Hair Theorem' in that section which says that black holes with independent scalar charge do not exist. Let's take a look at the obtained Dilaton solutions and see what changes.
In the one vector field case we see that for the solution is RN, as it should be. But for any value other than zero we see that the inner horizon () becomes a zero-point of the metric function meaning the area of the sphere is zero (area singularity). So we lost the inner horizon and the causal structure resembles SS (one regular event horizon, for , just as in SS), the difference only being that the area of spheres is smaller. Since remains finite at the (area) singularity there is no ''infinite stretching'' analogous to what happens when you hit the singularity in SS .
We mentioned earlier that it is suspected that slightly non-spherical charged collapse would make the inner horizon singular in the RN case. Now we have another example of the instability of the inner horizon; for every value other than zero the inner horizon becomes singular .
Also striking is the behaviour of the Dilaton field. It's singular at the area singularity in the electric but zero in the magnetic case (to go from electric to magnetic we only need to invert the Dilaton field). At the curvature singularity the Dilaton field behaves in the opposite way.
Also, it should be mentioned that the solution is in agreement with 'No Hair Theorem' because although we found a solution with scalar 'hair' this Dilaton charge is not an independent quantity, it depends on the electric/magnetic charge. The Dilaton field disappears completely when the charge is put zero. So the only parameters determining the black hole are it's charge and it's mass.
When we take a look at the dyonic solution we immediately see that things change drastically with respect to the singly charged solution. Suddenly, the causal structure isn't like SS but the inner and outer horizon make the causal structure to be similar to RN. The only difference being the shift of the curvature singularity to and directly connected with that the smaller area of spheres. The Dilaton field is either infinite or zero at the curvature singularity depending what charges (electric or magnetic) dominate ( can be either positive or negative).
The parameter relations again show that the only parameters determining the black hole are the four charges (constrained through eq.5.9 and eq.5.10) and the mass of the black hole. When and the solution reduces to the RN solution which is what we expect since the source of the Dilaton is which vanishes when , .
So it turns out that the presence of the Dilaton changes the properties of charged black holes considerably (especially in the singly charged case). The appearance of an area singularity is one of most notable features and, as we will see later on, it is this fact that changes the thermodynamic properties of the extremal solutions.