and the fixed values $Z^{i}_{\rm fix}(q)$ are not, there must be a singularity between $r=\infty$ and $r=0$, which will induce a curvature singularity. We will require both the asymptotic and the fixed values to be admissible. These aspects will be discussed in Section \[sec-n2d4\].

Black-hole solutions whose scalars take the asymptotic values $Z^{i}_{\infty}=Z^{i}_{\rm fix}$ have constant scalar fields, and are called *doubly extreme black holes*. These values are the ones that extremize, not the central charge, but the zero-Kähler-weight combination $e^{\mathcal{K}/2}\mathcal{Z}$:

$$\left. \mathfrak{D}_{i}\mathcal{Z}\right|_{Z^{i}=Z^{i}_{\rm fix}}=
\left.  
e^{-\mathcal{K}/2}
\partial_{i}\left(e^{\mathcal{K}/2}\mathcal{Z}\right)\right|_{Z^{i}=
Z^{i}_{\rm fix}}=0\, .$$

### Consequences of the existence of attractors

There are no more scalar fields in the theory, but in the timelike supersymmetric solutions there is another scalar object[^10] that satisfies a first-order differential equation: $X$. From the Killing spinor equation associated to the gravitino supersymmetry transformation rule it is possible to derive [@Meessen:2006tu]

$$\mathfrak{D}_{\mu}X = -i T^{+}{}_{\mu\nu}V^{\nu}\, ,$$

where $V^{\mu}$ is the timelike Killing vector constructed from the Killing spinor. The graviphoton field strength can be written in the form

$$T^{+} = \langle\, \mathcal{V} \mid F \, \rangle\, ,$$

and, together with

$$V^{\nu}F_{\nu\mu}=2 \nabla_{\mu}(|X|^{2}\mathcal{R})\, ,$$

the equation for $X$ becomes

$$\mathfrak{D}_{\mu}X =2 i  \langle\, \mathcal{V} \mid 
\nabla_{\mu}(|X|^{2}\mathcal{R}) \, \rangle\, .$$

Dividing 